That he spared an entire day was exceptional — usually his availability is "modulo nap time," a sacred daily ritual from 1 p.m.
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MOSE (an acronym for Modulo Sperimentale Elettromeccanico, or "Experimental Electromechanical Module") is one of the world's largest and highest-profile civil-engineering works.
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The 512S Modulo debuted in 1970 as a concept car from Italian car design company Pininfarina ... after it was passed on to them by Enzo Ferrari.
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The one-of-a-kind Ferrari 512S Modulo hit the road Tuesday for its first ever ride on public pavement ... nearly 50 years AFTER it was introduced.
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In informal contexts, mathematicians often use the word modulo (or simply "mod") for similar purposes, as in "modulo isomorphism".
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The primitive roots modulo n are the primitive \varphi(n)-roots of unity modulo n, where \varphi is Euler's totient function.
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The basic fact in this case is :if a is a residue modulo n, then a is a residue modulo pk for every prime power dividing n. :if a is a nonresidue modulo n, then a is a nonresidue modulo pk for at least one prime power dividing n. Modulo a composite number, the product of two residues is a residue. The product of a residue and a nonresidue may be a residue, a nonresidue, or zero.
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The algorithm completely ignores any numbers with remainder modulo 60 that is divisible by two, three, or five, since numbers with a modulo 60 remainder divisible by one of these three primes are themselves divisible by that prime. All numbers with modulo-sixty remainder 1, 13, 17, 29, 37, 41, 49, or 53 have a modulo-four remainder of 1. These numbers are prime if and only if the number of solutions to is odd and the number is squarefree (proven as theorem 6.1 of). All numbers with modulo-sixty remainder 7, 19, 31, or 43 have a modulo-six remainder of 1.
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By the rational root theorem this has no rational zeroes. Neither does it have linear factors modulo 2 or 3. The Galois group of modulo 2 is cyclic of order 6, because modulo 2 factors into polynomials of orders 2 and 3, .
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Since the only residue (mod 3) is 1, we see that −3 is a quadratic residue modulo every prime which is a residue modulo 3.
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Slots and tables are trivially computed from hashes. The target table is simply the lowest eight bits of the hash (i.e. hash modulo 256), and the slot within the table is the remaining bits of the hash modulo the table length (i.e. hash divided by 256 modulo table length).
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Modulo design sketch The Ferrari 512S Modulo is a concept sports car designed by Paolo Martin of the Italian carrozzeria Pininfarina, unveiled at the 1970 Geneva Motor Show.
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This is only an upper bound because not every matrix is invertible and thus usable as a key. The number of invertible matrices can be computed via the Chinese Remainder Theorem. I.e., a matrix is invertible modulo 26 if and only if it is invertible both modulo 2 and modulo 13.
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Narrow bars are 0 and wide bars are 1. This symbology is not self checking, though a modulo 10 or modulo 11 checksum (or some combination of both checksums, depending on application) is usually appended.
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Peugeot 104 Ferrari Modulo Fiat 130 Coupé Lancia Beta Montecarlo 1982 Rolls- Royce Camargue Pininfarina Ferrari Modulo Paolo Martin (born 1943) is an Italian car designer widely known for his career with Studio Tecnico Michelotti, Carrozzeria Bertone, Pininfarina and De Tomaso/Ghia where he styled the Ferrari Dino Berlinetta Competizione, Ferrari Modulo concept, Fiat 130 Coupé and the Rolls-Royce Camargue.
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Given an integer , called a modulus, two integers are said to be congruent modulo , if is a divisor of their difference (i.e., if there is an integer such that ). Congruence modulo is a congruence relation, meaning that it is an equivalence relation that is compatible with the operations of addition, subtraction, and multiplication. Congruence modulo is denoted: :a \equiv b \pmod n.
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Like any congruence relation, congruence modulo is an equivalence relation, and the equivalence class of the integer , denoted by , is the set . This set, consisting of all the integers congruent to modulo , is called the congruence class, residue class, or simply residue of the integer modulo . When the modulus is known from the context, that residue may also be denoted .
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In arithmetic topology, there is an analogy between knots and prime numbers in which one considers links between primes. The triple of primes are linked modulo 2 (the Rédei symbol is −1) but are pairwise unlinked modulo 2 (the Legendre symbols are all 1). Therefore, these primes have been called a "proper Borromean triple modulo 2" or "mod 2 Borromean primes".
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This is incremented for successive I-frames, modulo 8 or modulo 128. Depending on the number of bits in the sequence number, up to 7 or 127 I-frames may be awaiting acknowledgment at any time.
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The observations about −3 and 5 continue to hold: −7 is a residue modulo p if and only if p is a residue modulo 7, −11 is a residue modulo p if and only if p is a residue modulo 11, 13 is a residue (mod p) if and only if p is a residue modulo 13, etc. The more complicated-looking rules for the quadratic characters of 3 and −5, which depend upon congruences modulo 12 and 20 respectively, are simply the ones for −3 and 5 working with the first supplement. :Example. For −5 to be a residue (mod p), either both 5 and −1 have to be residues (mod p) or they both have to be non-residues: i.e., p ≡ ±1 (mod 5) and p ≡ 1 (mod 4) or p ≡ ±2 (mod 5) and p ≡ 3 (mod 4).
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Modulo a prime, the product of two nonresidues is a residue and the product of a nonresidue and a (nonzero) residue is a nonresidue. The first supplementGauss, DA, art 111 to the law of quadratic reciprocity is that if p ≡ 1 (mod 4) then −1 is a quadratic residue modulo p, and if p ≡ 3 (mod 4) then −1 is a nonresidue modulo p. This implies the following: If p ≡ 1 (mod 4) the negative of a residue modulo p is a residue and the negative of a nonresidue is a nonresidue. If p ≡ 3 (mod 4) the negative of a residue modulo p is a nonresidue and the negative of a nonresidue is a residue.
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Expanding on the last two examples, there is an analogy between knots and prime numbers in which one considers "links" between primes. The triple of primes are "linked" modulo 2 (the Rédei symbol is −1) but are "pairwise unlinked" modulo 2 (the Legendre symbols are all 1). Therefore these primes have been called a "proper Borromean triple modulo 2" or "mod 2 Borromean primes".
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Modulo Stepwgn Concept is a concept vehicle based on Stepwgn, with around-vehicle sensors. Stepwgn Modulo Concept X Final Room is a concept vehicle based on Stepwgn, with around-vehicle sensors. The vehicles were unveiled in the 2006 Tokyo Auto Salon.
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One says that relative homology is given by the relative cycles, chains whose boundaries are chains on A, modulo the relative boundaries (chains that are homologous to a chain on A, i.e., chains that would be boundaries, modulo A again).
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In August 2016, SAI Global acquired Modulo International for US$6.8mi, with exclusive worldwide licence, except in Brazil, Angola, and Mozambique, includes the right to modify, use, and sell the source code with Modulo Security Solutions SA as the owner.
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Modulo 2, every integer is a quadratic residue. Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues, by Euler's criterion. In this case, it is customary to consider 0 as a special case and work within the multiplicative group of nonzero elements of the field Z/pZ. (In other words, every congruence class except zero modulo p has a multiplicative inverse.
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The method of casting out nines offers a quick check of decimal arithmetic computations performed by hand. It is based on modular arithmetic modulo 9, and specifically on the crucial property that 10 ≡ 1 (mod 9). Arithmetic modulo 7 is used in algorithms that determine the day of the week for a given date. In particular, Zeller's congruence and the Doomsday algorithm make heavy use of modulo-7 arithmetic.
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Thus, to divide by x (modulo y) we need merely instead multiply by a.
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Two other characterizations of squares modulo a prime are Euler's criterion and Zolotarev's lemma.
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Kaplansky's proof uses the facts that 2 is a 4th power modulo p if and only if p is representable by x2 + 64y2, and that −4 is an 8th power modulo p if and only if p is representable by x2 + 32y2.
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A sequence (a1, a2, a3, ...) of real numbers is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if the sequence of the fractional parts of an, denoted by (an) or by an − ⌊an⌋, is equidistributed in the interval [0, 1].
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For an election with n candidates, the above procedure is followed using modulo-n equations.
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The residue class ring modulo a Gaussian integer is a field if and only if z_0 is a Gaussian prime. If is a decomposed prime or the ramified prime (that is, if its norm is a prime number, which is either 2 or a prime congruent to 1 modulo 4), then the residue class field has a prime number of elements (that is, ). It is thus isomorphic to the field of the integers modulo . If, on the other hand, is an inert prime (that is, is the square of a prime number, which is congruent to 3 modulo 4), then the residue class field has elements, and it is an extension of degree 2 (unique, up to an isomorphism) of the prime field with elements (the integers modulo ).
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101 If the modulus is pn, :then pka ::is a residue modulo pn if k ≥ n ::is a nonresidue modulo pn if k < n is odd ::is a residue modulo pn if k < n is even and a is a residue ::is a nonresidue modulo pn if k < n is even and a is a nonresidue.Gauss, DA, art. 102 Notice that the rules are different for powers of two and powers of odd primes. Modulo an odd prime power n = pk, the products of residues and nonresidues relatively prime to p obey the same rules as they do mod p; p is a nonresidue, and in general all the residues and nonresidues obey the same rules, except that the products will be zero if the power of p in the product ≥ n.
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There is a natural equivalence between # the monopoles of charge k for the group SU(2), modulo gauge transformations, and # the solutions of Nahm equations satisfying the additional conditions above, modulo the simultaneous conjugation of T1,T2, T3 by the group O(k,R).
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The only place where a direct reduction modulo N is necessary is in the precomputation of .
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Only one unit of the Ferrari 512 S Modulo was built, to a design by Pininfarina.
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Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose group structure is the direct sum of the 'reduced groups' obtained by performing the equations defining the group arithmetic modulo the unknown prime factors p1, p2, ... By the Chinese remainder theorem, arithmetic modulo N corresponds to arithmetic in all the reduced groups simultaneously. The aim is to find an element which is not the identity of the group modulo N, but is the identity modulo one of the factors, so a method for recognising such one-sided identities is required. In general, one finds them by performing operations that move elements around and leave the identities in the reduced groups unchanged. Once the algorithm finds a one-sided identity all future terms will also be one-sided identities, so checking periodically suffices.
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Modulo-N code is a lossy compression algorithm used to compress correlated data sources using modular arithmetic.
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1\. Euler's theorem can be proven using concepts from the theory of groups:Ireland & Rosen, corr. 1 to prop 3.3.2 The residue classes modulo that are coprime to form a group under multiplication (see the article Multiplicative group of integers modulo n for details). The order of that group is .
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Because this is equivalent to modulo-2 addition, this generates the modulo-2 version of Pascal's triangle. The diagram has a 1 wherever Pascal's triangle has an odd number, and a 0 wherever Pascal's triangle has an even number. This is a discrete version of the Sierpiński triangle., pp.
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Lifting idempotents also has major consequences for the category of R modules. All idempotents lift modulo I if and only if every R direct summand of R/I has a projective cover as an R module. Idempotents always lift modulo nil ideals and rings for which R/I is I-adically complete. Lifting is most important when , the Jacobson radical of R. Yet another characterization of semiperfect rings is that they are semilocal rings whose idempotents lift modulo J(R).
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If X is proper then its boundary is homeomorphic to the space of Busemann functions on X modulo translations.
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If using binary values for true (1) and false (0), then exclusive or works exactly like addition modulo 2.
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'Rabbits are mammals'. Thus, the stimulus meaning is less useful to approximate the intuitive meaning of standing sentences. However, the difference between occasion and standing sentences is only a gradual difference. This difference depends on the modulus because 'an occasion sentence modulo n seconds can be a standing sentence modulo n – 1'.
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Since both and are between and , they must be equal. Therefore, the terms , , ..., when reduced modulo must be distinct. To summarise: when we reduce the numbers , , ..., modulo , we obtain distinct members of the sequence , , ..., . Since there are exactly of these, the only possibility is that the former are a rearrangement of the latter.
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Notice how the modulo operator always guarantees that only the fractional sum will be kept. To calculate 16n−k mod (8k + 1) quickly and efficiently, the modular exponentiation algorithm is used. When the running product becomes greater than one, the modulo is taken, just as for the running total in each sum.
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We can do these calculations faster by using various modular arithmetic and Legendre symbol properties. If we keep calculating the values, we find: :(17/p) = +1 for p = {13, 19, ...} (17 is a quadratic residue modulo these values) :(17/p) = −1 for p = {3, 5, 7, 11, 23, ...} (17 is not a quadratic residue modulo these values). Example 2: Finding residues given a prime modulus p Which numbers are squares modulo 17 (quadratic residues modulo 17)? We can manually calculate it as: : 12 = 1 : 22 = 4 : 32 = 9 : 42 = 16 : 52 = 25 ≡ 8 (mod 17) : 62 = 36 ≡ 2 (mod 17) : 72 = 49 ≡ 15 (mod 17) : 82 = 64 ≡ 13 (mod 17). So the set of the quadratic residues modulo 17 is {1,2,4,8,9,13,15,16}. Note that we did not need to calculate squares for the values 9 through 16, as they are all negatives of the previously squared values (e.g. 9 ≡ −8 (mod 17), so 92 ≡ (−8)2 = 64 ≡ 13 (mod 17)).
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Even without knowledge that we are working in the multiplicative group of integers modulo n, we can show that a actually has an order by noting that the powers of a can only take a finite number of different values modulo n, so according to the pigeonhole principle there must be two powers, say s and t and without loss of generality s > t, such that as ≡ at (mod n). Since a and n are coprime, this implies that a has an inverse element a−1 and we can multiply both sides of the congruence with a−t, yielding as−t ≡ 1 (mod n). The concept of multiplicative order is a special case of the order of group elements. The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n.
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More generally, this equation can be satisfied only when v is congruent to 0, 3, 4, or 7 modulo 12 .
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29, no. 3, pp. 439–441, 1983 Tomlinson-Harashima precodingM. Tomlinson, New automatic equalizer employing modulo arithmetic, Electronics Letters, vol.
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Finally, we must explain why the sequence :a, 2a, 3a, \ldots, (p-1)a, when reduced modulo p, becomes a rearrangement of the sequence :1, 2, 3, \ldots, p-1. To start with, none of the terms , , ..., can be congruent to zero modulo , since if is one of the numbers , then is relatively prime with , and so is , so Euclid's lemma tells us that shares no factor with . Therefore, at least we know that the numbers , , ..., , when reduced modulo , must be found among the numbers . Furthermore, the numbers , , ..., must all be distinct after reducing them modulo , because if :ka \equiv ma \pmod p, where and are one of , then the cancellation law tells us that :k \equiv m \pmod p.
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That cipher can be regarded as a predecessor to the RSA (cryptosystem) since all that is needed to transform it into RSA is to change the arithmetic from modulo a prime number to modulo a composite number. In his spare time Stephen Pohlig was a keen kayaker known to many throughout the New England area.
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Software pipelining has been known to assembly language programmers of machines with instruction-level parallelism since such architectures existed. Effective compiler generation of such code dates to the invention of modulo scheduling by Rau and Glaeser.B.R. Rau and C.D. Glaeser, "Some scheduling techniques and an easily schedulable horizontal architecture for high performance scientific computing", In Proceedings of the Fourteenth Annual Workshop on Microprogramming (MICRO-14), December 1981, pages 183-198 Lam showed that special hardware is unnecessary for effective modulo scheduling. Her technique, modulo variable expansion is widely used in practice.
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The set of isomorphism classes of Legendrian knots modulo negative Legendrian stabilizations is in bijection with the set of transverse knots.
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Because a2 ≡ (n − a)2 (mod n), the list of squares modulo n is symmetrical around n/2, and the list only needs to go that high. This can be seen in the table below. Thus, the number of quadratic residues modulo n cannot exceed n/2 + 1 (n even) or (n + 1)/2 (n odd).Gauss, DA, art.
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Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over after it reaches 12, this is arithmetic modulo 12. In terms of the definition below, 15 is congruent to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock.
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Some of the proofs of Fermat's little theorem given below depend on two simplifications. The first is that we may assume that is in the range . This is a simple consequence of the laws of modular arithmetic; we are simply saying that we may first reduce modulo . This is consistent with reducing a^p modulo , as one can check.
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The sum of the six products is calculated. The calculated sum modulo 24 is subtracted from 24 to give an index number.
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The pipe in APL is the modulo or residue function between two operands and the absolute value function next to one operand.
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Pp. 51–70. 1973. According to Dickson, Tonelli's algorithm can take square roots of x modulo prime powers pλ apart from primes.
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In order to compute the DFT, we need to evaluate the remainder of x(z) modulo N degree-1 polynomials as described above. Evaluating these remainders one by one is equivalent to the evaluating the usual DFT formula directly, and requires O(N2) operations. However, one can combine these remainders recursively to reduce the cost, using the following trick: if we want to evaluate x(z) modulo two polynomials U(z) and V(z), we can first take the remainder modulo their product U(z) V(z), which reduces the degree of the polynomial x(z) and makes subsequent modulo operations less computationally expensive. The product of all of the monomials (z - \omega_N^k) for k=0..N-1 is simply z^N-1 (whose roots are clearly the N roots of unity).
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In mathematics, potential good reduction is a property of the reduction modulo a prime or, more generally, prime ideal, of an algebraic variety.
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In mathematics, rational reconstruction is a method that allows one to recover a rational number from its value modulo a sufficiently large integer.
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Excel has issues with modulo operations. In the case of excessively large results, Excel will return the error warning instead of an answer.
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In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic). Dirichlet's unit theorem shows that the unit group has rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a fundamental system of units.
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When R is a power of a small positive integer b, can be computed by Hensel's lemma: The inverse of N modulo b is computed by a naive algorithm (for instance, if then the inverse is 1), and Hensel's lemma is used repeatedly to find the inverse modulo higher and higher powers of b, stopping when the inverse modulo R is known; is the negation of this inverse. The constants and can be generated as and as . The fundamental operation is to compute REDC of a product. When standalone REDC is needed, it can be computed as REDC of a product with .
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The interval class class(i) modulo Ω depends only on i modulo ℘, hence we may also define a version of class, Class(i), for pitch classes modulo Ω, which are called generic intervals. The specific pitch classes belonging to Class(i) are then called specific intervals. The class of the unison, Class(0), consists solely of multiples of Ω and is typically excluded from consideration, so that the number of generic intervals is ℘ − 1\. Hence the generic intervals are numbered from 1 to ℘ − 1, and a scale is proper if for any two generic intervals i < j implies class(i) < class(j).
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In computer science, modular arithmetic is often applied in bitwise operations and other operations involving fixed-width, cyclic data structures. The modulo operation, as implemented in many programming languages and calculators, is an application of modular arithmetic that is often used in this context. The logical operator XOR sums 2 bits, modulo 2. In music, arithmetic modulo 12 is used in the consideration of the system of twelve-tone equal temperament, where octave and enharmonic equivalency occurs (that is, pitches in a 1∶2 or 2∶1 ratio are equivalent, and C-sharp is considered the same as D-flat).
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Binary Galois LFSRs like the ones shown above can be generalized to any q-ary alphabet {0, 1, ..., q − 1} (e.g., for binary, q = 2, and the alphabet is simply {0, 1}). In this case, the exclusive-or component is generalized to addition modulo-q (note that XOR is addition modulo 2), and the feedback bit (output bit) is multiplied (modulo-q) by a q-ary value, which is constant for each specific tap point. Note that this is also a generalization of the binary case, where the feedback is multiplied by either 0 (no feedback, i.e.
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The original technique for constructing -independent hash functions, given by Carter and Wegman, was to select a large prime number , choose random numbers modulo , and use these numbers as the coefficients of a polynomial of degree whose values modulo are used as the value of the hash function. All polynomials of the given degree modulo are equally likely, and any polynomial is uniquely determined by any -tuple of argument-value pairs with distinct arguments, from which it follows that any -tuple of distinct arguments is equally likely to be mapped to any -tuple of hash values.
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Kaplansky's theorem states that a prime p congruent to 1 modulo 16 is representable by both or none of x2 + 32y2 and x2 + 64y2, whereas a prime p congruent to 9 modulo 16 is representable by exactly one of these quadratic forms. This is remarkable since the primes represented by each of these forms individually are not describable by congruence conditions..
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For proving that there is no solution, one may reduce the equation modulo . For example, the Diophantine equation :x^2+y^2=3z^2, does not have any other solution than the trivial solution . In fact, by dividing and by their greatest common divisor, one may suppose that they are coprime. The squares modulo 4 are congruent to 0 and 1.
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Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order ideal (prime order ideal, maximal order ideal) of the Boolean algebra. The quotient ring of a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal.
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For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number , or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient understood to mean the sum of copies of . For example, over the integers modulo , the derivative of the polynomial is the polynomial .
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However, the reduction in size of the universe of possible checksum values acts against this and reduces performance slightly. One study showed that Fletcher-32 outperforms Adler-32 in both performance and in its ability to detect errors. As modulo-65,535 addition is considerably simpler and faster to implement than modulo-65,521 addition, the Fletcher-32 checksum is generally a faster algorithm.
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Wayne Snyder gave a generalization of both higher-order unification and E-unification, i.e. an algorithm to unify lambda-terms modulo an equational theory.
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Ambiguity resolution is used to find the value of a measurement that requires modulo sampling. This is required for pulse-Doppler radar signal processing.
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Case A for n = 5 can be proven immediately by Sophie Germain's theorem if the auxiliary prime θ = 11\. A more methodical proof is as follows. By Fermat's little theorem, : x5 ≡ x (mod 5) : y5 ≡ y (mod 5) : z5 ≡ z (mod 5) and therefore : x + y + z ≡ 0 (mod 5) This equation forces two of the three numbers x, y, and z to be equivalent modulo 5, which can be seen as follows: Since they are indivisible by 5, x, y and z cannot equal 0 modulo 5, and must equal one of four possibilities: ±1 or ±2. If they were all different, two would be opposites and their sum modulo 5 would be zero (implying contrary to the assumption of this case that the other one would be 0 modulo 5). Without loss of generality, x and y can be designated as the two equivalent numbers modulo 5. That equivalence implies that : x5 ≡ y5 (mod 25) (note change in modulo) : −z5 ≡ x5 \+ y5 ≡ 2 x5 (mod 25) However, the equation x ≡ y (mod 5) also implies that : −z ≡ x + y ≡ 2 x (mod 5) : −z5 ≡ 25 x5 ≡ 32 x5 (mod 25) Combining the two results and dividing both sides by x5 yields a contradiction : 2 ≡ 32 (mod 25) Thus, case A for n = 5 has been proven.
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The Fibonacci sequence itself is the first row, and a shift of the Lucas sequence is the second row.. See also Fibonacci integer sequences modulo .
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If an array is used to represent a cycle, it is convenient to obtain the index with a modulo function, which can result in zero.
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We can find quadratic residues or verify them using the above formula. To test if 2 is a quadratic residue modulo 17, we calculate 2(17 − 1)/2 = 28 ≡ 1 (mod 17), so it is a quadratic residue. To test if 3 is a quadratic residue modulo 17, we calculate 3(17 − 1)/2 = 38 ≡ 16 ≡ −1 (mod 17), so it is not a quadratic residue.
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A theorem of Johannes van der Corput states that if for each h the sequence sn+h − sn is uniformly distributed modulo 1, then so is sn.Kuipers & Niederreiter (2006) p. 26Montgomery (1994) p.18 A van der Corput set is a set H of integers such that if for each h in H the sequence sn+h − sn is uniformly distributed modulo 1, then so is sn.
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Prime numbers are frequently used for hash tables. For instance the original method of Carter and Wegman for universal hashing was based on computing hash functions by choosing random linear functions modulo large prime numbers. Carter and Wegman generalized this method to k-independent hashing by using higher-degree polynomials, again modulo large primes. For k-independent hashing see problem 11–4, p. 251.
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Trivially 1 is a quadratic residue for all primes. The question becomes more interesting for −1. Examining the table, we find −1 in rows 5, 13, 17, 29, 37, and 41 but not in rows 3, 7, 11, 19, 23, 31, 43 or 47. The former set of primes are all congruent to 1 modulo 4, and the latter are congruent to 3 modulo 4.
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When applied to two nodes in a network whose data are in close range of each other modulo-N code requires one node (say odd) to send the coded data value as the raw data M_o = D_o; the even node is required to send the coded data as the M_e = D_e \bmod N . Hence the name modulo-N code. Since at least \log_2 K bits are required to represent a number K in binary, the modulo coded data of the two nodes requires \log_2 M_o + \log_2 M_e bits. As we can generally expect \log_2 M_e \le \log_2 M_o always, because M_e \le N. This is the how compression is achieved.
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In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p2, p3, and so on.
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Therefore, VSH can be useful in embedded environments where code space is limited. Two major variants of VSH were proposed. For one, finding a collision is as difficult as finding a nontrivial modular square root of a very smooth number modulo n. The other one uses a prime modulus p (with no trapdoor), and its security proof relies on the hardness of finding discrete logarithms of very smooth numbers modulo p.
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For all primes up to , only in two cases: and , where is the number of vertices in the cycle of 1 in the doubling diagram modulo . Here the doubling diagram represents the directed graph with the non-negative integers less than m as vertices and with directed edges going from each vertex x to vertex 2x reduced modulo m. It was shown, that for all odd prime numbers either or .
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Some checksum methods are based on the mathematics of prime numbers. For instance the checksums used in International Standard Book Numbers are defined by taking the rest of the number modulo 11, a prime number. Because 11 is prime this method can detect both single-digit errors and transpositions of adjacent digits. Another checksum method, Adler-32, uses arithmetic modulo 65521, the largest prime number less than 2^{16}.
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Modulo a prime, there is only the subgroup of squares and a single coset. The fact that, e.g., modulo 15 the product of the nonresidues 3 and 5, or of the nonresidue 5 and the residue 9, or the two residues 9 and 10 are all zero comes from working in the full ring Z/nZ, which has zero divisors for composite n. For this reason some authorse.g.
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See also the sieve algorithm for all such primes: The terminology is ambiguous: "Euler's lucky numbers" are neither the same as, neither related to the "lucky numbers" defined by a sieve algorithm. In fact, the only number which is both lucky and Euler-lucky is 3, since all other Euler-lucky numbers are congruent to 2 modulo 3, but no lucky numbers are congruent to 2 modulo 3.
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A fraction in lowest terms with a prime denominator other than 2 or 5 (i.e. coprime to 10) always produces a repeating decimal. The length of the repetend (period of the repeating decimal segment) of is equal to the order of 10 modulo p. If 10 is a primitive root modulo p, the repetend length is equal to p − 1; if not, the repetend length is a factor of p − 1\.
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Umbertide is an important centre of automotive factories. There is the headquarters of Tiberina holding, a car components group. Other important companies are Proma, Modulo and Terex Genie.
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The even numbers form an ideal in the ring of integers,. but the odd numbers do not -- this is clear from the fact that the identity element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2.. All known perfect numbers are even; it is unknown whether any odd perfect numbers exist.. Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers.
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In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number , then this root corresponds to a unique root of the same equation modulo any higher power of , which can be found by iteratively "lifting" the solution modulo successive powers of . More generally it is used as a generic name for analogues for complete commutative rings (including p-adic fields in particular) of Newton's method for solving equations. Since p-adic analysis is in some ways simpler than real analysis, there are relatively neat criteria guaranteeing a root of a polynomial.
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Semi-log plot of solutions of x^3+y^3+z^3=n for integer x, y, and z, and 0\le n\le 100. Green bands denote values of n proven not to have a solution. In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum. A necessary condition for n to equal such a sum is that n cannot equal 4 or 5 modulo 9, because the cubes modulo 9 are 0, 1, and −1, and no three of these numbers can sum to 4 or 5 modulo 9.
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In fact any set of integers which are in distinct residue classes modulo may be used as -adic digits. In number theory, Teichmüller representatives are sometimes used as digits.
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Addition using the polynomial basis is as simple as addition modulo p. For example, in GF(3m): :(2\alpha^2 + 2\alpha + 1) + (2\alpha + 2) = 2\alpha^2 + 4\alpha + 3 = 2\alpha^2 + \alpha \pmod 3 In GF(2m), addition is especially easy, since addition and subtraction modulo 2 are the same thing (so like terms "cancel out"), and furthermore this operation can be done in hardware using the basic XOR logic gate.
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The most direct method of calculating a modular exponent is to calculate directly, then to take this number modulo . Consider trying to compute , given , , and : : One could use a calculator to compute 413; this comes out to 67,108,864. Taking this value modulo 497, the answer is determined to be 445. Note that is only one digit in length and that is only two digits in length, but the value is 8 digits in length.
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If the receiver finds y is neither x nor −x modulo N, the receiver will be able to factor N and therefore decrypt me to recover m (see Rabin encryption for more details). However, if y is x or −x mod N, the receiver will have no information about m beyond the encryption of it. Since every quadratic residue modulo N has four square roots, the probability that the receiver learns m is 1/2.
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The first few Eisenstein primes that equal a natural prime are: : 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ... . Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes: they admit nontrivial factorizations in Z[ω]. For example: : : . In general, if a natural prime p is 1 modulo 3 and can therefore be written as , then it factorizes over Z[ω] as : .
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It is a straightforward exercise to show that, under multiplication, the set of congruence classes modulo n that are coprime to n satisfy the axioms for an abelian group. Indeed, a is coprime to n if and only if . Integers in the same congruence class satisfy , hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined.
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Without the second modulo operation, the calculation could result in a check digit value of = 11, which is invalid. (Strictly speaking, the first "modulo 11" is not needed, but it may be considered to simplify the calculation.) For example, the check digit for the ISBN-10 of 0-306-40615-? is calculated as follows: Thus the check digit is 2. It is possible to avoid the multiplications in a software implementation by using two accumulators.
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The larger neighbors of vertex 1 are all vertices with numbers congruent to 2 or 3 modulo 4, because those are exactly the numbers with a nonzero bit at index 1.; .
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The following formulas provide the number of padding bytes required to align the start of a data structure (where mod is the modulo operator): padding = (align - (offset mod align)) mod align aligned = offset + padding = offset + ((align - (offset mod align)) mod align) For example, the padding to add to offset 0x59d for a 4-byte aligned structure is 3. The structure will then start at 0x5a0, which is a multiple of 4. However, when the alignment of offset is already equal to that of align, the second modulo in (align - (offset mod align)) mod align will return zero, therefore the original value is left unchanged. Since the alignment is by definition a power of two, the modulo operation can be reduced to a bitwise boolean AND operation.
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Informally, super Minkowski space can be thought of as the super Poincaré algebra modulo the algebra of the Lorentz group, in the same way that ordinary Minkowski spacetime can be viewed as the cosets of the ordinary Poincaré algebra modulo the action of the Lorentz algebra. The coset space is naturally affine, (lacking an origin) and a nilpotent anti-commuting behavior of the fermionic directions arises naturally from the Clifford algebra associated with the Lorentz group.
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The extended Euclidean algorithm is particularly useful when a and b are coprime. With that provision, x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Similarly, the polynomial extended Euclidean algorithm allows one to compute the multiplicative inverse in algebraic field extensions and, in particular in finite fields of non prime order. It follows that both extended Euclidean algorithms are widely used in cryptography.
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The map :x \bmod N \mapsto (x \bmod n_1, \ldots, x\bmod n_k) maps congruence classes modulo to sequences of congruence classes modulo . The proof of uniqueness shows that this map is injective. As the domain and the codomain of this map have the same number of elements, the map is also surjective, which proves the existence of the solution. This proof is very simple but does not provide any direct way for computing a solution.
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If the sign of the offset is alternated (e.g. +1, −4, +9, −16, etc.), and if the number of buckets is a prime number p congruent to 3 modulo 4 (e.g. 3, 7, 11, 19, 23, 31, etc.), then the first p offsets will be unique (modulo p). In other words, a permutation of 0 through p-1 is obtained, and, consequently, a free bucket will always be found as long as at least one exists.
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The Modulo has an extremely low and wedge- shaped body, with a canopy-style glass roof that slides forward to permit entry to the cabin of the car. All four wheels are partly covered. Another special feature of the design are 24 holes in the engine cover that reveal the Ferrari V12 engine which develops to propel the Modulo to a top speed of around and from 0–60 mph (97 km/h) in approximately 3.0 seconds.
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Another application that often involves DPLL is automated theorem proving or satisfiability modulo theories (SMT), which is a SAT problem in which propositional variables are replaced with formulas of another mathematical theory.
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It consists of a (modified) inner product between the message and a key modulo a prime p. The construction of MMH works in the finite field F_p for some prime integer p.
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The power of the AC classes can be affected by adding additional gates. If we add gates which calculate the modulo operation for some modulus m, we have the classes ACCi[m].
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Honda Cars Philippines Inc. introduced the CR-Z hybrid coupe to the Philippine market in August 2013. The Honda CR-Z is offered in three variants; a standard trim, Modulo, and Mugen.
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There are several common algorithms for hashing integers. The method giving the best distribution is data-dependent. One of the simplest and most common methods in practice is the modulo division method.
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A similar construction using the moment curve modulo a prime number, but in two dimensions rather than three, provides a linear bound for the no-three-in-line problem.Credited by to Paul Erdős.
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As is the case with many identification numbers, the TFN includes a check digit for detecting erroneous numbers. The algorithm is based on simple modulo 11 arithmetic per many other digit checksum schemes.
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This product of moduli is the largest of any of the n choose possible products, therefore any subset of equivalences can be any integer modulo its product, and no information from is leaked.
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Annals of Mathematics, Vol. 3, No. 4 (Aug., 1887), pp. 97–103 In particular, it follows that if and only if p is a prime and 10 is a primitive root modulo p.
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Constructing the sum not over all primes but only over primes which are in the same modulo class introduces further types of infinite series that are a reduction of the Dirichlet L-function.
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Consider the group of integers Z (under addition) and the subgroup 2Z consisting of all even integers. This is a normal subgroup, because Z is abelian. There are only two cosets: the set of even integers and the set of odd integers, and therefore the quotient group Z/2Z is the cyclic group with two elements. This quotient group is isomorphic with the set with addition modulo 2; informally, it is sometimes said that Z/2Z equals the set with addition modulo 2.
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The proof uses the fact that the residue classes modulo a prime number are a field. See the article prime field for more details. Because the modulus is prime, Lagrange's theorem applies: a polynomial of degree can only have at most roots. In particular, has at most 2 solutions for each . This immediately implies that besides 0, there are at least distinct quadratic residues modulo : each of the possible values of can only be accompanied by one other to give the same residue.
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The complete packet was thus 132 bytes long, containing 128 bytes of payload data, for a total channel efficiency of about 97%. The checksum was the sum of all bytes in the packet modulo 256. The modulo operation was easily computed by discarding all but the eight least significant bits of the result, or alternatively on an eight-bit machine, ignoring arithmetic overflow which would produce the same effect automatically. In this way, the checksum was restricted to an eight-bit quantity.
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The list of the number of quadratic residues modulo n, for n = 1, 2, 3 ..., looks like: :1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, 4, 9, 8, 10, 6, 8, 12, 12, 6, 11, 14, 11, 8, 15, 12, 16, 7, 12, 18, 12, 8, 19, 20, 14, 9, 21, 16, 22, 12, 12, 24, 24, 8, 22, 22, ... A formula to count the number of squares modulo n is given by Stangl.
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In set theory, the random algebra or random real algebra is the Boolean algebra of Borel sets of the unit interval modulo the ideal of measure zero sets. It is used in random forcing to add random reals to a model of set theory. The random algebra was studied by John von Neumann in 1935 (in work later published as ) who showed that it is not isomorphic to the Cantor algebra of Borel sets modulo meager sets. Random forcing was introduced by .
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The complete Cantor algebra is the complete Boolean algebra of Borel subsets of the reals modulo meager sets . It is isomorphic to the completion of the countable Cantor algebra. (The complete Cantor algebra is sometimes called the Cohen algebra, though "Cohen algebra" usually refers to a different type of Boolean algebra.) The complete Cantor algebra was studied by von Neumann in 1935 (later published as ), who showed that it is not isomorphic to the random algebra of Borel subsets modulo measure zero sets.
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The Dirichlet L-functions may be written as a linear combination of the Hurwitz zeta- function at rational values. Fixing an integer k ≥ 1, the Dirichlet L-functions for characters modulo k are linear combinations, with constant coefficients, of the ζ(s,q) where q = m/k and m = 1, 2, ..., k. This means that the Hurwitz zeta-function for rational q has analytic properties that are closely related to the Dirichlet L-functions. Specifically, let χ be a character modulo k.
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A de Bruijn's 3-dimensional graph Koorde is based on Chord but also on De Bruijn graph (De Bruijn sequence). In a d-dimensional de Bruijn graph, there are 2d nodes, each of which has a unique d-bit ID. The node with ID i is connected to nodes 2i modulo 2d and 2i+1 modulo 2d. Thanks to this property, the routing algorithm can route to any destination in d hops by successively "shifting in" the bits of the destination ID but only if the dimensions of the distance between modulo 1d and 3d are equal. Routing a message from node m to node k is accomplished by taking the number m and shifting in the bits of k one at a time until the number has been replaced by k.
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If n is composite, there exists a subgroup of the multiplicative group, called the "group of false witnesses", in which the elements, when raised to the power , are congruent to 1 modulo n (since the residue 1, to any power, is congruent to 1 modulo n, the set of such elements is nonempty). One could say, because of Fermat's Little Theorem, that such residues are "false positives" or "false witnesses" for the primality of n. The number 2 is the residue most often used in this basic primality check, hence is famous since 2340 is congruent to 1 modulo 341, and 341 is the smallest such composite number (with respect to 2). For 341, the false witnesses subgroup contains 100 residues and so is of index 3 inside the 300 element multiplicative group mod 341.
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Finding the points on an elliptic curve modulo a given prime p is conceptually straightforward, as there are only a finite number of possibilities to check. However, for large primes it is computationally intensive.
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Colombeau's construction: see Colombeau algebra. These are factor spaces :G = M / N of "moderate" modulo "negligible" nets of functions, where "moderateness" and "negligibility" refers to growth with respect to the index of the family.
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Thus arithmetic modulo 12 is used to represent octave equivalence. One advantage of this system is that it ignores the "spelling" of notes (B, C and D are all 0) according to their diatonic functionality.
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In the case of non-orientable manifolds, every homology class of H_n(X,\Z_2), where \Z_2 denotes the integers modulo 2, can be realized by a non-oriented manifold, f\colon M^n\to X.
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Conversely, because finding square roots modulo a composite number turns out to be probabilistic polynomial-time equivalent to factoring that number, any integer factorization algorithm can be used efficiently to identify a congruence of squares.
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The following laws can be verified using the properties of divisibility. They are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, and multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic.
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The check digit is chosen so that the sum of all digits in the bar code is a multiple of 10. Equivalently, the modulo-10 sum is 0. To calculate the check digit: #Add up the digits. For example, if a letter is sent to Young America, Minnesota, it might be sent to 55555-1237, which would have the sum of 38. #Find the remainder of this number when it is divided by 10, in this case 8. This is also known as the sum modulo 10.
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In cryptography, mod n cryptanalysis is an attack applicable to block and stream ciphers. It is a form of partitioning cryptanalysis that exploits unevenness in how the cipher operates over equivalence classes (congruence classes) modulo n. The method was first suggested in 1999 by John Kelsey, Bruce Schneier, and David Wagner and applied to RC5P (a variant of RC5) and M6 (a family of block ciphers used in the FireWire standard). These attacks used the properties of binary addition and bit rotation modulo a Fermat prime.
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He also studied, under the form of integer polynomials modulo both a prime number and an irreducible polynomial (remaining irreducible modulo that prime number), what can nowadays be recognized as finite fields (more general than those of prime order).David A. Cox, "Why Eisenstein proved the Eisenstein criterion and why Schönemann discovered it first", American Mathematical Monthly 118 Vol 1, January 2011, pp. 3–31. See p. 10. He was educated in Königsberg and Berlin, where among his teachers were Jakob Steiner and Carl Gustav Jacob Jacobi.
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The Lin–Tsien equation (named after C. C. Lin and H. S. Tsien) is an integrable partial differential equation : 2u_{tx}+u_xu_{xx}-u_{yy} = 0. Integrability of this equation follows from its being, modulo an appropriate linear change of dependent and independent variables, a potential form of the dispersionless KP equation. Namely, if u satisfies the Lin–Tsien equation, then v=u_x satisfies, modulo the said change of variables, the dispersionless KP equation. The Lin-Tsien equation admits a (3+1)-dimensional integrable generalization, see.
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5 is in rows 11, 19, 29, 31, and 41 but not in rows 3, 7, 13, 17, 23, 37, 43, or 47. The former are ≡ ±1 (mod 5) and the latter are ≡ ±2 (mod 5). Since the only residues (mod 5) are ±1, we see that 5 is a quadratic residue modulo every prime which is a residue modulo 5. −5 is in rows 3, 7, 23, 29, 41, 43, and 47 but not in rows 11, 13, 17, 19, 31, or 37.
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On a manifold that is sufficiently smooth, various kinds of jet bundles can also be considered. The (first-order) tangent bundle of a manifold is the collection of curves in the manifold modulo the equivalence relation of first-order contact. By analogy, the k-th order tangent bundle is the collection of curves modulo the relation of k-th order contact. Likewise, the cotangent bundle is the bundle of 1-jets of functions on the manifold: the k-jet bundle is the bundle of their k-jets.
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In mathematics, a measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the Boolean algebra of measurable sets modulo null sets.
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There are no more than cosets, because . The coset is the congruence class of modulo . The subgroup is normal in , and so, can be used to form the quotient group the group of integers mod m.
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The triple cover has a complex representation of dimension 783. When reduced modulo 3 this has 1-dimensional invariant subspaces and quotient spaces, giving an irreducible representation of dimension 781 over the field with 3 elements.
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A simple way to combine the two steps is to sum the digits without a tens column at all, but discard all carries. #Subtract the sum modulo 10 from 10. Continuing with the example, 10 − 8 = 2.
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The sequence generated is 1, 7, 21, 107, 273, 1911, 5189, 28123, ... . This can be obtained by taking the coefficients of the successive powers of (1+x+x2) modulo 2 and interpreting them as integers in binary.
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A and B are maximal length LFSRs. The modulo operations correspond to resets. Note that both are reset each millisecond (synchronized with C/A code epochs). In addition, the extra modulo operation in the description of A is due to the fact it is reset 1 cycle before its natural period (which is 8,191) so that the next repetition becomes offset by 1 cycle with respect to B (otherwise, since both sequences would repeat, I5 and Q5 would repeat within any 1 ms period as well, degrading correlation characteristics).
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Example 1: Finding primes for which a is a residue Let a = 17. For which primes p is 17 a quadratic residue? We can test prime p's manually given the formula above. In one case, testing p = 3, we have 17(3 − 1)/2 = 171 ≡ 2 ≡ −1 (mod 3), therefore 17 is not a quadratic residue modulo 3. In another case, testing p = 13, we have 17(13 − 1)/2 = 176 ≡ 1 (mod 13), therefore 17 is a quadratic residue modulo 13. As confirmation, note that 17 ≡ 4 (mod 13), and 22 = 4.
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These numbers are prime if and only if the number of solutions to is odd and the number is squarefree (proven as theorem 6.2 of). All numbers with modulo-sixty remainder 11, 23, 47, or 59 have a modulo-twelve remainder of 11. These numbers are prime if and only if the number of solutions to is odd and the number is squarefree (proven as theorem 6.3 of). None of the potential primes are divisible by 2, 3, or 5, so they can't be divisible by their squares.
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In theoretical mathematics, modular arithmetic is one of the foundations of number theory, touching on almost every aspect of its study, and it is also used extensively in group theory, ring theory, knot theory, and abstract algebra. In applied mathematics, it is used in computer algebra, cryptography, computer science, chemistry and the visual and musical arts. A very practical application is to calculate checksums within serial number identifiers. For example, International Standard Book Number (ISBN) uses modulo 11 (for 10 digit ISBN) or modulo 10 (for 13 digit ISBN) arithmetic for error detection.
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Every cycle graph is a circulant graph, as is every crown graph with vertices. The Paley graphs of order (where is a prime number congruent to ) is a graph in which the vertices are the numbers from 0 to and two vertices are adjacent if their difference is a quadratic residue modulo . Since the presence or absence of an edge depends only on the difference modulo of two vertex numbers, any Paley graph is a circulant graph. Every Möbius ladder is a circulant graph, as is every complete graph.
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The case of elliptic curves was worked out by Hasse in 1934. Since the genus is 1, the only possibilities for the matrix H are: H is zero, Hasse invariant 0, p-rank 0, the supersingular case; or H non-zero, Hasse invariant 1, p-rank 1, the ordinary case. Here there is a congruence formula saying that H is congruent modulo p to the number N of points on C over F, at least when q = p. Because of Hasse's theorem on elliptic curves, knowing N modulo p determines N for p ≥ 5.
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She can use her own private key to do so. She produces a hash value of the message, raises it to the power of d (modulo n) (as she does when decrypting a message), and attaches it as a "signature" to the message. When Bob receives the signed message, he uses the same hash algorithm in conjunction with Alice's public key. He raises the signature to the power of e (modulo n) (as he does when encrypting a message), and compares the resulting hash value with the message's actual hash value.
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Then the space of flat connections on \Sigma modulo gauge equivalence is a symplectic manifold M(\Sigma) of dimension 6g − 6, where g is the genus of the surface \Sigma. In the Heegaard splitting, \Sigma bounds two different 3-manifolds; the space of flat connections modulo gauge equivalence on each 3-manifold with boundary embeds into M(\Sigma) as a Lagrangian submanifold. One can consider the Lagrangian intersection Floer homology. Alternately, we can consider the Instanton Floer homology of the 3-manifold Y. The Atiyah–Floer conjecture asserts that these two invariants are isomorphic.
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Both I and S frames contain a receive sequence number N(R). N(R) provides a positive acknowledgement for the receipt of I-frames from the other side of the link. Its value is always the first frame not yet received; it acknowledges that all frames with N(S) values up to N(R)−1 (modulo 8 or modulo 128) have been received and indicates the N(S) of the next frame it expects to receive. N(R) operates the same way whether it is part of a command or response.
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The decision problem for free theories is particularly important, as many theories can be reduced to it. Free theories can be solved by searching for common subexpressions to form the congruence closure. Solvers include satisfiability modulo theories solvers.
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Every unital ring may be viewed as an additive category with a single object, and the quotient of additive categories defined above coincides in this case with the notion of a quotient ring modulo a two-sided ideal.
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Modulo 8, the product of the nonresidues 3 and 5 is the nonresidue 7, and likewise for permutations of 3, 5 and 7. In fact, the multiplicative group of the non-residues and 1 form the Klein four-group.
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However, it is possible that two previously unequal integers will be identical modulo 9 (on average, a ninth of the time). The operation does not work on fractions, since a given fractional number does not have a unique representation.
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However, it is not known whether such a polyhedron can be realized geometrically (rather than as an abstract polytope). More generally this equation can be satisfied precisely when f is congruent to 0, 3, 4, or 7 modulo 12.
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The coordinate σ is only defined modulo 2π, and is best taken to range from -π to π, by taking it as the negative of the acute angle F1 P F2 if P is in the lower half plane.
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Kunio MurasugiMurasugi, Kunio, The Arf Invariant for Knot Types, Proceedings of the American Mathematical Society, Vol. 21, No. 1. (Apr., 1969), pp. 69-72 proved that the Arf invariant is zero if and only if Δ(−1) \equiv ±1 modulo 8\.
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An automaton that counts the length of its input modulo can be used to distinguish the two strings from each other in this case. Therefore, strings of unequal lengths can always be distinguished from each other by automata with few states.
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The sequence generated is 1, 3, 5, 15, 17, 51, 85, 255, .... This can be obtained by taking successive rows of Pascal's triangle modulo 2 and interpreting them as integers in binary, which can be graphically represented by a Sierpinski triangle.
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6 (1925): 281–284. in 1928 at Bologna,Glenn, O. E. "The complex realm modulo n, an arbitrary integer." In Atti del Congresso Internazionale dei Matematici: Bologna del 3 al 10 de settembre di 1928, vol. 2, pp. 43–50. 1929.
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Unit fractions play an important role in modular arithmetic, as they may be used to reduce modular division to the calculation of greatest common divisors. Specifically, suppose that we wish to perform divisions by a value x, modulo y. In order for division by x to be well defined modulo y, x and y must be relatively prime. Then, by using the extended Euclidean algorithm for greatest common divisors we may find a and b such that :\displaystyle ax + by = 1, from which it follows that :\displaystyle ax \equiv 1 \pmod y, or equivalently :a \equiv \frac1x \pmod y.
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The caravan was specifically made for the Tokyo Auto Salon and is a cross between a commercial can and a leisure cruising vehicle. Nissan also showed up to the Auto Salon with race cars like the Motul Autech GT-R. Honda, a Japanese motor company, showed up to the Tokyo Auto Salon with a series of 10 vehicles in the N-One mini family. Models included the N-One Modulo style, Mugen Racing N-One Concept, and an N-One model created by the Japan Nailist Association. Honda also brought with them an “exhibition model” S2000 Modulo.
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First, there are elements \alpha_i representing small loops around the punctures P_i. Then there are elements \beta_j that are coming from the first homology of the compactification of each of the components. The one-cycle in X_k \subset X (k=1,2) corresponding to a cycle in the compactification of this component, is not canonical: these elements are determined modulo the span of \alpha_1,\dots ,\alpha_n. Finally, modulo the first two types, the group is generated by a combinatorial cycle \gamma which goes from Q_1 to Q_2along a path in one component X_1 and comes back along a path in the other component X_2.
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This is the original Lehmer RNG construction. The period is m−1 if the multiplier a is chosen to be a primitive element of the integers modulo m. The initial state must be chosen between 1 and m−1. One disadvantage of a prime modulus is that the modular reduction requires a double-width product and an explicit reduction step. Often a prime just less than a power of 2 is used (the Mersenne primes 231−1 and 261−1 are popular), so that the reduction modulo m = 2e − d can be computed as (ax mod 2e) + d .
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In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, and It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares. Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4). However, there are many values of v that are allowed, for which the existence of a conference graph is unknown.
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The first constraint is that only even permutations of the face centers are possible (e.g. it is impossible to have only two face centre pieces swapped); this divides the limit by 2. The second constraint is that all centre permutations are dependent on the orientation of the corner pieces. Some permutations of centres are only possible when the total number of clockwise rotations of corner pieces is divisible by 3; other permutations are only possible when the total number of clockwise rotations is equivalent to 1 modulo 3; others are only possible when the number is equivalent to 2 modulo 3.
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In fact, :2 + 3X = a + bX does not hold unless a = 2 and b = 3. This is because X is not, and does not designate, a number. The distinction is subtle, since a polynomial in X can be changed to a function in x by substitution. But the distinction is important because information may be lost when this substitution is made. For example, when working in modulo 2, we have that: :0 - 0^2 = 0, \quad 1 - 1^2 = 0, so the polynomial function x - x^2 is identically equal to 0 for x having any value in the modulo-2 system.
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Bézout's identity asserts that and are coprime if and only if there exist integers and such that :ns+at=1 Reducing this identity modulo gives :at \equiv 1 \mod n. Thus , or, more exactly, the remainder of the division of by , is the multiplicative inverse of modulo . To adapt the extended Euclidean algorithm to this problem, one should remark that the Bézout coefficient of is not needed, and thus does not need to be computed. Also, for getting a result which is positive and lower than n, one may use the fact that the integer provided by the algorithm satisfies .
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1; } In Java, `%` is the remainder operator (modulo), and in Java, if its first operand is negative, the result can also be negative (unlike the modulo used in mathematics). Here, the programmer has assumed that `total` is non-negative, so that the remainder of a division with 2 will always be 0 or 1. The assertion makes this assumption explicit: if `countNumberOfUsers` does return a negative value, the program may have a bug. A major advantage of this technique is that when an error does occur it is detected immediately and directly, rather than later through often obscure effects.
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András Ádám conjectured that these linear maps are the only ways of renumbering a circulant graph while preserving the circulant property: that is, if and are isomorphic circulant graphs, with different numberings, then there is a linear map that transforms the numbering for into the numbering for . However, Ádám's conjecture is now known to be false. A counterexample is given by graphs and with 16 vertices each; a vertex in is connected to the six neighbors , , and modulo 16, while in the six neighbors are , , and modulo 16. These two graphs are isomorphic, but their isomorphism cannot be realized by a linear map.
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If p and q are two prime divisors of n, then implies the same equation also and These two smaller elliptic curves with the \boxplus-addition are now genuine groups. If these groups have Np and Nq elements, respectively, then for any point P on the original curve, by Lagrange's theorem, is minimal such that kP=\infty on the curve modulo p implies that k divides Np; moreover, N_p P=\infty. The analogous statement holds for the curve modulo q. When the elliptic curve is chosen randomly, then Np and Nq are random numbers close to and respectively (see below).
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The Tonelli–Shanks algorithm (referred to by Shanks as the RESSOL algorithm) is used in modular arithmetic to solve for r in a congruence of the form r2 ≡ n (mod p), where p is a prime: that is, to find a square root of n modulo p. Tonelli–Shanks cannot be used for composite moduli: finding square roots modulo composite numbers is a computational problem equivalent to integer factorization.Oded Goldreich, Computational complexity: a conceptual perspective, Cambridge University Press, 2008, p. 588. An equivalent, but slightly more redundant version of this algorithm was developed by Alberto Tonelli in 1891.
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Consider the abelian group (that is, the set with addition modulo 4), and its subgroup . The quotient group is . This is a group with identity element , and group operations such as . Both the subgroup and the quotient group are isomorphic with Z2.
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Clube Atlético Tricordiano, or simply Tricordiano, is a currently inactive Brazilian football team from Três Corações, Minas Gerais, founded on May 13, 2008. In 2016, they competed in the Campeonato Mineiro - Módulo I, however in 2017 they were relegated to Modulo II.
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In the terminology of abstract algebra, the ability to perform division means that modular arithmetic modulo a prime number forms a field or, more specifically, a finite field, while other moduli only give a ring but not a field., pp. 27–28.
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The numerator of V_{2k+1} is handled in the same way. (Adding n does not change the result modulo n.) Observe that, for each term that we compute in the U sequence, we compute the corresponding term in the V sequence.
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In algebra, an SBI ring is a ring R (with identity) such that every idempotent of R modulo the Jacobson radical can be lifted to R. The abbreviation SBI was introduced by Irving Kaplansky and stands for "suitable for building idempotent elements" .
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Conversely, Alexander Beilinson proved that the existence of a category of motives implies the standard conjectures. Additionally, cycles are connected to algebraic K-theory by Bloch's formula, which expresses groups of cycles modulo rational equivalence as the cohomology of K-theory sheaves.
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The Tonelli–Shanks algorithm can (naturally) be used for any process in which square roots modulo a prime are necessary. For example, it can be used for finding points on elliptic curves. It is also useful for the computations in the Rabin cryptosystem.
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Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears. Furthermore, when the order is large, Fast Fourier Transformation can be used to solve for the coefficients of the interpolated polynomial.
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The known pattern to this sequence comes from algebraic number theory, specifically, this sequence is the set of primes p such that 10 is a primitive root modulo p. Artin's conjecture on primitive roots is that this sequence contains 37.395..% of the primes.
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In 2019, Karthikeyan raced with Nakajima Racing for one race, winning the Fuji Super GT x DTM Dream Race in Fuji, gaining the fastest lap during the race. He then raced with Modulo Epson NSX-GT Team for 8 races, gaining one podium.
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In cryptography, differential equations of addition (DEA) are one of the most basic equations related to differential cryptanalysis that mix additions over two different groups (e.g. addition modulo 232 and addition over GF(2)) and where input and output differences are expressed as XORs.
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For encryption, a square modulo n must be calculated. This is more efficient than RSA, which requires the calculation of at least a cube. For decryption, the Chinese remainder theorem is applied, along with two modular exponentiations. Here the efficiency is comparable to RSA.
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In unibit PLLs, the output frequency is defined by the input frequency and the modulo count of the two counters. In each counter, only the most significant bit (MSB) is used. The other output lines of the counters are ignored; this is wasted information.
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From the definition of , it follows that . For example, given , and , the solution is the remainder of dividing by . Modular exponentiation can be performed with a negative exponent by finding the modular multiplicative inverse of modulo using the extended Euclidean algorithm. That is: :, where and .
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An alternate (algebraic) view of this construction is as follows. The points of this projective plane are the equivalence classes of the set modulo the equivalence relation :x ~ kx, for all k in K×. Lines in the projective plane are defined exactly as above.
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Begin loop Break loop End loop; The algorithm first finds the largest value among the and then the supremum within the set of . Then it raises to the power , multiplies this value with , and then assigns the result of this computation and the value modulo .
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Dirichlet's theorem states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d.
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A theory is satisfiable when it is possible to present an interpretation in which all of its sentences are true. The study of algorithms to automatically discover interpretations of theories that render all sentences as being true is known as the satisfiability modulo theories problem.
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Each of the first nine digits of the 10-digit ISBN—excluding the check digit itself—is multiplied by its (integer) weight, descending from 10 to 2, and the sum of these nine products found. The value of the check digit is simply the one number between 0 and 10 which, when added to this sum, means the total is a multiple of 11. For example, the check digit for an ISBN-10 of 0-306-40615-? is calculated as follows: Adding 2 to 130 gives a multiple of 11 (because 132 = 12×11) – this is the only number between 0 and 10 which does so. Therefore, the check digit has to be 2, and the complete sequence is ISBN 0-306-40615-2. If the value of x_{10} required to satisfy this condition is 10, then an 'X' should be used. Alternatively, modular arithmetic is convenient for calculating the check digit using modulus 11. The remainder of this sum when it is divided by 11 (i.e. its value modulo 11), is computed. This remainder plus the check digit must equal either 0 or 11. Therefore, the check digit is (11 minus the remainder of the sum of the products modulo 11) modulo 11. Taking the remainder modulo 11 a second time accounts for the possibility that the first remainder is 0.
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This produces integer polynomials f_1(x),...,f_r(x) whose product matches f(x) mod p. Next, apply Hensel lifting; this updates the f_i(x) in such a way that their product matches f(x) mod p^a, where a is chosen in such a way that p^a is larger than 2B. Modulo p^a, the polynomial f(x) has (up to units) 2^r factors: for each subset of {f_1(x),...,f_r(x)}, the product is a factor of f(x) mod p^a. However, a factor modulo p^a need not correspond to a so-called "true factor": a factor of f(x) in Z[x].
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Both are written as exponentiation modulo a composite number, and both are related to the problem of prime factorization. Functions related to the hardness of the discrete logarithm problem (either modulo a prime or in a group defined over an elliptic curve) are not known to be trapdoor functions, because there is no known "trapdoor" information about the group that enables the efficient computation of discrete logarithms. A trapdoor in cryptography has the very specific aforementioned meaning and is not to be confused with a backdoor (these are frequently used interchangeably, which is incorrect). A backdoor is a deliberate mechanism that is added to a cryptographic algorithm (e.g.
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In field theory, a primitive element of a finite field is a generator of the multiplicative group of the field. In other words, is called a primitive element if it is a primitive th root of unity in ; this means that each non- zero element of can be written as for some integer . If is a prime number, the elements of can be identified with the integers modulo . In this case, a primitive element is also called a primitive root modulo For example, 2 is a primitive element of the field and , but not of since it generates the cyclic subgroup of order 3; however, 3 is a primitive element of .
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He extended the Dirac quantization condition to the dyon and used the model to predict the existence of a particle with the properties of the J/ψ meson prior to its discovery in 1974. The allowed charges of dyons are restricted by the Dirac quantization condition. This states in particular that their magnetic charge must be integral, and that their electric charges must all be equal modulo 1. The Witten effect, demonstrated by Edward Witten in his 1979 paper, states that the electric charges of dyons must all be equal, modulo one, to the product of their magnetic charge and the theta angle of the theory.
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Fermat's theorem on sums of two squares asserts that an odd prime number p can be expressed as : p = x^2 + y^2 with integer x and y if and only if p is congruent to 1 (mod 4). The statement was announced by Girard in 1625, and again by Fermat in 1640, but neither supplied a proof. The "only if" clause is easy: a perfect square is congruent to 0 or 1 modulo 4, hence a sum of two squares is congruent to 0, 1, or 2. An odd prime number is congruent to either 1 or 3 modulo 4, and the second possibility has just been ruled out.
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The cyclic group G = Z/3Z of congruence classes modulo 3 (see modular arithmetic) is simple. If H is a subgroup of this group, its order (the number of elements) must be a divisor of the order of G which is 3. Since 3 is prime, its only divisors are 1 and 3, so either H is G, or H is the trivial group. On the other hand, the group G = Z/12Z is not simple. The set H of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an abelian group is normal.
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A "harmonious labelling" on a graph G is an injection from the vertices of G to the group of integers modulo k, where k is the number of edges of G, that induces a bijection between the edges of G and the numbers modulo k by taking the edge label for an edge (x, y) to be the sum of the labels of the two vertices x, y (mod k). A "harmonious graph" is one that has a harmonious labelling. Odd cycles are harmonious, as are Petersen graphs. It is conjectured that trees are all harmonious if one vertex label is allowed to be reused.
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Martin and Monk proved in 1973 that AD implies the Wadge order for Baire space is well founded. Hence under AD, the Wadge classes modulo complements form a wellorder. The Wadge rank of a set A is the order type of the set of Wadge degrees modulo complements strictly below [A]W. The length of the Wadge hierarchy has been shown to be Θ. Wadge also proved that the length of the Wadge hierarchy restricted to the Borel sets is φω1(1) (or φω1(2) depending on the notation), where φγ is the γth Veblen function to the base ω1 (instead of the usual ω).
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There is a simple reduction from breaking this cryptosystem to the problem of determining whether a random value modulo N with Jacobi symbol +1 is a quadratic residue. If an algorithm A breaks the cryptosystem, then to determine if a given value x is a quadratic residue modulo N, we test A to see if it can break the cryptosystem using (x,N) as a public key. If x is a non- residue, then A should work properly. However, if x is a residue, then every "ciphertext" will simply be a random quadratic residue, so A cannot be correct more than half of the time.
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This method is faster if the moduli have been ordered by decreasing value, that is if n_1>n_2> \cdots > n_k. For the example, this gives the following computation. We consider first the numbers that are congruent to 4 modulo 5 (the largest modulus), which are 4, , , ... For each of them, compute the remainder by 4 (the second largest modulus) until getting a number congruent to 3 modulo 4. Then one can proceed by adding at each step, and computing only the remainders by 3. This gives :4 mod 4 → 0. Continue :4 + 5 = 9 mod 4 →1. Continue :9 + 5 = 14 mod 4 → 2. Continue :14 + 5 = 19 mod 4 → 3.
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In knot theory, a virtual knot is a generalization of knots in 3-dimensional Euclidean space, , to knots in thickened surfaces \Sigma \times [0,1] modulo an equivalence relation called stabilization/destabilization. Here \Sigma is required to be closed and oriented. Virtual knots were first introduced by .
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For an arbitrary integer n, the length λ(n) of the repetend of divides φ(n), where φ is the totient function. The length is equal to φ(n) if and only if 10 is a primitive root modulo n.William E. Heal. Some Properties of Repetends.
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Since the plane at infinity is a projective plane, it is homeomorphic to the surface of a "sphere modulo antipodes", i.e. a sphere in which antipodal points are equivalent: S2/{1,-1} where the quotient is understood as a quotient by a group action (see quotient space).
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Shift-register for the (7, [171, 133]) convolutional code polynomial. Branches: h^1 = 171_o = [1111001]_b, h^2 = 133_o = [1011011]_b. All of the math operations should be done by modulo 2. Theoretical bit-error rate curves of encoded QPSK (soft decision), additive white Gaussian noise channel.
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Although quadratic residues appear to occur in a rather random pattern modulo n, and this has been exploited in such applications as acoustics and cryptography, their distribution also exhibits some striking regularities. Using Dirichlet's theorem on primes in arithmetic progressions, the law of quadratic reciprocity, and the Chinese remainder theorem (CRT) it is easy to see that for any M > 0 there are primes p such that the numbers 1, 2, ..., M are all residues modulo p. > For example, if p ≡ 1 (mod 8), (mod 12), (mod 5) and (mod 28), then by the > law of quadratic reciprocity 2, 3, 5, and 7 will all be residues modulo p, > and thus all numbers 1-10 will be. The CRT says that this is the same as p ≡ > 1 (mod 840), and Dirichlet's theorem says there are an infinite number of > primes of this form. 2521 is the smallest, and indeed 12 ≡ 1, 10462 ≡ 2, > 1232 ≡ 3, 22 ≡ 4, 6432 ≡ 5, 872 ≡ 6, 6682 ≡ 7, 4292 ≡ 8, 32 ≡ 9, and 5292 ≡ > 10 (mod 2521).
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If a ciphertext is created this way, its creator would be aware, in some sense, of the plaintext. However, many cryptosystems are not plaintext-aware. As an example, consider the RSA cryptosystem without padding. In the RSA cryptosystem, plaintexts and ciphertexts are both values modulo N (the modulus).
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This follows from elementary group theory, because the exponent of any finite group must divide the order of the group. is the exponent of the multiplicative group of integers modulo while is the order of that group. We can thus view Carmichael's theorem as a sharpening of Euler's theorem.
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In this animation there are 8 ECT paths available from each highlighted source to destination and therefore services could be assigned to 8 different pools based on the VID. One such initial assignment in Figure 6 could therefore be (ISID modulo 8) with subsequent fine tuning as required.
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This follows from the fact that the sum of the labels of the vertices is twice the sum of the edges, modulo p. This is useful for disproving a graph is edge-graceful. For instance, one can apply this directly to the path and cycle examples given above.
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The number theoretic Hilbert transform is an extension () of the discrete Hilbert transform to integers modulo an appropriate prime number. In this it follows the generalization of discrete Fourier transform to number theoretic transforms. The number theoretic Hilbert transform can be used to generate sets of orthogonal discrete sequences().
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The quotients by congruence subgroups are of significant interest. Other important quotients are the triangle groups, which correspond geometrically to descending to a cylinder, quotienting the coordinate modulo , as . is the group of icosahedral symmetry, and the triangle group (and associated tiling) is the cover for all Hurwitz surfaces.
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The residues of a2 modulo p are distinct for every a between 0 and (p − 1)/2 (inclusive). To see this, take some a and define c as a2 mod p. a is a root of the polynomial over the field . So is (which is different from a).
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A stronger result was obtained: there is an infinity of E-irregular primes congruent to 1 modulo 8. As in the case of Kummer's B-regular primes, there is as yet no proof that there are infinitely many E-regular primes, though this seems likely to be true.
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OK, continue by considering remainders modulo 3 and adding 5×4 = 20 each time :19 mod 3 → 1. Continue :19 + 20 = 39 mod 3 → 0. OK, this is the result. This method works well for hand-written computation with a product of moduli that is not too big.
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The simplest nontrivial Vassiliev invariant of knots is given by the coefficient of the quadratic term of the Alexander–Conway polynomial. It is an invariant of order two. Modulo two, it is equal to the Arf invariant. Any coefficient of the Kontsevich invariant is a finite type invariant.
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In mathematics, modular forms are particular complex analytic functions on the upper half-plane of interest in complex analysis and number theory. When reduced modulo a prime p, there is an analogous theory to the classical theory of complex modular forms and the p-adic theory of modular forms.
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The sequence generated is 1, 5, 17, 85, 257, 1285, 4369, 21845, ... . This can be obtained by taking successive rows of Pascal's triangle modulo 2 and interpreting them as integers in base 4. Note that rules 18, 26, 82, 146, 154, 210 and 218 generate the same sequence.
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Multiplication in a finite field is multiplication modulo an irreducible reducing polynomial used to define the finite field. (I.e., it is multiplication followed by division using the reducing polynomial as the divisor--the remainder is the product.) The symbol "•" may be used to denote multiplication in a finite field.
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There are no official variations or sequels to the game. More mathematically skilled players may wish to include operations such as exponentiation, logarithms, and modulo, or incorporate sequences using any number of adjacent tiles in a line (such as 4 × 2 + 2 = 10, where only the 10 is played).
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If H forms a field under the usual addition and multiplication of functions then so will H modulo this equivalence relation under the induced addition and multiplication operations. Moreover, if every function in H is eventually differentiable and the derivative of any function in H is also in H then H modulo the above equivalence relation is called a Hardy field. Elements of a Hardy field are thus equivalence classes and should be denoted, say, [f]∞ to denote the class of functions that are eventually equal to the representative function f. However, in practice the elements are normally just denoted by the representatives themselves, so instead of [f]∞ one would just write f.
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If seventeen is excluded, then Theodorus's proof may have relied merely on considering whether numbers are even or odd. Indeed, Hardy and Wright and Knorr suggest proofs that rely ultimately on the following theorem: If x^2=ny^2 is soluble in integers, and n is odd, then n must be congruent to 1 modulo 8 (since x and y can be assumed odd, so their squares are congruent to 1 modulo 8). A possibility suggested earlier by Zeuthen is that Theodorus applied the so- called Euclidean algorithm, formulated in Proposition X.2 of the Elements as a test for incommensurability. In modern terms, the theorem is that a real number with an infinite continued fraction expansion is irrational.
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Some systems of PDEs have large symmetry groups. For example, the Yang–Mills equations are invariant under an infinite-dimensional gauge group, and many systems of equations (such as the Einstein field equations) are invariant under diffeomorphisms of the underlying manifold. Any such symmetry groups can usually be used to help study the equations; in particular if one solution is known one can trivially generate more by acting with the symmetry group. Sometimes equations are parabolic or hyperbolic "modulo the action of some group": for example, the Ricci flow equation is not quite parabolic, but is "parabolic modulo the action of the diffeomorphism group", which implies that it has most of the good properties of parabolic equations.
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The search of the solution may be made dramatically faster by sieving. For this method, we suppose, without loss of generality, that 0\le a_i (if it were not the case, it would suffice to replace each a_i by the remainder of its division by n_i). This implies that the solution belongs to the arithmetic progression :a_1, a_1 + n_1, a_1+2n_1, \ldots By testing the values of these numbers modulo n_2, one eventually finds a solution x_2 of the two first congruences. Then the solution belongs to the arithmetic progression :x_2, x_2 + n_1n_2, x_2+2n_1n_2, \ldots Testing the values of these numbers modulo n_3,, and continuing until every modulus has been tested gives eventually the solution.
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The choice of polynomial can dramatically affect the time to complete the remainder of the algorithm. The method of choosing polynomials based on the expansion of in base shown above is suboptimal in many practical situations, leading to the development of better methods. One such method was suggested by Murphy and Brent; they introduce a two-part score for polynomials, based on the presence of roots modulo small primes and on the average value that the polynomial takes over the sieving area. The best reported results were achieved by the method of Thorsten Kleinjung, which allows , and searches over composed of small prime factors congruent to 1 modulo 2 and over leading coefficients of which are divisible by 60.
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A congruence subgroup is (roughly) a subgroup of an arithmetic group defined by taking all matrices satisfying certain equations modulo an integer, for example the group of 2 by 2 integer matrices with diagonal (respectively off-diagonal) coefficients congruent to 1 (respectively 0) modulo a positive integer. These are always finite-index subgroups and the congruence subgroup problem roughly asks whether all subgroups are obtained in this way. The conjecture (usually attributed to Jean-Pierre Serre) is that this is true for (irreducible) arithmetic lattices in higher-rank groups and false in rank-one groups. It is still open in this generality but there are many results establishing it for specific lattices (in both its positive and negative cases).
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Although (0, 2, 4) is not admissible it does produce the single set of primes, (3, 5, 7). Some inadmissible k-tuples have more than one all-prime solution. This cannot happen for a k-tuple that includes all values modulo 3, so to have this property a k-tuple must cover all values modulo a larger prime, implying that there are at least five numbers in the tuple. The shortest inadmissible tuple with more than one solution is the 5-tuple (0, 2, 8, 14, 26), which has two solutions: (3, 5, 11, 17, 29) and (5, 7, 13, 19, 31) where all congruences (mod 5) are included in both cases.
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Gauss published the first and second proofs of the law of quadratic reciprocity on arts 125-146 and 262 of Disquisitiones Arithmeticae in 1801. In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: . }} This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form x^2\equiv a \bmod p for an odd prime p; that is, to determine the "perfect squares" modulo p.
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In algebraic geometry, the Néron–Severi group of a variety is the group of divisors modulo algebraic equivalence; in other words it is the group of components of the Picard scheme of a variety. Its rank is called the Picard number. It is named after Francesco Severi and André Néron.
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As one progresses from the MSB toward the least significant bit (LSB), the frequency increases. For a binary counter, each next bit is at twice the frequency of the previous one. For modulo counters, the relationship is more complicated. Only the MSB of the two counters are at the same frequency.
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Similar identities hold for these conventions. Many identities are used that are true modulo certain subgroups. These can be particularly useful in the study of solvable groups and nilpotent groups. For instance, in any group, second powers behave well: :(xy)^2 = x^2 y^2 [y, x] y, x], y].
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Tonelli; the algorithm requires O(log4n) steps. (in 1891) and Cipolla; the algorithm requires O(log3 n) steps and is also nondetermisitic. found efficient algorithms that work for all prime moduli. Both algorithms require finding a quadratic nonresidue modulo n, and there is no efficient deterministic algorithm known for doing that.
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Codablock A is based on the Code 39 barcode, consists of 2 to a maximum of 22 barcode lines of 1 to 61 data character each and can encode up to 1,340 characters. The checksum for the error correction is calculated according to modulo 43 over the entire code block.
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But it is wrong over the real numbers, since the series does not converge. However, there are other contexts (e.g. working with 2-adic numbers, or with integers modulo a power of 2), where the series does converge. The formal calculation implies that the last equation must be valid in those contexts.
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Let be a root of the primitive polynomial . The traditional representation of elements of this field is as polynomials in α of degree 2 or less. A table of Zech logarithms for this field are , , , , , , , and . The multiplicative order of α is 7, so the exponential representation works with integers modulo 7.
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Thus left rotation by a single bit has a simple description modulo 3. Analysis of other operations (data dependent rotation and modular addition) reveals similar, notable biases. Although there are some theoretical problems analysing the operations in combination, the bias can be detected experimentally for the entire cipher. In (Kelsey et al.
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Alt-Ergo is an automatic solver for mathematical formulas, specifically designed for program verification. It is based on satisfiability modulo theories (SMT). It is distributed under an open-source license (Cecill-C). Its original authors were Sylvain Conchon and Evelyne Contejean, at LRI, but it is now developed and maintained at OCamlPro.
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Postnikov On character sums modulo a prime power, Izvestia Akad. Nauka SSSR, Ser. Math. 19, 1955, 11–16 This was also the subject of his Russian doctorate (higher doctoral recognition) in 1956 (Investigation of the method of Vinogradov for trigonometric sums (in Russian)). He was later a senior scientist at the Steklov Institute.
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When (Z/nZ)× is cyclic, its generators are called primitive roots modulo n. For a prime number p, the group (Z/pZ)× is always cyclic, consisting of the non-zero elements of the finite field of order p. More generally, every finite subgroup of the multiplicative group of any field is cyclic..
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In fact, after 8, all the numbers listed above are odd, and after 167 all the numbers listed above are congruent to 29 modulo 30. The concept is somewhat analogous to that of highly composite numbers. Just as there are infinitely many highly composite numbers, there are also infinitely many highly cototient numbers.
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The prime decomposition of the number 2450 is given by 2450 = 257. Of the primes occurring in this decomposition, 2, 5, and 7, only 7 is congruent to 3 modulo 4. Its exponent in the decomposition, 2, is even. Therefore, the theorem states that it is expressible as the sum of two squares.
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" In 2008, Car Design News called the Modulo "iconic". The Modulo placed third in Jalopnik's 2008 competition for the top ten best "Wedge Car Designs Of The 60s, 70s and 80s." By contrast, the Rolls Royce Camargue ranked in 2010 as one of the "10 Worst Cars" as chosen by readers of The Globe and Mail; ranked 38 in the 2005 book Crap Cars by Richard Porter (the author saying the car "looked utterly terrible)" and ranked 92 in a 2008 poll of the 100 ugliest cars of all time by readers of The Daily Telegraph. Autoblog said the Camargue had been ranked "conspicuously low on the list," adding the Camargue "really was horrid, no matter how well it sold.
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The Fibonacci numbers are the numbers in the integer sequence: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, ... defined by the recurrence relation :F_0 = 0 :F_1 = 1 :F_i = F_{i-1} + F_{i-2}. For any integer n, the sequence of Fibonacci numbers Fi taken modulo n is periodic. The Pisano period, denoted '(n), is the length of the period of this sequence. For example, the sequence of Fibonacci numbers modulo 3 begins: :0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, ... This sequence has period 8, so '(3) = 8.
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Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with specific types of factors; it is the simplest example of an algebraic-group factorisation algorithm. The factors it finds are ones for which the number preceding the factor, p − 1, is powersmooth; the essential observation is that, by working in the multiplicative group modulo a composite number N, we are also working in the multiplicative groups modulo all of N's factors. The existence of this algorithm leads to the concept of safe primes, being primes for which p − 1 is two times a Sophie Germain prime q and thus minimally smooth.
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This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4. Since the Diophantus identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as the sum of two squares, by applying Fermat's theorem to the prime factorization of any positive integer n, we see that if all the prime factors of n congruent to 3 modulo 4 occur to an even exponent, then n is expressible as a sum of two squares. The converse also holds.For a proof of the converse see for instance 20.1, Theorems 367 and 368, in: G.H. Hardy and E.M. Wright.
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Possible (and useful) adequate equivalence relations include rational, algebraic, homological and numerical equivalence. They are called "adequate" because dividing out by the equivalence relation is functorial, i.e. push-forward (with change of co-dimension) and pull-back of cycles is well-defined. Codimension one cycles modulo rational equivalence form the classical group of divisors.
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In signal processing, the fast folding algorithm (Staelin, 1969) is an efficient algorithm for the detection of approximately-periodic events within time series data. It computes superpositions of the signal modulo various window sizes simultaneously. The FFA is best known for its use in the detection of pulsars, as popularised by SETI@home and Astropulse.
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The Digital Signature Algorithm (DSA) is a variant of the ElGamal signature scheme, which should not be confused with ElGamal encryption. ElGamal encryption can be defined over any cyclic group G, like multiplicative group of integers modulo n. Its security depends upon the difficulty of a certain problem in G related to computing discrete logarithms.
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An element of a ring that is equal to its own square is called an idempotent. In any ring, 0 and 1 are idempotents. There are no other idempotents in fields and more generally in integral domains. However, the ring of the integers modulo has idempotents, where is the number of distinct prime factors of .
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If the base a is coprime to the exponent p then Fermat's little theorem says that qp(a) will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then qp(a) will be a cyclic number, and p will be a full reptend prime.
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MAX-3LIN-EQN is a problem in Computational complexity theory where the input is a system of linear equations (modulo 2). Each equation contains at most 3 variables. The problem is to find an assignment to the variables that satisfies the maximum number of equations. This problem is closely related to the MAX-3SAT problem.
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Every probability measure is a continuous submeasure, so as the corresponding Boolean algebra of measurable sets modulo measure zero sets is complete, it is a Maharam algebra. solved a long-standing problem by constructing a Maharam algebra that is not a measure algebra, i.e., that does not admit any countably additive strictly positive finite measure.
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The above formulas lead to an efficient Cohen, pp. 29–31 algorithm for calculating the Jacobi symbol, analogous to the Euclidean algorithm for finding the gcd of two numbers. (This should not be surprising in light of rule 2.) # Reduce the "numerator" modulo the "denominator" using rule 2. # Extract any even "numerator" using rule 9.
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In algebraic number theory, the Gras conjecture relates the p-parts of the Galois eigenspaces of an ideal class group to the group of global units modulo cyclotomic units. It was proved by as a corollary of their work on the main conjecture of Iwasawa theory. later gave a simpler proof using Euler systems.
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Once again nZ is normal in Z because Z is abelian. The cosets are the collection {nZ, 1+nZ, ..., (n−2)+nZ, (n−1)+nZ}. An integer k belongs to the coset r+nZ, where r is the remainder when dividing k by n. The quotient Z/nZ can be thought of as the group of "remainders" modulo n.
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"Resultados del Modulo de Movilidad Social Intergeneracional" , INEGI, 16 June 2017, Retrieved on 30 April 2018. respectively. A study performed in hospitals of Mexico City reported that an average 51.8% of Mexican newborns presented the congenital skin birthmark known as the Mongolian spot."Alteraciones cutáneas del neonato en dos grupos de población de México", Scielo, March/April 2005.
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LCGs are fast and require minimal memory (one modulo-m number, often 32 or 64 bits) to retain state. This makes them valuable for simulating multiple independent streams. LCGs are not intended, and must not be used, for cryptographic applications; use a cryptographically secure pseudorandom number generator for such applications. Hyperplanes of a linear congruential generator in three dimensions.
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This generalization is commonly called satisfiability modulo theories. The question whether a sentence in propositional logic is satisfiable is a decidable problem. In general, the question whether sentences in first-order logic are satisfiable is not decidable. In universal algebra and equational theory, the methods of term rewriting, congruence closure and unification are used to attempt to decide satisfiability.
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It is shown by Stephen Pohlig and Martin Hellman in 1978 that if all the factors of p − 1 are less than log p, then the problem of solving discrete logarithm modulo p is in P. Therefore, for cryptosystems based on discrete logarithm, such as DSA, it is required that p − 1 have at least one large prime factor.
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29, No. 5 (Nov., 1998), pp. 354-370. (abstract) (JSTOR) One can represent the twelve equal-tempered pitch classes by the cyclic group of order twelve, or equivalently, the residue classes modulo twelve, Z/12Z. The group Z_{12} has four generators, which can be identified with the ascending and descending semitones and the ascending and descending perfect fifths.
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Serre introduced the idea of working in homotopy theory modulo some class C of abelian groups. This meant that groups A and B were treated as isomorphic, if for example A/B lay in C. Later Dennis Sullivan had the bold idea instead of using the localization of a topological space, which took effect on the underlying topological spaces.
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This approach has a modular error challenge: measured ranges are modulo the RF carrier wavelength. The Swiss Ranger is a compact, short-range device, with ranges of 5 or 10 meters and a resolution of 176 x 144 pixels. With phase unwrapping algorithms, the maximum uniqueness range can be increased. The PMD can provide ranges up to 60 m.
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The pattern produced by an elementary cellular automaton using rule 60 is exactly Pascal's triangle of binomial coefficients reduced modulo 2 (black cells correspond to odd binomial coefficients). Rule 102 also produces this pattern when trailing zeros are omitted. Rule 90 produces the same pattern but with an empty cell separating each entry in the rows.
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Plot of the first 10,000 Pisano periods. In number theory, the nth Pisano period, written '(n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774.
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Set e_i=s_i\cdot M/m_i. From the identities 1 = 1\cdot 221 - 20\cdot 11 = (-5)\cdot 187 + 72\cdot 13 = 5\cdot 143 + (-42)\cdot 17, we get that e_1 = 221, e_2=-935, e_3=715, and the unique solution modulo 11\cdot 13\cdot 17 is 155. Finally, S = 155 \equiv 2 \mod 3 .
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Thus, there was no socket mechanism, but a libsocket(3) existed that used asynchronous I/O to talk to the TCP/IP handler. The typical Berkeley-derived networking program could be compiled and run unchanged (modulo the usual Unix porting problems), though it might not be as efficient as an equivalent program that used native asynchronous I/O.
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A cyclically ordered group is a group together with a cyclic order preserved by the group structure. Every cyclic group can be given a structure as a cyclically ordered group, consistent with the ordering of the integers (or the integers modulo the order of the group). Every finite subgroup of a cyclically ordered group is cyclic..
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Entries congruent modulo N to 0, 1, or N − 1 do not appear in this sequence of numbers,. See Section 2. because they would correspond either to a loop or multiple adjacency, neither of which are permitted in simple graphs. Often the pattern repeats, and the number of repetitions can be indicated by a superscript in the notation.
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As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring ).See Euler's theorem.
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Many applications require one to consider the unification of typed lambda-terms instead of first-order terms. Such unification is often called higher-order unification. A well studied branch of higher-order unification is the problem of unifying simply typed lambda terms modulo the equality determined by αβη conversions. Such unification problems do not have most general unifiers.
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In mathematics, Kummer sum is the name given to certain cubic Gauss sums for a prime modulus p, with p congruent to 1 modulo 3. They are named after Ernst Kummer, who made a conjecture about the statistical properties of their arguments, as complex numbers. These sums were known and used before Kummer, in the theory of cyclotomy.
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Because Codabar is self-checking, most standards do not define a check digit. Some standards that use Codabar will define a check digit, but the algorithm is not universal. For purely numerical data, such as the library barcode pictured above, the Luhn algorithm is popular. When all 16 symbols are possible, a simple modulo-16 checksum is used.
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Since and implies , the set of classes coprime to n is closed under multiplication. Integer multiplication respects the congruence classes, that is, and implies . This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying .
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The smallest example with a nontrivial subgroup of false witnesses is . There are 6 residues coprime to 9: 1, 2, 4, 5, 7, 8. Since 8 is congruent to , it follows that 88 is congruent to 1 modulo 9. So 1 and 8 are false positives for the "primality" of 9 (since 9 is not actually prime).
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A second drawback is that it won't break up clustered keys. For example, the keys 123000, 456000, 789000, etc. modulo 1000 all map to the same address. This technique works well in practice because many key sets are sufficiently random already, and the probability that a key set will be cyclical by a large prime number is small.
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Likewise, International Bank Account Numbers (IBANs), for example, make use of modulo 97 arithmetic to spot user input errors in bank account numbers. In chemistry, the last digit of the CAS registry number (a unique identifying number for each chemical compound) is a check digit, which is calculated by taking the last digit of the first two parts of the CAS registry number times 1, the previous digit times 2, the previous digit times 3 etc., adding all these up and computing the sum modulo 10. In cryptography, modular arithmetic directly underpins public key systems such as RSA and Diffie–Hellman, and provides finite fields which underlie elliptic curves, and is used in a variety of symmetric key algorithms including Advanced Encryption Standard (AES), International Data Encryption Algorithm (IDEA), and RC4.
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The constructive existence proof shows that, in the case of two moduli, the solution may be obtained by the computation of the Bézout coefficients of the moduli, followed by a few multiplications, additions and reductions modulo n_1n_2 (for getting a result in the interval (0, n_1n_2-1)). As the Bézout's coefficients may be computed with the extended Euclidean algorithm, the whole computation, at most, has a quadratic time complexity of O((s_1+s_2)^2), where s_i denotes the number of digits of n_i. For more than two moduli, the method for two moduli allows the replacement of any two congruences by a single congruence modulo the product of the moduli. Iterating this process provides eventually the solution with a complexity, which is quadratic in the number of digits of the product of all moduli.
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The first difference between the two algorithms is that Adler-32 sums are calculated modulo a prime number, whereas Fletcher sums are calculated modulo 24−1, 28−1, or 216−1 (depending on the number of bits used), which are all composite numbers. Using a prime number makes it possible for Adler-32 to catch differences in certain combinations of bytes that Fletcher is unable to detect. The second difference, which has the largest effect on the speed of the algorithm, is that the Adler sums are computed over 8-bit bytes rather than 16-bit words, resulting in twice the number of loop iterations. This results in the Adler-32 checksum taking between one-and-a-half to two times as long as Fletcher's checksum for 16-bit word aligned data.
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Given an unramified finite extension of local fields, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of residue fields. Suppose is an unramified extension of local fields, with ring of integers OK of such that the residue field, the integers of modulo their unique maximal ideal , is a finite field of order , where is a power of a prime. If is a prime of lying over , that is unramified means by definition that the integers of modulo , the residue field of , will be a finite field of order extending the residue field of where is the degree of . We may define the Frobenius map for elements of the ring of integers of as an automorphism of such that :s_\Phi(x) \equiv x^q \mod \Phi.
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Lagrange's theorem states that the order of any subgroup of a finite group divides the order of the entire group, in this case . If is any number coprime to then is in one of these residue classes, and its powers modulo form a subgroup of the group of residue classes, with . Lagrange's theorem says must divide , i.e. there is an integer such that .
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In theoretical physics, geometrodynamics is an attempt to describe spacetime and associated phenomena completely in terms of geometry. Technically, its goal is to unify the fundamental forces and reformulate general relativity as a configuration space of three-metrics, modulo three-dimensional diffeomorphisms. It was enthusiastically promoted by John Wheeler in the 1960s, and work on it continues in the 21st century.
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The padding works as follows: first a single bit, 1, is appended to the end of the message. This is followed by as many zeros as are required to bring the length of the message up to 64 bits fewer than a multiple of 512. The remaining bits are filled up with 64 bits representing the length of the original message, modulo 264.
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Classical estimators of connectivity are correlation and coherence. The above measures provide information on the directionality of interactions in terms of delay (correlation) or coherence (phase), however the information does not imply causal interaction. Moreover it may be ambiguous, since phase is determined modulo 2π. It is also not possible to identify by means of correlation or coherence reciprocal connections.
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In 1892 Corrado Segre recalled the tessarine algebra as bicomplex numbers.G. Baley Price (1991) An introduction to multicomplex spaces and functions, Marcel Dekker Naturally the subalgebra of real tessarines arose and came to be called the bireal numbers. In 1946 U. Bencivenga published an essayBencivenga, U. (1946) "Sulla Rappresentazione Geometrica Della Algebre Doppie Dotate Di Modulo", Atti. Accad. Sci. Napoli Ser(3) v.
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For a continuous random variable \theta_i distributed about the unit circle, the Von Mises distribution maximizes the entropy when the real and imaginary parts of the first circular moment are specified or, equivalently, the circular mean and circular variance are specified. When the mean and variance of the angles \theta_i modulo 2\pi are specified, the wrapped normal distribution maximizes the entropy.
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Also the Colombia-Solidarity (Solidaridad) Bridge (located about NW of the city in Colombia, Nuevo León). There are no urban areas on either side of this bridge. Nuevo Laredo is a strategic investment point. On this site there are six recognized industrial parks: Oradel Industrial Center, Longoria Industrial Park, Rio Bravo Industrial Park, Modulo Industrial America, FINSA Industrial Park, and Industrial Park Pyme.
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Therefore, reducing t into the desired range requires at most a single subtraction, so the algorithm's output lies in the correct range. To use REDC to compute the product of 7 and 15 modulo 17, first convert to Montgomery form and multiply as integers to get 12 as above. Then apply REDC with , , , and . The first step sets m to .
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Not all Euclid numbers are prime. E6 = 13# + 1 = 30031 = 59 × 509 is the first composite Euclid number. Every Euclid number is congruent to 3 mod 4 since the primorial of which it is composed is twice the product of only odd primes and thus congruent to 2 modulo 4. This property implies that no Euclid number can be a square.
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Any ring of characteristic 0 is infinite. The ring Z/nZ of integers modulo n has characteristic n. If R is a subring of S, then R and S have the same characteristic. For instance, if q(X) is an irreducible polynomial with coefficients in the field Z/pZ where p is prime, then the factor ring is a field of characteristic p.
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Wichmann–Hill is a pseudorandom number generator proposed in 1982 by Brian Wichmann and David Hill. It consists of three linear congruential generators with different prime moduli, each of which is used to produce a uniformly distributed number between 0 and 1. These are summed, modulo 1, to produce the result. The problem occurs with single-precision floating point when rounding to zero.
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Schematic of a Barcode (Code 128B). 1:quiet zone, 2:start code, 3:data, 4:checksum, 5:stop code A Code 128 barcode has seven sections: # Quiet zone # Start symbol # Encoded data # Check symbol (mandatory) # Stop symbol # Final bar (often considered part of the stop symbol) # Quiet zone The check symbol is calculated from a weighted sum (modulo 103) of all the symbols.
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Stronger forms of Dirichlet's theorem state that for any such arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges and that different such arithmetic progressions with the same modulus have approximately the same proportions of primes. Equivalently, the primes are evenly distributed (asymptotically) among the congruence classes modulo d containing a's coprime to d.
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In 1877, Alexander von Brill determined the sign of the discriminant. Leopold Kronecker first stated Minkowski's theorem in 1882, though the first proof was given by Hermann Minkowski in 1891. In the same year, Minkowski published his bound on the discriminant. Near the end of the nineteenth century, Ludwig Stickelberger obtained his theorem on the residue of the discriminant modulo four.
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Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the dividend or the divisor is negative: rounding may be toward zero (so called T-division) or toward −∞ (F-division); rarer styles can occur – see Modulo operation for the details. Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another.
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Up to isometry, there is only one even unimodular lattice in 15 dimensions or less -- the E8 lattice. Up to dimension 24, there is only one even unimodular lattice without roots, the Leech lattice. Three of the sporadic simple groups were discovered by Conway while investigating the automorphism group of the Leech lattice. For example, Co1 is the automorphism group itself modulo ±1.
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Finally, in 2014 Imbrie presented a proof of MBL for certain one dimensional spin chains with strong disorder, with the localization being stable to arbitrary local perturbations – i.e. the systems were shown to be in a many body localized phase. It is now believed that MBL can arise also in periodically driven "Floquet" systems where energy is conserved only modulo the drive frequency.
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All-new Honda Freed launched in Japan, priced between RM76k to RM110k It is also offered in five- seater option as the Freed+, which replaced the Freed Spike in the previous generation. On 15 December 2017, the Freed Modulo X was released in Japan. It featured a full body kit, front and rear bumpers, side skirts, and a tailgate spoiler.
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There is a proof reducing its security to the computational difficulty of factoring. When the primes are chosen appropriately, and O(log log M) lower-order bits of each xn are output, then in the limit as M grows large, distinguishing the output bits from random should be at least as difficult as solving the Quadratic residuosity problem modulo M.
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In theoretical physics, a twisted sector is a subspace of the full Hilbert space of closed string states in a particular theory over a (good) orbifold. In the first quantized formalism of string theory (or in two-dimensional conformal field theory) the target space is an orbifold M/G if the observables of the string are only defined modulo G. Consequently, the value of the field after one cycle around the closed string need only be the same as its original value modulo some G transformation. i.e. there exists some g\in G such that :X(\sigma+2\pi,\tau)=g[X(\sigma,\tau)] For each conjugacy class of G, we have a different superselection sector (wrt the worldsheet). The conjugacy class consisting of the identity gives rise to the untwisted sector and all the other conjugacy classes give rise to twisted sectors.
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In fact, after n divisions, it can be proven that the remaining pile is divisible by n, a property made convenient use of by the creator of the problem. A formal way of stating the above argument is: The original pile of coconuts will be divided by 5 a total of 5 times with a remainder of 1, not considering the last division in the morning. Let N = number of coconuts in the original pile. Each division must leave the number of nuts in the same congruence class (mod 5). So, :N \equiv 4/5\cdot(N-1) (mod 5) (the –1 is the nut tossed to the monkey) :5N \equiv 4N - 4 (mod 5) :N \equiv -4 (mod 5) (–4 is the congruence class) So if we began in modulo class –4 nuts then we will remain in modulo class –4.
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It follows from the Chinese remainder theorem that there are at least four distinct square roots of 1 modulo N (since there are two roots for each modular equation). The aim of the algorithm is to find a square root b of 1 modulo N that is different from 1 and \- 1 , because then : b^2 - 1 = (b+1)(b-1) = mN for a non-zero integer m which gives us the non- trivial divisors \gcd(N, b+1) and \gcd(N, b-1) of N . This idea is similar to other factoring algorithms, like the quadratic sieve. In turn, finding such a b is reduced to finding an element a of even period with a certain additional property (as explained below, it is required that the condition of Step 6 of the classical part does not hold).
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Some programs predating the 80286 were designed to take advantage of the wrap-around (modulo) memory addressing behavior, so the 80286 presented a problem for backward compatibility. Forcing the 21st address line (the actual logic signal wire coming out of the chip) to a logic low, representing a zero, results in a modulo-2^20 effect to match the earlier processors' address arithmetic, but the 80286 has no internal capability to perform this function. When IBM used the 80286 in their IBM PC/AT, they solved this problem by including a software-settable gate to enable or disable (force to zero) the A20 address line, between the A20 pin on the 80286 and the system bus; this is known as Gate-A20 (the A20 gate), and it is still implemented in PC chipsets to this day. Most versions of the HIMEM.
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They have the same pattern of point-line intersections as the Euclidean version of the configuration. The finite projective plane PG(2,7) has 57 points and 57 lines, and can be given coordinates based on the integers modulo 7. In this space, every conic C (the set of solutions to a two-variable quadratic equation modulo 7) has 28 secant lines through pairs of its points, 8 tangent lines through a single point, and 21 nonsecant lines that are disjoint from C. Dually, there are 28 points where pairs of tangent lines meet, 8 points on C, and 21 interior points that do not belong to any tangent line. The 21 nonsecant lines and 21 interior points form an instance of the Grünbaum–Rigby configuration, meaning that again these points and lines have the same pattern of intersections.
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The Miller–Rabin primality test uses the following extension of Fermat's little theorem: > If is an odd prime number, and , with odd, then for every prime to , either > , or there exists such that and This result may be deduced from Fermat's little theorem by the fact that, if is an odd prime, then the integers modulo form a finite field, in which has exactly two square roots, 1 and −1. The Miller–Rabin test uses this property in the following way: given , with odd, an odd integer for which primality has to be tested, choose randomly such that ; then compute ; if is not 1 nor −1, then square it repeatedly modulo until you get 1, −1, or have squared times. If and −1 has not been obtained, then is not prime. Otherwise, may be prime or not.
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In cryptography, the RSA problem summarizes the task of performing an RSA private-key operation given only the public key. The RSA algorithm raises a message to an exponent, modulo a composite number N whose factors are not known. Thus, the task can be neatly described as finding the eth roots of an arbitrary number, modulo N. For large RSA key sizes (in excess of 1024 bits), no efficient method for solving this problem is known; if an efficient method is ever developed, it would threaten the current or eventual security of RSA- based cryptosystems—both for public-key encryption and digital signatures. More specifically, the RSA problem is to efficiently compute P given an RSA public key (N, e) and a ciphertext C ≡ P e (mod N). The structure of the RSA public key requires that N be a large semiprime (i.e.
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In order for a k-tuple to have infinitely many positions at which all of its values are prime, there cannot exist a prime p such that the tuple includes every different possible value modulo p. For, if such a prime p existed, then no matter which value of n was chosen, one of the values formed by adding n to the tuple would be divisible by p, so there could only be finitely many prime placements (only those including p itself). For example, the numbers in a k-tuple cannot take on all three values 0, 1, and 2 modulo 3; otherwise the resulting numbers would always include a multiple of 3 and therefore could not all be prime unless one of the numbers is 3 itself. A k-tuple that satisfies this condition (i.e.
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Therefore, RSA is not plaintext aware: one way of generating a ciphertext without knowing the plaintext is to simply choose a random number modulo N. In fact, plaintext- awareness is a very strong property. Any cryptosystem that is semantically secure and is plaintext-aware is actually secure against a chosen-ciphertext attack, since any adversary that chooses ciphertexts would already know the plaintexts associated with them.
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He introduced the van der Corput lemma, a technique for creating an upper bound on the measure of a set drawn from harmonic analysis, and the van der Corput theorem on equidistribution modulo 1. He became member of the Royal Netherlands Academy of Arts and Sciences in 1929, and foreign member in 1953. He was a Plenary Speaker of the ICM in 1936 in Oslo.
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The class of locally finite groups is closed under subgroups, quotients, and extensions . Locally finite groups satisfy a weaker form of Sylow's theorems. If a locally finite group has a finite p-subgroup contained in no other p-subgroups, then all maximal p-subgroups are finite and conjugate. If there are finitely many conjugates, then the number of conjugates is congruent to 1 modulo p.
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To decrypt a ciphertext c, we must find the subset of B which sums to c. We do this by transforming the problem into one of finding a subset of W. That problem can be solved in polynomial time since W is superincreasing. 1\. Calculate the modular inverse of r modulo q using the Extended Euclidean algorithm. The inverse will exist since r is coprime to q.
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A bit loosely, one might express this by saying that the smooth structure is (essentially) unique. The case for k = 0 is different. Namely, there exist topological manifolds which admit no C1−structure, a result proved by , and later explained in the context of Donaldson's theorem (compare Hilbert's fifth problem). Smooth structures on an orientable manifold are usually counted modulo orientation-preserving smooth homeomorphisms.
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One problem considered in the study of combinatorics on words in group theory is the following: for two elements x,y of a semigroup, does x=y modulo the defining relations of x and y. Post and Markov studied this problem and determined it undecidable. Undecidable means the theory cannot be proved. The Burnside question was proved using the existence of an infinite cube-free word.
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For example, in the case given above, the discriminant is so that is the only prime that has a chance of making it satisfy the criterion. Modulo , it becomes -- a repeated root is inevitable, since the discriminant is . Therefore the variable shift is actually something predictable. Again, for the cyclotomic polynomial, it becomes :; the discriminant can be shown to be (up to sign) , by linear algebra methods.
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In 2012, Vernay won the Porsche Carrera Cup France driving for Sébastien Loeb Racing. Vernay won the 24 Hours of Le Mans LMGTE Am class in 2013 while competing for IMSA Performance Matmut, driving a Porsche 997 GT3-RSR. He also contested in the full 2013 FIA World Endurance Championship season. In 2014, Vernay will drive for Weider Modulo Dome Racing in Super GT GT500 class.
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In 1994, he and Jerry Burch published an influential paper on microprocessor verification, inventing a technique known as the Burch-Dill verification method. He was also an early contributor to the research field known as satisfiability modulo theories (SMT), supervising the development of several early SMT solvers: the Stanford Validity Checker (SVC), C. Barrett, D. Dill, J. Levitt. 1996, Validity Checking for Combinations of Theories with Equality.
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This is indeed correct, because 7 is not a quadratic residue modulo 11. The above sequence of residues : 7, 3, 10, 6, 2 may also be written : −4, 3, −1, −5, 2. In this form, the integers larger than 11/2 appear as negative numbers. It is also apparent that the absolute values of the residues are a permutation of the residues : 1, 2, 3, 4, 5.
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In 2013, a new designed alloy wheels for both 1.3 and 1.5 variants and a color of the City has been replaced Habanero Red has been replaced with Carnelian Red for the City. A Modulo version is available in both variants and in 2013 a Mugen version is available for the 1.5 variant only. Honda Cars India, the Indian subsidiary of Honda Motor Co. Ltd.
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This is maximized when is prime and is a primitive element modulo . In this case, the period is q-1. In this case the output sequence is called an l-sequence (for "long sequence"). l-sequences have many excellent statistical properties that make them candidates for use in applications, including near uniform distribution of sub-blocks, ideal arithmetic autocorrelations, and the arithmetic shift and add property.
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This heuristic predicts that the number of Wolstenholme primes between K and N is roughly ln ln N − ln ln K. The Wolstenholme condition has been checked up to 109, and the heuristic says that there should be roughly one Wolstenholme prime between 109 and 1024. A similar heuristic predicts that there are no "doubly Wolstenholme" primes, for which the congruence would hold modulo p5.
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This is a trivial modular square root, because 3^2 ot\geq n and so the modulus is not involved when squaring. The integer b_2 = 15 is also Very Smooth Quadratic Residue modulo n. All prime factors are smaller than 7.37 and the Modular Square Root is x_2 = 20 since 20^2 = 400 \equiv 15 (mod n). This is thus a non-trivial root.
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The same uniqueness result was used by David Mumford for discrete Heisenberg groups, in his theory of equations defining abelian varieties. This is a large generalization of the approach used in Jacobi's elliptic functions, which is the case of the modulo 2 Heisenberg group, of order 8. The simplest case is the theta representation of the Heisenberg group, of which the discrete case gives the theta function.
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Every Hanoi graph contains a Hamiltonian cycle. The Hanoi graph H^1_k is a complete graph on k vertices. Because they contain complete graphs, all larger Hanoi graphs H^n_k require at least k colors in any graph coloring. They may be colored with exactly k colors by summing the indexes of the towers containing each disk, and using the sum modulo k as the color.
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It was common to encipher a message after first encoding it, to increase the difficulty of cryptanalysis. With a numerical code, this was commonly done with an "additive" - simply a long key number which was digit-by-digit added to the code groups, modulo 10. Unlike the codebooks, additives would be changed frequently. The famous Japanese Navy code, JN-25, was of this design.
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It is a family of declarative "ultra high-level" languages. It features abstract types, generic modules, subsorts (subtypes with multiple inheritance), pattern-matching modulo equations, E-strategies (user control over laziness), module expressions (for combining modules), theories and views (for describing module interfaces) for the massively parallel RRM (rewrite rule machine). Members of the OBJ family of languages include CafeOBJ, Eqlog, FOOPS, Kumo, Maude, OBJ2, and OBJ3.
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Note that octaves are usually ignored in constructing periodicity blocks (as they are in scale theory generally) because it is assumed that for any pitch in the tuning system, all pitches differing from it by some number of octaves are also available in principle. In other words, all pitches and intervals can be considered as residues modulo octave. This simplification is commonly known as octave equivalence.
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A two-dimensional visualisation of the Hamming distance. The color of each pixel indicates the Hamming distance between the binary representations of its x and y coordinates, modulo 16, in the 16-color system. At the Bell Labs Hamming shared an office for a time with Claude Shannon. The Mathematical Research Department also included John Tukey and Los Alamos veterans Donald Ling and Brockway McMillan.
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The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible. The modules so constructed are called Specht modules, and every irreducible does arise inside some such module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For example, even their dimensions are not known in general.
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Modern computers compute division by methods that are faster than long division, with the more efficient ones relying on approximation techniques from numerical analysis. For division with remainder, see Division algorithm. In modular arithmetic (modulo a prime number) and for real numbers, nonzero numbers have a multiplicative inverse. In these cases, a division by may be computed as the product by the multiplicative inverse of .
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The theorem is frequently referred to as the (although it is a theorem and not an algorithm), because its proof as given below lends itself to a simple division algorithm for computing and (see the section Proof for more). Division is not defined in the case where ; see division by zero. For the remainder and the modulo operation, there are conventions other than , see .
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The prefix checksum is the 8-bit sum of the four-bit hexadecimal value of the six digits that make up the address and byte count. # Data -- contains the data to be transferred, followed by a 2 character (1 byte) checksum. The data checksum is the 8-bit sum, modulo 256, of the 4-bit hexadecimal values of the digits that make up the data bytes.
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In these cases, additional 0-probability place holders must be added. This is because the tree must form an n to 1 contractor; for binary coding, this is a 2 to 1 contractor, and any sized set can form such a contractor. If the number of source words is congruent to 1 modulo n-1, then the set of source words will form a proper Huffman tree.
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If Heath-Brown's conjecture is true, the problem is decidable. In this case, an algorithm could correctly solve the problem by computing n modulo 9, returning false when this is 4 or 5, and otherwise returning true. Heath-Brown's research also includes more precise conjectures on how far an algorithm would have to search to find an explicit representation rather than merely determining whether one exists.
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For instance, if one wishes to know what the result of the evaluation of a mathematical expression involving only integers +, -, ×, is worth modulo n, then one needs only perform all operations modulo n (a familiar form of this abstraction is casting out nines). Abstractions, however, though not necessarily exact, should be sound. That is, it should be possible to get sound answers from them—even though the abstraction may simply yield a result of undecidability. For instance, students in a class may be abstracted by their minimal and maximal ages; if one asks whether a certain person belongs to that class, one may simply compare that person's age with the minimal and maximal ages; if his age lies outside the range, one may safely answer that the person does not belong to the class; if it does not, one may only answer "I don't know".
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Discourse status is determined via the entailments of the context. This is achieved through the definition in (16): (16) Definition of given: An utterance of U counts as given if it has a salient antecedent A and ::a. if U is type e, then A and U corefer; ::b. otherwise: modulo \exists-type-shifting, A entails the existential F-closure of U. The operation in (16b) can apply to any constituent.
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Finite fields (also called Galois fields) are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example is a field with four elements. Its subfield is the smallest field, because by definition a field has at least two distinct elements . In modular arithmetic modulo 12, 9 + 4 = 1 since 9 + 4 = 13 in , which divided by 12 leaves remainder 1\.
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The Purdy Polynomial hash algorithm was developed for the ARPANET to protect passwords in 1971 at the request of Larry Roberts, head of ARPA at that time. It computed a polynomial of degree 224 \+ 17 modulo the 64-bit prime p = 264 − 59. The algorithm was later used by Digital Equipment Corporation (DEC) to hash passwords in the VMS operating system and is still being used for this purpose.
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A closed manifold is called essential if its fundamental class defines a nonzero element in the homology of its fundamental group, or more precisely in the homology of the corresponding Eilenberg–MacLane space. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise. Examples of essential manifolds include aspherical manifolds, real projective spaces, and lens spaces.
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In the above formula in terms of exponential and trigonometric functions, the primitive th roots of unity are those for which and are coprime integers. Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see . For the case of roots of unity in rings of modular integers, see Root of unity modulo n.
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Shifting all characters by one position to the left requires multiplying the entire sum H by a. Shifting all characters by one position to the right requires dividing the entire sum H by a. Note that in modulo arithmetic, a can be chosen to have a multiplicative inverse a^{-1} by which H can be multiplied to get the result of the division without actually performing a division.
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Suppose that the vertices of the quadrilateral Q are given by Q_1,Q_2,Q_3,Q_4 . Let b_1,b_2,b_3,b_4 be the perpendicular bisectors of sides Q_1Q_2,Q_2Q_3,Q_3Q_4,Q_4Q_1 respectively. Then their intersections Q_i^{(2)}=b_{i+2}b_{i+3} , with subscripts considered modulo 4, form the consequent quadrilateral Q^{(2)} . The construction is then iterated on Q^{(2)} to produce Q^{(3)} and so on.
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In planar graphs, colorings with k distinct colors are dual to nowhere zero flows over the ring \Z_k of integers modulo k. In this duality, the difference between the colors of two adjacent regions is represented by a flow value across the edge separating the regions. In particular, the existence of nowhere zero 4-flows is equivalent to the four color theorem. The snark theorem generalizes this result to nonplanar graphs.
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Hash(m) = xm mod n where n is hard to factor composite number, and x is some prespecified base value. A collision xm1 congruent to xm2 reveals a multiple m1 - m2 of the order of x. Such information can be used to factor n in polynomial time assuming certain properties of x. But the algorithm is quite inefficient because it requires on average 1.5 multiplications modulo n per message-bit.
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In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algebra of polynomials, the remainder is the polynomial "left over" after dividing one polynomial by another. The modulo operation is the operation that produces such a remainder when given a dividend and divisor.
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If x is a pth root of unity in k, then it satisfies x^p = 1. But consider the expression (x-1)^p = 0. By expanding using binomial coefficients we see that the operation of raising to the pth power, known here as the Frobenius homomorphism, introduces the factor p to every coefficient except the first and the last, and so modulo p these equations are the same. Therefore x = 1.
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A complex affine space A has a canonical projective completion P(A), defined as follows. Form the vector space F(A) which is the free vector space on A modulo the relation that affine combination in F(A) agrees with affine combination in A. Then , where n is the dimension of A. The projective completion of A is the projective space of one-dimensional complex linear subspaces of F(A).
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This is a separate game based on a similar rules set. The other one is Hacker which is also similar to the original Illuminati (modulo terminology), but the players fight for the control of computer networks. It is more loose, and based primarily on interlocking access to different computer systems in the web. Players are not set directly towards each other, and several players can share access to a system.
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To allow user selectable PINs it is possible to store a PIN offset value. The offset is found by subtracting natural PIN from the customer selected PIN using modulo 10. For example, if the natural PIN is 1234, and the user wishes to have a PIN of 2345, the offset is 1111. The offset can be stored either on the card track data, or in a database at the card issuer.
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This implies that there are more quadratic residues than nonresidues among the numbers 1, 2, ..., (q − 1)/2. > For example, modulo 11 there are four residues less than 6 (namely 1, 3, 4, > and 5), but only one nonresidue (2). An intriguing fact about these two theorems is that all known proofs rely on analysis; no-one has ever published a simple or direct proof of either statement.
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The fact that finding a square root of a number modulo a large composite n is equivalent to factoring (which is widely believed to be a hard problem) has been used for constructing cryptographic schemes such as the Rabin cryptosystem and the oblivious transfer. The quadratic residuosity problem is the basis for the Goldwasser-Micali cryptosystem. The discrete logarithm is a similar problem that is also used in cryptography.
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One of the most general types of exponential sum is the Weyl sum, with exponents 2πif(n) where f is a fairly general real-valued smooth function. These are the sums involved in the distribution of the values :ƒ(n) modulo 1, according to Weyl's equidistribution criterion. A basic advance was Weyl's inequality for such sums, for polynomial f. There is a general theory of exponent pairs, which formulates estimates.
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Suppose that and are both solutions to all the congruences. As and give the same remainder, when divided by , their difference is a multiple of each . As the are pairwise coprime, their product divides also , and thus and are congruent modulo . If and are supposed to be non negative and less than (as in the first statement of the theorem), then their difference may be a multiple of only if .
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Toida's conjecture refines Ádám's conjecture by considering only a special class of circulant graphs, in which all of the differences between adjacent graph vertices are relatively prime to the number of vertices. According to this refined conjecture, these special circulant graphs should have the property that all of their symmetries come from symmetries of the underlying additive group of numbers modulo . It was proven by two groups in 2001 and 2002.
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Coach operators National Express, Park's of Hamilton, Shearings and Wallace Arnold all purchased large quantities of B10Ms.Shearings builds up Commercial Motor 13 December 1990 In the 1990s, Stagecoach standardised on the bus version of the B10M as their full-size single decker. Most received Alexander PS bodies but some received Northern Counties Paladin bodywork. Stagecoach also took numerous examples of the coach version with Plaxton's Interurban bodywork and Jonckheere's Modulo bodywork.
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Dirichlet's theorem is proved by showing that the value of the Dirichlet L-function (of a non-trivial character) at 1 is nonzero. The proof of this statement requires some calculus and analytic number theory . In the particular case a = 1 (i.e., concerning the primes that are congruent to 1 modulo some n) can be proven by analyzing the splitting behavior of primes in cyclotomic extensions, without making use of calculus .
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For k = F3, the theorem holds except for G of type A1.Tits (1964), section 1.2. For a k-simple group G, in order to understand the whole group G(k), one can consider the Whitehead group W(k,G)=G(k)/G(k)+. For G simply connected and quasi-split, the Whitehead group is trivial, and so the whole group G(k) is simple modulo its center.
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In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane, who published it in 1937. It states that a finite undirected graph is planar if and only if the cycle space of the graph (taken modulo 2) has a cycle basis in which each edge of the graph participates in at most two basis vectors.
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In mathematics, the plactic monoid is the monoid of all words in the alphabet of positive integers modulo Knuth equivalence. Its elements can be identified with semistandard Young tableaux. It was discovered by (who called it the tableau algebra), using an operation given by in his study of the longest increasing subsequence of a permutation. It was named the "monoïde plaxique" by , who allowed any totally ordered alphabet in the definition.
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Since x and y are different (modulo p), (x - y) eq 0, and since both variables are greater than zero, (x + y) eq 0. Thus, by contradiction, the first p / 2 alternative locations after h(k) must be unique, and subsequently, an empty space can always be found so long as at most p / 2 locations are filled (i.e., the hash table is not more than half full).
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The check digit is an additional digit, used to verify that a barcode has been scanned correctly. It is computed modulo 10, where the weights in the checksum calculation alternate 3 and 1. In particular, since the weights are relatively prime to 10, the EAN-13 system will detect all single digit errors. It also recognizes 90% of transposition errors (all cases, where the difference between adjacent digits is not 5).
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The checksum is calculated as sum of products - taking an alternating weight value (3 or 1) times the value of each data digit. The checksum digit is the digit, which must be added to this checksum to get a number divisible by 10 (i.e. the additive inverse of the checksum, modulo 10).Check Digit Calculator, at GS1 US. See ISBN-13 check digit calculation for a more extensive description and algorithm.
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Zhegalkin (also Žegalkin, Gégalkine or Shegalkin) polynomials form one of many possible representations of the operations of Boolean algebra. Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927, they are the polynomial ring over the integers modulo 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient.
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When done with integers, the operation is typically exact (computed modulo some power of two). However, floating-point numbers have only a certain amount of mathematical precision. That is, digital floating-point arithmetic is generally not associative or distributive. (See .) Therefore, it makes a difference to the result whether the multiply–add is performed with two roundings, or in one operation with a single rounding (a fused multiply–add).
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150px Michele Cipolla (28 October 1880, Palermo - 7 September 1947, Palermo) was an Italian mathematician, mainly specializing in number theory. He was a professor of Algebraic Analysis at the University of Catania and, later, the University of Palermo. He developed (among other things) a theory for sequences of sets and Cipolla's algorithm for finding square roots modulo a prime number. He also solved the problem of binomial congruence.
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He left two conjectures, both known as Artin's conjecture. The first concerns Artin L-functions for a linear representation of a Galois group; and the second the frequency with which a given integer a is a primitive root modulo primes p, when a is fixed and p varies. These are unproven; in 1967, Hooley published a conditional proof for the second conjecture, assuming certain cases of the Generalized Riemann hypothesis.
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Using K-theory, they constructed an example of a torsion cohomology class—that is, a cohomology class such that for some positive integer —which is not the class of an algebraic cycle. Such a class is necessarily a Hodge class. reinterpreted their result in the framework of cobordism and found many examples of such classes. The simplest adjustment of the integral Hodge conjecture is: ::Integral Hodge conjecture modulo torsion.
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The wheels measuring 21 inches at the front and 22 inches at the rear were especially designed for the car. The Birdcage also lacks doors; instead, a bubble canopy composed of much of the front bodywork can be raised, in a similar manner to the Bond Bug, Ferrari Modulo and Saab's 2006 Aero X concept car. However, the lone demonstrator model lacks air-conditioning or any form of climate control.
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Middle school classes are given through distance education. Schools in the municipality include Escuela Telesecundario Estv16 723 (middle school), General Franicisco Villa (primary), Heroinas de la Independencia (middle school), Ignacio Lopez Rayon (primary), Ignacio Zaragoza (primary), Jardin de Niños (preschool), Jose Maria Morelos y Pavon (primary), Modulo I (kindergarten), Robert Owen (preschool) and Valentin Gomez Farias (primary). All are public institutions. The municipality has a growth rate of one percent.
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Either way, the end result is reached in a fairly small number of steps. Note that the numbers \alpha and \beta have the same digit sum and hence the same remainder modulo b - 1. Therefore, each number in a Kaprekar sequence of base b numbers (other than possibly the first) is a multiple of b - 1. When leading zeroes are retained, only repdigits lead to the trivial Kaprekar's constant.
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The CKKS scheme basically consists of those algorithms: key Generation, encryption, decryption, homomorphic addition and multiplication, and rescaling. For a positive integer q, let R_q := R/qR be the quotient ring of R modulo q. Let \chi_s, \chi_r and \chi_e be distributions over R which output polynomials with small coefficients. These distributions, the initial modulus Q , and the ring dimension n are predetermined before the key generation phase.
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The Rado graph may also be formed by a construction resembling that for Paley graphs, taking as the vertices of a graph all the prime numbers that are congruent to 1 modulo 4, and connecting two vertices by an edge whenever one of the two numbers is a quadratic residue modulo the other. By quadratic reciprocity and the restriction of the vertices to primes congruent to 1 mod 4, this is a symmetric relation, so it defines an undirected graph, which turns out to be isomorphic to the Rado graph. Another construction of the Rado graph shows that it is an infinite circulant graph, with the integers as its vertices and with an edge between each two integers whose distance (the absolute value of their difference) belongs to a particular set S. To construct the Rado graph in this way, S may be chosen randomly, or by choosing the indicator function of S to be the concatenation of all finite binary sequences., Section 1.2.
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When counting in binary, the digit sum modulo 2 is the Thue-Morse sequence To compute the nth element tn, write the number n in binary. If the number of ones in this binary expansion is odd then tn = 1, if even then tn = 0. For this reason John H. Conway et al. call numbers n satisfying tn = 1 odious (for odd) numbers and numbers for which tn = 0 evil (for even) numbers.
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Fix an integer d and let D be the discriminant of the imaginary quadratic field Q(√-d). The Zimmert set Z(d) is the set of positive integers n such that 4n2 < -D-3 and n ≠ 2; D is a quadratic non-residue of all odd primes in d; n is odd if D is not congruent to 5 modulo 8. The cardinality of Z(d) may be denoted by z(d).
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For applying the above general construction of finite fields in the case of , one has to find an irreducible polynomial of degree 2. For , this has been done in the preceding section. If is an odd prime, there are always irreducible polynomials of the form , with in . More precisely, the polynomial is irreducible over if and only if is a quadratic non-residue modulo (this is almost the definition of a quadratic non-residue).
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The L5 CNAV data includes SV ephemerides, system time, SV clock behavior data, status messages and time information, etc. The 50 bit/s data is coded in a rate 1/2 convolution coder. The resulting 100 symbols per second (sps) symbol stream is modulo-2 added to the I5-code only; the resultant bit-train is used to modulate the L5 in-phase (I5) carrier. This combined signal is called the L5 Data signal.
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The quadratic residuosity problem in computational number theory is to decide, given integers a and N, whether a is a quadratic residue modulo N or not. Here N = p_1 p_2 for two unknown primes p_1 and p_2, and a is among the numbers which are not obviously quadratic non-residues (see below). The problem was first described by Gauss in his Disquisitiones Arithmeticae in 1801. This problem is believed to be computationally difficult.
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Let G be the cyclic group on 6 elements {0,1,2,3,4,5} with modular addition, H be the cyclic on 2 elements {0,1} with modular addition, and f the homomorphism that maps each element g in G to the element g modulo 2 in H. Then ker f = {0, 2, 4}, since all these elements are mapped to 0H. The quotient group G/(ker f) has two elements: {0,2,4} and {1,3,5}. It is indeed isomorphic to H.
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If it is not, we toss another biased coin, Coin 3, with probability of winning P_3=(3/4)-\epsilon. The role of modulo M provides the periodicity as in the ratchet teeth. It is clear that by playing Game A, we will almost surely lose in the long run. Harmer and Abbott show via simulation that if M=3 and \epsilon = 0.005, Game B is an almost surely losing game as well.
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It was the third largest primality proof by ECPP from its discovery until March 2009. Currently, the fastest known algorithm for proving the primality of Wagstaff numbers is ECPP. The LLR (Lucas-Lehmer-Riesel) tool by Jean Penné is used to find Wagstaff probable primes by means of the Vrba- Reix test. It is a PRP test based on the properties of a cycle of the digraph under x^2-2 modulo a Wagstaff number.
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The proof of this generalization is similar to the one for the original statement, considering the reduction of the coefficients modulo ; the essential point is that a single-term polynomial over the integral domain cannot decompose as a product in which at least one of the factors has more than one term (because in such a product there can be no cancellation in the coefficient either of the highest or the lowest possible degree).
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The most common result of an overflow is that the least significant representable digits of the result are stored; the result is said to wrap around the maximum (i.e. modulo a power of the radix, usually two in modern computers, but sometimes ten or another radix). An overflow condition may give results leading to unintended behavior. In particular, if the possibility has not been anticipated, overflow can compromise a program's reliability and security.
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In a fraction, the numerator is occasionally referred to as upstairs and the denominator downstairs, as in "bringing a term upstairs". ; up to, modulo, mod out by: An extension to mathematical discourse of the notions of modular arithmetic. A statement is true up to a condition if the establishment of that condition is the only impediment to the truth of the statement. Also used when working with members of equivalence classes, esp.
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There are several types of instruction scheduling: #Local (basic block) scheduling: instructions can't move across basic block boundaries. #Global scheduling: instructions can move across basic block boundaries. #Modulo scheduling: an algorithm for generating software pipelining, which is a way of increasing instruction level parallelism by interleaving different iterations of an inner loop. #Trace scheduling: the first practical approach for global scheduling, trace scheduling tries to optimize the control flow path that is executed most often.
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For lower ranks, fewer such groups are Hurwitz. For np the order of p modulo 7, one has that PSL(2,q) is Hurwitz if and only if either q=7 or q = pnp. Indeed, PSL(3,q) is Hurwitz if and only if q = 2, PSL(4,q) is never Hurwitz, and PSL(5,q) is Hurwitz if and only if q = 74 or q = pnp, . Similarly, many groups of Lie type are Hurwitz.
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The cipher's strength rests on a strong mixing of its inner state between two consecutive iterations. The mixing function is entirely based on arithmetical operations that are available on a modern processor, i.e., no S-boxes or lookup tables are required to implement the cipher. The mixing function uses a g-function based on arithmetical squaring, and the ARX operations -- logical XOR, bit-wise rotation with hard-wired rotation amounts, and addition modulo 232.
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In mathematics, Legendre's equation is the Diophantine equation :ax^2+by^2+cz^2=0. The equation is named for Adrien Marie Legendre who proved in 1785 that it is solvable in integers x, y, z, not all zero, if and only if −bc, −ca and −ab are quadratic residues modulo a, b and c, respectively, where a, b, c are nonzero, square-free, pairwise relatively prime integers, not all positive or all negative .
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This is done in two parts: −1·xn−r is added to xn−r, resulting in a least significant word of zero. And second, a·xn−r is added to the carry. This makes the integer one word longer, producing two new most significant words xn and cn. So far, this has simply added a multiple of p to the state, resulting in a different representative of the same residue class modulo p.
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A quartic or biquadratic residue (mod p) is any number congruent to the fourth power of an integer (mod p). If x4 ≡ a (mod p) does not have an integer solution, a is a quartic or biquadratic nonresidue (mod p).Gauss, BQ § 2 As is often the case in number theory, it is easiest to work modulo prime numbers, so in this section all moduli p, q, etc., are assumed to positive, odd primes.
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Each device adds its own secret keys together (using unsigned addition modulo 256) according to a KSV received from another device. Depending on the order of the bits set to 1 in the KSV, a corresponding secret key is used or ignored in the addition. The generation of keys and KSVs gives both devices the same 56-bit number, which is later used to encrypt data. Encryption is done by a stream cipher.
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The ten steps of the mixing function. Two of the five dimensions are unrolled. CubeHash round function consists of the following ten steps: # Add x[0jklm] into x[1jklm] modulo 2, for each (j,k,l,m). # Rotate x[0jklm] upwards by 7 bits, for each (j,k,l,m). # Swap x[00klm] with x[01klm], for each (k,l,m). # Xor x[1jklm] into x[0jklm], for each (j,k,l,m).
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Similar to the fan, equipment found in the mass production manufacturing industry demonstrate rotation around a fixed axis effectively. For example, a multi-spindle lathe is used to rotate the material on its axis to effectively increase production of cutting, deformation and turning. The angle of rotation is a linear function of time, which modulo 360° is a periodic function. An example of this is the two-body problem with circular orbits.
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To mitigate the computational penalty of modular arithmetic, three tricks are used in practice: # One chooses the prime p to be close to a power of two, such as a Mersenne prime. This allows arithmetic modulo p to be implemented without division (using faster operations like addition and shifts). For instance, on modern architectures one can work with p = 2^{61}-1, while x_i's are 32-bit values. # One can apply vector hashing to blocks.
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Development of Axigen Mail Server began in 2003 by GeCAD Technologies, a Romanian company established in 2001 and part of the GECAD Group. It was initially launched in September 2005. In 2012 the Axigen product and technology were spun-off into a new company called Axigen Messaging, along with the original development team. In January 2014, Axigen Messaging was sold to an investor group affiliated with Romanian I.T. services company Modulo Consulting.
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In characteristic 0 the 4371-dimensional representation of the baby monster does not have a nontrivial invariant algebra structure analogous to the Griess algebra, but showed that it does have such an invariant algebra structure if it is reduced modulo 2. The smallest faithful matrix representation of the Baby Monster is of size 4370 over the finite field of order 2. constructed a vertex operator algebra acted on by the baby monster.
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Prime powers are powers of prime numbers. Every prime power (except powers of 2) has a primitive root; thus the multiplicative group of integers modulo pn (i.e. the group of units of the ring Z/pnZ) is cyclic. The number of elements of a finite field is always a prime power and conversely, every prime power occurs as the number of elements in some finite field (which is unique up to isomorphism).
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The symbol can be annotated to denote various sets, with varying usage amongst different authors: , or for the positive integers, or for non-negative integers, and for non-zero integers. Some authors use for non-zero integers, while others use it for non-negative integers, or for }. Additionally, is used to denote either the set of integers modulo (i.e., the set of congruence classes of integers), or the set of -adic integers.
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However, is not a field because 12 is not a prime number. The simplest finite fields, with prime order, are most directly accessible using modular arithmetic. For a fixed positive integer , arithmetic "modulo " means to work with the numbers : The addition and multiplication on this set are done by performing the operation in question in the set of integers, dividing by and taking the remainder as result. This construction yields a field precisely if is a prime number.
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The quotient ring is a field, because is irreducible over , so the ideal it generates is maximal. The formulas for addition and multiplication in the ring , modulo the relation , correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs. So the two definitions of the field are isomorphic (as fields). Accepting that is algebraically closed, since it is an algebraic extension of in this approach, is therefore the algebraic closure of .
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The list above includes the perfect numbers 28 and 496. All even perfect numbers are triangular numbers whose index is an odd Mersenne prime.. Since every Mersenne prime greater than 3 is congruent to 1 modulo 3, it follows that every even perfect number greater than 6 is a centered nonagonal number. In 1850, Sir Frederick Pollock conjectured that every natural number is the sum of at most eleven centered nonagonal numbers, which has been neither proven nor disproven..
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A number of refinements and variants have followed. This article highlights the fundamental mathematical structure of RLWE signatures and follows the original Lyubashevsky work and the work of Guneysu, Lyubashevsky and Popplemann (GLP). This presentation is based on a 2017 update to the GLP scheme called GLYPH. A RLWE-SIG works in the quotient ring of polynomials modulo a degree n polynomial Φ(x) with coefficients in the finite field Zq for an odd prime q ( i.e.
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Fermat primes are particularly useful in generating pseudo-random sequences of numbers in the range 1 … N, where N is a power of 2. The most common method used is to take any seed value between 1 and P − 1, where P is a Fermat prime. Now multiply this by a number A, which is greater than the square root of P and is a primitive root modulo P (i.e., it is not a quadratic residue).
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In the area of mathematics called combinatorial group theory, the Schreier coset graph is a graph associated with a group G, a generating set {xi : i in I} of G, and a subgroup H ≤ G. The Schreier graph encodes the abstract structure of a group modulo an equivalence relation formed by the coset. The graph is named after Otto Schreier, who used the term “Nebengruppenbild”. An equivalent definition was made in an early paper of Todd and Coxeter.
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Macintyre developed a first-order model theory for intersection theory and showed connections to Alexander Grothendieck's standard conjectures on algebraic cycles. Macintyre has proved many results on the model theory of real and complex exponentiation. With Alex Wilkie he proved the decidability of real exponential fields (solving a problem of Alfred Tarski) modulo Schanuel's conjecture from transcendental number theory. With Lou van den Dries he initiated and studied the model theory of logarithmic-exponential series and Hardy fields.
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Sylow's test: Let n be a positive integer that is not prime, and let p be a prime divisor of n. If 1 is the only divisor of n that is equal to 1 modulo p, then there does not exist a simple group of order n. Proof: If n is a prime-power, then a group of order n has a nontrivial centerSee the proof in p-group, for instance. and, therefore, is not simple.
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Image:goldbachs comet.gif The coloring of points in the above image is based on the value of E/2 modulo 3 with red points corresponding to 0 mod 3, blue points corresponding to 1 mod 3 and green points corresponding to 2 mod 3. In other words, the red points are multiples of 6; the blue points are of the form "a multiple of 6, plus 2"; and the green points are multiples of 6 plus 4.
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Starting with W4 = 63 and W5 = 159, every sixth Woodall number is divisible by 3; thus, in order for Wn to be prime, the index n cannot be congruent to 4 or 5 (modulo 6). Also, for a positive integer m, the Woodall number W2m may be prime only if 2m \+ m is prime. As of January 2019, the only known primes that are both Woodall primes and Mersenne primes are W2 = M3 = 7, and W512 = M521.
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The assertion that the Wadge lemma holds for sets in Γ is the semilinear ordering principle for Γ or SLO(Γ). Any defines a linear order on the equivalence classes modulo complements. Wadge's lemma can be applied locally to any pointclass Γ, for example the Borel sets, Δ1n sets, Σ1n sets, or Π1n sets. It follows from determinacy of differences of sets in Γ. Since Borel determinacy is proved in ZFC, ZFC implies Wadge's lemma for Borel sets.
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A closed manifold M is called essential if its fundamental class [M] defines a nonzero element in the homology of its fundamental group , or more precisely in the homology of the corresponding Eilenberg–MacLane space K(, 1), via the natural homomorphism :H_n(M)\to H_n(K(\pi,1)), where n is the dimension of M. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise.
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The Coppersmith method, proposed by Don Coppersmith, is a method to find small integer zeroes of univariate or bivariate polynomials modulo a given integer. The method uses the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL) to find a polynomial that has the same zeroes as the target polynomial but smaller coefficients. In cryptography, the Coppersmith method is mainly used in attacks on RSA when parts of the secret key are known and forms a base for Coppersmith's attack.
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Wieferich primes can be defined by other equivalent congruences. If p is a Wieferich prime, one can multiply both sides of the congruence by 2 to get . Raising both sides of the congruence to the power p shows that a Wieferich prime also satisfies , and hence for all . The converse is also true: for some implies that the multiplicative order of 2 modulo p2 divides gcd, φ, that is, and thus p is a Wieferich prime.
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In its full generality, partitioning cryptanalysis works by dividing the sets of possible plaintexts and ciphertexts into efficiently-computable partitions such that the distribution of ciphertexts is significantly non-uniform when the plaintexts are chosen uniformly from a given block of the partition. Partitioning cryptanalysis has been shown to be more effective than linear cryptanalysis against variants of DES and CRYPTON. A specific partitioning attack called mod n cryptanalysis uses the congruence classes modulo some integer for partitions.
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These concepts can even assist with in number- theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the existence of square roots modulo integer prime numbers. Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected with the failure of unique factorization in the cyclotomic integers., Section I.7, p.
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The algorithm allows for efficient in software implementations; to encrypt bytes of plaintext do: All arithmetic is performed modulo 256. i := 0 while GeneratingOutput: a := S[i] j := S[j + a] output S[S[S[j] + 1 swap S[i] and S[j] (b := S[j]; S[i] := b; S[j] := a)) i := i + 1 endwhile Where 256-element permutation and integer value are obtained from the encryption password using the VMPC-KSA (Key Scheduling Algorithm).
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In § VI of the Disquisitiones ArithmeticaeGauss, DA, arts 329-334 Gauss discusses two factoring algorithms that use quadratic residues and the law of quadratic reciprocity. Several modern factorization algorithms (including Dixon's algorithm, the continued fraction method, the quadratic sieve, and the number field sieve) generate small quadratic residues (modulo the number being factorized) in an attempt to find a congruence of squares which will yield a factorization. The number field sieve is the fastest general-purpose factorization algorithm known.
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If the field K has characteristic equal to zero or greater than n then by Maschke's theorem the group algebra KSn is semisimple. In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary). However, the irreducible representations of the symmetric group are not known in arbitrary characteristic. In this context it is more usual to use the language of modules rather than representations.
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The answer is known classically as Schwarz's list. In monodromy terms, the question is of identifying the cases of finite monodromy group. By reformulation and passing to a larger system, the essential case is for rational functions in A and rational number coefficients. Then a necessary condition is that for almost all prime numbers p, the system defined by reduction modulo p should also have a full set of algebraic solutions, over the finite field with p elements.
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For instance, one applies vector hashing to each 16-word block of the string, and applies string hashing to the \lceil k/16 \rceil results. Since the slower string hashing is applied on a substantially smaller vector, this will essentially be as fast as vector hashing. # One chooses a power-of-two as the divisor, allowing arithmetic modulo 2^w to be implemented without division (using faster operations of bit masking). The NH hash-function family takes this approach.
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Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups. Every cyclic group of prime order is a simple group which cannot be broken down into smaller groups.
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An Adler-32 checksum is obtained by calculating two 16-bit checksums A and B and concatenating their bits into a 32-bit integer. A is the sum of all bytes in the stream plus one, and B is the sum of the individual values of A from each step. At the beginning of an Adler-32 run, A is initialized to 1, B to 0. The sums are done modulo 65521 (the largest prime number smaller than 216).
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There are further invariants of compact complex surfaces that are not used so much in the classification. These include algebraic invariants such as the Picard group Pic(X) of divisors modulo linear equivalence, its quotient the Néron–Severi group NS(X) with rank the Picard number ρ, topological invariants such as the fundamental group π1 and the integral homology and cohomology groups, and invariants of the underlying smooth 4-manifold such as the Seiberg–Witten invariants and Donaldson invariants.
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Interval class . In musical set theory, an interval class (often abbreviated: ic), also known as unordered pitch-class interval, interval distance, undirected interval, or "(even completely incorrectly) as 'interval mod 6'" (; ), is the shortest distance in pitch class space between two unordered pitch classes. For example, the interval class between pitch classes 4 and 9 is 5 because 9 − 4 = 5 is less than 4 − 9 = −5 ≡ 7 (mod 12). See modular arithmetic for more on modulo 12.
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The de Rham cohomology has inspired many mathematical ideas, including Dolbeault cohomology, Hodge theory, and the Atiyah–Singer index theorem. However, even in more classical contexts, the theorem has inspired a number of developments. Firstly, the Hodge theory proves that there is an isomorphism between the cohomology consisting of harmonic forms and the de Rham cohomology consisting of closed forms modulo exact forms. This relies on an appropriate definition of harmonic forms and of the Hodge theorem.
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Soon after, Valiant found holographic algorithms with reductions to matchgates for #7Pl-Rtw-Mon-3CNF and #7Pl-3/2Bip-VC. These problems may appear somewhat contrived, especially with respect to the modulus. Both problems were already known to be #P-hard when ignoring the modulus and Valiant supplied proofs of #P-hardness modulo 2, which also used holographic reductions. Valiant found these two problems by a computer search that looked for problems with holographic reductions to matchgates.
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Goresky and MacPherson introduced a class of "allowable" cycles for which general position does make sense. They introduced an equivalence relation for allowable cycles (where only "allowable boundaries" are equivalent to zero), and called the group :IH_i(X) of i-dimensional allowable cycles modulo this equivalence relation "intersection homology". They furthermore showed that the intersection of an i- and an (n-i)-dimensional allowable cycle gives an (ordinary) zero-cycle whose homology class is well- defined.
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A position was transcribed by cycling through the pieces in a position, indexing the corresponding random numbers (vacant spaces were not included in the calculation), and XORing them together (the starting value could be 0, the identity value for XOR, or a random seed). The resulting value was reduced by modulo, folding or some other operation to produce a hash table index. The original Zobrist hash was stored in the table as the representation of the position.
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Aho, Sethi, Ullman, 1986, Compilers: Principles, Techniques and Tools, pp.435. Addison-Wesley, Reading, MA. This hash function offsets the bytes 4 bits before ADDing them together. When the quantity wraps, the high 4 bits are shifted out and if non-zero, XORed back into the low byte of the cumulative quantity. The result is a word size hash code to which a modulo or other reducing operation can be applied to produce the final hash index.
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The Klein bottle can be seen as a fiber bundle over the circle S1, with fibre S1, as follows: one takes the square (modulo the edge identifying equivalence relation) from above to be E, the total space, while the base space B is given by the unit interval in y, modulo 1~0. The projection π:E→B is then given by . The Klein bottle can be constructed (in a four dimensional space, because in three dimensional space it cannot be done without allowing the surface to intersect itself) by joining the edges of two Möbius strips together, as described in the following limerick by Leo Moser: The initial construction of the Klein bottle by identifying opposite edges of a square shows that the Klein bottle can be given a CW complex structure with one 0-cell P, two 1-cells C1, C2 and one 2-cell D. Its Euler characteristic is therefore . The boundary homomorphism is given by and , yielding the homology groups of the Klein bottle K to be , and for .
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The period of a lag-r MWC generator is the order of b in the multiplicative group of numbers modulo p = abr − 1\. While it is theoretically possible to choose a non-prime modulus, a prime modulus eliminates the possibility of the initial seed sharing a common divisor with the modulus, which would reduce the generator's period. Because 2 is a quadratic residue of numbers of the form 8k±1, b = 2k cannot be a primitive root of p = abr − 1\. Therefore, MWC generators with base 2k have their parameters chosen so their period is (abr−1)/2. This is one of the difficulties that use of b = 2k − 1 overcomes. The basic form of an MWC generator has parameters a, b and r, and r+1 words of state. The state consists of r residues modulo b : 0 ≤ x0, x1, x2,..., xr−1 < b, and a carry cr−1 < a. The initial state ("seed") values are arbitrary, except that they must not be all zero, nor all at the maximum permitted values (xi = b−1 and cr−1 = a−1).
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In nonparametric statistics, a 1977 paper by Persi Diaconis and Graham studied the statistical properties of Pearson's footrule, a measure of rank correlation that compares two permutations by summing, over each item, the distance between the positions of the item in the two permutations. They compared this measure to other rank correlation methods, resulting in the "Diaconis–Graham inequalities" :I+E\le D\le 2I where D is Pearson's footrule, I is the number of inversions between the two permutations (a non-normalized version of the Kendall rank correlation coefficient), and E is the minimum number of two-element swaps needed to obtain one permutation from the other. The Chung–Diaconis–Graham random process is a random walk on the integers modulo an odd integer p, in which at each step one doubles the previous number and then randomly adds zero, 1, or -1 (modulo p). In a 1987 paper, Fan Chung, Diaconis, and Graham studied the mixing time of this process, motivated by the study of pseudorandom number generators.
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In this example, having the inverse of e modulo φ(n), the Euler's totient function of n, is the trapdoor: : f(x) = x^e \mod n If the factorization is known, φ(n) can be computed, so then the inverse of can be computed = e−1 mod φ(n), and then given y = f(x) we can find x = yd mod n = xed mod n = x mod n. Its hardness follows from RSA assumption.Goldwasser's lecture notes, 2.3.2; Lindell's notes, pp.
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Given such a clique, one can form a covering of space by cubes of side two whose centers have coordinates that, when taken modulo four, are vertices of the clique. The condition that any two vertices of the clique have a coordinate that differs by two implies that cubes corresponding to these vertices do not overlap. The condition that the clique has size 2n implies that the cubes within any period of the tiling have the same total volume as the period itself.
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Let \alpha, \beta be two quasi- isometric embeddings of [0, +\infty[ into X ("quasi-geodesic rays"). They are considered equivalent if and only if the function t \mapsto d(\alpha(t), \beta(t)) is bounded on [0, +\infty[. If the space X is proper then the set of all such embeddings modulo equivalence with its natural topology is homeomorphic to \partial X as defined above. A similar realisation is to fix a basepoint and consider only quasi-geodesic rays originating from this point.
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Ferrari has produced a number of concept cars, such as the Mythos. While some of these were quite radical (such as the Modulo) and never intended for production, others such as the Mythos have shown styling elements which were later incorporated into production models. The most recent concept car to be produced by Ferrari themselves was the 2010 Millechili. A number of one-off special versions of Ferrari road cars have also been produced, commissioned to coachbuilders by wealthy owners.
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The apparent angle between two stars in the combined fields of view, modulo the grid period, was obtained from the phase difference of the two star pulse trains. Originally targeting the observation of some 100,000 stars, with an astrometric accuracy of about 0.002 arc-sec, the final Hipparcos Catalogue comprised nearly 120,000 stars with a median accuracy of slightly better than 0.001 arc-sec (1 milliarc-sec). Optical micrograph of part of the main modulating grid (top) and the star mapper grid (bottom).
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For morphisms to Spec(R), the fiber above the special point is the special fiber, an important concept for example in reduction modulo p, monodromy theory and other theories about degeneration. The generic fiber, equally, is the fiber above the generic point. Geometry of degeneration is largely then about the passage from generic to special fibers, or in other words how specialization of parameters affects matters. (For a discrete valuation ring the topological space in question is the Sierpinski space of topologists.
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Primary pseudoperfect numbers were first investigated and named by Butske, Jaje, and Mayernik (2000). Using computational search techniques, they proved the remarkable result that for each positive integer r up to 8, there exists exactly one primary pseudoperfect number with precisely r (distinct) prime factors, namely, the rth known primary pseudoperfect number. Those with 2 ≤ r ≤ 8, when reduced modulo 288, form the arithmetic progression 6, 42, 78, 114, 150, 186, 222, as was observed by Sondow and MacMillan (2017).
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Program repair is performed with respect to an oracle, encompassing the desired functionality of the program which is used for validation of the generated fix. A simple example is a test-suite—the input/output pairs specify the functionality of the program. A variety of techniques are employed, most notably using satisfiability modulo theories (SMT) solvers, and genetic programming, using evolutionary computing to generate and evaluate possible candidates for fixes. The former method is deterministic, while the latter is randomized.
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A primefree sequence found by Herbert Wilf has initial terms :a1 = 20615674205555510, a2 = 3794765361567513 . The proof that every term of this sequence is composite relies on the periodicity of Fibonacci-like number sequences modulo the members of a finite set of primes. For each prime p, the positions in the sequence where the numbers are divisible by p repeat in a periodic pattern, and different primes in the set have overlapping patterns that result in a covering set for the whole sequence.
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This has often been cited as a reason that software pipelining cannot be effectively implemented on conventional architectures. In fact, Monica Lam presents an elegant solution to this problem in her thesis, A Systolic Array Optimizing Compiler (1989) (). She calls it modulo variable expansion. The trick is to replicate the body of the loop after it has been scheduled, allowing different registers to be used for different values of the same variable when they have to be live at the same time.
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The Bernoulli numbers can be expressed in terms of the Riemann zeta function as for integers provided for the expression is understood as the limiting value and the convention is used. This intimately relates them to the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties. For example, the Agoh–Giuga conjecture postulates that is a prime number if and only if is congruent to −1 modulo .
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For some applications, such as timers and clocks, wrapping on overflow can be desirable. The C11 standard states that for unsigned integers modulo wrapping is the defined behavior and the term overflow never applies: "a computation involving unsigned operands can never overflow." On some processors like graphics processing units (GPUs) and digital signal processors (DSPs) which support saturation arithmetic, overflowed results would be "clamped", i.e. set to the minimum or the maximum value in the representable range, rather than wrapped around.
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The product of two pitch classes is the product of their pitch-class numbers modulo 12. Since complementation and multiplication are not isometries of pitch-class space, they do not necessarily preserve the musical character of the objects they transform. Other writers, such as Allen Forte, have emphasized the Z-relation, which obtains between two sets that share the same total interval content, or interval vector—but are not transpositionally or inversionally equivalent . Another name for this relationship, used by Howard , is "isomeric" .
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Two vectors of Rn are in the same congruence class modulo the subspace if and only if they are identical in the last n−m coordinates. The quotient space Rn/ Rm is isomorphic to Rn−m in an obvious manner. More generally, if V is an (internal) direct sum of subspaces U and W, :V=U\oplus W then the quotient space V/U is naturally isomorphic to W . An important example of a functional quotient space is a Lp space.
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Alina Carmen Cojocaru is a Romanian mathematician who works in number theory and is known for her research on elliptic curves, arithmetic geometry, and sieve theory. She is a professor of mathematics at the University of Illinois at Chicago and a researcher in the Institute of Mathematics of the Romanian Academy. Cojocaru earned her Ph.D. from Queen's University in Kingston, Ontario, in 2002. Her dissertation, Cyclicity of Elliptic Curves Modulo p, was jointly supervised by M. Ram Murty and Ernst Kani.
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In mathematics, in particular in computational algebra, the Berlekamp- Zassenhaus algorithm is an algorithm for factoring polynomials over the integers, named after Elwyn Berlekamp and Hans Zassenhaus. As a consequence of Gauss's lemma, this amounts to solving the problem also over the rationals. The algorithm starts by finding factorizations over suitable finite fields using Hensel's lemma to lift the solution from modulo a prime p to a convenient power of p. After this the right factors are found as a subset of these.
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The braid group is the universal central extension of the modular group. The braid group is the universal central extension of the modular group, with these sitting as lattices inside the (topological) universal covering group . Further, the modular group has a trivial center, and thus the modular group is isomorphic to the quotient group of modulo its center; equivalently, to the group of inner automorphisms of . The braid group in turn is isomorphic to the knot group of the trefoil knot.
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The field F is algebraically closed if and only if it has no proper algebraic extension. If F has no proper algebraic extension, let p(x) be some irreducible polynomial in F[x]. Then the quotient of F[x] modulo the ideal generated by p(x) is an algebraic extension of F whose degree is equal to the degree of p(x). Since it is not a proper extension, its degree is 1 and therefore the degree of p(x) is 1\.
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While extremely useful, casting out nines does not catch all errors made while doing calculations. For example, the casting-out-nines method would not recognize the error in a calculation of 5 × 7 which produced any of the erroneous results 8, 17, 26, etc. (that is, any result congruent to 8 modulo 9). In other words, the method only catches erroneous results whose digital root is one of the 8 digits that is different from that of the correct result.
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The last word in an ANC packet is the Checksum word. It is computed by computing the sum (modulo 512) of bits 0-8 (not bit 9), of all the other words in the ANC packet, excluding the packet start sequence. Bit 9 of the checksum word is then defined as the inverse of bit 8. Note that the checksum word does not contain a parity bit; instead, the parity bits of other words are included in the checksum calculations.
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The second-generation Fit was launched as the Jazz in Malaysia in August 2008. Two models were available: Grade S and Grade V. In June 2009, a limited edition Grade S Modulo variant was launched limited to 100 units. In April 2011, the facelift model was officially launched with a sole Grade V variant being available. In August 2012, the sole Grade V variant was replaced with a Grade S variant that saw a reduction in price and a reduction in equipment.
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To go back to Alexander's original work, it is assumed that X is a simplicial complex. Alexander had little of the modern apparatus, and his result was only for the Betti numbers, with coefficients taken modulo 2. What to expect comes from examples. For example the Clifford torus construction in the 3-sphere shows that the complement of a solid torus is another solid torus; which will be open if the other is closed, but this does not affect its homology.
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More specifically, for an ideal I in the ring k[x1, ..., xn] over a field k, a (Ritt) characteristic set C of I is composed of a set of polynomials in I, which is in triangular shape: polynomials in C have distinct main variables (see the formal definition below). Given a characteristic set C of I, one can decide if a polynomial f is zero modulo I. That is, the membership test is checkable for I, provided a characteristic set of I.
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This 4-fold and 8-fold periodicity in the structure of manifolds is related to the 4-fold periodicity of L-theory and the 8-fold periodicity of real topological K-theory, which is known as Bott periodicity. If a compact oriented smooth spin manifold has dimension , or exactly, then its signature is an integer multiple of 16.Ochanine, Serge, "Signature modulo 16, invariants de Kervaire généralisés et nombres caractéristiques dans la K-théorie réelle", Mém. Soc. Math. France 1980/81, no.
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This is the group of units of the ring Zn; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or U(Zn). As a consequence of Lagrange's theorem, ordn(a) always divides φ(n). If ordn(a) is actually equal to φ(n), and therefore as large as possible, then a is called a primitive root modulo n. This means that the group U(n) is cyclic and the residue class of a generates it.
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The exterior product of a -form and an -form is a ()-form denoted . At each point of the manifold , the forms and are elements of an exterior power of the cotangent space at . When the exterior algebra is viewed as a quotient of the tensor algebra, the exterior product corresponds to the tensor product (modulo the equivalence relation defining the exterior algebra). The antisymmetry inherent in the exterior algebra means that when is viewed as a multilinear functional, it is alternating.
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The latter refers to the fact that k is the solution to the Chinese remainder problem k = k1 mod N1 and k = k2 mod N2. (One could instead use the Ruritanian mapping for the output k and the CRT mapping for the input n, or various intermediate choices.) A great deal of research has been devoted to schemes for evaluating this re-indexing efficiently, ideally in-place, while minimizing the number of costly modulo (remainder) operations (Chan, 1991, and references).
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A header check sequence (HCS) is an error checking feature for various header data structures, such as in the Media Access Control (MAC) header of Ethernet. It may consist of a cyclic redundancy check (CRC) of the frame, obtained as the remainder of the division (modulo 2) by the generator polynomial multiplied by the content of the header excluding the HCS field. The HCS can be one octet long, as in WiMAX, or a 16-bit value for cable modems.
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Two integers whose difference is a multiple of have the same representation in the residue numeral system defined by the s. More precisely, the Chinese remainder theorem asserts that each of the different sets of possible residues represents exactly one residue class modulo . That is, each set of residues represents exactly one integer in the interval . In applications where one is also interested with negative integers, it is often more convenient to represent integers belonging to an interval centered at 0.
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The paradigmatic example of folding by characters is to add up the integer values of all the characters in the string. A better idea is to multiply the hash total by a constant, typically a sizeable prime number, before adding in the next character, ignoring overflow. Using exclusive 'or' instead of add is also a plausible alternative. The final operation would be a modulo, mask, or other function to reduce the word value to an index the size of the table.
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In algebraic geometry, a reflexive sheaf is a coherent sheaf that is isomorphic to its second dual (as a sheaf of modules) via the canonical map. The second dual of a coherent sheaf is called the reflexive hull of the sheaf. A basic example of a reflexive sheaf is a locally free sheaf and, in practice, a reflexive sheaf is thought of as a kind of a vector bundle modulo some singularity. The notion is important both in scheme theory and complex algebraic geometry.
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For example, consider the group with addition modulo 6: G = {0, 1, 2, 3, 4, 5}. Consider the subgroup N = {0, 3}, which is normal because G is abelian. Then the set of (left) cosets is of size three: : G/N = { a+N : a ∈ G } = { {0, 3}, {1, 4}, {2, 5} } = { 0+N, 1+N, 2+N }. The binary operation defined above makes this set into a group, known as the quotient group, which in this case is isomorphic to the cyclic group of order 3.
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This leads to various local rigidity results for actions on Hermitian symmetric spaces. With John Parker, he examined the complex hyperbolic ideal triangle group representations. These are representations of hyperbolic ideal triangle groups to the group of holomorphic isometries of the complex hyperbolic plane such that each standard generator of the triangle group maps to a complex reflection and the products of pairs of generators to parabolics. The space of representations for a given triangle group (modulo conjugacy) is parametrized by a half-open interval.
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Leiva moved from Cuba to the US as a teenager in the 1950s. She did her undergraduate studies at Guilford College, graduating in 1961, and was initially denied admission for graduate study in mathematics at the University of North Carolina for being a woman. Nevertheless, she persisted, and earned a master's degree there in 1966 under the mentorship of Alfred Brauer, with a thesis on Elementary estimates for the least positive primitive root modulo pr. After finishing her master's degree, she became a secondary school mathematics teacher.
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In this case, the first player that has both feet removed is "it" or "out". In theory a counting rhyme is determined entirely by the starting selection (and would result in a modulo operation), but in practice they are often accepted as random selections because the number of words has not been calculated beforehand, so the result is unknown until someone is selected. A variant of counting-out game, known as the Josephus problem, represents a famous theoretical problem in mathematics and computer science.
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Players can request additional cards which are dealt face up; if it is a ten or a face card, they can reject it and ask for another. In an early version of this game, going over 9 with extra cards amounts to a "bust" as in blackjack, later versions use modulo 10 arithmetic as in the other games. Beating the banker with a pair only awards an equal amount to the bet. When the deck is exhausted, the player to the banker's left becomes the new banker.
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An L-function L(E, s) can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime p. This L-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form. It is a special case of a Hasse–Weil L-function. The natural definition of L(E, s) only converges for values of s in the complex plane with Re(s) > 3/2.
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The pentagram map is also defined on the larger space of twisted polygons. A twisted N-gon is a bi-infinite sequence of points in the projective plane that is N-periodic modulo a projective transformation That is, some projective transformation M carries P_k to P_{N+k} for all k. The map M is called the monodromy of the twisted N-gon. When M is the identity, a twisted N-gon can be interpreted as an ordinary N-gon whose vertices have been listed out repeatedly.
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The following protocol was suggested by David Chaum. A group, G, is chosen in which the discrete logarithm problem is intractable, and all operation in the scheme take place in this group. Commonly, this will be the finite cyclic group of order p contained in Z/nZ, with p being a large prime number; this group is equipped with the group operation of integer multiplication modulo n. An arbitrary primitive element (or generator), g, of G is chosen; computed powers of g then combine obeying fixed axioms.
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To work with inequalities, it is convenient to consider R-divisors, meaning finite linear combinations of Cartier divisors with real coefficients. The R-divisors modulo numerical equivalence form a real vector space N^1(X) of finite dimension, the Néron–Severi group tensored with the real numbers.Lazarsfeld (2004), Example 1.3.10. (Explicitly: two R-divisors are said to be numerically equivalent if they have the same intersection number with all curves in X.) An R-divisor is called nef if it has nonnegative degree on every curve.
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The nef R-divisors form a closed convex cone in N^1(X), the nef cone Nef(X). The cone of curves is defined to be the convex cone of linear combinations of curves with nonnegative real coefficients in the real vector space N_1(X) of 1-cycles modulo numerical equivalence. The vector spaces N^1(X) and N_1(X) are dual to each other by the intersection pairing, and the nef cone is (by definition) the dual cone of the cone of curves.Lazarsfeld (2004), Definition 1.4.25.
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The conventional term Hodge cycle therefore is slightly inaccurate, in that x is considered as a class (modulo boundaries); but this is normal usage. The importance of Hodge cycles lies primarily in the Hodge conjecture, to the effect that Hodge cycles should always be algebraic cycles, for V a complete algebraic variety. This is an unsolved problem, ; it is known that being a Hodge cycle is a necessary condition to be an algebraic cycle that is rational, and numerous particular cases of the conjecture are known.
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A semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type.Husemöller (1987) pp.116-117 Suppose E is an elliptic curve defined over the rational number field Q. It is known that there is a finite, non-empty set S of prime numbers p for which E has bad reduction modulo p. The latter means that the curve Ep obtained by reduction of E to the prime field with p elements has a singular point.
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When Windows later transitioned to Unicode, there was a desire to extend the Alt codes to allow entry of any Unicode code point. Some applications (RichEdit- based) like Word 2010, Wordpad, and PSPad will display the characters corresponding to many Alt codes larger than 255 when they have fonts available with the relevant glyphs. Other Windows applications, including Notepad, Chrome, Firefox, and Microsoft Edge interpret all numbers greater than 255 modulo 256. Numbers less than 256 work as before, using the OEM code page.
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In that case, u is a renaming of t, too, since a renaming substitution σ has an inverse σ−1, and t = uσ−1. Both terms are then also said to be equal modulo renaming. In many contexts, the particular variable names in a term don't matter, e.g. the commutativity axiom for addition can be stated as x+y=y+x or as a+b=b+a; in such cases the whole formula may be renamed, while an arbitrary subterm usually may not, e.g.
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Safe primes are also important in cryptography because of their use in discrete logarithm-based techniques like Diffie–Hellman key exchange. If is a safe prime, the multiplicative group of numbers modulo has a subgroup of large prime order. It is usually this prime- order subgroup that is desirable, and the reason for using safe primes is so that the modulus is as small as possible relative to p. A prime number p = 2q + 1 is called a safe prime if q is prime.
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Mears helped Janos Edvard Hanson and Joseph Blum earn their Ph.D.s from the George Washington University.PhD's Awarded by the Department of Mathematics, The George Washington UniversityJoseph Blum Papers, American University Joseph Blum earned his Ph.D. in 1958 following his completion of his dissertation on Banach Spaces Functionals and Matrix Summability Method. Two years later, Mears would also aid Janos Edvard Hanson in earning his Ph.D. in 1960, after writing his final dissertation on Linear Sequence Spaces, which permit omission and adjunction and have Finite Dimension Modulo Convergence.
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In mathematics, a transverse knot is a smooth embedding of a circle into a three-dimensional contact manifold such that the tangent vector at every point of the knot is transverse to the contact plane at that point. Any Legendrian knot can be C0-perturbed in a direction transverse to the contact planes to obtain a transverse knot. This yields a bijection between the set of isomorphism classes of transverse knots and the set of isomorphism classes of Legendrian knots modulo negative Legendrian stabilization.
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If is a positive integer, the ring may be identified with the set } of the remainders of Euclidean division by , the addition and the multiplication consisting in taking the remainder by of the result of the addition and the multiplication of integers. An element of has a multiplicative inverse (that is, it is a unit) if it is coprime to . In particular, if is prime, has a multiplicative inverse if it is not zero (modulo ). Thus is a field if and only if is prime.
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Starting with a prime integer q, the Ring-LWE key exchange works in the ring of polynomials modulo a polynomial \Phi(x) with coefficients in the field of integers mod q (i.e. the ring R_q := Z_q[x] / \Phi(x)). Multiplication and addition of polynomials will work in the usual fashion with results of a multiplication reduced mod \Phi(x). The idea of using LWE and Ring LWE for key exchange was first proposed and filed at the University of Cincinnati in 2011 by Jintai Ding.
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The square function is defined in any field or ring. An element in the image of this function is called a square, and the inverse images of a square are called square roots. The notion of squaring is particularly important in the finite fields Z/pZ formed by the numbers modulo an odd prime number . A non-zero element of this field is called a quadratic residue if it is a square in Z/pZ, and otherwise, it is called a quadratic non-residue.
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Integer powers of nonzero complex numbers are defined by repeated multiplication or division as above. If i is the imaginary unit and n is an integer, then in equals 1, i, −1, or −i, according to whether the integer n is congruent to 0, 1, 2, or 3 modulo 4. Because of this, the powers of i are useful for expressing sequences of period 4. Complex powers of positive reals are defined via ex as in section Complex exponents with positive real bases above.
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In a Hamiltonian graph, the vertices can be arranged in a cycle, which accounts for two edges per vertex. The third edge from each vertex can then be described by how many positions clockwise (positive) or counter-clockwise (negative) it leads. The basic form of the LCF notation is just the sequence of these numbers of positions, starting from an arbitrarily chosen vertex and written in square brackets. The numbers between the brackets are interpreted modulo N, where N is the number of vertices.
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Transcontinental Media primarily published specialty business-to-business publications, particularly within the construction and financial industries (such as Les Affaires, and the former B2B publications of Rogers Media). It also publishes French-language educational resources under the Éditions Transcontinental, Éditions Caractère, and Groupe Modulo imprints. On September 19, 2019, the company announced that it would sell its financial industry publications to two companies. Several, including Les Affaires and Benefits Canada, were divested to Contex Media—a new company led by Transcontinential president Pierre Marcoux.
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A linear subspace that contains all elements but one of a basis of the ambient space is a vector hyperplane. In a vector space of finite dimension , a vector hyperplane is thus a subspace of dimension . The counterpart to subspaces are quotient vector spaces. Given any subspace , the quotient space V/W ("V modulo W") is defined as follows: as a set, it consists of where v is an arbitrary vector in V. The sum of two such elements and is and scalar multiplication is given by .
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The order of a finite field is always a prime or a power of prime. For each prime power q = pr, there exists exactly one finite field with q elements, up to isomorphism. This field is denoted GF(q) or Fq. If p is prime, GF(p) is the prime field of order p; it is the field of residue classes modulo p, and its p elements are denoted 0, 1, ..., p−1. Thus a = b in GF(p) means the same as a ≡ b (mod p).
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250px According to the truth table, it is easy to calculate the individual coefficients of the Zhegalkin polynomial. To do this, sum up modulo 2 the values of the function in those rows of the truth table where variables that are not in the conjunction (that corresponds to the coefficient being calculated) take zero values. Suppose, for example, that we need to find the coefficient of the xz conjunction for the function of three variables f(x, y, z). There is no variable y in this conjunction.
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In mathematics, a Kleinian group is a discrete subgroup of PSL(2, C). The group PSL(2, C) of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic space H3, and as orientation preserving conformal maps of the open unit ball B3 in R3 to itself. Therefore, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.
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One then wishes to find a recursive factorization of z^N-1 into polynomials of few terms and smaller and smaller degree. To compute the DFT, one takes x(z) modulo each level of this factorization in turn, recursively, until one arrives at the monomials and the final result. If each level of the factorization splits every polynomial into an O(1) (constant-bounded) number of smaller polynomials, each with an O(1) number of nonzero coefficients, then the modulo operations for that level take O(N) time; since there will be a logarithmic number of levels, the overall complexity is O (N log N). More explicitly, suppose for example that z^N-1 = F_1(z) F_2(z) F_3(z), and that F_k(z) = F_{k,1}(z) F_{k,2}(z), and so on. The corresponding FFT algorithm would consist of first computing xk(z) = x(z) mod Fk(z), then computing xk,j(z) = xk(z) mod Fk,j(z), and so on, recursively creating more and more remainder polynomials of smaller and smaller degree until one arrives at the final degree-0 results.
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He has been summoned several times to the work modules of the Venezuela national under-20 football team,"Zambrano y Koufatty convocados al modulo". lavinotinto.com and summoned to the list of "Good Faith" of the Venezuelan National Team for the Copa América CentenarioCopa América Centenario website where he was discarded in the last Cut of the National Team (Rafael Dudamel). However, he made a preparation tour with his team for that competition, which had an opportunity to play against the Galicia national football team on May 20, 2016 and against Panama on May 24, 2016.
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In particular, the term well-defined is used with respect to (binary) operations on cosets. In this case one can view the operation as a function of two variables and the property of being well-defined is the same as that for a function. For example, addition on the integers modulo some n can be defined naturally in terms of integer addition. :[a]\oplus[b] = [a+b] The fact that this is well-defined follows from the fact that we can write any representative of [a] as a+kn, where k is an integer.
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Circuit complexity goes back to Shannon (1949), who proved that almost all Boolean functions on n variables require circuits of size Θ(2n/n). Despite this fact, complexity theorists have only been able to prove superpolynomial circuit lower bounds on functions explicitly constructed for the purpose of being hard to calculate. More commonly, superpolynomial lower bounds have been proved under certain restrictions on the family of circuits used. The first function for which superpolynomial circuit lower bounds were shown was the parity function, which computes the sum of its input bits modulo 2.
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For a mod-8 code, we have Encoder D_o=43,D_e=47 M_o=43,M_e=47 mod(8) = 7, Decoder M_o=43,M_e=47 mod(8) = 7, D_o=43,D_e=CLOSEST(43,8⋅k + 7) 43 \simeq 8 \cdot 5 + 7 D_o=43,D_e=47 Modulo-N decoding is similar to phase unwrapping and has the same limitation: If the difference from one node to the next is more than N/2 (if the phase changes from one sample to the next more than \pi), then decoding leads to an incorrect value.
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In group theory, one may have a group G acting on a set X, in which case, one might say that two elements of X are equivalent "up to the group action"—if they lie in the same orbit. Another typical example is the statement that "there are two different groups of order 4 up to isomorphism", or "modulo isomorphism, there are two groups of order 4". This means that there are two equivalence classes of groups of order 4—assuming that one considers groups to be equivalent if they are isomorphic.
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Pilen had a Porsche Carrera 6 race car like Corgi and Solido, but the details of the Pilen model match the Solido. A couple other models appear to be copies of previous Politoys models of Italy: the Ferrari Modulo Pininfarina and the Lancia Stratos Bertone. The American Ford GT Mark II is very similar to the Mebetoys (also of Italy) version, and the SEAT (FIAT) 850 Spyder is like the earlier Mercury, another Italian model producer. The company's Chevrolet Corvair Monza open cockpit concept is a doppelganger of the Danish Tekno issue.
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The Ferrari 512 S Modulo showcar was a casting from Italian Mercury (History of Mercury 2013). Bickford (2009) also reports that the Javelin, SEAT 124 Coupe and the Monteverdi Hai castings made their way to the Venezuelan company of Juguinsa when Pilen was through with them. Around 1980 there was a Pilen connection with Holland OTO, which had taken over Dutch Efsi Toys. A 1980 Auto Pilen catalog shows many of the revered Efsi vehicles like the Model T series and many Efsi trucks continued as a line Pilen 1980 (Bras 2012).
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This structure is what the spectral test measures. Although LCGs have a few specific weaknesses, many of their flaws come from having too small a state. The fact that people have been lulled for so many years into using them with such small moduli can be seen as a testament to strength of the technique. A LCG with large enough state can pass even stringent statistical tests; a modulo-2 LCG which returns the high 32 bits passes TestU01's SmallCrush suite, and a 96-bit LCG passes the most stringent BigCrush suite.
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A benefit of LCGs is that with appropriate choice of parameters, the period is known and long. Although not the only criterion, too short a period is a fatal flaw in a pseudorandom number generator. While LCGs are capable of producing pseudorandom numbers which can pass formal tests for randomness, the quality of the output is extremely sensitive to the choice of the parameters m and a. For example, a = 1 and c = 1 produces a simple modulo-m counter, which has a long period, but is obviously non-random.
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Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax2 \+ bx + c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups, which are the groups of integers modulo n. Set theory is a branch of logic and not technically a branch of algebra. Binary operations: The notion of addition (+) is abstracted to give a binary operation, ∗ say. The notion of binary operation is meaningless without the set on which the operation is defined.
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Reduction of an abelian variety A modulo a prime ideal of (the integers of) K -- say, a prime number p -- to get an abelian variety Ap over a finite field, is possible for almost all p. The 'bad' primes, for which the reduction degenerates by acquiring singular points, are known to reveal very interesting information. As often happens in number theory, the 'bad' primes play a rather active role in the theory. Here a refined theory of (in effect) a right adjoint to reduction mod p -- the Néron model -- cannot always be avoided.
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A counterclockwise rotation of more than one complete turn is normally measured modulo 360°, meaning that 360° is subtracted off as many times as possible to leave a non-negative measurement less than 360°. For example, the carts on a Ferris wheel move along a circle around the center point of that circle. If a cart moves around the wheel once, the angle of rotation is 360°. If the cart was stuck halfway, at the top of the wheel, at that point its angle of rotation was only 180°.
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When operating on 2 symbols at once, a Hill cipher offers no particular advantage over Playfair or the bifid cipher, and in fact is weaker than either, and slightly more laborious to operate by pencil-and-paper. As the dimension increases, the cipher rapidly becomes infeasible for a human to operate by hand. A Hill cipher of dimension 6 was implemented mechanically. Hill and a partner were awarded a patent () for this device, which performed a 6 × 6 matrix multiplication modulo 26 using a system of gears and chains.
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The output sequence is strictly periodic if and only if u is between -q and 0. It is possible to express u as a simple quadratic polynomial involving the initial state and the qi. There is also an exponential representation of FCSRs: if g is the inverse of N \pmod q, and the output sequence is strictly periodic, then a_i = (A g_i \bmod q) \bmod N, where A is an integer. It follows that the period is at most the order of in the multiplicative group of units modulo .
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The quadratic sieve is an optimization of Dixon's method. It selects values of x close to the square root of such that x2 modulo N is small, thereby largely increasing the chance of obtaining a smooth number. Other ways to optimize Dixon's method include using a better algorithm to solve the matrix equation, taking advantage of the sparsity of the matrix: a number z cannot have more than \log_2 z factors, so each row of the matrix is almost all zeros. In practice, the block Lanczos algorithm is often used.
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When used on their own without such context, the codes are often assumed to refer to the class of elementary cellular automata, two-state one- dimensional cellular automata with a (contiguous) three-cell neighbourhood, which Wolfram extensively investigates in his book. Notable rules in this class include rule 30, rule 110, and rule 184. Rule 90 is also interesting because it creates Pascal's Triangle modulo 2. A code of this type suffixed by an R, such as "Rule 37R", indicates a second-order cellular automaton with the same neighborhood structure.
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If the final difference is 10, then the check digit becomes 0. To ensure that this does not happen, the standard recommends that serial numbers should not be used which produce a final difference of 10; however, there are containers in the market which do not follow this recommendation, so handling this case has to be included if a check digit calculator is programmed. Notice that step (ii) to (v) is a calculation of the remainder found after division of (i) by 11. Most programming languages have a modulo operator for this.
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If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal. Since it is a proper, non-identity subgroup, the group is not simple. Burnside: A non-Abelian finite simple group has order divisible by at least three distinct primes.
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Anti-unification is the process of constructing a generalization common to two given symbolic expressions. As in unification, several frameworks are distinguished depending on which expressions (also called terms) are allowed, and which expressions are considered equal. If variables representing functions are allowed in an expression, the process is called "higher-order anti-unification", otherwise "first-order anti-unification". If the generalization is required to have an instance literally equal to each input expression, the process is called "syntactical anti-unification", otherwise "E-anti-unification", or "anti-unification modulo theory".
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Counting modulo permutations of N or X (or both) is reflected by calling the balls or the boxes, respectively, "indistinguishable". This is an imprecise formulation (in practice individual balls and boxes can always be distinguished by their location, and one could not assign different balls to different boxes without distinguishing them), intended to indicate that different configurations are not to be counted separately if one can be transformed into the other by some interchange of balls or of boxes. This possibility of transformation is formalized by the action by permutations.
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Network coding is a field of research founded in a series of papers from the late 1990s to the early 2000s. However, the concept of network coding, in particular linear network coding, appeared much earlier. In a 1978 paper, a scheme for improving the throughput of a two-way communication through a satellite was proposed. In this scheme, two users trying to communicate with each other transmit their data streams to a satellite, which combines the two streams by summing them modulo 2 and then broadcasts the combined stream.
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His thesis work influenced both program verification and automated theorem proving, especially in the area now known as satisfiability modulo theories, where he contributed techniques for combining decision procedures, as well as efficient decision procedures for quantifier-free constraints in first-order logic and term algebra. He received the Herbrand Award in 2013: He was instrumental in the development of the Simplify theorem prover used by ESC/Java. He also made significant contributions in several other areas. He contributed to the area of programming language design as a member of the Modula-3 committee.
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A prime modulus requires the computation of a double-width product and an explicit reduction step. If a modulus just less than a power of 2 is used (the Mersenne primes 231−1 and 261−1 are popular, as are 232−5 and 264−59), reduction modulo can be implemented more cheaply than a general double-width division using the identity . The basic reduction step divides the product into two e-bit parts, multiplies the high part by d, and adds them: . This can be followed by subtracting m until the result is in range.
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One of the easiest examples is the Foucault pendulum. An easy explanation in terms of geometric phases is given by Wilczek and Shapere : How does the pendulum precess when it is taken around a general path C? For transport along the equator, the pendulum will not precess. [...] Now if C is made up of geodesic segments, the precession will all come from the angles where the segments of the geodesics meet; the total precession is equal to the net deficit angle which in turn equals the solid angle enclosed by C modulo 2π.
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Spectrum: ku for connective K-theory, ko for connective real K-theory. Coefficient ring: For ku, the coefficient ring is the ring of polynomials over Z on a single class v1 in dimension 2. For ko, the coefficient ring is the quotient of a polynomial ring on three generators, η in dimension 1, x4 in dimension 4, and v14 in dimension 8, the periodicity generator, modulo the relations that 2η = 0, x42 = 4v14, η3 = 0, and ηx = 0\. Roughly speaking, this is K-theory with the negative dimensional parts killed off.
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This is because the Pell equation implies that −1 is a quadratic residue modulo n. Thus, for example, x2 − 3ny2 = −1 is never solvable, but x2 − 5ny2 = −1 may be. The first few numbers n for which x2 − ny2 = −1 is solvable are :1, 2, 5, 10, 13, 17, 26, 29, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, ... . The proportion of square-free n divisible by k primes of the form 4m + 1 for which the negative Pell equation is solvable is at least 40%.
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Phelix is a high-speed stream cipher with a built-in single-pass message authentication code (MAC) functionality, submitted in 2004 to the eSTREAM contest by Doug Whiting, Bruce Schneier, Stefan Lucks, and Frédéric Muller. The cipher uses only the operations of addition modulo 232, exclusive or, and rotation by a fixed number of bits. Phelix uses a 256-bit key and a 128-bit nonce, claiming a design strength of 128 bits. Concerns have been raised over the ability to recover the secret key if the cipher is used incorrectly.
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Fermat, Euler, Lagrange, Legendre, and other number theorists of the 17th and 18th centuries established theoremsLemmemeyer, Ch. 1 and formed conjecturesLemmermeyer, pp 6-8, p. 16 ff about quadratic residues, but the first systematic treatment is § IV of Gauss's Disquisitiones Arithmeticae (1801). Article 95 introduces the terminology "quadratic residue" and "quadratic nonresidue", and states that if the context makes it clear, the adjective "quadratic" may be dropped. For a given n a list of the quadratic residues modulo n may be obtained by simply squaring the numbers 0, 1, ..., .
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Compare the values 6 and 4 for Euler's totient function, the multiplicative group of integers modulo n for n = 9 and 10, respectively. This triples and doubles the number of automorphisms compared with the two automorphisms as isometries (keeping the order of the rotations the same or reversing the order). The only values of n for which φ(n) = 2 are 3, 4, and 6, and consequently, there are only three dihedral groups that are isomorphic to their own automorphism groups, namely (order 6), (order 8), and (order 12).
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In some applications, and in particular in Pollard's rho algorithm for integer factorization, the algorithm has much more limited access to and to . In Pollard's rho algorithm, for instance, is the set of integers modulo an unknown prime factor of the number to be factorized, so even the size of is unknown to the algorithm. To allow cycle detection algorithms to be used with such limited knowledge, they may be designed based on the following capabilities. Initially, the algorithm is assumed to have in its memory an object representing a pointer to the starting value .
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Although the first five terms are prime, no known methods can prove that any further terms are prime (in any reasonable time) simply because they are too huge. However, if c_5 is not prime, there is a chance to discover this by computing c_5 modulo some small prime p (using recursive modular exponentiation). If the resulting residue is zero, p represents a factor of c_5 and thus would disprove its primality. Since c_5 is a Mersenne number, such prime factor p must be of the form 2kc_4 +1.
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Time-keeping on this clock uses arithmetic modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00.
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The sum of a number and its ones' complement is an -bit word with all 1 bits, which is (reading as an unsigned binary number) . Then adding a number to its one's complement results in the lowest bits set to 0 and the carry bit 1, where the latter has the weight (reading it as an unsigned binary number) of . Hence, in the unsigned binary arithmetic the value of two's-complement negative number of a positive satisfies the equality .For we have , which is equivalent to modulo (i.e.
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A finite field is a set of numbers with four generalized operations. The operations are called addition, subtraction, multiplication and division and have their usual properties, such as commutativity, associativity and distributivity. An example of a finite field is the set of 13 numbers {0, 1, 2, ..., 12} using modular arithmetic. In this field, the results of any mathematical operation (addition, subtraction, multiplication, or division) is reduced modulo 13; that is, multiples of 13 are added or subtracted until the result is brought within the range 0–12.
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Diego Fumaça spent his early years playing in the lower divisions of Campeonato Mineiro. He represented Araxá and Valeriodoce, won the Modulo II with Patrocinense in 2017 and won the 2nd division with Ipatinga, also in 2017. In 2019 he competed in Campeonato Goiano with Goiânia, and earned himself a transfer to Atlético Goianiense on 15 April 2019. He made his national league debut for Atlético Goianiense in the Campeonato Brasileiro Série B match against Vitória on 26 May 2019, coming on as a late substitute in the 1–1 draw.
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In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J(χ, ψ) for Dirichlet characters χ, ψ modulo a prime number p, defined by : J(\chi,\psi) = \sum \chi(a) \psi(1 - a) \,, where the summation runs over all residues (for which neither a nor is 0). Jacobi sums are the analogues for finite fields of the beta function. Such sums were introduced by C. G. J. Jacobi early in the nineteenth century in connection with the theory of cyclotomy.
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CSR produces the Quatro family of SoCs that contain one or more custom Imaging DSPs optimized for processing document image data for scanner and copier applications. Microchip Technology produces the PIC24 based dsPIC line of DSPs. Introduced in 2004, the dsPIC is designed for applications needing a true DSP as well as a true microcontroller, such as motor control and in power supplies. The dsPIC runs at up to 40MIPS, and has support for 16 bit fixed point MAC, bit reverse and modulo addressing, as well as DMA.
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Those address-generation calculations involve different integer arithmetic operations, such as addition, subtraction, modulo operations, or bit shifts. Often, calculating a memory address involves more than one general-purpose machine instruction, which do not necessarily decode and execute quickly. By incorporating an AGU into a CPU design, together with introducing specialized instructions that use the AGU, various address-generation calculations can be offloaded from the rest of the CPU, and can often be executed quickly in a single CPU cycle. Capabilities of an AGU depend on a particular CPU and its architecture.
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3673–3686 Bowditch's work relied on extracting various discrete tree-like structures from the action of a word-hyperbolic group on its boundary. Bowditch also proved that (modulo a few exceptions) the boundary of a one- ended word-hyperbolic group G has local cut-points if and only if G admits an essential splitting, as an amalgamated free product or an HNN extension, over a virtually infinite cyclic group. This allowed Bowditch to produceB. H. Bowditch, "Cut points and canonical splittings of hyperbolic groups" Acta Mathematica, vol. 180 (1998), no.
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Adi Shamir, co-inventor of RSA (the others are Ron Rivest and Leonard Adleman) The idea of an asymmetric public-private key cryptosystem is attributed to Whitfield Diffie and Martin Hellman, who published this concept in 1976. They also introduced digital signatures and attempted to apply number theory. Their formulation used a shared-secret-key created from exponentiation of some number, modulo a prime number. However, they left open the problem of realizing a one-way function, possibly because the difficulty of factoring was not well-studied at the time.
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Analogously, the alternating group is a subgroup of index 2 in the symmetric group on n letters. The elements of the alternating group, called even permutations, are the products of even numbers of transpositions. The identity map, an empty product of no transpositions, is an even permutation since zero is even; it is the identity element of the group.; The rule "even × integer = even" means that the even numbers form an ideal in the ring of integers, and the above equivalence relation can be described as equivalence modulo this ideal.
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Donald Dines Wall (August 13, 1921 – November 28, 2000) was a mathematician working primarily on number theory. He obtained his Ph.D. on normal numbers from University of California, Berkeley in 1949, where his adviser was Derrick Henry Lehmer. His better known papers include the first modern analysis of Fibonacci sequence modulo a positive integer. Drawing on Wall's work, Zhi-Hong Sun and his twin brother Zhi-Wei Sun proved a theorem about what are now known as the Wall–Sun–Sun primes that guided the search for counterexamples to Fermat's last theorem.
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The first three-pass protocol was the Shamir three-pass protocol developed circa in 1980. It is also called the Shamir No-Key Protocol because the sender and the receiver do not exchange any keys, however the protocol requires the sender and receiver to have two private keys for encrypting and decrypting messages. The Shamir algorithm uses exponentiation modulo a large prime as both the encryption and decryption functions. That is E(e,m) = me mod p and D(d,m) = md mod p where p is a large prime.
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White Mexicans are Mexican citizens who are considered white, typically due to European or other Western Eurasian descent. While the Mexican government does conduct ethnic censuses in which a Mexican has the option of identifying as "White""Resultados del Modulo de Movilidad Social Intergeneracional" , INEGI, 16 June 2017, Retrieved on 30 April 2018. the results obtained from these censuses are not published. What Mexico's government publishes instead is the percentage of "light-skinned Mexicans" there are in the country, with it being 47% in 2010 and 49% in 2017.
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In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N/H, the quotient of a nilpotent Lie group N modulo a closed subgroup H. This notion was introduced by Anatoly Mal'cev in 1951. In the Riemannian category, there is also a good notion of a nilmanifold. A Riemannian manifold is called a homogeneous nilmanifold if there exist a nilpotent group of isometries acting transitively on it.
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If G is cyclic then the transfer takes any element y of G to y[G:H]. A simple case is that seen in the Gauss lemma on quadratic residues, which in effect computes the transfer for the multiplicative group of non-zero residue classes modulo a prime number p, with respect to the subgroup {1, −1}. One advantage of looking at it that way is the ease with which the correct generalisation can be found, for example for cubic residues in the case that p − 1 is divisible by three.
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However, the values from this table may be combined by a more complicated function than bitwise exclusive or. Lemire shows that no scheme of this type can be 3-independent. Nevertheless, he shows that it is still possible to achieve 2-independence. In particular, a tabulation scheme that interprets the values T[xi] (where xi is, as before, the ith block of the input) as the coefficients of a polynomial over a finite field and then takes the remainder of the resulting polynomial modulo another polynomial, gives a 2-independent hash function.
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Meyer's theorem is best possible with respect to the number of variables: there are indefinite rational quadratic forms Q in four variables which do not represent zero. One family of examples is given by :Q(x1,x2,x3,x4) = x12 \+ x22 − p(x32 \+ x42), where p is a prime number that is congruent to 3 modulo 4. This can be proved by the method of infinite descent using the fact that if the sum of two perfect squares is divisible by such a p then each summand is divisible by p.
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Rear view The car's exterior, according to Okuyama is intended to be smooth while showing supporting components from the outside. A feature seen on the birdcage racing cars. The car has a clear engine cover from which the engine is visible as well as a large greenhouse area on the canopy to allow for better outward visibility. The Birdcage's design, especially the canopy, was inspired by the Ferrari Modulo concept car along with the Birdcage racing cars which were known for their protruding wheel arches, unusually low body lines and tall but extremely raked windshields.
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In computer science and mathematical logic, the satisfiability modulo theories (SMT) problem is a decision problem for logical formulas with respect to combinations of background theories expressed in classical first-order logic with equality. Examples of theories typically used in computer science are the theory of real numbers, the theory of integers, and the theories of various data structures such as lists, arrays, bit vectors and so on. SMT can be thought of as a form of the constraint satisfaction problem and thus a certain formalized approach to constraint programming.
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International standard ISO 1155ISO 1155:1978 Information processing -- Use of longitudinal parity to detect errors in information messages. states that a longitudinal redundancy check for a sequence of bytes may be computed in software by the following algorithm: lrc := 0 for each byte b in the buffer do lrc := (lrc + b) and 0xFF lrc := (((lrc XOR 0xFF) + 1) and 0xFF) which can be expressed as "the 8-bit two's-complement value of the sum of all bytes modulo 28" (`x AND 0xFF` is equivalent to `x MOD 28`).
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The most important open question in the theory of Hadamard matrices is that of existence. The Hadamard conjecture proposes that a Hadamard matrix of order 4k exists for every positive integer k. The Hadamard conjecture has also been attributed to Paley, although it was considered implicitly by others prior to Paley's work.. A generalization of Sylvester's construction proves that if H_n and H_m are Hadamard matrices of orders n and m respectively, then H_n \otimes H_m is a Hadamard matrix of order nm. This result is used to produce Hadamard matrices of higher order once those of smaller orders are known. Sylvester's 1867 construction yields Hadamard matrices of order 1, 2, 4, 8, 16, 32, etc. Hadamard matrices of orders 12 and 20 were subsequently constructed by Hadamard (in 1893). In 1933, Raymond Paley discovered the Paley construction, which produces a Hadamard matrix of order q + 1 when q is any prime power that is congruent to 3 modulo 4 and that produces a Hadamard matrix of order 2(q + 1) when q is a prime power that is congruent to 1 modulo 4. His method uses finite fields. The smallest order that cannot be constructed by a combination of Sylvester's and Paley's methods is 92.
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The walks were lined with a hundred allegorical marble sculptures, executed by Francesco Penso, Pietro Baratta, Marino Gropelli, Alvise Tagliapietra, Bartolomeo Modulo and other Venetian sculptors that were acquired by Sava Vladislavich. In the late 20th century, 90 surviving statues were moved indoors, while modern replicas took their place in the park. The sequence of patterned parterres, originally more formal than the current landscape, were the site of Imperial assemblies, or lavish parties which often included balls, feasts, and fireworks. Apart from the statuary, a major park attraction were the fountains, the oldest in Russia, representing scenes from Aesop's fables.
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In order for this generator to be secure, the prime number p needs to be large enough so that computing discrete logarithms modulo p is infeasible. To be more precise, any method that predicts the numbers generated will lead to an algorithm that solves the discrete logarithm problem for that prime. There is a paper discussing possible examples of the quantum permanent compromise attack to the Blum–Micali construction. This attacks illustrate how a previous attack to the Blum–Micali generator can be extended to the whole Blum–Micali construction, including the Blum Blum Shub and Kaliski generators.
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The Lucas–Lehmer primality test (LLT) is an efficient primality test that greatly aids this task, making it much easier to test the primality of Mersenne numbers than that of most other numbers of the same size. The search for the largest known prime has somewhat of a cult following. Consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing. Arithmetic modulo a Mersenne number is particularly efficient on a binary computer, making them popular choices when a prime modulus is desired, such as the Park–Miller random number generator.
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His algorithms include: Baby-step giant-step algorithm for computing the discrete logarithm, which is useful in public-key cryptography; Shanks's square forms factorization, an integer factorization method that generalizes Fermat's factorization method; and the Tonelli–Shanks algorithm that finds square roots modulo a prime, which is useful for the quadratic sieve method of integer factorization. In 1974, Shanks and John Wrench did some of the first computer work on estimating the value of Brun's constant, the sum of the reciprocals of the twin primes, calculating it over the twin primes among the first two million primes.
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The standard decimation-in-frequency (DIF) radix-r Cooley–Tukey algorithm corresponds closely to a recursive factorization. For example, radix-2 DIF Cooley–Tukey factors z^N-1 into F_1 = (z^{N/2}-1) and F_2 = (z^{N/2}+1). These modulo operations reduce the degree of x(z) by 2, which corresponds to dividing the problem size by 2. Instead of recursively factorizing F_2 directly, though, Cooley–Tukey instead first computes x2(z ωN), shifting all the roots (by a twiddle factor) so that it can apply the recursive factorization of F_1 to both subproblems.
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Every automorphism of \Q(\omega) is obtained in this way, and these automorphisms form the Galois group of \Q(\omega) over the field of the rationals. The rules of exponentiation imply that the composition of two such automorphisms is obtained by multiplying the exponents. It follows that the map :k\mapsto \left(\omega \mapsto \omega^k\right) defines a group isomorphism between the units of the ring of integers modulo and the Galois group of \Q(\omega). This shows that this Galois group is abelian, and implies thus that the primitive roots of unity may be expressed in terms of radicals.
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The fact that parity is not contained in AC0 was first established independently by Ajtai (1983) and by Furst, Saxe and Sipser (1984). Later improvements by Håstad (1987) in fact establish that any family of constant-depth circuits computing the parity function requires exponential size. Extending a result of Razborov, Smolensky (1987) proved that this is true even if the circuit is augmented with gates computing the sum of its input bits modulo some odd prime p. The k-clique problem is to decide whether a given graph on n vertices has a clique of size k.
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Whereas expressions denote mainly numbers in elementary algebra, in Boolean algebra, they denote the truth values false and true. These values are represented with the bits (or binary digits), namely 0 and 1. They do not behave like the integers 0 and 1, for which 1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. Addition and multiplication then play the Boolean roles of XOR (exclusive-or) and AND (conjunction), respectively, with disjunction x∨y (inclusive-or) definable as x + y - xy.
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Therefore, the check digit value is 7. i.e. (53 / 10) = 5 remainder 3; 10 - 3 = 7. Another example: to calculate the check digit for the following food item "01010101010x". # Add the odd number digits: 0+0+0+0+0+0 = 0. # Multiply the result by 3: 0 x 3 = 0. # Add the even number digits: 1+1+1+1+1=5. # Add the two results together: 0 + 5 = 5. # To calculate the check digit, take the remainder of (5 / 10), which is also known as (5 modulo 10), and if not 0, subtract from 10: i.e.
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The definition of ideals is such that "dividing" I "out" gives another ring, the factor ring R / I: it is the set of cosets of I together with the operations :(a + I) + (b + I) = (a + b) + I and (a + I)(b + I) = ab + I. For example, the ring Z/nZ (also denoted Zn), where n is an integer, is the ring of integers modulo n. It is the basis of modular arithmetic. An ideal is proper if it is strictly smaller than the whole ring. An ideal that is not strictly contained in any proper ideal is called maximal.
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For any integer k > 0 and any n−dimensional Ck−manifold, the maximal atlas contains a C∞−atlas on the same underlying set by a theorem due to Hassler Whitney. It has also been shown that any maximal Ck−atlas contains some number of distinct maximal C∞−atlases whenever n > 0, although for any pair of these distinct C∞−atlases there exists a C∞−diffeomorphism identifying the two. It follows that there is only one class of smooth structures (modulo pairwise smooth diffeomorphism) over any topological manifold which admits a differentiable structure, i.e. The C∞−, structures in a Ck−manifold.
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In computer science, software pipelining is a technique used to optimize loops, in a manner that parallels hardware pipelining. Software pipelining is a type of out-of-order execution, except that the reordering is done by a compiler (or in the case of hand written assembly code, by the programmer) instead of the processor. Some computer architectures have explicit support for software pipelining, notably Intel's IA-64 architecture. It is important to distinguish software pipelining, which is a target code technique for overlapping loop iterations, from modulo scheduling, the currently most effective known compiler technique for generating software pipelined loops.
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Salvatore and T. Shima, "Of coconuts and integrity," Crux Mathematicorum, 4 (1978) 182–185 Clever and succinct solutions using modulo congruences, sieves, and alternate number bases have been devised based partly or mostly on the recursive definition of the problem, a structure that won't be applicable in the general case. The smallest positive solutions to both versions are sufficiently large that trial and error is very likely to be fruitless. An ingenious concept of negative coconuts was introduced that fortuitously solves the original problem. Formalistic solutions are based on Euclid's algorithm applied to the Diophantine coefficients.
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The machine originally used electrostatic tubes (CRT or Williams tube) for memory which stored 2000 words, with an access time of 8 microseconds. Each word consisted of 16 decimal digits, using four bits to represent each digit, plus two modulo-4 error-checking bits. A word could store a 13-digit number with sign and 2-digit index, or one instruction. NORC used four sets of 66 electrostatic tubes in parallel for memory. Each of the tubes in a set of 66 stored one bit of each of 500 words, so each of the four sets of 66 tubes stored 2000 words.
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Integer interval complementation: 5 + 7 = 0 mod 12 Using integer notation and modulo 12 (in which the numbers "wrap around" at 12, 12 and its multiples therefore being defined as 0), any two intervals which add up to 0 (mod 12) are complements (mod 12). In this case the unison, 0, is its own complement, while for other intervals the complements are the same as above (for instance a perfect fifth, or 7, is the complement of the perfect fourth, or 5, 7 + 5 = 12 = 0 mod 12). Thus the #Sum of complementation is 12 (= 0 mod 12).
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Langford pairings exist only when n is congruent to 0 or 3 modulo 4; for instance, there is no Langford pairing when n = 1, 2, or 5. The numbers of different Langford pairings for n = 1, 2, …, counting any sequence as being the same as its reversal, are :0, 0, 1, 1, 0, 0, 26, 150, 0, 0, 17792, 108144, … . As describes, the problem of listing all Langford pairings for a given n can be solved as an instance of the exact cover problem, but for large n the number of solutions can be calculated more efficiently by algebraic methods.
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Retrieved on 9 May 2017. surveys that use as reference skin color such as those made by Mexico's National Council to Prevent Discrimination and Mexico's National Institute of Statistics and Geography reported a percentages of 47% in 2010 and 49% in 2017" Visión INEGI 2021 Dr. Julio Santaella Castell", INEGI, 03 July 2017, Retrieved on 30 April 2018."Resultados del Modulo de Movilidad Social Intergeneracional" , INEGI, 16 June 2017, Retrieved on 30 April 2018. respectively. Another survey published in 2018 reported a percentage significantly lower at 29%,"Encuesta Nacional sobre Discriminación 2017", CNDH, 6 August 2018, Retrieved on 10 August 2018.
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RSA-130 has 130 decimal digits (430 bits), and was factored on April 10, 1996 by a team led by Arjen K. Lenstra and composed of Jim Cowie, Marije Elkenbracht-Huizing, Wojtek Furmanski, Peter L. Montgomery, Damian Weber and Joerg Zayer.Arjen K. Lenstra (1996-04-12), Factorization of RSA-130. Retrieved on 2008-03-10. The value and factorization are as follows: RSA-130 = 1807082088687404805951656164405905566278102516769401349170127021450056662540244048387341127590812303371781887966563182013214880557 RSA-130 = 39685999459597454290161126162883786067576449112810064832555157243 × 45534498646735972188403686897274408864356301263205069600999044599 The factorization was found using the Number Field Sieve algorithm and the polynomial 5748302248738405200 x5 \+ 9882261917482286102 x4 \- 13392499389128176685 x3 \+ 16875252458877684989 x2 \+ 3759900174855208738 x1 \- 46769930553931905995 which has a root of 12574411168418005980468 modulo RSA-130.
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Sequence numbers modulo 4, with wr=1. Initially, nt=nr=0 So far, the protocol has been described as if sequence numbers are of unlimited size, ever-increasing. However, rather than transmitting the full sequence number x in messages, it is possible to transmit only x mod N, for some finite N. (N is usually a power of 2.) For example, the transmitter will only receive acknowledgments in the range na to nt, inclusive. Since it guarantees that nt−na ≤ wt, there are at most wt+1 possible sequence numbers that could arrive at any given time.
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The dimensions hp,q of spaces of harmonic (p,q)-differential forms (equivalently, the cohomology, i.e., closed forms modulo exact forms) are conventionally arranged in a diamond shape called the Hodge Diamond. These (p,q)-betti numbers can be computed for complete intersections using a generating function described by Friedrich Hirzebruch. For a three-dimensional manifold, for example, the Hodge diamond has p and q ranging from 0 to 3: Mirror symmetry translates the dimension number of the (p, q)-th differential form hp,q for the original manifold into hn-p,q of that for the counter pair manifold.
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However, the definition of a cup product generalizes to complexes and topological manifolds. This is an advantage for mathematicians who are interested in complexes and topological manifolds (not only in PL and smooth manifolds). When the 4-manifold is smooth, then in de Rham cohomology, if a and b are represented by 2-forms \alpha and \beta, then the intersection form can be expressed by the integral : Q(a,b)= \int_M \alpha \wedge \beta where \wedge is the wedge product. The definition using cup product has a simpler analogue modulo 2 (which works for non-orientable manifolds).
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The Blum Blum Shub pseudorandom number generator, published jointly by Blum, Manuel Blum, and Michael Shub, is based on the operation of squaring numbers modulo the products of two large primes. Its security can be reduced to the computational hardness assumption that integer factorization is infeasible. Blum is also known for the Blum–Shub–Smale machine, a theoretical model of computation over the real numbers. Blum and her co-authors, Michael Shub and Stephen Smale, showed that (analogously to the theory of Turing machines) one can define analogues of NP- completeness, undecidability, and universality for this model.
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The total percentage dedicated to commerce, tourism and services is 53%. The municipality has several preschools, primary schools along with center for study at the high school level. Villa de Etla is the head of the twenty eight schools of Scholastic Zone Number 5. Preschools include Ignacio José Allende, Jaime Torres Bodet, Leona Vicario, Niños Heroes de Chapultepec, Ovidio Decroly, Porfirio Díaz and Prodei Modulo 2. Primary schools include Basilio e Zarate, Cuauhtémoc, Iep 13 General de Division Ignacio Mejia, José María Morelos, Juan de la Barrera, Margarita Maza de Juarez, Ricardo Flores Magon and Valentin Gomez Farias.
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Therefore, the application using these random numbers must use the most significant bits; reducing to a smaller range using a modulo operation with an even modulus will produce disastrous results. To achieve this period, the multiplier must satisfy a ≡ ±3 (mod 8) and the seed X must be odd. Using a composite modulus is possible, but the generator must be seeded with a value coprime to m, or the period will be greatly reduced. For example, a modulus of F = 2+1 might seem attractive, as the outputs can be easily mapped to a 32-bit word 0 ≤ X−1 < 2\.
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In this context, between 1988 and 1992, experiments were carried out on a prototype gate (MOdulo Sperimentale Elettromeccanico, hence the name MOSE) and in 1989, a conceptual design for the mobile barriers was drawn up. This was completed in 1992 and subsequently approved by the Higher Council of Public Works then subjected to an Environmental Impact Assessment procedure and further developed as requested by the Comitatone. In 2002 the final design was presented and on 3April 2003, the Comitatone gave the go-ahead for its implementation. The same year, construction sites opened at the three lagoon inlets of Lido, Malamocco and Chioggia.
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In the case of a nonlinear equation, it will only rarely be possible to obtain the general solution in closed form. However, if the equation is quasilinear (linear in the highest order derivatives), then we can still obtain approximate information similar to the above: specifying a member of the solution space will be "modulo nonlinear quibbles" equivalent to specifying a certain number of functions in a smaller number of variables. The number of these functions is the Einstein strength of the p.d.e. In the simple example above, the strength is two, although in this case we were able to obtain more precise information.
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In mathematics, the Chang number of an irreducible representation of a simple complex Lie algebra is its dimension modulo 1 + h, where h is the Coxeter number. Chang numbers are named after , who rediscovered an element of order h + 1 found by . showed that there is a unique class of regular elements σ of order h + 1, in the complex points of the corresponding Chevalley group. He showed that the trace of σ on an irreducible representation is −1, 0, or +1, and if h + 1 is prime then the trace is congruent to the dimension mod h+1.
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Rational equivalence of divisors (known as linear equivalence) was studied in various forms during the 19th century, leading to the ideal class group in number theory and the Jacobian variety in the theory of algebraic curves. For higher-codimension cycles, rational equivalence was introduced by Francesco Severi in the 1930s. In 1956, Wei-Liang Chow gave an influential proof that the intersection product is well-defined on cycles modulo rational equivalence for a smooth quasi-projective variety, using Chow's moving lemma. Starting in the 1970s, Fulton and MacPherson gave the current standard foundation for Chow groups, working with singular varieties wherever possible.
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Applying the Doomsday algorithm involves three steps: Determination of the anchor day for the century, calculation of the anchor day for the year from the one for the century, and selection of the closest date out of those that always fall on the doomsday, e.g., 4/4 and 6/6, and count of the number of days (modulo 7) between that date and the date in question to arrive at the day of the week. The technique applies to both the Gregorian calendar and the Julian calendar, although their doomsdays are usually different days of the week.
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All odd squares are ≡ 1 (mod 8) and thus also ≡ 1 (mod 4). If a is an odd number and m = 8, 16, or some higher power of 2, then a is a residue modulo m if and only if a ≡ 1 (mod 8).Gauss, DA, art. 103 > For example, mod (32) the odd squares are :12 ≡ 152 ≡ 1 :32 ≡ 132 ≡ 9 :52 ≡ > 112 ≡ 25 :72 ≡ 92 ≡ 49 ≡ 17 and the even ones are :02 ≡ 82 ≡ 162 ≡ 0 :22 ≡ > 62≡ 102 ≡ 142≡ 4 :42 ≡ 122 ≡ 16\. So a nonzero number is a residue mod 8, 16, etc.
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The method works because the original numbers are 'decimal' (base 10), the modulus is chosen to differ by 1, and casting out is equivalent to taking a digit sum. In general any two 'large' integers, x and y, expressed in any smaller modulus as x' and y' (for example, modulo 7) will always have the same sum, difference or product as their originals. This property is also preserved for the 'digit sum' where the base and the modulus differ by 1. If a calculation was correct before casting out, casting out on both sides will preserve correctness.
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Since modular arithmetic has such a wide range of applications, it is important to know how hard it is to solve a system of congruences. A linear system of congruences can be solved in polynomial time with a form of Gaussian elimination, for details see linear congruence theorem. Algorithms, such as Montgomery reduction, also exist to allow simple arithmetic operations, such as multiplication and exponentiation modulo , to be performed efficiently on large numbers. Some operations, like finding a discrete logarithm or a quadratic congruence appear to be as hard as integer factorization and thus are a starting point for cryptographic algorithms and encryption.
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There exist planar non- Hamiltonian graphs in which all faces have five or eight sides. For these graphs, Grinberg's formula taken modulo three is always satisfied by any partition of the faces into two subsets, preventing the application of his theorem to proving non-Hamiltonicity in this case . It is not possible to use Grinberg's theorem to find counterexamples to Barnette's conjecture, that every cubic bipartite polyhedral graph is Hamiltonian. Every cubic bipartite polyhedral graph has a partition of the faces into two subsets satisfying Grinberg's theorem, regardless of whether it also has a Hamiltonian cycle .
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In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour. The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether,For example L'émergence de la notion de groupe d'homologie, Nicolas Basbois (PDF), note 41, explicitly names Noether as inventing homology groups.
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When -2\leq k\leq 2 \pmod 8 there is a real Majorana spinor representation, whose dimension is half that of the Dirac representation. When k is even there is a Weyl spinor representation, whose real dimension is again half that of the Dirac spinor. Finally when k is divisible by eight, that is, when k is zero modulo eight, there is a Majorana–Weyl spinor, whose real dimension is one quarter that of the Dirac spinor. Occasionally one also considers symplectic Majorana spinor which exist when 3\leq k\leq 5, which have half has many components as Dirac spinors.
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In the theory of higher reciprocity laws, Alderson published necessary and sufficient conditions for 2 and 3 to be seventh powers, in modular arithmetic modulo a given prime number p. According to , "plain quasigroups were first studied by Helen Popova- Alderson, in a series of papers dating back to the early fifties". Smith cites in particular a posthumous paper and its references. In this context, a quasigroup is a mathematical structure consisting of a set of elements and a binary operation that does not necessarily obey the associative law, but where (like a group) this operation can be inverted.
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Nim has been mathematically solved for any number of initial heaps and objects, and there is an easily calculated way to determine which player will win and what winning moves are open to that player. The key to the theory of the game is the binary digital sum of the heap sizes, that is, the sum (in binary) neglecting all carries from one digit to another. This operation is also known as "bitwise xor" or "vector addition over GF(2)" (bitwise addition modulo 2). Within combinatorial game theory it is usually called the nim-sum, as it will be called here.
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In quantum chemistry electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either \rho(\textbf r) or n(\textbf r). The density is determined, through definition, by the normalized N-electron wavefunction which itself depends upon 4N variables (3N spatial and N spin coordinates). Conversely, the density determines the wave function modulo up to a phase factor, providing the formal foundation of density functional theory.
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The polynomial : has discriminant :, and so is unramified at the prime 3; it is also irreducible mod 3. Hence adjoining a root of it to the field of -adic numbers gives an unramified extension of . We may find the image of under the Frobenius map by locating the root nearest to , which we may do by Newton's method. We obtain an element of the ring of integers in this way; this is a polynomial of degree four in with coefficients in the -adic integers . Modulo this polynomial is :\rho^3 + 3(460+183\rho-354\rho^2-979\rho^3-575\rho^4).
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One reason to distinguish between polynomials and polynomial functions is that, over some rings, different polynomials may give rise to the same polynomial function (see Fermat's little theorem for an example where is the integers modulo ). This is not the case when is the real or complex numbers, whence the two concepts are not always distinguished in analysis. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like Euclidean division) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for .
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When hex dumps are intended to be manually entered into a computer, such as was the case with print magazine articles of the home computer era, a checksum byte (or two) would be added at the end of each row, commonly calculated as simple 256 modulo of sum of all values in the row or a more sophisticated CRC. This checksum would be used to determine whether users entered the row correctly or not. A variety of hex dump file formats including S-record, Intel HEX, and Tektronix extended HEX have a similar checksum value at the end of each row.
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Suppose that are a set of generators for the unit group modulo roots of unity. If is an algebraic number, write for the different embeddings into or , and set to 1 or 2 if the corresponding embedding is real or complex respectively. Then the matrix whose entries are , has the property that the sum of any row is zero (because all units have norm 1, and the log of the norm is the sum of the entries in a row). This implies that the absolute value of the determinant of the submatrix formed by deleting one column is independent of the column.
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Variably Modified Permutation Composition (VMPC) is another RC4 variant. It uses similar key schedule as RC4, with iterating 3 × 256 = 768 times rather than 256, and with an optional additional 768 iterations to incorporate an initial vector. The output generation function operates as follows: All arithmetic is performed modulo 256. i := 0 while GeneratingOutput: a := S[i] j := S[j + a] output S[S[S[j] + 1 Swap S[i] and S[j] (b := S[j]; S[i] := b; S[j] := a)) i := i + 1 endwhile This was attacked in the same papers as RC4A, and can be distinguished within 238 output bytes.
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Because of this uniqueness, Euclidean division is often considered without referring to any method of computation, and without explicitly computing the quotient and the remainder. The methods of computation are called integer division algorithms, the best known of which being long division. Euclidean division, and algorithms to compute it, are fundamental for many questions concerning integers, such as the Euclidean algorithm for finding the greatest common divisor of two integers, and modular arithmetic, for which only remainders are considered. The operation consisting of computing only the remainder is called the modulo operation, and is used often in both mathematics and computer science.
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We choose these conditions so as to ensure that they reduce modulo the maximal ideal to Frobenius and the derivative at the origin is the prime element. For each element a in Zp there is a unique endomorphism f of the Lubin–Tate formal group law such that f(x) = ax + higher-degree terms. This gives an action of the ring Zp on the Lubin–Tate formal group law. There is a similar construction with Zp replaced by any complete discrete valuation ring with finite residue class field, where p is replaced by a choice of uniformizer.
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When expressing Easter algorithms without using tables, it has been customary to employ only the integer operations addition, subtraction, multiplication, division, modulo, and assignment (`plus, minus, times, div, mod, assign`) as it is compatible with the use of simple mechanical or electronic calculators. That restriction is undesirable for computer programming, where conditional operators and statements, as well as look-up tables, are available. One can easily see how conversion from day-of-March (22 to 56) to day-and-month (22 March to 25 April) can be done as . More importantly, using such conditionals also simplifies the core of the Gregorian calculation.
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In geometric situations when X is a closed manifold, the importance of the Betti numbers may arise from a different direction, namely that they predict the dimensions of vector spaces of closed differential forms modulo exact differential forms. The connection with the definition given above is via three basic results, de Rham's theorem and Poincaré duality (when those apply), and the universal coefficient theorem of homology theory. There is an alternate reading, namely that the Betti numbers give the dimensions of spaces of harmonic forms. This requires also the use of some of the results of Hodge theory, about the Hodge Laplacian.
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The RSA problem is defined as the task of taking eth roots modulo a composite n: recovering a value m such that , where is an RSA public key and c is an RSA ciphertext. Currently the most promising approach to solving the RSA problem is to factor the modulus n. With the ability to recover prime factors, an attacker can compute the secret exponent d from a public key , then decrypt c using the standard procedure. To accomplish this, an attacker factors n into p and q, and computes which allows the determination of d from e.
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A crescent Moon over Kingman, Arizona Each of the four intermediate phases lasts approximately seven days (7.38 days on average), but varies slightly due to lunar apogee and perigee. The number of days counted from the time of the new moon is the Moon's "age". Each complete cycle of phases is called a "lunation". The approximate age of the Moon, and hence the approximate phase, can be calculated for any date by calculating the number of days since a known new moon (such as January 1, 1900 or August 11, 1999) and reducing this modulo 29.530588853 (the length of a synodic month).
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Range ambiguity resolution is a technique used with medium Pulse repetition frequency (PRF) radar to obtain range information for distances that exceed the distance between transmit pulses. This signal processing technique is required with pulse-Doppler radar. The raw return signal from a reflection will appear to be arriving from a distance less than the true range of the reflection when the wavelength of the pulse repetition frequency (PRF) is less than the range of the reflection. This causes reflected signals to be folded, so that the apparent range is a modulo function of true range.
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However that is disputed: shows using a computer search and proofs that there are precisely two 8_4 that are actually "theorems": the Möbius configuration and one other. The latter (which corresponds to the conjugacy class (12)(34) above) is also a theorem for all three-dimensional projective spaces over a field, but not over a general division ring. There are other close similarities between the two configurations, including the fact that both are self-dual under Matroid duality. In abstract terms, the latter configuration has "points" 0,...,7 and "planes" 0125+i, (i = 0,...,7), where these integers are modulo eight.
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Because the exterior derivative has the property that , it can be used as the differential (coboundary) to define de Rham cohomology on a manifold. The -th de Rham cohomology (group) is the vector space of closed -forms modulo the exact -forms; as noted in the previous section, the Poincaré lemma states that these vector spaces are trivial for a contractible region, for . For smooth manifolds, integration of forms gives a natural homomorphism from the de Rham cohomology to the singular cohomology over . The theorem of de Rham shows that this map is actually an isomorphism, a far-reaching generalization of the Poincaré lemma.
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In contrast, this equation has no solution in the finite fields Z/p where p is an odd prime but is not Pythagorean., p. 100. The Paley graph with 13 vertices For every Pythagorean prime p, there exists a Paley graph with p vertices, representing the numbers modulo p, with two numbers adjacent in the graph if and only if their difference is a quadratic residue. This definition produces the same adjacency relation regardless of the order in which the two numbers are subtracted to compute their difference, because of the property of Pythagorean primes that −1 is a quadratic residue..
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DFAO generating the Thue–Morse sequence The Thue–Morse sequence t(n) () is the fixed point of the morphism 0 → 01, 1 → 10. Since the n-th term of the Thue–Morse sequence counts the number of ones modulo 2 in the base-2 representation of n, it is generated by the two-state deterministic finite automaton with output pictured here, where being in state q0 indicates there are an even number of ones in the representation of n and being in state q1 indicates there are an odd number of ones. Hence, the Thue–Morse sequence is 2-automatic.
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You may want to know, in what integer residue class rings you have a primitive k-th root of unity. You need it for instance if you want to compute a Discrete Fourier Transform (more precisely a Number theoretic transform) of a k-dimensional integer vector. In order to perform the inverse transform, you also need to divide by k, that is, k shall also be a unit modulo n. A simple way to find such an n is to check for primitive k-th roots with respect to the moduli in the arithmetic progression k+1, 2k+1, 3k+1, \dots.
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One important application of the Cantor–Zassenhaus algorithm is in computing discrete logarithms over finite fields of prime-power order. Computing discrete logarithms is an important problem in public key cryptography. For a field of prime-power order, the fastest known method is the index calculus method, which involves the factorisation of field elements. If we represent the prime-power order field in the usual way – that is, as polynomials over the prime order base field, reduced modulo an irreducible polynomial of appropriate degree – then this is simply polynomial factorisation, as provided by the Cantor–Zassenhaus algorithm.
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For large values of , the th telephone number is divisible by a large power of two, . More precisely, the 2-adic order (the number of factors of two in the prime factorization) of and of is ; for it is , and for it is .. For any prime number , one can test whether there exists a telephone number divisible by by computing the recurrence for the sequence of telephone numbers, modulo , until either reaching zero or detecting a cycle. The primes that divide at least one telephone number are :2, 5, 13, 19, 23, 29, 31, 43, 53, 59, ...
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Similarly, a convenient modulus would be 255, although, again, others could be chosen. So, the simple checksum is computed by adding together all the 8-bit bytes of the message, dividing by 255 and keeping only the remainder. (In practice, the modulo operation is performed during the summation to control the size of the result.) The checksum value is transmitted with the message, increasing its length to 137 bytes, or 1096 bits. The receiver of the message can re-compute the checksum and compare it to the value received to determine whether the message has been altered by the transmission process.
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In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity. There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert norm residue symbols (a,b/p), taking values in roots of unity, is equal to 1.
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In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its integral homology groups: : completely determine its homology groups with coefficients in , for any abelian group : : Here might be the simplicial homology, or more generally the singular homology: the result itself is a pure piece of homological algebra about chain complexes of free abelian groups. The form of the result is that other coefficients may be used, at the cost of using a Tor functor. For example it is common to take to be , so that coefficients are modulo 2.
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Modern microprocessors will allow for much faster processing, if 8-bit character strings are not hashed by processing one character at a time, but by interpreting the string as an array of 32 bit or 64 bit integers and hashing/accumulating these "wide word" integer values by means of arithmetic operations (e.g. multiplication by constant and bit-shifting). The final word, which may have unoccupied byte positions, is filled with zeros or a specified "randomizing" value before being folded into the hash. The accumulated hash code is reduced by a final modulo or other operation to yield an index into the table.
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DVB Additive scramblers (they are also referred to as synchronous) transform the input data stream by applying a pseudo-random binary sequence (PRBS) (by modulo-two addition). Sometimes a pre-calculated PRBS stored in the read-only memory is used, but more often it is generated by a linear-feedback shift register (LFSR). In order to assure a synchronous operation of the transmitting and receiving LFSR (that is, scrambler and descrambler), a sync-word must be used. A sync-word is a pattern that is placed in the data stream through equal intervals (that is, in each frame).
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The field of fractions K of an integral domain R is the set of fractions a/b with a and b in R and b ≠ 0 modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations. It is "the smallest field containing R " in the sense that there is an injective ring homomorphism such that any injective ring homomorphism from R to a field factors through K. The field of fractions of the ring of integers \Z is the field of rational numbers \Q. The field of fractions of a field is isomorphic to the field itself.
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This shows that every polynomial over the rationals is associated with a unique primitive polynomial over the integers, and that the Euclidean algorithm allows the computation of this primitive polynomial. A consequence is that factoring polynomials over the rationals is equivalent to factoring primitive polynomials over the integers. As polynomials with coefficients in a field are more common than polynomials with integer coefficients, it may seem that this equivalence may be used for factoring polynomials with integer coefficients. In fact, the truth is exactly the opposite: every known efficient algorithm for factoring polynomials with rational coefficient uses this equivalence for reducing the problem modulo some prime number (see Factorization of polynomials).
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If f(x) is a univariate polynomial over the integers, assumed to be content-free and square-free, one starts by computing a bound B such that any factor g(x) has coefficients of absolute value bounded by B. This way, if m is an integer larger than 2B, and if g(x) is known modulo m, then g(x) can be reconstructed from its image mod m. The Zassenhaus algorithm proceeds as follows. First, choose a prime number p such that the image of f(x) mod p remains square-free, and of the same degree as f(x). Then factor f(x) mod p.
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The MOSE project (which stands for Modulo Sperimentale Elettromeccanico, i.e. "Experimental Electromechanical Module") has been under construction since 2003, the long time period partly because of budget constraints, and partly because of the sheer complexity of the undertaking. The project should significantly reduce the effects of "exceptional high waters" (but not those of lesser, yet detrimental, tidal events) by completing the installation of 79 separate 300-ton flaps hinged on the seabed between the lagoon and the Adriatic sea. While normally fully submerged and invisible, the flaps can be raised preemptively to create a temporary barrier, which is expected to protect the city from exceptional acqua alta.
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The resulting quotient is written , where G is the original group and N is the normal subgroup. (This is pronounced "G mod N", where "mod" is short for modulo.) Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of G. Specifically, the image of G under a homomorphism is isomorphic to where ker(φ) denotes the kernel of φ. The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one.
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In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.
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In a finite field, the product of two non-squares is a square; this implies that the polynomial x^4 + 1, which is irreducible over the integers, is reducible modulo every prime number. For example, :x^4 + 1 \equiv (x+1)^4 \pmod 2; :x^4 + 1 \equiv (x^2+x-1)(x^2-x-1) \pmod 3,\qquadsince 1^2 \equiv -2 \pmod 3; :x^4 + 1 \equiv (x^2+2)(x^2-2) \pmod 5,\qquadsince 2^2 \equiv -1 \pmod 5; :x^4 + 1 \equiv (x^2+3x+1)(x^2-3x+1) \pmod 7,\qquadsince 3^2 \equiv 2 \pmod 7.
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Watson, G.S. (1961) "Goodness-Of-Fit Tests on a Circle", Biometrika, 48 (1/2), 109–114 Another test statistic having this property is the Watson statistic, which is related to the Cramér–von Mises test. However, if failures occur mostly on weekends, many uniform-distribution tests such as K-S and Kuiper would miss this, since weekends are spread throughout the year. This inability to distinguish distributions with a comb-like shape from continuous uniform distributions is a key problem with all statistics based on a variant of the K-S test. Kuiper's test, applied to the event times modulo one week, is able to detect such a pattern.
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The Rabin–Karp string search algorithm is often explained using a rolling hash function that only uses multiplications and additions: :H = c_1 a^{k-1} + c_2 a^{k-2} + c_3 a^{k-3} + ... + c_k a^{0}, where a is a constant, and c_1, ..., c_k are the input characters (but this function is not a Rabin fingerprint, see below). In order to avoid manipulating huge H values, all math is done modulo n. The choice of a and n is critical to get good hashing; see linear congruential generator for more discussion. Removing and adding characters simply involves adding or subtracting the first or last term.
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Multiplicative number theory deals primarily in asymptotic estimates for arithmetic functions. Historically the subject has been dominated by the prime number theorem, first by attempts to prove it and then by improvements in the error term. The Dirichlet divisor problem that estimates the average order of the divisor function d(n) and Gauss's circle problem that estimates the average order of the number of representations of a number as a sum of two squares are also classical problems, and again the focus is on improving the error estimates. The distribution of primes numbers among residue classes modulo an integer is an area of active research.
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In computational number theory and computational algebra, Pollard's kangaroo algorithm (also Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced in 1978 by the number theorist J. M. Pollard, in the same paper J. Pollard, Monte Carlo methods for index computation (mod p), Mathematics of Computation, Volume 32, 1978 as his better-known Pollard's rho algorithm for solving the same problem. Although Pollard described the application of his algorithm to the discrete logarithm problem in the multiplicative group of units modulo a prime p, it is in fact a generic discrete logarithm algorithm—it will work in any finite cyclic group.
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The Indonesian model top trim (Prestige) is equipped with 5-spoke 18-inch Modulo alloy wheels not seen in other markets. In the Philippines, the 1.6-liter i-DTEC diesel engine manages 120 PS and 300 Nm of torque. The CR-V Hybrid was unveiled at the 2017 Auto Shanghai in China during April 2017. The fifth generation CR-V was also launched in Japan on 30 August 2018 and went on sale on the following day, making it the return of the CR-V for the Japanese domestic market after a two-year hiatus since the fourth generation CR-V was discontinued there in August 2016.
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Under EFM rules, the data to be stored is first broken into eight-bit blocks (bytes). Each eight-bit block is translated into a corresponding fourteen-bit codeword using a lookup table. The 14-bit words are chosen such that binary ones are always separated by a minimum of two and a maximum of ten binary zeroes. This is because bits are encoded with NRZI encoding, or modulo-2 integration, so that a binary one is stored on the disc as a change from a land to a pit or a pit to a land, while a binary zero is indicated by no change.
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In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer. The existence of congruence subgroups in an arithmetic group provides it with a wealth of subgroups, in particular it shows that the group is residually finite.
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More generally, differential invariants can be considered for mappings from any smooth manifold X into another smooth manifold Y for a Lie group acting on the Cartesian product X×Y. The graph of a mapping X -> Y is a submanifold of X×Y that is everywhere transverse to the fibers over X. The group G acts, locally, on the space of such graphs, and induces an action on the k-th prolongation Y(k) consisting of graphs passing through each point modulo the relation of k-th order contact. A differential invariant is a function on Y(k) that is invariant under the prolongation of the group action.
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For example, since the 1970s fractals have been studied also as models for algorithmic composition. As an example of deterministic compositions through mathematical models, the On-Line Encyclopedia of Integer Sequences provides an option to play an integer sequence as 12-tone equal temperament music. (It is initially set to convert each integer to a note on an 88-key musical keyboard by computing the integer modulo 88, at a steady rhythm. Thus 123456, the natural numbers, equals half of a chromatic scale.) As another example, the all-interval series has been used for computer-aided composition Mauricio Toro, Carlos Agon, Camilo Rueda, Gerard Assayag.
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The classification of complex manifolds is much more subtle than that of differentiable manifolds. For example, while in dimensions other than four, a given topological manifold has at most finitely many smooth structures, a topological manifold supporting a complex structure can and often does support uncountably many complex structures. Riemann surfaces, two dimensional manifolds equipped with a complex structure, which are topologically classified by the genus, are an important example of this phenomenon. The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a moduli space, the structure of which remains an area of active research.
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The following presentation incorporates a slight simplification of his argument, due to Gauss, which appears in article 182 of the Disquisitiones Arithmeticae. An (integral binary) quadratic form is an expression of the form ax^2 + bxy + cy^2 with a,b,c integers. A number n is said to be represented by the form if there exist integers x,y such that n = ax^2 + bxy + cy^2. Fermat's theorem on sums of two squares is then equivalent to the statement that a prime p is represented by the form x^2 + y^2 (i.e., a=c=1, b=0) exactly when p is congruent to 1 modulo 4.
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The REDC algorithm requires products modulo R, and typically so that REDC can be used to compute products. However, when R is a power of B, there is a variant of REDC which requires products only of machine word sized integers. Suppose that positive multi-precision integers are stored little endian, that is, x is stored as an array such that for all i and . The algorithm begins with a multiprecision integer T and reduces it one word at a time. First an appropriate multiple of N is added to make T divisible by B. Then a multiple of N is added to make T divisible by B2, and so on.
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In Rabin's oblivious transfer protocol, the sender generates an RSA public modulus N=pq where p and q are large prime numbers, and an exponent e relatively prime to λ(N) = (p − 1)(q − 1). The sender encrypts the message m as me mod N. # The sender sends N, e, and me mod N to the receiver. # The receiver picks a random x modulo N and sends x2 mod N to the sender. Note that gcd(x,N) = 1 with overwhelming probability, which ensures that there are 4 square roots of x2 mod N. # The sender finds a square root y of x2 mod N and sends y to the receiver.
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ACORN is particularly simple to implement in exact integer arithmetic, in various computer languages, using only a few lines of code.R.S. Wikramaratna, Theoretical and empirical convergence results for additive congruential random number generators, Journal of Computational and Applied Mathematics (2009), doi:10.1016/j.cam.2009.10.015 Integer arithmetic is preferred to the real arithmetic modulo 1 in the original presentation, as the algorithm is then reproducible, producing exactly the same sequence on any machine and in any language, and its periodicity is mathematically provable. The ACORN generator has not seen the wide adoption of some other PRNGs, although it is included in the Fortran and C library routines of NAG Numerical Library.
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In 1977, a generalization of Cocks' scheme was independently invented by Ron Rivest, Adi Shamir and Leonard Adleman, all then at MIT. The latter authors published their work in 1978, and the algorithm came to be known as RSA, from their initials. RSA uses exponentiation modulo a product of two very large primes, to encrypt and decrypt, performing both public key encryption and public key digital signature. Its security is connected to the extreme difficulty of factoring large integers, a problem for which there is no known efficient general technique (though prime factorization may be obtained through brute-force attacks; that may be harder the larger the prime factors are).
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Computation proceeds by picking an arbitrary element x of the group modulo N and computing a large and smooth multiple Ax of it; if the order of at least one but not all of the reduced groups is a divisor of A, this yields a factorisation. It need not be a prime factorisation, as the element might be an identity in more than one of the reduced groups. Generally, A is taken as a product of the primes below some limit K, and Ax is computed by successive multiplication of x by these primes; after each multiplication, or every few multiplications, the check is made for a one-sided identity.
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Cauchy used an infinitesimal \alpha to write down a unit impulse, infinitely tall and narrow Dirac-type delta function \delta_\alpha satisfying \int F(x)\delta_\alpha(x) = F(0) in a number of articles in 1827, see Laugwitz (1989). Cauchy defined an infinitesimal in 1821 (Cours d'Analyse) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and Lazare Carnot's terminology. Modern set- theoretic approaches allow one to define infinitesimals via the ultrapower construction, where a null sequence becomes an infinitesimal in the sense of an equivalence class modulo a relation defined in terms of a suitable ultrafilter.
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BN-pairs can be used to prove that many groups of Lie type are simple modulo their centers. More precisely, if G has a BN-pair such that B is a solvable group, the intersection of all conjugates of B is trivial, and the set of generators of W cannot be decomposed into two non-empty commuting sets, then G is simple whenever it is a perfect group. In practice all of these conditions except for G being perfect are easy to check. Checking that G is perfect needs some slightly messy calculations (and in fact there are a few small groups of Lie type which are not perfect).
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All representation of the symmetric groups are real (and in fact rational), since we can build a complete set of irreducible representations using Young tableaux. All representations of the rotation groups on odd-dimensional spaces are real, since they all appear as subrepresentations of tensor products of copies of the fundamental representation, which is real. Further examples of real representations are the spinor representations of the spin groups in 8k−1, 8k, and 8k+1 dimensions for k = 1, 2, 3 ... This periodicity modulo 8 is known in mathematics not only in the theory of Clifford algebras, but also in algebraic topology, in KO-theory; see spin representation.
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In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression :a,\ a+d,\ a+2d,\ a+3d,\ \dots,\ and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem, named after Peter Gustav Lejeune Dirichlet, extends Euclid's theorem that there are infinitely many prime numbers.
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Another important question is the existence of automorphisms in recursion-theoretic structures. One of these structures is that one of recursively enumerable sets under inclusion modulo finite difference; in this structure, A is below B if and only if the set difference B − A is finite. Maximal sets (as defined in the previous paragraph) have the property that they cannot be automorphic to non-maximal sets, that is, if there is an automorphism of the recursive enumerable sets under the structure just mentioned, then every maximal set is mapped to another maximal set. Soare (1974) showed that also the converse holds, that is, every two maximal sets are automorphic.
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In logic and computer science, unification is an algorithmic process of solving equations between symbolic expressions. Depending on which expressions (also called terms) are allowed to occur in an equation set (also called unification problem), and which expressions are considered equal, several frameworks of unification are distinguished. If higher-order variables, that is, variables representing functions, are allowed in an expression, the process is called higher-order unification, otherwise first-order unification. If a solution is required to make both sides of each equation literally equal, the process is called syntactic or free unification, otherwise semantic or equational unification, or E-unification, or unification modulo theory.
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For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring over the real numbers by factoring out the ideal of multiples of the polynomial . Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring (see modular arithmetic). If is commutative, then one can associate with every polynomial in a polynomial function with domain and range equal to . (More generally, one can take domain and range to be any same unital associative algebra over .) One obtains the value by substitution of the value for the symbol in .
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The CADE ATP System Competition (CASC) is a yearly competition of fully automated theorem provers for classical logic CASC is associated with the Conference on Automated Deduction and the International Joint Conference on Automated Reasoning organized by the Association for Automated Reasoning. It has inspired similar competition in related fields, in particular the successful SMT-COMP competition for Satisfiability Modulo Theories, the SAT Competition for propositional reasoners, and the modal logic reasoning competition. The first CASC, CASC-13, was held as part of the 13th Conference on Automated Deduction at Rutgers University, New Brunswick, NJ, in 1996. Among the systems competing were Otter and SETHEO.
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Many stream ciphers are based on linear-feedback shift registers (LFSRs), which, while efficient in hardware, are less so in software. The design of RC4 avoids the use of LFSRs and is ideal for software implementation, as it requires only byte manipulations. It uses 256 bytes of memory for the state array, S[0] through S[255], k bytes of memory for the key, key[0] through key[k-1], and integer variables, i, j, and K. Performing a modular reduction of some value modulo 256 can be done with a bitwise AND with 255 (which is equivalent to taking the low-order byte of the value in question).
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Polynomial factoring algorithms use basic polynomial operations such as products, divisions, gcd, powers of one polynomial modulo another, etc. A multiplication of two polynomials of degree at most n can be done in O(n2) operations in Fq using "classical" arithmetic, or in O(nlog(n) log(log(n)) ) operations in Fq using "fast" arithmetic. A Euclidean division (division with remainder) can be performed within the same time bounds. The cost of a polynomial greatest common divisor between two polynomials of degree at most n can be taken as O(n2) operations in Fq using classical methods, or as O(nlog2(n) log(log(n)) ) operations in Fq using fast methods.
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By an elementary consideration of the equation ap \+ bp = cp, it is clear that one of a, b, c is even and hence so is N. By the Taniyama–Shimura conjecture, E is a modular elliptic curve. Since all odd primes dividing a, b, c in N appear to a pth power in the minimal discriminant Δ, by Ribet's theorem one can perform level descent modulo p repetitively to strip off all odd primes from the conductor. However, there are no newforms of level 2 as the genus of the modular curve X0(2) is zero (and newforms of level N are differentials on X0(N)).
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A residue numeral system (RNS) is a numeral system representing integers by their values modulo several pairwise coprime integers called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if is the product of the moduli, there is, in an interval of length , exactly one integer having any given set of modular values. The arithmetic of a residue numeral system is also called multi-modular arithmetic. Multi- modular arithmetic is widely used for computation with large integers, typically in linear algebra, because it provides faster computation than with the usual numeral systems, even when the time for converting between numeral systems is taken into account.
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The idea of hashing is to distribute the entries (key/value pairs) across an array of buckets. Given a key, the algorithm computes an index that suggests where the entry can be found: index = f(key, array_size) Often this is done in two steps: hash = hashfunc(key) index = hash % array_size In this method, the hash is independent of the array size, and it is then reduced to an index (a number between `0` and `array_size − 1`) using the modulo operator (`%`). In the case that the array size is a power of two, the remainder operation is reduced to masking, which improves speed, but can increase problems with a poor hash function.
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Computational complexity varies with the number of instructions required and latency of individual instructions, with the simplest being the bitwise methods (folding), followed by the multiplicative methods, and the most complex (slowest) are the division-based methods. Because collisions should be infrequent, and cause a marginal delay but are otherwise harmless, it's usually preferable to choose a faster hash function over one that needs more computation but saves a few collisions. Division-based implementations can be of particular concern, because division is microprogrammed on nearly all chip architectures. Divide (modulo) by a constant can be inverted to become a multiply by the word-size multiplicative-inverse of the constant.
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In 1992, Roger Heath-Brown conjectured that every n unequal to 4 or 5 modulo 9 has infinitely many representations as sums of three cubes. The case n=33 of this problem was used by Bjorn Poonen as the opening example in a survey on undecidable problems in number theory, of which Hilbert's tenth problem is the most famous example. Although this particular case has since been resolved, it is unknown whether representing numbers as sums of cubes is decidable. That is, it is not known whether an algorithm can, for every input, test in finite time whether a given number has such a representation.
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Picard's little theorem states that every nonconstant entire function takes every value in the complex plane, with perhaps one exception. Picard's great theorem states that an analytic function with an essential singularity takes every value infinitely often, with perhaps one exception, in any neighborhood of the singularity. He made important contributions in the theory of differential equations, including work on Picard–Vessiot theory, Painlevé transcendents and his introduction of a kind of symmetry group for a linear differential equation. He also introduced the Picard group in the theory of algebraic surfaces, which describes the classes of algebraic curves on the surface modulo linear equivalence.
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Computation of a cyclic redundancy check is derived from the mathematics of polynomial division, modulo two. In practice, it resembles long division of the binary message string, with a fixed number of zeroes appended, by the "generator polynomial" string except that exclusive or operations replace subtractions. Division of this type is efficiently realised in hardware by a modified shift register, and in software by a series of equivalent algorithms, starting with simple code close to the mathematics and becoming faster (and arguably more obfuscated) through byte-wise parallelism and space–time tradeoffs. CRC. The generator is a Galois-type shift register with XOR gates placed according to powers (white numbers) of x in the generator polynomial.
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A face of an embedded graph is an open 2-cell in the surface that is disjoint from the graph, but whose boundary is the union of some of the edges of the embedded graph. Let F be a face of an embedded graph G and let v0, v1, ..., vn−1,vn = v0 be the vertices lying on the boundary of F (in that circular order). A circular interval for F is a set of vertices of the form {va, va+1, ..., va+s} where a and s are integers and where subscripts are reduced modulo n. Let Λ be a finite list of circular intervals for F. We construct a new graph as follows.
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It can be defined in graph theoretic terms by choosing an arbitrary orientation of the graph, and defining an integral cycle of a graph G to be an assignment of integers to the edges of G (an element of the free abelian group over the edges) with the property that, at each vertex, the sum of the numbers assigned to incoming edges equals the sum of the numbers assigned to outgoing edges.. A member of H_1(G,\Z) or of H_1(G,\Z_k) (the cycle space modulo k) with the additional property that all of the numbers assigned to the edges are nonzero is called a nowhere-zero flow or a nowhere-zero k-flow..
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The 8086, 8088, and 80186 have a 20-bit address bus, but the unusual segmented addressing scheme Intel chose for these processors actually produces effective addresses which can have 21 significant bits. This scheme shifts a 16-bit segment number left four bits (making a 20-bit number with four least-significant zeros) before adding to it a 16-bit address offset; the maximum sum occurs when both the segment and offset are 0xFFFF, yielding 0xFFFF0 + 0xFFFF = 0x10FFEF. On the 8086, 8088, and 80186, the result of an effective address that overflows 20 bits is that the address "wraps around" to the zero end of the address range, i.e. it is taken modulo 2^20 (2^20 = 1048576 = 0x100000).
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Discrete logarithm records are the best results achieved to date in solving the discrete logarithm problem, which is the problem of finding solutions x to the equation gx = h given elements g and h of a finite cyclic group G. The difficulty of this problem is the basis for the security of several cryptographic systems, including Diffie–Hellman key agreement, ElGamal encryption, the ElGamal signature scheme, the Digital Signature Algorithm, and the elliptic curve cryptography analogs of these. Common choices for G used in these algorithms include the multiplicative group of integers modulo p, the multiplicative group of a finite field, and the group of points on an elliptic curve over a finite field.
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For example, the ISID modulo the number of ECT-VIDs could be used to decide on the actual relative VID to use. In the event that the ECT paths are not sufficiently diverse the operator has the option of adjusting the inputs to the distributed ECT algorithms to apply attraction or repulsion from a given node by adjusting that node's Bridge Priority. This can be experimented with via offline tools until the desired routes are achieved at which point the bias can be applied to the real network and then ISIDs can be moved to the resulting routes. Looking at the animations in Figure 6 shows the diversity available for traffic engineering in a 66 node network.
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One year later, experiments in Super-Kamiokande and Sudbury Neutrino Observatory began to show that solar and atmospheric neutrinos change flavors and therefore are massive, ruling out the Spectral Standard Model. Only in 2006 a solution to the latter problem was proposed, independently by John W. Barrett and Alain Connes , almost at the same time. They show that massive neutrinos can be incorporated into the model by disentangling the KO-dimension (which is defined modulo 8) from the metric dimension (which is zero) for the finite space. By setting the KO-dimension to be 6, not only massive neutrinos were possible, but the see-saw mechanism was imposed by the formalism and the fermion doubling problem was also addressed.
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When multiplied, these produce , and the following Montgomery reduction produces , the Montgomery form of the desired product. (A final second Montgomery reduction converts out of Montgomery form.) Converting to and from Montgomery form makes this slower than the conventional or Barrett reduction algorithms for a single multiply. However, when performing many multiplications in a row, as in modular exponentiation, intermediate results can be left in Montgomery form, and the initial and final conversions become a negligible fraction of the overall computation. Many important cryptosystems such as RSA and Diffie–Hellman key exchange are based on arithmetic operations modulo a large number, and for these cryptosystems, the computation by Montgomery multiplication is faster than the available alternatives.
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Given any two smooth submanifolds, it is possible to perturb either of them by an arbitrarily small amount such that the resulting submanifold intersects transversally with the fixed submanifold. Such perturbations do not affect the homology class of the manifolds or of their intersections. For example, if manifolds of complementary dimension intersect transversally, the signed sum of the number of their intersection points does not change even if we isotope the manifolds to another transverse intersection. (The intersection points can be counted modulo 2, ignoring the signs, to obtain a coarser invariant.) This descends to a bilinear intersection product on homology classes of any dimension, which is Poincaré dual to the cup product on cohomology.
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In mathematics, in semigroup theory, a Rees factor semigroup (also called Rees quotient semigroup or just Rees factor), named after David Rees, is a certain semigroup constructed using a semigroup and an ideal of the semigroup. Let S be a semigroup and I be an ideal of S. Using S and I one can construct a new semigroup by collapsing I into a single element while the elements of S outside of I retain their identity. The new semigroup obtained in this way is called the Rees factor semigroup of S modulo I and is denoted by S/I. The concept of Rees factor semigroup was introduced by David Rees in 1940.
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Let K be a global field, i.e. a number field or a function field in one variable over a finite field and let E be an elliptic curve. If v is a non-archimedean place of norm qv of K and a ∈ K, with v(a) = 0 then ≥ 1. v is called a Wieferich place for base a, if > 1, an elliptic Wieferich place for base P ∈ E, if NvP ∈ E2 and a strong elliptic Wieferich place for base P ∈ E if nvP ∈ E2, where nv is the order of P modulo v and Nv gives the number of rational points (over the residue field of v) of the reduction of E at v.
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Several interference patterns are recorded for different shifts of the reference beam and by analyzing them the phase information modulo 2 can be extracted. This ambiguity of the phase is called the phase wrapping effect and can be removed by so-called "phase unwrapping techniques". These techniques can be used when the signal-to-noise ratio of the image is sufficiently high and phase variation is not too abrupt. As an alternative to the fringe scanning method, the Fourier-transform method can be used to extract the phase shift information with only one interferogram, thus shortening the exposure time, but this has the disadvantage of limiting the spatial resolution by the spacing of the carrier fringes.
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The winners of three Nobel Prizes: John Kendrew and Max Perutz (Chemistry, 1962), Andrew Huxley (Medicine, 1963) and Martin Ryle (Physics, 1974) benefitted from EDSAC's revolutionary computing power. In their acceptance prize speeches, each acknowledged the role that EDSAC had played in their research. In the early 1960s Peter Swinnerton-Dyer used the EDSAC computer to calculate the number of points modulo p (denoted by Np) for a large number of primes p on elliptic curves whose rank was known. Based on these numerical results conjectured that Np for a curve E with rank r obeys an asymptotic law, the Birch and Swinnerton-Dyer conjecture, considered one of the top unsolved problems in mathematics as of 2016.
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Many types of sums are used in formulating particular problems; applications require usually a reduction to some known type, often by ingenious manipulations. Partial summation can be used to remove coefficients an, in many cases. A basic distinction is between a complete exponential sum, which is typically a sum over all residue classes modulo some integer N (or more general finite ring), and an incomplete exponential sum where the range of summation is restricted by some inequality. Examples of complete exponential sums are Gauss sums and Kloosterman sums; these are in some sense finite field or finite ring analogues of the gamma function and some sort of Bessel function, respectively, and have many 'structural' properties.
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Carmichael numbers can be generalized using concepts of abstract algebra. The above definition states that a composite integer n is Carmichael precisely when the nth-power-raising function pn from the ring Zn of integers modulo n to itself is the identity function. The identity is the only Zn-algebra endomorphism on Zn so we can restate the definition as asking that pn be an algebra endomorphism of Zn. As above, pn satisfies the same property whenever n is prime. The nth-power-raising function pn is also defined on any Zn- algebra A. A theorem states that n is prime if and only if all such functions pn are algebra endomorphisms.
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The fact that univalent foundations are inherently constructive was discovered in the process of writing the Foundations library (now part of UniMath). At present, in univalent foundations, classical mathematics is considered to be a "retract" of constructive mathematics, i.e., classical mathematics is both a subset of constructive mathematics consisting of those theorems and constructions that use the law of the excluded middle as their assumption and a "quotient" of constructive mathematics by the relation of being equivalent modulo the axiom of the excluded middle. In the formalization system for univalent foundations that is based on Martin-Löf type theory and its descendants such as Calculus of Inductive Constructions, the higher dimensional analogs of sets are represented by types.
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For every positive integer n, the set of the integers modulo n that are relatively prime to n is written as (Z/nZ)×; it forms a group under the operation of multiplication. This group is not always cyclic, but is so whenever n is 1, 2, 4, a power of an odd prime, or twice a power of an odd prime ... This is the multiplicative group of units of the ring Z/nZ; there are φ(n) of them, where again φ is the Euler totient function. For example, (Z/6Z)× = {1,5}, and since 6 is twice an odd prime this is a cyclic group. In contrast, (Z/8Z)× = {1,3,5,7} is a Klein 4-group and is not cyclic.
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In mathematics, a Jónsson–Tarski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set X onto the product X×X. They were introduced by . , named them after Georg Cantor because of Cantor's pairing function and Cantor's theorem that an infinite set X has the same number of elements as X×X; the term "Cantor algebra" is also occasionally used to mean the Boolean algebra of all clopen subsets of the Cantor set, or the Boolean algebra of Borel subsets of the reals modulo meager sets (sometimes called the Cohen algebra). The group of order preserving automorphisms of the free Jónsson–Tarski algebra on one generator is the Thompson group F.
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For any cycle in a graph , one can form an -dimensional 0-1 vector that has a 1 in the coordinate positions corresponding to edges in and a 0 in the remaining coordinate positions. The cycle space of the graph is the vector space formed by all possible linear combinations of vectors formed in this way. In Mac Lane's characterization, is a vector space over the finite field with two elements; that is, in this vector space, vectors are added coordinatewise modulo two. A 2-basis of is a basis of with the property that, for each edge in , at most two basis vectors have nonzero coordinates in the position corresponding to .
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There were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of equations, and geometry. Leonhard Euler considered algebraic operations on numbers modulo an integer—modular arithmetic—in his generalization of Fermat's little theorem. These investigations were taken much further by Carl Friedrich Gauss, who considered the structure of multiplicative groups of residues mod n and established many properties of cyclic and more general abelian groups that arise in this way. In his investigations of composition of binary quadratic forms, Gauss explicitly stated the associative law for the composition of forms, but like Euler before him, he seems to have been more interested in concrete results than in general theory.
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The contents of the left upper blocks are transferred without change into the corresponding cells of the lower block (green arrows in the figure). Then, the operation "addition modulo two" is performed bitwise over the right upper and left upper blocks and the result is transferred to the corresponding cells of the right side of the lower block (red arrows in the figure). This operation is performed with all lines from top to bottom and with all blocks in each line. After the construction is completed, the bottom line contains a string of numbers, which are the coefficients of the Zhegalkin polynomial, written in the same sequence as in the triangle method described above.
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The first cohomology group is the quotient of the so-called crossed homomorphisms, i.e. maps (of sets) f : G → M satisfying f(ab) = f(a) + af(b) for all a, b in G, modulo the so-called principal crossed homomorphisms, i.e. maps f : G → M given by f(a) = am−m for some fixed m ∈ M. This follows from the definition of cochains above. If the action of G on M is trivial, then the above boils down to H1(G,M) = Hom(G, M), the group of group homomorphisms G → M. Consider the case of H^1(\Z/2, \Z_-), where \Z_- denotes the non-trivial \Z/2-structure on the group of integers.
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Starting from the 1960s, Iliprandi worked as art director for numerous magazines including Popular Photography Italiana, Phototeca, Sci nautico and Interni. He drew the covers for the records of the label , and cured the corporate identity for the companies Cucine RB, Ankerfarm and Stilnovo. During his career he received several awards including: the Grand Prize at the XIII Triennale (1964); a prize at the first International Poster Biennale in Warsaw (1966); and four Compasso d'Oro. The Compasso d'Oro were in 1979 for the font Modulo and for the instrumental graphic of the Fiat 131 Mirafiori, in 2004 for the cover design of the magazine Arca and in 2011 as a career award.
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The figure groups between which differences were to be made were on punched tape. A duplicate of the tape was made, with one blank group added with the two tapes looped and read at the same time. The calculating relays computed the difference (modulo 10) between the two groups and the teleprinter recorded it; the two tapes then stepped simultaneously and the difference between the second and third was computed and recorded; then between the third and fourth; and so on. On the second time around, since the duplicate tape was one group longer than the original, the offset was automatically changed so that the first group was now differenced with the third group, the second with the fourth, and so on.
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In combinatorial game theory, an evil number is a number that has an even number of 1's in its binary representation, and an odious number is a number that has an odd number of 1's in its binary representation; these numbers play an important role in the strategy for the game Kayles.. See in particular p. 68. The parity function maps a number to the number of 1's in its binary representation, modulo 2, so its value is zero for evil numbers and one for odious numbers. The Thue–Morse sequence, an infinite sequence of 0's and 1's, has a 0 in position i when i is evil, and a 1 in that position when i is odious..
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In a plane cubic model three points sum to zero in the group if and only if they are collinear. For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions. The intersection of two quadric surfaces is, in general, a nonsingular curve of genus one and degree four, and thus an elliptic curve, if it has a rational point. In special cases, the intersection either may be a rational singular quartic or is decomposed in curves of smaller degrees which are not always distinct (either a cubic curve and a line, or two conics, or a conic and two lines, or four lines).
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In the random oracle model, hash-then-sign (an idealized version of that practice where hash and padding combined have close to N possible outputs), this form of signature is existentially unforgeable, even against a chosen-plaintext attack. There are several reasons to sign such a hash (or message digest) instead of the whole document. ;For efficiency: The signature will be much shorter and thus save time since hashing is generally much faster than signing in practice. ;For compatibility: Messages are typically bit strings, but some signature schemes operate on other domains (such as, in the case of RSA, numbers modulo a composite number N). A hash function can be used to convert an arbitrary input into the proper format.
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For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold. Loosely speaking, a cycle is a closed submanifold, a boundary is a cycle which is also the boundary of a submanifold, and a homology class (which represents a hole) is an equivalence class of cycles modulo boundaries.
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The sphere, before being transformed, is not homeomorphic to the real projective plane, RP2. But the sphere centered at the origin has this property, that if point (x,y,z) belongs to the sphere, then so does the antipodal point (-x,-y,-z) and these two points are different: they lie on opposite sides of the center of the sphere. The transformation T converts both of these antipodal points into the same point, : T : (x, y, z) \rightarrow (y z, z x, x y), : T : (-x, -y, -z) \rightarrow ((-y) (-z), (-z) (-x), (-x) (-y)) = (y z, z x, x y). Since this is true of all points of S2, then it is clear that the Roman surface is a continuous image of a "sphere modulo antipodes".
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Address generation unit (AGU), sometimes also called address computation unit (ACU), is an execution unit inside the CPU that calculates addresses used by the CPU to access main memory. By having address calculations handled by separate circuitry that operates in parallel with the rest of the CPU, the number of CPU cycles required for executing various machine instructions can be reduced, bringing performance improvements. While performing various operations, CPUs need to calculate memory addresses required for fetching data from the memory; for example, in-memory positions of array elements must be calculated before the CPU can fetch the data from actual memory locations. Those address-generation calculations involve different integer arithmetic operations, such as addition, subtraction, modulo operations, or bit shifts.
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However, the symmetric difference of two Eulerian subgraphs (the graph consisting of the edges that belong to exactly one of the two given graphs) is again Eulerian. This follows from the fact that the symmetric difference of two sets with an even number of elements is also even. Applying this fact separately to the neighborhoods of each vertex shows that the symmetric difference operator preserves the property of being Eulerian. A family of sets closed under the symmetric difference operation can be described algebraically as a vector space over the two-element finite field \Z_2.. This field has two elements, 0 and 1, and its addition and multiplication operations can be described as the familiar addition and multiplication of integers, taken modulo 2\.
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Combined slotted/pozidriv heads are so ubiquitous in electrical switchgear to have earned the nickname "electrician's screws" (the first screwdriver out of the toolbox is used, and the user does not have to waste valuable time searching for the correct driver). Their rise to popular use has been in spite of the fact that neither a flat screwdriver or Pozidriv screwdriver is fully successful in driving these screws to the required torque. Some screwdriver manufacturers offer matching screwdrivers and call them "Modulo", "Plus-minus", or "contractor screwdrivers", although the original concept of not needing to search for a particular driver being defeated as a contractor screwdriver is useless for non-combination heads. Slotted/Phillips (as opposed to slotted/pozidriv) heads occur in some North American-made switchgear.
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A five-vertex cycle, modeling a set of five values that can be transmitted across a noisy communications channel and the pairs of values that can be confused with each other The Shannon capacity models the amount of information that can be transmitted across a noisy communication channel in which certain signal values can be confused with each other. In this application, the confusion graph or confusability graph describes the pairs of values that can be confused. For instance, suppose that a communications channel has five discrete signal values, any one of which can be transmitted in a single time step. These values may be modeled mathematically as the five numbers 0, 1, 2, 3, or 4 in modular arithmetic modulo 5\.
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The Euler products of the Hasse–Weil zeta-function for an algebraic variety V over a number field K, formed by reducing modulo prime ideals to get local zeta-functions, are conjectured to have a global functional equation; but this is currently considered out of reach except in special cases. The definition can be read directly out of étale cohomology theory, again; but in general some assumption coming from automorphic representation theory seems required to get the functional equation. The Taniyama–Shimura conjecture was a particular case of this as general theory. By relating the gamma-factor aspect to Hodge theory, and detailed studies of the expected ε factor, the theory as empirical has been brought to quite a refined state, even if proofs are missing.
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Now, λ-1 = 1 and the product rule in terms of the α's is αα = α, where = + -1 (mod -1). The Latin squares are constructed as follows, the ()th entry in Latin square L (with ≠ 0) is L() = α \+ αα, where all the operations occur in GF(). In the case that the field is a prime field ( = a prime), where the field elements are represented in the usual way, as the integers modulo , the naming convention above can be dropped and the construction rule can be simplified to L() = + , where ≠ 0 and , and are elements of GF() and all operations are in GF(). The MOLS(4) and MOLS(5) examples above arose from this construction, although with a change of alphabet.
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G-theory had been defined early in the development of the subject by Grothendieck. Grothendieck defined G0(X) for a variety X to be the free abelian group on isomorphism classes of coherent sheaves on X, modulo relations coming from exact sequences of coherent sheaves. In the categorical framework adopted by later authors, the K-theory of a variety is the K-theory of its category of vector bundles, while its G-theory is the K-theory of its category of coherent sheaves. Not only could Quillen prove the existence of a localization exact sequence for G-theory, he could prove that for a regular ring or variety, K-theory equaled G-theory, and therefore K-theory of regular varieties had a localization exact sequence.
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The loop space is dual to the suspension of the same space; this duality is sometimes called Eckmann–Hilton duality. The basic observation is that :[\Sigma Z,X] \approxeq [Z, \Omega X] where [A,B] is the set of homotopy classes of maps A \rightarrow B, and \Sigma A is the suspension of A, and \approxeq denotes the natural homeomorphism. This homeomorphism is essentially that of currying, modulo the quotients needed to convert the products to reduced products. In general, [A, B] does not have a group structure for arbitrary spaces A and B. However, it can be shown that [\Sigma Z,X] and [Z, \Omega X] do have natural group structures when Z and X are pointed, and the aforementioned isomorphism is of those groups.
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There are other uses of the term "interpretation" that are commonly used, which do not refer to the assignment of meanings to formal languages. In model theory, a structure A is said to interpret a structure B if there is a definable subset D of A, and definable relations and functions on D, such that B is isomorphic to the structure with domain D and these functions and relations. In some settings, it is not the domain D that is used, but rather D modulo an equivalence relation definable in A. For additional information, see Interpretation (model theory). A theory T is said to interpret another theory S if there is a finite extension by definitions T′ of T such that S is contained in T′.
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Because Presburger arithmetic is decidable, automatic theorem provers for Presburger arithmetic exist. For example, the Coq proof assistant system features the tactic omega for Presburger arithmetic and the Isabelle proof assistant contains a verified quantifier elimination procedure by . The double exponential complexity of the theory makes it infeasible to use the theorem provers on complicated formulas, but this behavior occurs only in the presence of nested quantifiers: Oppen and Nelson (1980) describe an automatic theorem prover which uses the simplex algorithm on an extended Presburger arithmetic without nested quantifiers to prove some of the instances of quantifier-free Presburger arithmetic formulas. More recent satisfiability modulo theories solvers use complete integer programming techniques to handle quantifier-free fragment of Presburger arithmetic theory ().
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We show by contradiction that m equals 1: supposing it is not the case, we prove the existence of a positive integer r less than m, for which rp is also the sum of four squares (this is in the spirit of the infinite descentHere the argument is a direct proof by contradiction. With the initial assumption that m > 2, m < p, is some integer such that mp is the sum of four squares (not necessarily the smallest), the argument could be modified to become an infinite descent argument in the spirit of Fermat. method of Fermat). For this purpose, we consider for each xi the yi which is in the same residue class modulo m and between and m/2 (included).
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In a field K, any polynomial of degree n has at most n distinct roots (Lagrange's theorem (number theory)), so there are no other a with this property, in particular not among 0 to . Similarly, for b taking integral values between 0 and (inclusive), the are distinct. By the pigeonhole principle, there are a and b in this range, for which a2 and are congruent modulo p, that is for which :a^2 + b^2 + 1^2 + 0^2 = np. Now let m be the smallest positive integer such that mp is the sum of four squares, (we have just shown that there is some m (namely n) with this property, so there is a least one m, and it is smaller than p).
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For proving bounds on this problem, it may be assumed without loss of generality that the inputs are strings over a two-letter alphabet. For, if two strings over a larger alphabet differ then there exists a string homomorphism that maps them to binary strings of the same length that also differ. Any automaton that distinguishes the binary strings can be translated into an automaton that distinguishes the original strings, without any increase in the number of states.. It may also be assumed that the two strings have equal length. For strings of unequal length, there always exists a prime number whose value is logarithmic in the smaller of the two input lengths, such that the two lengths are different modulo .
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To decrypt a Tunny message required knowledge not only of the logical functioning of the machine, but also the start positions of each rotor for the particular message. The search was on for a process that would manipulate the ciphertext or key to produce a frequency distribution of characters that departed from the uniformity that the enciphering process aimed to achieve. While on secondment to the Research Section in July 1942, Alan Turing worked out that the XOR combination of the values of successive characters in a stream of ciphertext and key emphasised any departures from a uniform distribution. The resultant stream (symbolised by the Greek letter "delta" Δ) was called the difference because XOR is the same as modulo 2 subtraction.
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RC4+ is a modified version of RC4 with a more complex three-phase key schedule (taking about three times as long as RC4, or the same as RC4-drop512), and a more complex output function which performs four additional lookups in the S array for each byte output, taking approximately 1.7 times as long as basic RC4. All arithmetic modulo 256. << and >> are left and right shift, ⊕ is exclusive OR while GeneratingOutput: i := i + 1 a := S[i] j := j + a Swap S[i] and S[j] (b := S[j]; S[i] := b; S[j] := a) c := S[i<<5 ⊕ j>>3] + S[j<<5 ⊕ i>>3] output (S[a+b] + S[c⊕0xAA]) ⊕ S[j+b] endwhile This algorithm has not been analyzed significantly.
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It follows that every sheaf E has an injective resolution: :0\to E\to I_0\to I_1\to I_2\to \cdots. Then the sheaf cohomology groups Hi(X,E) are the cohomology groups (the kernel of one homomorphism modulo the image of the previous one) of the complex of abelian groups: : 0\to I_0(X) \to I_1(X) \to I_2(X)\to \cdots. Standard arguments in homological algebra imply that these cohomology groups are independent of the choice of injective resolution of E. The definition is rarely used directly to compute sheaf cohomology. It is nonetheless powerful, because it works in great generality (any sheaf on any topological space), and it easily implies the formal properties of sheaf cohomology, such as the long exact sequence above.
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In the category of sets, every monomorphism (injective function) with a non-empty domain is a section, and every epimorphism (surjective function) is a retraction; the latter statement is equivalent to the axiom of choice. In the category of vector spaces over a field K, every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis. In the category of abelian groups, the epimorphism Z → Z/2Z which sends every integer to its remainder modulo 2 does not split; in fact the only morphism Z/2Z → Z is the zero map. Similarly, the natural monomorphism Z/2Z → Z/4Z doesn't split even though there is a non-trivial morphism Z/4Z → Z/2Z.
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Isabelle is generic: it provides a meta- logic (a weak type theory), which is used to encode object logics like first- order logic (FOL), higher-order logic (HOL) or Zermelo–Fraenkel set theory (ZFC). The most widely used object logic is Isabelle/HOL, although significant set theory developments were completed in Isabelle/ZF. Isabelle's main proof method is a higher-order version of resolution, based on higher-order unification. Though interactive, Isabelle features efficient automatic reasoning tools, such as a term rewriting engine and a tableaux prover, various decision procedures, and, through the Sledgehammer proof-automation interface, external satisfiability modulo theories (SMT) solvers (including CVC4) and resolution-based automated theorem provers (ATPs), including E and SPASS (the Metis proof method reconstructs resolution proofs generated by these ATPs).
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The search was on for a process that would manipulate the ciphertext or key to produce a frequency distribution of characters that departed from the uniformity that the enciphering process aimed to achieve. Turing worked out that the XOR combination of the values of successive (adjacent) characters in a stream of ciphertext or key, emphasised any departures from a uniform distribution. The resultant stream was called the difference (symbolised by the Greek letter "delta" Δ) because XOR is the same as modulo 2 subtraction. So, for a stream of characters S, the difference ΔS was obtained as follows, where _underline_ indicates the succeeding character: :::: ΔS = S ⊕ _S_ The stream S may be ciphertext Z, plaintext P, key K or either of its two components \chi and \psi.
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The Hesse configuration has the same incidence relations as the lines and points of the affine plane over the field of 3 elements. That is, the points of the Hesse configuration may be identified with ordered pairs of numbers modulo 3, and the lines of the configuration may correspondingly be identified with the triples of points satisfying a linear equation . Alternatively, the points of the configuration may be identified by the squares of a tic-tac-toe board, and the lines may be identified with the lines and broken diagonals of the board. Each point belongs to four lines: in the tic tac toe interpretation of the configuration, one line is horizontal, one vertical, and two are diagonals or broken diagonals.
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For the special purpose of searching for a counterexample to the Collatz conjecture, this precomputation leads to an even more important acceleration, used by Tomás Oliveira e Silva in his computational confirmations of the Collatz conjecture up to large values of . If, for some given and , the inequality : holds for all , then the first counterexample, if it exists, cannot be modulo . For instance, the first counterexample must be odd because , smaller than ; and it must be 3 mod 4 because , smaller than . For each starting value which is not a counterexample to the Collatz conjecture, there is a for which such an inequality holds, so checking the Collatz conjecture for one starting value is as good as checking an entire congruence class.
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Cryptographic hash functions are believed to provide good hash functions for any table size, either by modulo reduction or by bit masking. They may also be appropriate if there is a risk of malicious users trying to sabotage a network service by submitting requests designed to generate a large number of collisions in the server's hash tables. However, the risk of sabotage can also be avoided by cheaper methods (such as applying a secret salt to the data, or using a universal hash function). A drawback of cryptographic hashing functions is that they are often slower to compute, which means that in cases where the uniformity for any size is not necessary, a non-cryptographic hashing function might be preferable.
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In general, the ISBN-13 check digit is calculated as follows. Let Then This check system – similar to the UPC check digit formula – does not catch all errors of adjacent digit transposition. Specifically, if the difference between two adjacent digits is 5, the check digit will not catch their transposition. For instance, the above example allows this situation with the 6 followed by a 1. The correct order contributes 3×6+1×1 = 19 to the sum; while, if the digits are transposed (1 followed by a 6), the contribution of those two digits will be 3×1+1×6 = 9. However, 19 and 9 are congruent modulo 10, and so produce the same, final result: both ISBNs will have a check digit of 7.
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In 2014, Honda Philippines unveiled a facelift version of the Civic that includes new sportier front grille while the 1.8 EXi is no longer available and the 1.8 S now receives front fog lights. The 1.8 E receives new alloy wheels and is available in standard or Modulo kit and the 2.0 EL now receives side curtain airbags and a choice between standard model or with Mugen kit. The ECON button is available in all models while the 1.8 E and 2.0 EL receives the touchscreen audio panel with USB and HDMI connectivity and push start or stop engine. Starting from September 2015, models for the Turkish market were available with factory converted LPG version with the commencement of a new LPG-only assembly line in Honda's Turkey plant.
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The disproof of Keller's conjecture, for sufficiently high dimensions, has progressed through a sequence of reductions that transform it from a problem in the geometry of tilings into a problem in group theory, and from there into a problem in graph theory. first reformulated Keller's conjecture in terms of factorizations of abelian groups. He shows that, if there is a counterexample to the conjecture, then it can be assumed to be a periodic tiling of cubes with an integer side length and integer vertex positions; thus, in studying the conjecture, it is sufficient to consider tilings of this special form. In this case, the group of integer translations, modulo the translations that preserve the tiling, forms an abelian group, and certain elements of this group correspond to the positions of the tiles.
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A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. In a quotient of a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup.
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For 128 bits of security in the supersingular isogeny Diffie-Hellman (SIDH) method, De Feo, Jao and Plut recommend using a supersingular curve modulo a 768-bit prime. If one uses elliptic curve point compression the public key will need to be no more than 8x768 or 6144 bits in length. A March 2016 paper by authors Azarderakhsh, Jao, Kalach, Koziel, and Leonardi showed how to cut the number of bits transmitted in half, which was further improved by authors Costello, Jao, Longa, Naehrig, Renes and Urbanik resulting in a compressed-key version of the SIDH protocol with public keys only 2640 bits in size. This makes the number of bits transmitted roughly equivalent to the non-quantum secure RSA and Diffie- Hellman at the same classical security level.
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For applications of monotone priority queues such as Dijkstra's algorithm in which the minimum priorities form a monotonic sequence, the sum of these differences is at most , so the total time for a sequence of operations is , rather than the slower time bound that would result without this optimization. Another optimization (already given by ) can be used to save space when the priorities are monotonic and, at any point in time, fall within a range of values rather than extending over the whole range from 0 to . In this case, one can index the array by the priorities modulo rather than by their actual values. The search for the minimum priority element should always begin at the previous minimum, to avoid priorities that are higher than the minimum but have lower moduli.
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Joseph Schillinger embraced not only contrapuntal inverse, retrograde, and retrograde-inverse—operations of matrix multiplication in Euclidean vector space—but also their rhythmic counterparts as well. Thus he could describe a variation of theme using the same pitches in same order, but employing its original time values in retrograde order. He saw the scope of this multiplicatory universe beyond simple reflection, to include transposition and rotation (possibly with projection back to source), as well as dilation which had formerly been limited in use to the time dimension (via augmentation and diminution) (). Thus he could describe another variation of theme, or even of a basic scale, by multiplying the halfstep counts between each successive pair of notes by some factor, possibly normalizing to the octave via Modulo-12 operation ().
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In additive number theory, the Skolem–Mahler–Lech theorem states that if a sequence of numbers is generated by a linear recurrence relation, then with finitely many exceptions the positions at which the sequence is zero form a regularly repeating pattern. More precisely, this set of positions can be decomposed into the union of a finite set and finitely many full arithmetic progressions. Here, an infinite arithmetic progression is full if there exist integers a and b such that the progression consists of all positive integers equal to b modulo a. This result is named after Thoralf Skolem (who proved the theorem for sequences of rational numbers), Kurt Mahler (who proved it for sequences of algebraic numbers), and Christer Lech (who proved it for sequences whose elements belong to any field of characteristic 0).
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One construction of the localization is done by declaring that its objects are the same as those in C, but the morphisms are enhanced by adding a formal inverse for each morphism in W. Under suitable hypotheses on W, the morphisms between two objects X, Y are given by roofs :X \stackrel f \leftarrow X' \rightarrow Y (where X' is an arbitrary object of C and f is in the given class W of morphisms), modulo certain equivalence relations. These relations turn the map going in the "wrong" direction into an inverse of f. This procedure, however, in general yields a proper class of morphisms between X and Y. Typically, the morphisms in a category are only allowed to form a set. Some authors simply ignore such set-theoretic issues.
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The structure is similar to the Mersenne Twister, a large state made up of previous output words (32 bits each), from which a new output word is generated using linear recurrences modulo 2 over a finite binary field F_2. However, a more complex recurrence produces a denser generator polynomial, producing better statistical properties. Each step of the generator reads five words of state: the oldest 32 bits (which may straddle a word boundary if the state size is not a multiple of 32), the newest 32 bits, and three other words in between. Then a series of eight single-word transformations (mostly of the form `x := x ⊕ (x >> k)`) and six exclusive-or operations combine those into two words, which become the newest two words of state, one of which will be the output.
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In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683. It may be defined conveniently by means of field theory, as the transformation on minimal polynomials implied by a different choice of primitive element. This is the most general transformation of an irreducible polynomial that takes a root to some rational function applied to that root. In detail, let K be a field, and P(t) a polynomial over K. If P is irreducible, then the quotient ring of the polynomial ring K[t] by the principal ideal generated by P, :K[t]/(P(t)) = L, is a field extension of K. We have :L = K(α) where α is t modulo (P).
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In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. In general, it is the map which preserves the widest amount of structure, and it tends to be unique. In the rare cases where latitude in choices remains, the map is either conventionally agreed upon to be the most useful for further analysis, or sometimes the most elegant map known up to date. A standard form of canonical map involves some function mapping a set X to the set X/R (X modulo R), where R is an equivalence relation on X. A closely related notion is a structure map or structure morphism; the map or morphism that comes with the given structure on the object.
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Miguel Onorato, Lara Vozella, Davide Proment, Yuri V. Lvov, (2015) A route to thermalization in the α-Fermi–Pasta–Ulam system ArXiv 1402.1603. Rewriting the FPUT model in terms of normal modes, the non-linear term expresses itself as a three-mode interaction (using the language of statistical mechanics, this could be called a "three-phonon interaction".) It is, however, not a resonant interaction,A resonant interaction is one where all of the wave-vectors add/subtract to zero, modulo the Brillouin zone, as well as the corresponding frequencies obtained from the dispersion relation. Since they sum to zero, there is no preferred vector basis for the corresponding vector space, and so all amplitudes can be re-arranged freely. In effect, this places all modes into the same ergodic component, where they can mix "instantly".
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It can be demonstrated that multiple top-to-bottom (non-Scarne's) cuts are equivalent to some single cut. In fact, knowing the size of the deck and the size of the cuts, the formula for the composite single cut is given as the sum of the sizes of the cuts modulo the size of the deck. For example, in a 10 card deck, if a 7 card cut and a 4 card cut are made, that is, 7 cards are moved from the top of the deck to the bottom and then the resulting top 4 cards are also moved to the bottom, then those two consecutive cuts are equivalent to a cut the size of (7 + 4 = 11 (mod 10)) = 1. The deck will be in the order (2,3,...,10,1).
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The extremely popular HDLC protocol uses a 3-bit sequence number, and has optional provision for selective repeat. However, if selective repeat is to be used, the requirement that nt+nr ≤ 8 must be maintained; if wr is increased to 3, wt must be decreased to 6. Suppose that wr =2, but an unmodified transmitter is used with wt =7, as is typically used with the go-back-N variant of HDLC. Further suppose that the receiver begins with nr =ns =0. Now suppose that the receiver sees the following series of packets (all modulo 8): : 0 1 2 3 4 5 6 (pause) 0 Because wr =2, the receiver will accept and store the final packet 0 (thinking it is packet 8 in the series), while requesting a retransmission of packet 7.
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In mathematics, the Vogel plane is a method of parameterizing simple Lie algebras by eigenvalues α, β, γ of the Casimir operator on the symmetric square of the Lie algebra, which gives a point (α: β: γ) of P2/S3, the projective plane P2 divided out by the symmetric group S3 of permutations of coordinates. It was introduced by , and is related by some observations made by . generalized Vogel's work to higher symmetric powers. The point of the projective plane (modulo permutations) corresponding to a simple complex Lie algebra is given by three eigenvalues α, β, γ of the Casimir operator acting on spaces A, B, C, where the symmetric square of the Lie algebra (usually) decomposes as a sum of the complex numbers and 3 irreducible spaces A, B, C.
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In general, for any finite group G of order n, it is straightforward to determine the signature of the permutation πg made by left-multiplication by the element g of G. The permutation πg will be even, unless there are an odd number of orbits of even size. Assuming n even, therefore, the condition for πg to be an odd permutation, when g has order k, is that n/k should be odd, or that the subgroup generated by g should have odd index. We will apply this to the group of nonzero numbers mod p, which is a cyclic group of order p − 1\. The jth power of a primitive root modulo p will by index calculus have index the greatest common divisor :i = (j, p − 1).
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In detail, the question applies to the 'round spheres' and to their tangent bundles: in fact since all exotic spheres have isomorphic tangent bundles, the Radon–Hurwitz numbers ρ(N) determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere. The case of N odd is taken care of by the Poincaré–Hopf index theorem (see hairy ball theorem), so the case N even is an extension of that. Adams showed that the maximum number of continuous (smooth would be no different here) pointwise linearly-independent vector fields on the (N − 1)-sphere is exactly ρ(N) − 1\. The construction of the fields is related to the real Clifford algebras, which is a theory with a periodicity modulo 8 that also shows up here.
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For m=0 the generalized Jacobian Jm is just the usual Jacobian J, an abelian variety of dimension g, the genus of C. For m a nonzero effective divisor the generalized Jacobian is an extension of J by a connected commutative affine algebraic group Lm of dimension deg(m)−1. So we have an exact sequence :0 → Lm → Jm → J → 0 The group Lm is a quotient :0 → Gm → ΠRi → Lm → 0 of a product of groups Ri by the multiplicative group Gm of the underlying field. The product runs over the points Pi in the support of m, and the group Ri is the group of invertible elements of the local ring modulo those that are 1 mod m. The group Ri has dimension ni, the number of times Pi occurs in m.
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A generalization of the Sierpinski triangle can also be generated using Pascal's triangle if a different Modulo is used. Iteration n can be generated by taking a Pascal's triangle with Pn rows and coloring numbers by their value for x mod P. As n approaches infinity, a fractal is generated. The same fractal can be achieved by dividing a triangle into a tessellation of P2 similar triangles and removing the triangles that are upside-down from the original, then iterating this step with each smaller triangle. Conversely, the fractal can also be generated by beginning with a triangle and duplicating it and arranging of the new figures in the same orientation into a larger similar triangle with the vertices of the previous figures touching, then iterating that step.
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If and , then the sequence in question is :3, 6, 9, 12, 15, 18; reducing modulo 7 gives :3, 6, 2, 5, 1, 4, which is just a rearrangement of :1, 2, 3, 4, 5, 6. Multiplying them together gives :3 \times 6 \times 9 \times 12 \times 15 \times 18 \equiv 3 \times 6 \times 2 \times 5 \times 1 \times 4 \equiv 1 \times 2 \times 3 \times 4 \times 5 \times 6 \pmod 7; that is, :3^6 (1 \times 2 \times 3 \times 4 \times 5 \times 6) \equiv (1 \times 2 \times 3 \times 4 \times 5 \times 6) \pmod 7. Canceling out 1 × 2 × 3 × 4 × 5 × 6 yields :3^6 \equiv 1 \pmod 7, which is Fermat's little theorem for the case and .
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If the coefficients do not belong to Fp, the p-th root of a polynomial with zero derivative is obtained by the same substitution on x, completed by applying the inverse of the Frobenius automorphism to the coefficients. This algorithm works also over a field of characteristic zero, with the only difference that it never enters in the blocks of instructions where pth roots are computed. However, in this case, Yun's algorithm is much more efficient because it computes the greatest common divisors of polynomials of lower degrees. A consequence is that, when factoring a polynomial over the integers, the algorithm which follows is not used: one compute first the square-free factorization over the integers, and to factor the resulting polynomials, one chooses a p such that they remain square-free modulo p.
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DFC can actually use a key of any size up to 256 bits; the key schedule uses another 4-round Feistel network to generate a 1024-bit "expanded key". The arbitrary constants, including all entries of the S-box, are derived using the binary expansion of e as a source of "nothing up my sleeve numbers". Soon after DFC's publication, Ian Harvey raised the concern that reduction modulo a 65-bit number was beyond the native capabilities of most platforms, and that careful implementation would be required to protect against side-channel attacks, especially timing attacks. Although DFC was designed using Vaudenay's decorrelation theory to be provably secure against ordinary differential and linear cryptanalysis, in 1999 Lars Knudsen and Vincent Rijmen presented a differential chosen-ciphertext attack that breaks 6 rounds faster than exhaustive search.
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Doing a Fisher–Yates shuffle involves picking uniformly distributed random integers from various ranges. Most random number generators, however — whether true or pseudorandom — will only directly provide numbers in a fixed range from 0 to RAND_MAX, and in some libraries, RAND_MAX may be as low as 32767.The GNU C Library: ISO Random A simple and commonly used way to force such numbers into a desired range is to apply the modulo operator; that is, to divide them by the size of the range and take the remainder. However, the need in a Fisher–Yates shuffle to generate random numbers in every range from 0–1 to 0–n pretty much guarantees that some of these ranges will not evenly divide the natural range of the random number generator.
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Appendix 1 of the International ISBN Agency's official user manual describes how the 13-digit ISBN check digit is calculated. The ISBN-13 check digit, which is the last digit of the ISBN, must range from 0 to 9 and must be such that the sum of all the thirteen digits, each multiplied by its (integer) weight, alternating between 1 and 3, is a multiple of 10. Formally, using modular arithmetic, this is rendered: The calculation of an ISBN-13 check digit begins with the first twelve digits of the 13-digit ISBN (thus excluding the check digit itself). Each digit, from left to right, is alternately multiplied by 1 or 3, then those products are summed modulo 10 to give a value ranging from 0 to 9.
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The Rado graph, as numbered by and . In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of . The Rado graph can also be constructed non- randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other.
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The free abelian group on a set S is defined via its universal property in the analogous way, with obvious modifications: Consider a pair (F, φ), where F is an abelian group and φ: S → F is a function. F is said to be the free abelian group on S with respect to φ if for any abelian group G and any function ψ: S → G, there exists a unique homomorphism f: F → G such that :f(φ(s)) = ψ(s), for all s in S. The free abelian group on S can be explicitly identified as the free group F(S) modulo the subgroup generated by its commutators, [F(S), F(S)], i.e. its abelianisation. In other words, the free abelian group on S is the set of words that are distinguished only up to the order of letters.
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Until January 1, 2007, all ISBNs were allocated as 9-digit numbers followed by a modulo 11 checksum character that was either a decimal digit or the letter "X". A Bookland EAN was generated by concatenating the Bookland UCC 978, the 9 digits of the book's ISBN other than its checksum, and the EAN checksum digit. Since parts of the 10-character ISBN space are nearly full, all books published from 2007 on have been allocated a 13-digit ISBN, which is identical to the Bookland EAN. Most of UCC 979 (formerly "Musicland") has now been assigned for the expansion of Bookland, and was first used by publishers in the French language, which can now use the additional prefix "979-10-" in addition to the nearly full "978-2-" prefix (onto which legacy 10-character ISBNs starting with "2-" have been remapped).
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Since there are 5 known Fermat primes, we know of 31 numbers that are products of distinct Fermat primes, and hence 31 constructible odd-sided regular polygons. These are 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295 . As John Conway commented in The Book of Numbers, these numbers, when written in binary, are equal to the first 32 rows of the modulo-2 Pascal's triangle, minus the top row, which corresponds to a monogon. (Because of this, the 1s in such a list form an approximation to the Sierpiński triangle.) This pattern breaks down after this, as the next Fermat number is composite (4294967297 = 641 × 6700417), so the following rows do not correspond to constructible polygons.
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Every pair a, b of friendly numbers gives rise to a positive proportion of all natural numbers being friendly (but in different clubs), by considering pairs na, nb for multipliers n with gcd(n, ab) = 1. For example, the "primitive" friendly pair 6 and 28 gives rise to friendly pairs 6n and 28n for all n that are congruent to 1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, or 41 modulo 42. This shows that the natural density of the friendly numbers (if it exists) is positive. Anderson and Hickerson proposed that the density should in fact be 1 (or equivalently that the density of the solitary numbers should be 0).. According to the MathWorld article on Solitary Number (see References section below), this conjecture has not been resolved, although Pomerance thought at one point he had disproved it.
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The three octatonic collections are transpositionally and inversionally symmetric—that is, they are related by a variety of transposition and inversion operations: They are each closed under transpositions by 3, 6, or 9 semitones. A transposition by 1, 4, 7, or 10 semitones will transform the E scale into the D scale, the D scale into the D scale, and the D scale into the E scale. Conversely, transpositions by 2, 5, 8, or 11 semitones acts in the reverse way; the E scale goes to the D scale, D to D and D to E. Thus, the set of transpositions acts on the set of diminished collections as the integers modulo 3. If the transposition is congruent to 0 mod 3 the pitch collection is unchanged and the transpositions by 1 semitone or by 2 semitones are inverse to one another.
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For instance, the number of strong orientations is and the number of acyclic orientations is .. For bridgeless planar graphs, graph colorings with colors correspond to nowhere- zero flows modulo on the dual graph. For instance, the four color theorem (the existence of a 4-coloring for every planar graph) can be expressed equivalently as stating that the dual of every bridgeless planar graph has a nowhere-zero 4-flow. The number of -colorings is counted (up to an easily computed factor) by the Tutte polynomial value and dually the number of nowhere-zero -flows is counted by . An st-planar graph is a connected planar graph together with a bipolar orientation of that graph, an orientation that makes it acyclic with a single source and a single sink, both of which are required to be on the same face as each other.
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The family of all cut sets of an undirected graph is known as the cut space of the graph. It forms a vector space over the two- element finite field of arithmetic modulo two, with the symmetric difference of two cut sets as the vector addition operation, and is the orthogonal complement of the cycle space... If the edges of the graph are given positive weights, the minimum weight basis of the cut space can be described by a tree on the same vertex set as the graph, called the Gomory–Hu tree.. Each edge of this tree is associated with a bond in the original graph, and the minimum cut between two nodes s and t is the minimum weight bond among the ones associated with the path from s to t in the tree.
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A 2012 aerial photograph of the Lido inlet and the worksites where MOSE was constructed Acqua alta floods in Piazza San Marco. This ever more frequent event causes considerable damage and has driven the authorities to seek solutions such as MOSE MOSE (MOdulo Sperimentale Elettromeccanico, Experimental Electromechanical Module) is a project intended to protect the city of Venice, Italy, and the Venetian Lagoon from flooding. The project is an integrated system consisting of rows of mobile gates installed at the Lido, Malamocco, and Chioggia inlets that are able to isolate the Venetian Lagoon temporarily from the Adriatic Sea during acqua alta high tides. Together with other measures, such as coastal reinforcement, the raising of quaysides, and the paving and improvement of the lagoon, MOSE is designed to protect Venice and the lagoon from tides of up to .
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Optical light is still the primary means by which astronomy occurs, and in the context of cosmology, this means observing distant galaxies and galaxy clusters in order to learn about the large scale structure of the Universe as well as galaxy evolution. Redshift surveys have been a common means by which this has been accomplished with some of the most famous including the 2dF Galaxy Redshift Survey, the Sloan Digital Sky Survey, and the upcoming Large Synoptic Survey Telescope. These optical observations generally use either photometry or spectroscopy to measure the redshift of a galaxy and then, via Hubble's Law, determine its distance modulo redshift distortions due to peculiar velocities. Additionally, the position of the galaxies as seen on the sky in celestial coordinates can be used to gain information about the other two spatial dimensions.
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The algorithm attempts to set up a congruence of squares modulo n (the integer to be factorized), which often leads to a factorization of n. The algorithm works in two phases: the data collection phase, where it collects information that may lead to a congruence of squares; and the data processing phase, where it puts all the data it has collected into a matrix and solves it to obtain a congruence of squares. The data collection phase can be easily parallelized to many processors, but the data processing phase requires large amounts of memory, and is difficult to parallelize efficiently over many nodes or if the processing nodes do not each have enough memory to store the whole matrix. The block Wiedemann algorithm can be used in the case of a few systems each capable of holding the matrix.
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The basic approach used is as follows. Calling the input character x, divide x-1 by 32, keeping quotient and remainder. Unless the quotient is 2 or 3, just output x, having kept a copy of it during the division. If the quotient is 2 or 3, divide the remainder ((x-1) modulo 32) by 13; if the quotient here is 0, output x+13; if 1, output x-13; if 2, output x. Regarding the division algorithm, when dividing y by z to get a quotient q and remainder r, there is an outer loop which sets q and r first to the quotient and remainder of 1/z, then to those of 2/z, and so on; after it has executed y times, this outer loop terminates, leaving q and r set to the quotient and remainder of y/z.
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The goal is to find an inverse to 10 modulo the prime under consideration (does not work for 2 or 5) and use that as a multiplier to make the divisibility of the original number by that prime depend on the divisibility of the new (usually smaller) number by the same prime. Using 31 as an example, since 10 × (−3) = −30 = 1 mod 31, we get the rule for using y − 3x in the table above. Likewise, since 10 × (28) = 280 = 1 mod 31 also, we obtain a complementary rule y + 28x of the same kind - our choice of addition or subtraction being dictated by arithmetic convenience of the smaller value. In fact, this rule for prime divisors besides 2 and 5 is really a rule for divisibility by any integer relatively prime to 10 (including 33 and 39; see the table below).
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The parity sequence is the same as the sequence of operations. Using this form for , it can be shown that the parity sequences for two numbers and will agree in the first terms if and only if and are equivalent modulo . This implies that every number is uniquely identified by its parity sequence, and moreover that if there are multiple Hailstone cycles, then their corresponding parity cycles must be different. Applying the function times to the number will give the result , where is the result of applying the function times to , and is how many increases were encountered during that sequence (e.g. for there are 3 increases as 1 iterates to 2, 1, 2, 1, and finally to 2 so the result is ; for there is only 1 increase as 1 rises to 2 and falls to 1 so the result is ).
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Unlike basic Nihilist, the additive was added by non-carrying addition (digit-wise addition modulo 10), thus producing a more uniform output which doesn't leak as much information. More importantly, the additive was generated not through a keyword, but by selecting lines at random from almanacs of industrial statistics. Such books were deemed dull enough to not arouse suspicion if an agent was searched (particularly as the agents' cover stories were as businessmen), and to have such high entropy density as to provide a very secure additive. Of course the figures from such a book are not actually uniformly distributed (there is an excess of "0" and "1" (see Benford's Law), and sequential numbers are likely to be somewhat similar), but nevertheless they have much higher entropy density than passphrases and the like; at any rate, in practice they seem never to have been successfully cryptanalysed.
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The Lucas–Lehmer–Riesel test is a particular case of group-order primality testing; we demonstrate that some number is prime by showing that some group has the order that it would have were that number prime, and we do this by finding an element of that group of precisely the right order. For Lucas-style tests on a number N, we work in the multiplicative group of a quadratic extension of the integers modulo N; if N is prime, the order of this multiplicative group is N2 − 1, it has a subgroup of order N + 1, and we try to find a generator for that subgroup. We start off by trying to find a non- iterative expression for the u_i. Following the model of the Lucas–Lehmer test, put u_i = a^{2^i} + a^{-2^i}, and by induction we have u_i = u_{i-1}^2 - 2.
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Some of Roth's earliest works included a 1949 paper on sums of powers, showing that almost all positive integers could be represented as a sum of a square, a cube, and a fourth power, and a 1951 paper on the gaps between squarefree numbers, describes as "quite sensational" and "of considerable importance" respectively by Chen and Vaughan. His inaugural lecture at Imperial College concerned the large sieve: bounding the size of sets of integers from which many congruence classes of numbers modulo prime numbers have been forbidden. Roth had previously published a paper on this problem in 1965. The optimal square packing in a square can sometimes involve tilted squares; Roth and Bob Vaughan showed that non-constant area must be left uncovered Another of Roth's interests was the Heilbronn triangle problem, of placing points in a square to avoid triangles of small area.
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The pentagram map, when acting on the moduli space X of convex polygons, has an invariant volume form. At the same time, as was already mentioned, the function f=O_NE_N has compact level sets on X. These two properties combine with the Poincaré recurrence theorem to imply that the action of the pentagram map on X is recurrent: The orbit of almost any equivalence class of convex polygon P returns infinitely often to every neighborhood of P. This is to say that, modulo projective transformations, one typically sees nearly the same shape, over and over again, as one iterates the pentagram map. (It is important to remember that one is considering the projective equivalence classes of convex polygons. The fact that the pentagram map visibly shrinks a convex polygon is irrelevant.) It is worth mentioning that the recurrence result is subsumed by the complete integrability results discussed below.
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However, more complicated coding schemes allow a greater amount of information to be sent across the same channel, by using codewords of length greater than one. For instance, suppose that in two consecutive steps the sender transmits one of the five code words "11", "23", "35", "54", or "42". (Here, the quotation marks indicate that these words should be interpreted as strings of symbols, not as decimal numbers.) Each pair of these code words includes at least one position where its values differ by two or more modulo 5; for instance, "11" and "23" differ by two in their second position, while "23" and "42" differ by two in their first position. Therefore, a recipient of one of these code words will always be able to determine unambiguously which one was sent: no two of these code words can be confused with each other.
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Souradyuti Paul and Bart Preneel have proposed an RC4 variant, which they call RC4A. RC4A uses two state arrays and , and two indexes and . Each time is incremented, two bytes are generated: # First, the basic RC4 algorithm is performed using and , but in the last step, is looked up in . # Second, the operation is repeated (without incrementing again) on and , and is output. Thus, the algorithm is: All arithmetic is performed modulo 256 i := 0 j1 := 0 j2 := 0 while GeneratingOutput: i := i + 1 j1 := j1 + S1[i] swap values of S1[i] and S1[j1] output S2[S1[i] + S1[j1 j2 := j2 + S2[i] swap values of S2[i] and S2[j2] output S1[S2[i] + S2[j2 endwhile Although the algorithm required the same number of operations per output byte, there is greater parallelism than RC4, providing a possible speed improvement.
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The attack on the distribution question leads quickly to problems that are now seen to be special cases of those on local zeta-functions, for the particular case of some special hyperelliptic curves such as Y^2 = X(X-1)(X-2)\ldots (X-k). Bounds for the zeroes of the local zeta-function immediately imply bounds for sums \sum \chi(X(X-1)(X-2)\ldots (X-k)), where χ is the Legendre symbol modulo a prime number p, and the sum is taken over a complete set of residues mod p. In the light of this connection it was appropriate that, with a Trinity research fellowship, Davenport in 1932-1933 spent time in Marburg and Göttingen working with Helmut Hasse, an expert on the algebraic theory. This produced the work on the Hasse–Davenport relations for Gauss sums, and contact with Hans Heilbronn, with whom Davenport would later collaborate.
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There exist other representations, and other parameterized families of representations, for 1. For 2, the other known representations are :1\ 214\ 928^3 + 3\ 480\ 205^3 + (-3\ 528\ 875)^3 = 2, :37\ 404\ 275\ 617^3 + (-25\ 282\ 289\ 375)^3 + (-33\ 071\ 554\ 596)^3 = 2, :3\ 737\ 830\ 626\ 090^3 + 1\ 490\ 220\ 318\ 001^3 + (-3\ 815\ 176\ 160\ 999)^3 = 2. However, 1 and 2 are the only numbers with representations that can be parameterized by quartic polynomials in this way. Even in the case of representations of 3, Louis J. Mordell wrote in 1953 "I do not know anything" more than its small solutions :1^3+1^3+1^3=4^3+4^3+(-5)^3=3, and more than the fact that in this case each of the three cubed numbers must be equal modulo 9.
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Mathematically, the points of the diamond cubic structure can be given coordinates as a subset of a three- dimensional integer lattice by using a cubic unit cell four units across. With these coordinates, the points of the structure have coordinates (x, y, z) satisfying the equations :x = y = z (mod 2), and :x + y + z = 0 or 1 (mod 4).. There are eight points (modulo 4) that satisfy these conditions: :(0,0,0), (0,2,2), (2,0,2), (2,2,0), :(3,3,3), (3,1,1), (1,3,1), (1,1,3) All of the other points in the structure may be obtained by adding multiples of four to the x, y, and z coordinates of these eight points. Adjacent points in this structure are at distance apart in the integer lattice; the edges of the diamond structure lie along the body diagonals of the integer grid cubes. This structure may be scaled to a cubical unit cell that is some number a of units across by multiplying all coordinates by .
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Moshe Lerman suggested a background to Birkat Hachama by pointing out a possible connection between the traditional Hebrew dating and the two machzorim ("cycles") that are observed in Jewish tradition—the "small" 19-year cycle which is the basis of the Jewish calendar, and the "big" 28-year cycle which determines the year in which Birkat Hachama is recited. Mathematically, if one knows the position of a certain year in both cycles, one can compute the number associated to the year modulo 532 (19 times 28), given that the starting point of both cycles is year 1. Because the astronomical year is slightly shorter than 365.25 days, the date of Birkat Hachama shifts away from the spring equinox as history proceeds. A simple astronomical calculation shows that 84 cycles of 28 years before 5769, in the Jewish year 3417, the spring equinox was in the beginning of the night before the fourth day of the week as stipulated by the Talmud.
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Lerman takes this as a hint that the astronomically astute Jewish sages of the time concluded that the Jewish year 3417 was a first year in the cycle of 28 years. Moreover, Lerman suggests that these same Jewish sages would have reasoned that year 3421 was a first year in the 19-year cycle, in accordance with an ancient tradition that the world was created in the first week of the month of Nissan, and thus the sun was created on the fourth day of Nissan. Since every 19 years the solar and lunar calendars align, and the Spring equinox of 3421 occurred early in the night leading to the fourth day of the Jewish month of Nissan, it follows that 3421 was the first year of a 19-year cycle. Lerman surmises that the Jewish sages at the time could argue for a determination of the position of their years in both cycles and could therefore compute the absolute year- count modulo 532 years.
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All additions are modulo 256. The following Standard Forth version uses Core and Core Extension words only. 0 value ii 0 value jj 0 value KeyAddr 0 value KeyLen create SArray 256 allot \ state array of 256 bytes : KeyArray KeyLen mod KeyAddr ; : get_byte + c@ ; : set_byte + c! ; : as_byte 255 and ; : reset_ij 0 TO ii 0 TO jj ; : i_update 1 + as_byte TO ii ; : j_update ii SArray get_byte + as_byte TO jj ; : swap_s_ij jj SArray get_byte ii SArray get_byte jj SArray set_byte ii SArray set_byte ; : rc4_init ( KeyAddr KeyLen -- ) 256 min TO KeyLen TO KeyAddr 256 0 DO i i SArray set_byte LOOP reset_ij BEGIN ii KeyArray get_byte jj + j_update swap_s_ij ii 255 < WHILE ii i_update REPEAT reset_ij ; : rc4_byte ii i_update jj j_update swap_s_ij ii SArray get_byte jj SArray get_byte + as_byte SArray get_byte xor ; This is one of many ways to test the code: hex create AKey 61 c, 8A c, 63 c, D2 c, FB c, : test cr 0 DO rc4_byte .
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The final character of a ten-digit International Standard Book Number is a check digit computed so that multiplying each digit by its position in the number (counting from the right) and taking the sum of these products modulo 11 is 0. The digit the farthest to the right (which is multiplied by 1) is the check digit, chosen to make the sum correct. It may need to have the value 10, which is represented as the letter X. For example, take the : The sum of products is 0×10 + 2×9 + 0×8 + 1×7 + 5×6 + 3×5 + 0×4 + 8×3 + 2×2 + 1×1 = 99 ≡ 0 (mod 11). So the ISBN is valid. Note that positions can also be counted from left, in which case the check digit is multiplied by 10, to check validity: 0×1 + 2×2 + 0×3 + 1×4 + 5×5 + 3×6 + 0×7 + 8×8 + 2×9 + 1×10 = 143 ≡ 0 (mod 11).
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The Chebotarev density theorem may be viewed as a generalisation of Dirichlet's theorem on arithmetic progressions. A quantitative form of Dirichlet's theorem states that if N≥2 is an integer and a is coprime to N, then the proportion of the primes p congruent to a mod N is asymptotic to 1/n, where n=φ(N) is the Euler totient function. This is a special case of the Chebotarev density theorem for the Nth cyclotomic field K. Indeed, the Galois group of K/Q is abelian and can be canonically identified with the group of invertible residue classes mod N. The splitting invariant of a prime p not dividing N is simply its residue class because the number of distinct primes into which p splits is φ(N)/m, where m is multiplicative order of p modulo N; hence by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to N.
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Most commonly, the modulus is chosen as a prime number, making the choice of a coprime seed trivial (any 0 < X < m will do). This produces the best quality output, but introduces some implementation complexity and the range of the output is unlikely to match the desired application; converting to the desired range requires an additional multiplication. Using a modulus m which is a power of two makes for a particularly convenient computer implementation, but comes at a cost: the period is at most m/4, and the low bits have periods shorter than that. This is because the low k bits form a modulo-2 generator all by themselves; the higher-order bits never affect lower-order bits. The values X are always odd (bit 0 never changes), bits 2 and 1 alternate (the low 3 bits repeat with a period of 2), the low 4 bits repeat with a period of 4, and so on.
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Ada supports numerical types defined by a range, modulo types, aggregate types (records and arrays), and enumeration types. Access types define a reference to an instance of a specified type; untyped pointers are not permitted. Special types provided by the language are task types and protected types. For example, a date might be represented as: type Day_type is range 1 .. 31; type Month_type is range 1 .. 12; type Year_type is range 1800 .. 2100; type Hours is mod 24; type Weekday is (Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday); type Date is record Day : Day_type; Month : Month_type; Year : Year_type; end record; Types can be refined by declaring subtypes: subtype Working_Hours is Hours range 0 .. 12; -- at most 12 Hours to work a day subtype Working_Day is Weekday range Monday .. Friday; -- Days to work Work_Load: constant array(Working_Day) of Working_Hours -- implicit type declaration := (Friday => 6, Monday => 4, others => 10); -- lookup table for working hours with initialization Types can have modifiers such as limited, abstract, private etc.
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Ideally one would like to describe the (moduli) space of all solutions explicitly, and for some very special PDEs this is possible. (In general this is a hopeless problem: it is unlikely that there is any useful description of all solutions of the Navier–Stokes equation for example, as this would involve describing all possible fluid motions.) If the equation has a very large symmetry group, then one is usually only interested in the moduli space of solutions modulo the symmetry group, and this is sometimes a finite- dimensional compact manifold, possibly with singularities; for example, this happens in the case of the Seiberg–Witten equations. A slightly more complicated case is the self dual Yang–Mills equations, when the moduli space is finite-dimensional but not necessarily compact, though it can often be compactified explicitly. Another case when one can sometimes hope to describe all solutions is the case of completely integrable models, when solutions are sometimes a sort of superposition of solitons; this happens e.g.
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However, when the coefficients are integers, rational numbers or polynomials, these arithmetic operations imply a number of GCD computations of coefficients which is of the same order and make the algorithm inefficient. The subresultant pseudo- remainder sequences were introduced to solve this problem and avoid any fraction and any GCD computation of coefficients. A more efficient algorithm is obtained by using the good behavior of the resultant under a ring homomorphism on the coefficients: to compute a resultant of two polynomials with integer coefficients, one computes their resultants modulo sufficiently many prime numbers and then reconstructs the result with the Chinese remainder theorem. The use of fast multiplication of integers and polynomials allows algorithms for resultants and greatest common divisors that have a better time complexity, which is of the order of the complexity of the multiplication, multiplied by the logarithm of the size of the input (\log(s(d+e)), where is an upper bound of the number of digits of the input polynomials).
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The quantization forced by the Dirac string can be understood in terms of the cohomology of the fibre bundle representing the gauge fields over the base manifold of space-time. The magnetic charges of a gauge field theory can be understood to be the group generators of the cohomology group H^2(M) for the fiber bundle M. The cohomology arises from the idea of classifying all possible gauge field strengths F=dA, which are manifestly exact forms, modulo all possible gauge transformations, given that the field strength F must be a closed form: dF=0. Here, A is the vector potential and d represents the gauge- covariant derivative, and F the field strength or curvature form on the fiber bundle. Informally, one might say that the Dirac string carries away the "excess curvature" that would otherwise prevent F from being a closed form, as one has that dF=0 everywhere except at the location of the monopole.
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It is also necessary to solve the quadratic equation modulo small powers of p in order to recognise numbers divisible by the square of a factor-base prime. At the end of the factor base, any A[] containing a value above a threshold of roughly log(x2-n) will correspond to a value of y(x) which splits over the factor base. The information about exactly which primes divide y(x) has been lost, but it has only small factors, and there are many good algorithms for factoring a number known to have only small factors, such as trial division by small primes, SQUFOF, Pollard rho, and ECM, which are usually used in some combination. There are many y(x) values that work, so the factorization process at the end doesn't have to be entirely reliable; often the processes misbehave on say 5% of inputs, requiring a small amount of extra sieving.
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However, both Lars Knudsen and Sean Murphy found minor weaknesses in this version, prompting a redesign of the key schedule to one suggested by Knudsen; these variants were named SAFER SK-64 and SAFER SK-128 respectively -- the "SK" standing for "Strengthened Key schedule", though the RSA FAQ reports that, "one joke has it that SK really stands for 'Stop Knudsen', a wise precaution in the design of any block cipher". Another variant with a reduced key size was published, SAFER SK-40, to comply with 40-bit export restrictions. All of these ciphers use the same round function consisting of four stages, as shown in the diagram: a key-mixing stage, a substitution layer, another key-mixing stage, and finally a diffusion layer. In the first key-mixing stage, the plaintext block is divided into eight 8-bit segments, and subkeys are added using either addition modulo 256 (denoted by a "+" in a square) or XOR (denoted by a "+" in a circle).
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In 2014, Ronald Rivest gave a talk and co-wrote a paper on an updated redesign called Spritz. A hardware accelerator of Spritz was published in Secrypt, 2016 and shows that due to multiple nested calls required to produce output bytes, Spritz performs rather slowly compared to other hash functions such as SHA-3 and the best known hardware implementation of RC4. The algorithm is: All arithmetic is performed modulo 256 while GeneratingOutput: i := i + w j := k + S[j + S[i k := k + i + S[j] swap values of S[i] and S[j] output z := S[j + S[i + S[z + k ] endwhile The value , is relatively prime to the size of the S array. So after 256 iterations of this inner loop, the value (incremented by every iteration) has taken on all possible values 0...255, and every byte in the S array has been swapped at least once.
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In application of homological algebra techniques to algebraic geometry, it has been traditional since David Hilbert (though modern terminology is different) to apply free resolutions of R, considered as a graded module over the polynomial ring. This yields information about syzygies, namely relations between generators of the ideal I. In a classical perspective, such generators are simply the equations one writes down to define V. If V is a hypersurface there need only be one equation, and for complete intersections the number of equations can be taken as the codimension; but the general projective variety has no defining set of equations that is so transparent. Detailed studies, for example of canonical curves and the equations defining abelian varieties, show the geometric interest of systematic techniques to handle these cases. The subject also grew out of elimination theory in its classical form, in which reduction modulo I is supposed to become an algorithmic process (now handled by Gröbner bases in practice).
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The reverse operation, decoding a Gray-coded value into a binary number, is more complicated, but can be expressed as the prefix sum of the bits of , where each summation operation within the prefix sum is performed modulo two. A prefix sum of this type may be performed efficiently using the bitwise Boolean operations available on modern computers, by computing the exclusive or of with each of the numbers formed by shifting to the left by a number of bits that is a power of two.. Parallel prefix (using multiplication as the underlying associative operation) can also be used to build fast algorithms for parallel polynomial interpolation. In particular, it can be used to compute the divided difference coefficients of the Newton form of the interpolation polynomial.. This prefix based approach can also be used to obtain the generalized divided differences for (confluent) Hermite interpolation as well as for parallel algorithms for Vandermonde systems.
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Proceedings of the 1st International Roberto Gerhard Conference, pg. 69. As for what temporality respects, the same tone- row allows deriving from itself the duration of the notes: it is achieved by making a simple subtraction in modulo 12, that is, subtracting and considering the results in the interval [1, 12] so that if a number is left out, you can add or subtract as many 12 as necessary. This procedure greatly influences the structure of the work, as it determines blocks of 78 (78= 1 + 2 + \ldots + 12) eight notes where a single row acts. In addition, he divides the sequence into two hexachords so that their respective durations are 33 and 45. The 33:45 ratio has an essential implication throughout the movement, as he will divide it into 4 structured sections as follows: There are 3 sections of 11 bars (33 x 3 x 11) and the last one is 12 bars (12 x 45 – 33). (12=45-33).
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Generally, no such prime exists when b is congruent to 0 or 1 modulo 4. The values of p less than 1000 for which this formula produces cyclic numbers in decimal are: :7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863, 887, 937, 941, 953, 971, 977, 983, ... For example, the case b = 10, p = 7 gives the cyclic number 142857; thus 7 is a full reptend prime. Furthermore, 1 divided by 7 written out in base 10 is 0.142857 142857 142857 142857... Not all values of p will yield a cyclic number using this formula; for example p = 13 gives 076923 076923. These failed cases will always contain a repetition of digits (possibly several) over the course of p − 1 digits.
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One digital signature scheme (of many) is based on RSA. To create signature keys, generate a RSA key pair containing a modulus, N, that is the product of two random secret distinct large primes, along with integers, e and d, such that e d ≡ 1 (mod φ(N)), where φ is the Euler phi-function. The signer's public key consists of N and e, and the signer's secret key contains d. To sign a message, m, the signer computes a signature, σ, such that σ ≡ md (mod N). To verify, the receiver checks that σe ≡ m (mod N). Several early signature schemes were of a similar type: they involve the use of a trapdoor permutation, such as the RSA function, or in the case of the Rabin signature scheme, computing square modulo composite, N. A trapdoor permutation family is a family of permutations, specified by a parameter, that is easy to compute in the forward direction, but is difficult to compute in the reverse direction without already knowing the private key ("trapdoor").
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The great advantage of model checking is that it is often fully automatic; its primary disadvantage is that it does not in general scale to large systems; symbolic models are typically limited to a few hundred bits of state, while explicit state enumeration requires the state space being explored to be relatively small. Another approach is deductive verification. It consists of generating from the system and its specifications (and possibly other annotations) a collection of mathematical proof obligations, the truth of which imply conformance of the system to its specification, and discharging these obligations using either proof assistants (interactive theorem provers) (such as HOL, ACL2, Isabelle, Coq or PVS), automatic theorem provers, including in particular satisfiability modulo theories (SMT) solvers. This approach has the disadvantage that it typically requires the user to understand in detail why the system works correctly, and to convey this information to the verification system, either in the form of a sequence of theorems to be proved or in the form of specifications (invariants, preconditions, postconditions) of system components (e.g.
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In contrast, Armaiti is identified with "fruitfulness".. In the Counsels of Adarbad Mahraspandan the author advises his readership not to take medicine on the day of the month dedicated to Zam.. In the Pazend Afrin-i haft Amshespand ("Blessings of the seven Amesha Spenta"), Zam is joined by Amardad, Rashn and Ashtad (Ameretat, Rashnu and Arshtat) in withstanding the demons of hunger and thirst.. The last hymn recited in the procedure for the establishment of a Fire temple is the Zamyad Yasht. This is done because the required 91 recitals in honor of the Yazatas would in principle require each of the 30 hymns associated with the divinities of the 30 days to be recited thrice with one additional one. However, the first three recited are dedicated to Ahura Mazda, leaving 88, and 88 modulo 30 is 28, the day-number dedication of Zam.. From among the flowers associated with the yazatas, Zam's is the Basil (Bundahishn 27).. According to Xenophon (Cyropaedia, 8.24), Cyrus sacrificed animals to the earth as the Magians directed.
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Tait's motivation for studying the ménage problem came from trying to find a complete listing of mathematical knots with a given number of crossings, say n. In Dowker notation for knot diagrams, an early form of which was used by Tait, the 2n points where a knot crosses itself, in consecutive order along the knot, are labeled with the 2n numbers from 1 to 2n. In a reduced diagram, the two labels at a crossing cannot be consecutive, so the set of pairs of labels at each crossing, used in Dowker notation to represent the knot, can be interpreted as a perfect matching in a graph that has a vertex for every number in the range from 1 to 2n and an edge between every pair of numbers that has different parity and are non-consecutive modulo 2n. This graph is formed by removing a Hamiltonian cycle (connecting consecutive numbers) from a complete bipartite graph (connecting all pairs of numbers with different parity), and so it has a number of matchings equal to a ménage number.
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Informally, ACC0 models the class of computations realised by Boolean circuits of constant depth and polynomial size, where the circuit gates includes "modular counting gates" that compute the number of true inputs modulo some fixed constant. More formally, a language belongs to AC0[m] if it can be computed by a family of circuits C1, C2, ..., where Cn takes n inputs, the depth of every circuit is constant, the size of Cn is a polynomial function of n, and the circuit uses the following gates: AND gates and OR gates of unbounded fan-in, computing the conjunction and disjunction of their inputs; NOT gates computing the negation of their single input; and unbounded fan-in MOD-m gates, which compute 1 if the number of input 1s is a multiple of m. A language belongs to ACC0 if it belongs to AC0[m] for some m. In some texts, ACCi refers to a hierarchy of circuit classes with ACC0 at its lowest level, where the circuits in ACCi have depth O(login) and polynomial size.
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Chinese models went on sale on October 29, 2011. Early models include a choice of two engines: 1.8L and 2.0L; 5-speed manual (1.8L EXi), 5-speed automatic (1.8L EXi, 1.8L VTi) or 5-speed Tiptronic transmission (2.0L TYPE-S) and navigation system for (1.8L VTi, 2.0L TYPE-S).官方指导价13.18-18.58万元 第九代CIVIC（思域）上市销售 It was not sold in Japan because sales were affected by the fact the width dimension exceeded 1,700mm, an important distinction according to Japanese Government dimension regulations. In the Philippines, the ninth generation Civic was initially launched in early 2012 and sold in 4 variants: 1.8 S which is the base model of the Civic with a choice of 5-speed manual or 5-speed automatic, 1.8 E and the 1.8 EXi have the same alloy wheels and mirror with side turning lights while the EXi received fog lights and the top-of-the- line 2.0 EL with unique 17-inch style wheels, HID headlights, automatic climate control and leather seats, while the 1.8 E and 2.0 EL variants comes with the optional Modulo kit.
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Signals are transmitted in a 38° cone, using right-hand circular polarization, at an EIRP between 25 and 27 dBW (316 to 500 watts). Note that the 24-satellite constellation is accommodated with only 15 channels by using identical frequency channels to support antipodal (opposite side of planet in orbit) satellite pairs, as these satellites are never both in view of an earth-based user at the same time. The L2 band signals use the same FDMA as the L1 band signals, but transmit straddling 1246 MHz with the center frequency 1246 MHz + n×0.4375 MHz, where n spans the same range as for L1.GLONASS transmitter specs In the original GLONASS design, only obfuscated high-precision signal was broadcast in the L2 band, but starting with GLONASS-M, an additional civil reference signal L2OF is broadcast with an identical standard-precision code to the L1OF signal. A combined GLONASS/GPS Personal Radio Beacon The open standard-precision signal is generated with modulo-2 addition (XOR) of 511 kbit/s pseudo-random ranging code, 50 bit/s navigation message, and an auxiliary 100 Hz meander sequence (Manchester code), all generated using a single time/frequency oscillator.
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Separable polynomials occur frequently in Galois theory. For example, let P be an irreducible polynomial with integer coefficients and p be a prime number which does not divide the leading coefficient of P. Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable (which is the case for every p but a finite number) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P. Another example: P being as above, a resolvent R for a group G is a polynomial whose coefficients are polynomials in the coefficients of P, which provides some information on the Galois group of P. More precisely, if R is separable and has a rational root then the Galois group of P is contained in G. For example, if D is the discriminant of P then X^2-D is a resolvent for the alternating group. This resolvent is always separable (assuming the characteristic is not 2) if P is irreducible, but most resolvents are not always separable.
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Celso Reyes Daza began his professional wrestling career in 1982, initially working under the ring name Ovni, but soon changed his name to Pegasso I (Spanish for Pegasus I), an enmascarado (masked wrestler) character based on the mythical Pegasus. Initially he teamed with Pegasso II, but the Pegasso team did not last long as Reyes struck out on his own as a singles wrestler. On March 17, 1985 Pegaso I defeated El Modulo to win the Mexican National Lightweight Championship, his first professional wrestling championship. Pegaso I held the title for 118 days before losing it to El Khalifa. Reyes continued to work as Pegaso I until 1988 where he assumed a new enmascarado identity, Ciclón Ramírez, a fictional brother of Huracán Ramírez, a legendary Luchador. Ciclón Ramírez defeated Bestia Salvaje in the final of a tournament to win the Mexican National Welterweight Championship on May 21, 1980. Ramírez held the title for 430 days, defending the title on several occasions before losing it to Canelo Casas on July 25, 1990 a full 430 days after winning it. Ramírez regained the Welterweight title from Canelo Casas on February 13, 1991.
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It was realized that extraction of water from the aquifer was the cause. The sinking has slowed markedly since artesian wells were banned in the 1960s. However, the city is still threatened by more frequent low-level floods—the Acqua alta, that rise to a height of several centimetres over its quays—regularly following certain tides. In many old houses, staircases once used to unload goods are now flooded, rendering the former ground floor uninhabitable. Studies indicate that the city continues sinking at a relatively slow rate of 1–2mm per annum; therefore, the state of alert has not been revoked. In May 2003, Italian Prime Minister Silvio Berlusconi inaugurated the MOSE Project (Modulo Sperimentale Elettromeccanico), an experimental model for evaluating the performance of hollow floatable gates; the idea is to fix a series of 78 hollow pontoons to the sea bed across the three entrances to the lagoon. When tides are predicted to rise above 110 cm, the pontoons will be filled with air, causing them to float and block the incoming water from the Adriatic Sea. This engineering work was due to be completed by 2018.
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The linking number of two closed curves in three-dimensional space is a topological invariant of the curves: it is a number, defined from the curves in any of several equivalent ways, that does not change if the curves are moved continuously without passing through each other. The version of the linking number used for defining linkless embeddings of graphs is found by projecting the embedding onto the plane and counting the number of crossings of the projected embedding in which the first curve passes over the second one, modulo 2.. The projection must be "regular", meaning that no two vertices project to the same point, no vertex projects to the interior of an edge, and at every point of the projection where the projections of two edges intersect, they cross transversally; with this restriction, any two projections lead to the same linking number. The linking number of the unlink is zero, and therefore, if a pair of curves has nonzero linking number, the two curves must be linked. However, there are examples of curves that are linked but that have zero linking number, such as the Whitehead link.
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With a similar methodology, the American Sociological Association obtained a percentage of 18.8% having its higher frequency on the North region (22.3%–23.9%) followed by the Center region (18.4%–21.3%) and the South region (11.9%)."Stratification by Skin Color in Contemporary Mexico", Jstor org, available creating a free account , Retrieved on 27 January 2018. Another study made by the University College London in collaboration with Mexico's National Institute of Anthropology and History found that the frequencies of blond hair and light eyes in Mexicans are of 18% and 28% respectively, surveys that use as reference skin color such as those made by Mexico's National Council to Prevent Discrimination and Mexico's National Institute of Statistics and Geography reported a percentages of 47% in 2010"Encuesta Nacional Sobre Discriminación en Mexico", "CONAPRED", Mexico DF, June 2011. Retrieved on 28 April 2017."Documento Informativo Sobre Discriminación Racial en México", CONAPRED, Mexico, 21 March 2011, retrieved on 28 April 2017. and 49% in 2017"Visión INEGI 2021 Dr. Julio Santaella Castell", INEGI, 3 July 2017, Retrieved on 30 April 2018."Resultados del Modulo de Movilidad Social Intergeneracional" , INEGI, 16 June 2017, Retrieved on 30 April 2018. respectively.
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If the application software in use does not provide the ability to handle integers of this size, the modulo operation can be performed in a piece-wise manner (as is the case with the UN CEFACT TBG5 Javascript program). Piece-wise calculation can be done in many ways. One such way is as follows: # Starting from the leftmost digit of D, construct a number using the first 9 digits and call it N.231 is approximately equal to , making it possible for any 9-digit integer to be handled using 32 bit integer arithmetic # Calculate N mod 97. # Construct a new 9-digit N by concatenating above result (step 2) with the next 7 digits of D. If there are fewer than 7 digits remaining in D but at least one, then construct a new N, which will have less than 9 digits, from the above result (step 2) followed by the remaining digits of D # Repeat steps 2–3 until all the digits of D have been processed The result of the final calculation in step 2 will be D mod 97 = N mod 97.
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Over fields of characteristic greater than 3, all pseudo-reductive groups can be obtained from reductive groups by the "standard construction", a generalization of the construction above. The standard construction involves an auxiliary choice of a commutative pseudo-reductive group, which turns out to be a Cartan subgroup of the output of the construction, and the main complication for a general pseudo-reductive group is that the structure of Cartan subgroups (which are always commutative and pseudo-reductive) is mysterious. The commutative pseudo-reductive groups admit no useful classification (in contrast with the connected reductive case, for which they are tori and hence are accessible via Galois lattices), but modulo this one has a useful description of the situation away from characteristics 2 and 3 in terms of reductive groups over some finite (possibly inseparable) extensions of the ground field. Over imperfect fields of characteristics 2 and 3 there are some extra pseudo-reductive groups (called exotic) coming from the existence of exceptional isogenies between groups of types B and C in characteristic 2, between groups of type F₄ in characteristic 2, and between groups of type G₂ in characteristic 3, using a construction analogous to that of the Ree groups.
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Vieta's formulas are frequently used with polynomials with coefficients in any integral domain . Then, the quotients a_i/a_n belong to the ring of fractions of (and possibly are in itself if a_n happens to be invertible in ) and the roots r_i are taken in an algebraically closed extension. Typically, is the ring of the integers, the field of fractions is the field of the rational numbers and the algebraically closed field is the field of the complex numbers. Vieta's formulas are then useful because they provide relations between the roots without having to compute them. For polynomials over a commutative ring which is not an integral domain, Vieta's formulas are only valid when a_n is a non zero-divisor and P(x) factors as a_n(x-r_1)(x-r_2)\dots(x-r_n). For example, in the ring of the integers modulo 8, the polynomial P(x)=x^2-1 has four roots: 1, 3, 5, and 7. Vieta's formulas are not true if, say, r_1=1 and r_2=3, because P(x) eq (x-1)(x-3). However, P(x) does factor as (x-1)(x-7) and as (x-3)(x-5), and Vieta's formulas hold if we set either r_1=1 and r_2=7 or r_1=3 and r_2=5.
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Giovanni Battista Rizza (born 7 February 1924) (at the registry office Giambattista Rizza)See the list of the recipients of the medal "Benemeriti della Scuola, della Cultura, dell'Arte" and the Decreto ministeriale 17 febbraio 1999 conferring him the title of "Professor Emeritus". is an Italian mathematician, working in the fields of complex analysis of several variables and in differential geometry: he is known for his contribution to hypercomplex analysis, notably for extending Cauchy's integral theorem and Cauchy's integral formula to complex functions of a hypercomplex variable,According to the motivation for the award of the "Premio Ottorino Pomini", reported on the , "Sono particolarmente degni di nota i risultati sui teoremi integrali per le funzioni regolari, sulle estensioni della formula integrale di Cauchy alle funzioni monogene sulle algebre complesse dotate di modulo commutative e sul conseguente sviluppo della relativa teoria, ed infine sulla struttura delle algebre di Clifford" ("Particularly notable results are the ones on the integral theorems for regular functions, the ones on the extension of Cauchy integral formula to complex commutative algebras with modulus, and lastly the ones on the structure of Clifford algebras"). the theory of pluriharmonic functions and for the introduction of the now called Rizza manifolds.
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Currently the barcodes on a book's back cover (or inside a mass-market paperback book's front cover) are EAN-13; they may have a separate barcode encoding five digits called an EAN-5 for the currency and the recommended retail price. — including a detailed description of the EAN-13 format. For 10-digit ISBNs, the number "978", the Bookland "country code", is prefixed to the ISBN in the barcode data, and the check digit is recalculated according to the EAN-13 formula (modulo 10, 1x and 3x weighting on alternating digits). Partly because of an expected shortage in certain ISBN categories, the International Organization for Standardization (ISO) decided to migrate to a 13-digit ISBN (ISBN-13). The process began on 1 January 2005 and was planned to conclude on 1 January 2007. , all the 13-digit ISBNs began with 978. As the 978 ISBN supply is exhausted, the 979 prefix was introduced. Part of the 979 prefix is reserved for use with the Musicland code for musical scores with an ISMN. The 10-digit ISMN codes differed visually as they began with an "M" letter; the bar code represents the "M" as a zero (0), and for checksum purposes it counted as a 3.
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The numerical values of corresponding message and key letters are added together, modulo 26. So, if key material begins with "XMCKL" and the message is "HELLO", then the coding would be done as follows: H E L L O message 7 (H) 4 (E) 11 (L) 11 (L) 14 (O) message \+ 23 (X) 12 (M) 2 (C) 10 (K) 11 (L) key = 30 16 13 21 25 message + key = 4 (E) 16 (Q) 13 (N) 21 (V) 25 (Z) (message + key) mod 26 E Q N V Z → ciphertext If a number is larger than 25, then the remainder after subtraction of 26 is taken in modular arithmetic fashion. This simply means that if the computations "go past" Z, the sequence starts again at A. The ciphertext to be sent to Bob is thus "EQNVZ". Bob uses the matching key page and the same process, but in reverse, to obtain the plaintext. Here the key is subtracted from the ciphertext, again using modular arithmetic: E Q N V Z ciphertext 4 (E) 16 (Q) 13 (N) 21 (V) 25 (Z) ciphertext \- 23 (X) 12 (M) 2 (C) 10 (K) 11 (L) key = -19 4 11 11 14 ciphertext – key = 7 (H) 4 (E) 11 (L) 11 (L) 14 (O) ciphertext – key (mod 26) H E L L O → message Similar to the above, if a number is negative, then 26 is added to make the number zero or higher.
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These failed cases will always contain a repetition of digits (possibly several). The first values of p for which this formula produces cyclic numbers in decimal (b = 10) are :7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863, 887, 937, 941, 953, 971, 977, 983, ... For b = 12 (duodecimal), these ps are :5, 7, 17, 31, 41, 43, 53, 67, 101, 103, 113, 127, 137, 139, 149, 151, 163, 173, 197, 223, 257, 269, 281, 283, 293, 317, 353, 367, 379, 389, 401, 449, 461, 509, 523, 547, 557, 569, 571, 593, 607, 617, 619, 631, 641, 653, 691, 701, 739, 751, 761, 773, 787, 797, 809, 821, 857, 881, 929, 953, 967, 977, 991, ... For b = 2 (binary), these ps are :3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619, 653, 659, 661, 677, 701, 709, 757, 773, 787, 797, 821, 827, 829, 853, 859, 877, 883, 907, 941, 947, ... For b = 3 (ternary), these ps are :2, 5, 7, 17, 19, 29, 31, 43, 53, 79, 89, 101, 113, 127, 137, 139, 149, 163, 173, 197, 199, 211, 223, 233, 257, 269, 281, 283, 293, 317, 331, 353, 379, 389, 401, 449, 461, 463, 487, 509, 521, 557, 569, 571, 593, 607, 617, 631, 641, 653, 677, 691, 701, 739, 751, 773, 797, 809, 811, 821, 823, 857, 859, 881, 907, 929, 941, 953, 977, ... There are no such ps in the hexadecimal system. The known pattern to this sequence comes from algebraic number theory, specifically, this sequence is the set of primes p such that b is a primitive root modulo p. A conjecture of Emil Artin is that this sequence contains 37.395..% of the primes (for b in ).
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