You know, you multiply a plus-integer with a plus-integer, you're gonna get a plus result.
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It might appear that multiplying these numbers should give an integer product—ie, a whole number—since the mean is the result of dividing one integer by another.
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As BBC News points out, "Gangnam Style"'s view count eventually got so big that in 2014 YouTube had to upgrade its counter from a 32-bit integer to a 64-bit integer.
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Chinese export volumes have started to dip, however, according to Integer Research.
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Hope faded that I would ever find a phrase that started with INTEGER.
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My federal tax rate over the last five years is a negative integer.
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Pack for a Denver speaking engagement with the Integer Group, a marketing agency.
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In C++, for example, it's illegal to assign some text to an integer variable.
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The Integer plant in Tijuana that Hernandez oversees looks like a vast scientific laboratory.
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AX * Boyuan announces acquistion of integer securities Source text for Eikon: Further company coverage:
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They can be written in the form 217n – 23 where n is an integer.
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It measures a number of tasks, including 3D cube rotation, integer math, and library parsing.
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It occurs when we play notes together that can be reduced to simple integer ratios.
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That rounding clearly affects whether the product of it and the sample size will be an integer.
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Mersenne primes are those that can be written in the form 2ⁿ-1 where n is an integer.
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Luckily, the fine folks at the Online Encyclopedia of Integer Sequences have put both sets of numbers online.
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There are scales underlying it, and within those scales are relations that likewise resolve to simple integer ratios.
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As we explained then: A prime number is not divisible by any positive integer except 1 and itself.
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Tracks run between each integer allowing the output train to run to any of the 200 possible output combinations.
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If the product is not an integer (as in this case, where the answer is 290.57), something looks wrong.
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But a few months later, in October, when E.P.A. officials visited the Integer medical devices factory in Minneapolis, Minn.
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Pixel art fans will be pleased to hear that Nvidia is introducing GPU integer scaling with this new driver.
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Until now, the angular momentum resulting from that rotation was always thought to be an integer multiple of Planck's constant.
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Peak performance for the P4 is 260 TOPS for 4-bit integer operations and 65 TOPS for floating point operations.
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Quest To Learn in New York Students practice absolute value and integer operations using a card game designed by the Institute!
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Bennett conjectured that these eleven solutions (the eight with integer exponents and the three with fractional exponents) are the only ones.
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Write this set as: ... , -2, -1, 0, 1, 2, ... Next, state a theorem, for example, that there is no largest integer.
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And you can almost expect, if there are more planets, the next one would be somewhere further out, at another integer ratio.
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Mathematicians are particularly interested in rational numbers that solve what are called "Diophantine equations" — polynomial equations with integer coefficients, like x2 + y19803 = 1.
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The catch is that the data in the temp_f field from the JSON data-wad is an integer number and not a string.
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It then checked that this number was not divisible by any positive integer except 1 and itself — the definition of a prime number.
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The 2038 epochalypse will occur when the Unix time counter overflows the limits of the 32-bit integer data type (at 2,147,483,647 seconds).
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To start, the team reworked mediaserver to scan for integer overflow attacks up front, preventing the attack that's at the heart of Stagefright.
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Guessing also becomes cumbersome for quadratics with large numbers, and it only works neatly for problems that are contrived to have integer answers.
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Dubbed the "Gamescom Game Ready Driver," the 26 update includes a new ultra-low latency mode, integer scaling, and a new Freestyle sharpening filter.
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Several American companies have established factories in Mexico in recent years, including Medtronic, the large device maker, and Integer, which makes components for defibrillators.
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The CNN-AQI is given as an integer value, corresponding to five concentrations bands that are specific to each pollutant as defined in the table below.
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"According to Integer, there are no effective alternatives," read an E.P.A. memo about the visit, which was arranged by a chemicals group, the Halogenated Solvents Industry Alliance.
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Global consumption of DEF may reach 10 million tonnes of urea equivalent annually by 2027 from 2 million currently, said Adam Panayi, research manager at Integer Research.
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That is the essence of topology, a branch of geometry which deals in "invariants", such as holes, that can exist in geometric shapes only in discrete, integer numbers.
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To understand the GRIM test, consider an experiment in which participants were asked to assess something (someone else's friendliness, say) on an integer scale of one to seven.
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The first step is to redesign all of the key components with a clear focus on significantly lower power – aim for N times lower, where N is an integer!
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The most notable is Early Game Rating, a team-based integer assigned to the total value of its members' collective play in the first 15 minutes of a game.
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Let S be the area of P. An odd positive integer n is given such that the squares of the side lengths of P are integers divisible by n.
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"After the December meeting, Draghi has made himself an unpredictable integer," said Mark Farrington, managing director of $7.7 billion hedge-fund style currency managers Macro Currency Fund in London.
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You don't get that from the Steinitz-type proof, or the other way around—the circle packing proof will not prove that you can do it with integer coordinates.
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"The ultimate dream, which is maybe a bit naive, is if you understand the finite field world well enough, this might shed light on the integer world," Maynard said.
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After finding three-cubes solutions for each integer less than 213, mathematicians set their sites on another milestone: finding another sum-of-three-cubes solution for the number 243.
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But these exotic influences are channelled through the composer's signature use of the "infinity series," an integer sequence through which he organizes looping patterns of melody, harmony, and rhythm.
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Fermat asserted that equations of the form aⁿ + bⁿ = cⁿ do not have solutions when n is an integer greater than 2 and a, b and c are positive integers.
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Most of the exceptions come from a separate proof that all cubes are either multiples of nine or one integer away from a multiple of nine on the number line.
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The battery manufacturer was identified in the letter as Greatbatch, now known as Integer Holdings; it is not the case that the agency's letter did not identify the battery manufacturer.
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Amat Cama and Richard Zhu, who make up Team Fluoroacetate, scored $60,000 in bug bounties for their integer overflow exploit against the latest Amazon Echo Show 5, an Alexa-powered smart display.
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Scott gives a quick lesson in Unix time to explain why he thinks the bricking happens: for your iPhone, time is displayed as one single integer, representing the number of seconds since 1970.
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Because the P4s come with 13GB of DDR5 memory and can handle up to 22 tera-operations per second for integer operations, these cards can handle pretty much anything you throw at them.
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When certain integer variables were large enough, they would overwrite data in other parts of the phone, giving attackers a crucial foothold for breaking out of mediaserver and onto the rest of the phone.
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That, though, will be harder in the future—at least for sets of data that consist of integer numbers in a known range, as do, for example, the answers to many questionnaires in psychology experiments.
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Having even numbers of electrons (or the same number of electrons as protons) means the atoms that have a whole number spin value, a quantum mechanical property that can either assume half or integer values.
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The maths in question is topology, a branch of geometry which deals in "invariants", such as holes, that can exist in geometric shapes only in discrete, integer numbers (nothing can have half a hole in it).
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I don't buy it / want / not / them, whee That guy says The impotence of whiteness is the cause of the spread of facism: for a (rich) white audience, integer lone in my checking, diffident (as Bad).
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Contestants can use only a pencil and paper and have 45 seconds to solve word problems such as this one answered by the winner in 2014: The smallest integer of a set of consecutive integers is -32.
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Cramer's rule can be used to prove that an integer programming problem whose constraint matrix is totally unimodular and whose right-hand side is integer, has integer basic solutions. This makes the integer program substantially easier to solve.
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Meyer, p. 96. Consider the following routine definition: sum (addend1: INTEGER; addend2: INTEGER): INTEGER do Result := addend1 + addend2 end The routine `sum` takes two arguments `addend1` and `addend2`, which are called the routine's formal arguments. A call to `sum` specifies actual arguments, as shown below with `value1` and `value2`. sum_value: INTEGER value1: INTEGER = 40 value2: INTEGER = 2 … sum_value := sum (value1, value2) Parameters are also thought of as either formal or actual.
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If, however, its coefficients are actually all integers, f is called an algebraic integer. Any (usual) integer z ∈ Z is an algebraic integer, as it is the zero of the linear monic polynomial: :p(t) = t − z. It can be shown that any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer.
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Algebraic integer: An algebraic number that is the root of a monic polynomial with integer coefficients.
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Compilation is performed, for example, by: gcc -x objective-c main.m Integer.m Integer+Arithmetic.m Integer+Display.
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For sums involving the zeta-function at integer and half-integer values, see rational zeta series.
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Microprocessor Report. One of the AltiVec units executes integer and floating-point instructions, and the other only permute instructions. The latter has three subunits for simple integer, complex integer and floating- point instructions. These units have pipelines of varying lengths: 10 stages for simple integer and permute instructions, 13 stages for complex integer instructions and 16 stage for floating-point instructions.
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Advanced Video Coding (AVC) uses the integer DCT (IntDCT), an integer approximation of the DCT. It uses 4x4 and 8x8 integer DCT blocks. High Efficiency Video Coding (HEVC) and the High Efficiency Image Format (HEIF) use varied integer DCT block sizes between 4x4 and 32x32 pixels.
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In the context of integers, subtraction of one also plays a special role: for any integer a, the integer is the largest integer less than a, also known as the predecessor of a.
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The integer unit consists of the integer register file and three pipelines, two integer, one load store. The integer register file is 64 bits wide and contains 64 entries, of which 32 are architectural registers and 32 are rename registers which implement register renaming. The register file has seven read ports and three write ports. Both integer pipelines have an adder and a logic unit.
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Related to numeric representation is the size and precision of integer numbers that a CPU can represent. In the case of a binary CPU, this is measured by the number of bits (significant digits of a binary encoded integer) that the CPU can process in one operation, which is commonly called word size, bit width, data path width, integer precision, or integer size. A CPU's integer size determines the range of integer values it can directly operate on. For example, an 8-bit CPU can directly manipulate integers represented by eight bits, which have a range of 256 (28) discrete integer values.
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JPEG XT includes a lossless integer-to-integer DCT transform mode based on wavelet compression from JPEG 2000.
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In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought.
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An integer sequence is a computable sequence if there exists an algorithm which, given n, calculates an, for all n > 0\. The set of computable integer sequences is countable. The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable. Although some integer sequences have definitions, there is no systematic way to define what it means for an integer sequence to be definable in the universe or in any absolute (model independent) sense.
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A FRACTRAN program is an ordered list of positive fractions together with an initial positive integer input n. The program is run by multiplying the integer n by the first fraction f in the list for which nf is an integer. The integer n is then replaced by nf and the rule is repeated. If no fraction in the list produces an integer when multiplied by n then the program halts.
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The integer unit is responsible for executing integer instructions. It consists of the integer register file (IRF) and the E-box. The IRF contains thirty-two 64-bit registers and has four read ports and two write ports that are equally divided between the integer unit and the branch unit.Dobberpuhl 1992, pp.
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The Hilbert basis of a convex cone C is a minimal set of integer vectors such that every integer vector in C is a conical combination of the vectors in the Hilbert basis with integer coefficients.
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Two classes of particles with very different behaviour are bosons which have integer spin (), and fermions possessing half-integer spin ().
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The (integer) and (integer) variables are internal state variables. The internal state is retained across invocations. The (integer) constant is a configuration option. `get_next_hi` is a function that retrieves a new high value from a database server.
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A subset of the integer broom The integer broom space X is a subset of the plane R2. Assume that the plane is parametrised by polar coordinates. The integer broom contains the origin and the points such that n is a non-negative integer, and }. The image on the right gives an illustration for and .
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Ceylon is strongly and statically typed, but also has support for type inference. The `value` keyword is used to infer the type of a variable, and the `function` keyword is used to infer the type of a function. The following two definition pairs are each equivalent: Integer i = 3; value i = 3; Integer add(Integer i1, Integer i2) { return i1 + i2; } function add(Integer i1, Integer i2) { return i1 + i2; } However, to make single-pass type inference possible, type inference is only allowed for non-toplevel and unshared declarations.
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Eventually T is divisible by R, and after division by R the algorithm is in the same place as REDC was after the computation of t. function MultiPrecisionREDC is Input: Integer N with , stored as an array of p words, Integer , --thus, r = logB R Integer N′ in such that , Integer T in the range , stored as an array of words. Output: Integer S in such that , stored as an array of p words.
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The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10 In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.
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Fmin` function which uses F-bounded quantification. In mathematical notation, the types of the two functions are :min: ∀ T, ∀ S ⊆ {compareTo: T → int}. S → S → S :Fmin: ∀ T ⊆ Comparable[T]. T → T → T where :Comparable[T] = {compareTo: T → int} interface Comparable { public int compareTo(T other); } class Integer implements Comparable { @Override public int compareTo(Integer other) { //... } } class String implements Comparable { @Override public int compareTo(String other) { //... } } class Test { public static void main(String[] args) { Comparable a = min("cat", "dog"); Comparable b = min(new Integer(10), new Integer(3)); String str = Fmin("cat", "dog"); Integer i = Fmin(new Integer(10), new Integer(3)); } public static ~~S min(S a, S b) { if (a.
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For every integer t there is an integer N such that every graph G with no K_t minor is (t-1,N)-colourable.
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ANTIGONE (Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Equations), is a deterministic global optimization solver for general Mixed- Integer Nonlinear Programs (MINLP).
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There are two main reasons for using integer variables when modeling problems as a linear program: #The integer variables represent quantities that can only be integer. For example, it is not possible to build 3.7 cars. #The integer variables represent decisions (e.g. whether to include an edge in a graph) and so should only take on the value 0 or 1.
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List v = new ArrayList(); v.add("test"); // A String that cannot be cast to an Integer Integer i = (Integer)v.get(0); // Run time error Although the code is compiled without error, it throws a runtime exception (`java.lang.ClassCastException`) when executing the third line of code.
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A signed permutation matrix is a generalized permutation matrix whose nonzero entries are ±1, and are the integer generalized permutation matrices with integer inverse.
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Ichthyosaurus integer Bronn 1844 was also reassigned to the genus by Maisch to create the new combination Suevoleviathan integer. Based on the relocation of the holotype of Suevoleviathan integer and an updated description of the specimen, Maxwell (2018) concluded that the two Suevoleviathan species are growth stages of the same species, meaning that S. integer has priority and becomes the epithet of the Suevoleviathan type species.Maxwell, E. E. 2018. Redescription of the ‘lost’ holotype of Suevoleviathan integer (Bronn, 1844) (Reptilia: Ichthyosauria).
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An integer relation between a set of real numbers x1, x2, ..., xn is a set of integers a1, a2, ..., an, not all 0, such that :a_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\, An integer relation algorithm is an algorithm for finding integer relations. Specifically, given a set of real numbers known to a given precision, an integer relation algorithm will either find an integer relation between them, or will determine that no integer relation exists with coefficients whose magnitudes are less than a certain upper bound.Since the set of real numbers can only be specified up to a finite precision, an algorithm that did not place limits on the size of its coefficients would always find an integer relation for sufficiently large coefficients.
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The `DEFINE-statement` introduces user-specified names (e.g. `INTEGER`) for combinations of built-in attributes (e.g. `FIXED BINARY(31,0)`). Thus `DEFINE ALIAS INTEGER FIXED BINARY(31.0)` creates the `TYPE` name `INTEGER` as an alias for the set of built-in attributes FIXED BINARY(31.0).
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In general topology, a branch of mathematics, the integer broom topology, is an example of a topology on the so-called integer broom space X.
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For prime p, every integer-sided equiangular p-gon is regular. Moreover, every integer-sided equiangular pk-gon has p-fold rotational symmetry.McLean, K. Robin.
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In integer notation, or the integer model of pitch, all pitch classes and intervals between pitch classes are designated using the numbers 0 through 11.
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The Ebox executes integer, load and store instructions. It has two integer units, two load store units and two integer register files. Each integer register file contained 80 entries, of which 32 are architectural registers, 40 are rename registers and 8 are PAL shadow registers. There was no entry for register R31 because in the Alpha architecture, R31 is hardwired to zero and is read-only.
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The R8000 issues at most one integer store per cycle, and one final read port delivers the integer store data. Two register file write ports are used to write results from the two integer functional units. The R8000 issues two integer loads per cycle, and the other two write ports are used to write the results of integer loads to the register file. The level 1 data cache was organized as two redundant arrays, each of which had one read port and one write port.
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Integer BASIC is not in ROM on the II Plus and is instead loaded by DOS 3.x during boot up into the RAM at $D000 (if a language card is present, otherwise this step is skipped--II Pluses without a language card cannot run Integer BASIC). Normally, the RAM containing Integer BASIC is banked out and the Applesoft ROM is present at $D000. If the user types "INT", Integer BASIC is activated by swapping out the Applesoft ROM and switching in the RAM with Integer BASIC.
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An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear. Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP- complete problems.
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Although efficient algorithms exist to give integer solutions in some situations (see network flow models), most problems that require integer solutions cannot yet be solved efficiently.
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In number theory, the integer complexity of an integer is the smallest number of ones that can be used to represent it using ones and any number of additions, multiplications, and parentheses. It is always within a constant factor of the logarithm of the given integer.
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Three cuboid conjectures are three mathematical propositions claiming irreducibility of three univariate polynomials with integer coefficients depending on several integer parameters. They are neither proved nor disproved.
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In computer science, integer sorting is the algorithmic problem of sorting a collection of data values by integer keys. Algorithms designed for integer sorting may also often be applied to sorting problems in which the keys are floating point numbers, rational numbers, or text strings.. The ability to perform integer arithmetic on the keys allows integer sorting algorithms to be faster than comparison sorting algorithms in many cases, depending on the details of which operations are allowed in the model of computing and how large the integers to be sorted are. Integer sorting algorithms including pigeonhole sort, counting sort, and radix sort are widely used and practical. Other integer sorting algorithms with smaller worst-case time bounds are not believed to be practical for computer architectures with 64 or fewer bits per word.
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Floor and ceiling functions are examples of an integer-valued function of a real variable, but on real numbers and generally, on (non-disconnected) topological spaces integer-valued functions are not especially useful. Any such function on a connected space either has discontinuities or is constant. On the other hand, on discrete and other totally disconnected spaces integer-valued functions have roughly the same importance as real-valued functions have on non-discrete spaces. Any function with natural, or non-negative integer values is a partial case of integer-valued function.
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An integer is often a primitive data type in computer languages. However, integer data types can only represent a subset of all integers, since practical computers are of finite capacity. Also, in the common two's complement representation, the inherent definition of sign distinguishes between "negative" and "non- negative" rather than "negative, positive, and 0". (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.).
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In the third step, the fractional solution must be converted into an integer solution (and thus a solution to the original problem). This is called rounding the fractional solution. The resulting integer solution should (provably) have cost not much larger than the cost of the fractional solution. This will ensure that the cost of the integer solution is not much larger than the cost of the optimal integer solution.
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Shor's algorithm is a polynomial-time quantum computer algorithm for integer factorization. Informally, it solves the following problem: Given an integer N , find its prime factors. It was invented in 1994 by the American mathematician Peter Shor. On a quantum computer, to factor an integer N , Shor's algorithm runs in polynomial time (the time taken is polynomial in \log N , the size of the integer given as input).
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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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SCIP (Solving Constraint Integer Programs) is a mixed integer programming solver and a framework for Branch and cut and Branch and price, developed primarily at Zuse Institute Berlin. Unlike most commercial solvers, SCIP gives the user low-level control of and information about the solving process. Run as a standalone solver, it is one of the fastest non-commercial solvers for mixed integer programs.Mixed Integer Linear Programming Benchmark Mar 18, 2012.
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Storing values that are too low in an integer variable (e.g., attempting to store −1 in an unsigned integer) is properly referred to as integer , or more broadly, integer wraparound. The term underflow normally refers to floating point numbers only, which is a separate issue. It is not possible in most floating-point designs to store a too-low value, as usually they are signed and have a negative infinity value.
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Conditions are known in terms of elliptic curves for an integer triangle to have an integer ratio N of the circumradius to the inradius.MacLeod, Allan J., "Integer triangles with R/r = N", Forum Geometricorum 10, 2010: pp. 149−155.Goehl, John F. Jr., "More integer triangles with R/r = N", Forum Geometricorum 12, 2012: pp. 27−28 The smallest case, that of the equilateral triangle, has N=2.
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For example, the assignment `x:=1` is only valid if the variable x can contain an integer. Therefore, the context-free syntax `variable := value` is incomplete. In a two-level grammar, this might be specified in a context-sensitive manner as `REF TYPE variable := TYPE value`. Then `ref integer variable := integer value` could be a production rule but `ref Boolean variable := integer value` is not a possible production rule.
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In the same way that compilers emulate e.g. 64-bit integer arithmetic on architectures with register sizes less than 64 bits, some compilers also support 128-bit integer arithmetic. For example, the GCC C compiler 4.6 and later has a 128-bit integer type `__int128` for some architectures.
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Execution begins in stage four. The instruction queues dispatch up to eight instructions to the execution units. Integer instructions are executed in three integer execution units (termed "fixed-point units" by IBM). Two of the units are identical and execute all integer instructions except for multiply and divide.
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Square numbers are non-negative. Another way of saying that a (non-negative) integer is a square number is that its square root is again an integer. For example, = 3, so 9 is a square number. A positive integer that has no perfect square divisors except 1 is called square-free.
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The orbit of a rational map may contain infinitely many integers. For example, if is a polynomial with integer coefficients and if is an integer, then it is clear that the entire orbit consists of integers. Similarly, if is a rational map and some iterate is a polynomial with integer coefficients, then every -th entry in the orbit is an integer. An example of this phenomenon is the map , whose second iterate is a polynomial.
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When the numerator is squared that prime will still not divide into it because of the unique factorization. Therefore, the square of a rational non-integer is always a non-integer; by contrapositive, the square root of an integer is always either another integer, or irrational. Euclid used a restricted version of the fundamental theorem and some careful argument to prove the theorem. His proof is in Euclid's Elements Book X Proposition 9.
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MINTO (Mixed Integer Optimizer) is an integer programming solver which uses branch and bound algorithm. MINTO is a software system that solves mixed integer programming problem by a branch and bound algorithm with linear programming relaxations. It also provides automatic constraint classification, preprocessing, primal heuristics and constraint generation. It also has inbuilt cut generation and can create knapsack cuts, GUB cuts, clique cuts, implication cuts, flow cuts, mixed integer rounding and Gomory cuts.
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Binary scaling is a computer programming technique used typically in embedded C, DSP and assembler programs to implement non-integer operations by using the native integer arithmetic of the processor.
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Integer stores were written to both arrays. Two loads could be processed in parallel, one on each array. Integer functional units consisted of two integer units, a shift unit, a multiply-divide unit, and two address generator units. Multiply and divide instructions are executed in the multiply-divide unit, which is not pipelined.
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The SPARC64 V has separate update buffers for integer and floating-point units. Both have 32 entries each. The integer register has eight read ports and four write ports. Half of the write ports are used for results from the integer execution units and the other half by data returned by loads.
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In probability and statistics, the generalized integer gamma distribution (GIG) is the distribution of the sum of independent gamma distributed random variables, all with integer shape parameters and different rate parameters. This is a special case of the generalized chi-squared distribution. A related concept is the generalized near-integer gamma distribution (GNIG).
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To decode an Elias gamma-coded integer: #Read and count 0s from the stream until you reach the first 1. Call this count of zeroes N. #Considering the one that was reached to be the first digit of the integer, with a value of 2N, read the remaining N digits of the integer.
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Since integer polynomials must factor into integer polynomial factors, and evaluating integer polynomials at integer values must produce integers, the integer values of a polynomial can be factored in only a finite number of ways, and produce only a finite number of possible polynomial factors. For example, consider :f(x) = x^5 + x^4 + x^2 + x + 2. If this polynomial factors over Z, then at least one of its factors must be of degree two or less. We need three values to uniquely fit a second degree polynomial. We'll use f(0) = 2, f(1) = 6 and f(-1) = 2.
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This is the integer to which the above expression refers. But the above expression is only fifty-seven letters long, therefore it is definable in under sixty letters, and is not the smallest positive integer not definable in under sixty letters, and is not defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self- contradictory (any integer it defines is definable in under sixty letters), there cannot be any integer defined by it. Perhaps another helpful analogy to Berry's Paradox would be the phrase "indescribable feeling".
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The proof above for the square root of two can be generalized using the fundamental theorem of arithmetic. This asserts that every integer has a unique factorization into primes. Using it we can show that if a rational number is not an integer then no integral power of it can be an integer, as in lowest terms there must be a prime in the denominator that does not divide into the numerator whatever power each is raised to. Therefore, if an integer is not an exact th power of another integer, then that first integer's th root is irrational.
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According to the spin–statistics theorem, particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics. In relativistic quantum field theory, the Pauli principle follows from applying a rotation operator in imaginary time to particles of half-integer spin. In one dimension, bosons, as well as fermions, can obey the exclusion principle. A one-dimensional Bose gas with delta- function repulsive interactions of infinite strength is equivalent to a gas of free fermions.
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Nuclear spin is an intrinsic angular momentum that is quantized. This means that the magnitude of this angular momentum is quantized (i.e. S can only take on a restricted range of values), and also that the x, y, and z-components of the angular momentum are quantized, being restricted to integer or half-integer multiples of ħ. The integer or half- integer quantum number associated with the spin component along the z-axis or the applied magnetic field is known as the magnetic quantum number, m, and can take values from +S to −S, in integer steps.
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Trial division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer n, the integer to be factored, can be divided by each number in turn that is less than n. For example, for the integer , the only numbers that divide it are 1, 2, 3, 4, 6, 12. Selecting only the largest powers of primes in this list gives that .
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Integer-valued functions defined on the domain of all real numbers include the floor and ceiling functions, the Dirichlet function, the sign function and the Heaviside step function (except possibly at 0). Integer-valued functions defined on the domain of non-negative real numbers include the integer square root function and the prime-counting function.
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If one side is negative, the other positive, the comparison is trivial. Otherwise, square both sides, to get an integer comparison, reversing the comparison direction if both sides were negative. On squaring both sides, the is replaced with the integer 5. So, using integer values only, we can also compare numbers of the form (a + bφ).
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The linear system Ax=b has an integer solution x if and only if the system Hy=b has an integer solution y where Uy=x and H is the column-style Hermite normal form of A. Checking that Hy=b has an integer solution is easier than Ax=b because the matrix H is triangular.
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This is the same as the integer partition of n into k parts. For instance, in the above examples, the integer 3 can be partitioned into two parts as 2+1 only. Thus, there is only one monomial in B3,2. However, the integer 6 can be partitioned into two parts as 5+1, 4+2, and 3+3.
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It is a nontotient since there is no integer with exactly 122 coprimes below it. Nor is there an integer with exactly 122 integers with common factors below it, making 122 a noncototient.
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In mathematics, a power of three is a number of the form where is an integer, that is, the result of exponentiation with number three as the base and integer as the exponent.
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Data types supported include INTEGER, DECIMAL, TIMESTAMP, BLOB and VARCHAR.
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Additional registers can be added to provide additional integer divisors.
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Note that in this case all corners have integer coordinates.
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Convert each integer and mixed number into an improper fraction.
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Instruction decoding and register reads occur during stage two, and branch instructions are resolved as well, leading to a one-cycle branch mispredict penalty. Load and store instructions begin execution in stage three, and integer instructions in stage four. Integer execution was delayed until stage four so that integer instructions which use the result of a load as an operand may be issued in the cycle after the load. Results are written to the integer register file in stage five.
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For operations that have well defined rounding behavior, overflow classification may need to be postponed until after rounding is applied. The C11 standard defines that conversions from floating point to integer must round toward zero. If C is used to convert the floating point value 127.25 to integer, then rounding should be applied first to give an ideal integer output of 127. Since the rounded integer is in the outputs range, the C standard would not classify this conversion as an overflow.
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An even number is an integer that is "evenly divisible" by two, that is divisible by two without remainder; an odd number is an integer that is not even. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) Any odd number n may be constructed by the formula for a suitable integer k. Starting with the first non-negative odd numbers are {1, 3, 5, 7, ...}. Any even number m has the form where k is again an integer.
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In addition to the multiplication of two elements of F, it is possible to define the product of an arbitrary element of by a positive integer to be the -fold sum : (which is an element of .) If there is no positive integer such that :, then is said to have characteristic 0. For example, the field of rational numbers has characteristic 0 since no positive integer is zero. Otherwise, if there is a positive integer satisfying this equation, the smallest such positive integer can be shown to be a prime number. It is usually denoted by and the field is said to have characteristic then.
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It was shown in Jesus A. De Loera, Shmuel Onn: All linear and integer programs are slim 3-way transportation programs, SIAM Journal on Optimization, 17:806–821, 2006 that every (bounded) integer programming problem is precisely equivalent to the 3 × m × n table problem discussed above for some m and n and some line sums. Thus, such 3 × m × n table problems are universal for integer programming. Moreover, for each fixed m, the resulting class of integer programs can be solved in polynomial time by the iterative Graver basis method described above. So the table width m provides a parametrization of all integer programming problems.
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Suppose the set M is a transitive model of ZFC set theory. The transitivity of M implies that the integers and integer sequences inside M are actually integers and sequences of integers. An integer sequence is a definable sequence relative to M if there exists some formula P(x) in the language of set theory, with one free variable and no parameters, which is true in M for that integer sequence and false in M for all other integer sequences. In each such M, there are definable integer sequences that are not computable, such as sequences that encode the Turing jumps of computable sets.
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The 21064 could issue: an integer operate with a floating-point operate, any load/store instruction with any operate instruction, an integer operate with an integer branch, or a floating-point operate with a floating-point branch. Two combinations were not permitted: an integer operate and a floating-point store, and a floating-point operate and an integer store. If one of the two instructions cannot be issued together, the first four stages are stalled until the remaining instruction is issued. The first four stages are also stalled in the event that no instruction can be issued due to resource unavailability, dependencies, or similar conditions.
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For instance, using a binary heap as a priority queue in selection sort leads to the heap sort algorithm, a comparison sorting algorithm that takes time. Instead, using selection sort with a bucket queue gives a form of pigeonhole sort, and using van Emde Boas trees or other integer priority queues leads to other fast integer sorting algorithms. Instead of using an integer priority queue in a sorting algorithm, it is possible to go the other direction, and use integer sorting algorithms as subroutines within an integer priority queue data structure. used this idea to show that, if it is possible to perform integer sorting in time per key, then the same time bound applies to the time per insertion or deletion operation in a priority queue data structure.
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If is an even integer, then . If is an odd integer, then . Because of this, powers of are useful for expressing alternating sequences. For a similar discussion of powers of the complex number , see .
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All particles of ordinary matter have integer-value charge numbers, with the exception of quarks, which cannot exist in isolation under ordinary circumstances (the strong force keeps them bound into hadrons of integer charge numbers).
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Alternate-bracket notations like or are rarely used for integer intervals.
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It is the only case where the output is an integer.
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The many applications of the algorithmic theory of Graver bases also include stochastic integer programming, N-fold integer programming, N-fold 4-block decomposable integer programming,Raymond Hemmecke, Matthias Köppe, Robert Weismantel: A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs, IPCO 14, 2010 clustering, and disclosure control in statistical databases. In some of these applications the relevant Graver basis cannot be computed in polynomial time, but can be accessed in an indirect way that allows to use it in polynomial time.
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Higher powers give correspondingly better rational approximations. This property stems from the fact that for each n, the sum of nth powers of an algebraic integer x and its conjugates is exactly an integer; this follows from an application of Newton's identities. When x is a Pisot number, the nth powers of the other conjugates tend to 0 as n tends to infinity. Since the sum is an integer, the distance from xn to the nearest integer tends to 0 at an exponential rate.
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All known elementary and composite particles are bosons or fermions, depending on their spin: Particles with half-integer spin are fermions; particles with integer spin are bosons. In the framework of nonrelativistic quantum mechanics, this is a purely empirical observation. In relativistic quantum field theory, the spin–statistics theorem shows that half-integer spin particles cannot be bosons and integer spin particles cannot be fermions. In large systems, the difference between bosonic and fermionic statistics is only apparent at large densities – when their wave functions overlap.
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Demonstration, with Cuisenaire rods, of the perfect power nature of 4, 8, and 9 In mathematics, a perfect power is a positive integer that can be resolved into equal factors, and whose root can be exactly extracted, i.e., a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = n. In this case, n may be called a perfect kth power.
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Prime decomposition of n = 864 as By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. (By convention, 1 is the empty product.) Testing whether the integer is prime can be done in polynomial time, for example, by the AKS primality test. If composite, however, the polynomial time tests give no insight into how to obtain the factors. Given a general algorithm for integer factorization, any integer can be factored into its constituent prime factors by repeated application of this algorithm.
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The default size of an AutoNumber is a 4-byte (long) integer. This is the size used for start+increment and random AutoNumbers. For replication ID AutoNumbers, the `FieldSize` property of the field is changed from long integer to `Replication ID`. If an AutoNumber is a long integer, the `NewValues` property determines whether it is of the start+increment or random form.
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Assume the square root of D is a rational number p/q, assume the q here is the smallest for which this is true, hence the smallest number for which q is also an integer. Then: :( − n)q = qD − nq is also an integer. But 0 < ( − n) < 1 so ( − n)q < q. Hence ( − n)q is an integer smaller than q.
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Integers are not closed under division. Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example, 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches: # Say that 26 cannot be divided by 11; division becomes a partial function.
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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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LINDO (Linear, Interactive, and Discrete Optimizer) is a software package for linear programming, integer programming, nonlinear programming, stochastic programming and global optimization.Linus E. Schrage, Linear, Integer, and Quadratic Programming with Lindo, Scientific Press, 1986, Lindo also creates "What'sBest!" which is an add-in for linear, integer and nonlinear optimization. First released for Lotus 1-2-3 and later also for Microsoft Excel.
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The case n = −2 also has an infinitude of solutions, and these have a geometric interpretation in terms of right triangles with integer sides and an integer altitude to the hypotenuse. All primitive solutions to a^{-2} + b^{-2} = d^{-2} are given by : a = (v^2 - u^2)(v^2 + u^2), : b = 2uv(v^2 + u^2), : d = 2uv(v^2 - u^2), for coprime integers u, v with v > u. The geometric interpretation is that a and b are the integer legs of a right triangle and d is the integer altitude to the hypotenuse. Then the hypotenuse itself is the integer : c = (v^2 + u^2)^2, so (a, b, c) is a Pythagorean triple.
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If the data to be hashed is small enough, one can use the data itself (reinterpreted as an integer) as the hashed value. The cost of computing this identity hash function is effectively zero. This hash function is perfect, as it maps each input to a distinct hash value. The meaning of "small enough" depends on the size of the type that is used as the hashed value. For example, in Java, the hash code is a 32-bit integer. Thus the 32-bit integer `Integer` and 32-bit floating-point `Float` objects can simply use the value directly; whereas the 64-bit integer `Long` and 64-bit floating-point `Double` cannot use this method.
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Hence this ideal norm is always a positive integer. When I is a principal ideal αOK then N(I) is equal to the absolute value of the norm to Q of α, for α an algebraic integer.
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Goniodiscaster integer is a species of sea stars in the family Oreasteridae.
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Each of these triples gives the integer sides of a right triangle.
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All observable subatomic particles have their electric charge an integer multiple of the elementary charge. The Standard Model's quarks have "non-integer" electric charges, namely, multiple of e, but quarks (and other combinations with non-integer electric charge) cannot be isolated due to color confinement. For baryons, mesons, and their antiparticles the constituent quarks' charges sum up to an integer multiple of e. Through the work of Albert Einstein, Satyendra Nath Bose, Louis de Broglie, and many others, current scientific theory holds that all particles also have a wave nature.
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A short proof of the irrationality of can be obtained from the rational root theorem, that is, if is a monic polynomial with integer coefficients, then any rational root of is necessarily an integer. Applying this to the polynomial , it follows that is either an integer or irrational. Because is not an integer (2 is not a perfect square), must therefore be irrational. This proof can be generalized to show that any square root of any natural number that is not the square of a natural number is irrational.
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Any triple of positive integers can serve as the side lengths of an integer triangle as long as it satisfies the triangle inequality: the longest side is shorter than the sum of the other two sides. Each such triple defines an integer triangle that is unique up to congruence. So the number of integer triangles (up to congruence) with perimeter p is the number of partitions of p into three positive parts that satisfy the triangle inequality. This is the integer closest to when p is even and to when p is odd.
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From the work of Hardy and Littlewood, the related quantity G(k) was studied with g(k). G(k) is defined to be the least positive integer s such that every sufficiently large integer (i.e. every integer greater than some constant) can be represented as a sum of at most s positive integers to the power of k. Clearly, G(1) = 1. Since squares are congruent to 0, 1, or 4 (mod 8), no integer congruent to 7 (mod 8) can be represented as a sum of three squares, implying that .
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This means that if \sigma is a permutation, then for any positive integer b, there exists exactly one positive integer a such that \sigma (a) = b. In particular, if x e y, then \sigma (x) e \sigma (y).
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The integer register file has nine read ports and four write ports. Four read ports supply operands to the two integer execution units (the branch unit was considered part of an integer unit). Another four read ports supply operands to the two address generators. Four ports are needed, rather than two, because of the base(register) + index(register) address style added in the MIPS IV ISA.
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Within the context of integers, addition of one also plays a special role: for any integer a, the integer is the least integer greater than a, also known as the successor of a.Hempel, C.G. (2001). The philosophy of Carl G. Hempel: studies in science, explanation, and rationality. p. 7 For instance, 3 is the successor of 2 and 7 is the successor of 6.
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Half-integers occur frequently enough in mathematics that a distinct term is convenient. Note that a halving an integer does not always produce a half- integer; this is only true for odd integers. For this reason, half-integers are also sometimes called half-odd-integers. Half-integers are a special case of the dyadic rationals (numbers produced by dividing an integer by a power of two)..
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The combined instructions were then decoded in the next pipeline stage. The back-end was also heavily modified. The number of reservation station entries for branch and integer instructions were reduced to six and ten, respectively. Both the integer and floating-point register files had registers added to them: the integer register file gained 32, and there were a total of 256 floating-point registers.
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To eliminate ambiguity, the terms wrapping overflow and saturating overflow can be used. The term underflow is most commonly used for floating-point math and not for integer math.Arithmetic underflow But, many references can be found to integer underflow. When the term integer underflow is used, it means the ideal result was closer to minus infinity than the output type's representable value closest to minus infinity.
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The design of SCIP is based on the notion of constraints. It supports about 20 constraint types for mixed-integer linear programming, mixed-integer nonlinear programming, mixed-integer all- quadratic programming and Pseudo-Boolean Pseudo-Boolean challenge 2009 Feb 11, 2011. optimization. It can also solve Steiner Trees and multi-objective optimization problems.A Generic Approach to Solving the Steiner Tree Problem and Variants Nov 9, 2015.
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146 is an octahedral number as well as a composite number. It is a nontotient since there is no integer with 146 coprimes below it, noncototient since there is no integer with 146 natural numbers below it which are not coprime to it, and an untouchable number since there is no integer whose proper divisors add up to 146. 146 is a repdigit in base 8 (222).
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Today, Diophantine analysis is the area of study where integer (whole-number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in Arithmetica lead to quadratic equations. Diophantus looked at 3 different types of quadratic equations: , , and .
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A positive integer n is idoneal if and only if it cannot be written as ab + bc + ac for distinct positive integer a, b, and c.Eric Rains, Comments on A000926, December 2007. It is sufficient to consider the set }; if all these numbers are of the form , , or 2s for some integer s, where is a prime, then is idoneal.Roberts, Joe: The Lure of the Integers.
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Consider the matrix with integer entries, such that the entries of its th column are the coefficients of the th generator of the kernel. Then, the abelian group is isomorphic to the cokernel of linear map defined by . Conversely every integer matrix defines a finitely generated abelian group. It follows that the study of finitely generated abelian groups is totally equivalent with the study of integer matrices.
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An alternative way of stating the theorem is that a non- collinear set of points in the plane with integer distances can only be extended by adding finitely many additional points, before no more points can be added. A set of points with both integer coordinates and integer distances, to which no more can be added while preserving both properties, forms an Erdős–Diophantine graph.
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The output of the matched filter will have a maximum peak value if its coefficients are modulated by the carrier with the correct integer CFO. It is possible to deploy one such matched filter for each possible integer CFO. In this case, seven matched filters are needed. However, we can use only one set of matched filter hardware that handles different integer CFOs sequentially.
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Some computer languages define integer sizes in a machine-independent way; others have varying definitions depending on the underlying processor word size. Not all language implementations define variables of all integer sizes, and defined sizes may not even be distinct in a particular implementation. An integer in one programming language may be a different size in a different language or on a different processor.
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Each algebraic integer belongs to the ring of integers of some number field. A number is an algebraic integer if and only if the ring is finitely generated as an Abelian group, which is to say, as a -module.
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Support may also extend to other operations, such as formatting, rounding to an integer or floating point value, etc.. As in mathematics, those languages often interpret an integer value as equivalent to a rational value with a unit denominator.
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In Pascal, a similar end is performed by declaring a subrange of integer (a compiler may then choose to allocate a smaller amount of storage for the declared variable): type a = 1..100; b = -20..20; c = 0..100000; This subrange feature is not supported by C. A major, if subtle, difference between C and Pascal is how they promote integer operations. In Pascal, the result of an operation is defined for all integer/subrange types, even if intermediate results do not fit into an integer. The result is undefined only if it does not fit into the integer/subrange on the left hand side of the assignment. This may imply an artificial restriction on the range of integer types, or may require slow execution to handle the intermediate results: However, the compiler may take advantage of restricted subranges to produce more efficient code.
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For some transitive models M of ZFC, every sequence of integers in M is definable relative to M; for others, only some integer sequences are (Hamkins et al. 2013). There is no systematic way to define in M itself the set of sequences definable relative to M and that set may not even exist in some such M. Similarly, the map from the set of formulas that define integer sequences in M to the integer sequences they define is not definable in M and may not exist in M. However, in any model that does possess such a definability map, some integer sequences in the model will not be definable relative to the model (Hamkins et al. 2013). If M contains all integer sequences, then the set of integer sequences definable in M will exist in M and be countable and countable in M.
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The Pentium Pro has a total of six execution units: two integer units, one floating-point unit (FPU), a load unit, store address unit, and a store data unit. One of the integer units shares the same ports as the FPU, and therefore the Pentium Pro can only dispatch one integer micro-op and one floating-point micro-op, or two integer micro-ops per a cycle, in addition to micro-ops for the other three execution units. Of the two integer units, only the one that shares the path with the FPU on port 0 has the full complement of functions such as a barrel shifter, multiplier, divider, and support for LEA instructions. The second integer unit, which is connected to port 1, does not have these facilities and is limited to simple operations such as add, subtract, and the calculation of branch target addresses.
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This construction extends the usual algorithm to possibly non-integer values of β.
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These are of the form K = Q(), for a square-free integer d.
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Computations become harder, since integer factorization becomes harder as the numbers get larger.
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Execution begins during stage nine. There are six execution units, two for integer, two for loads and stores, and two for floating-point."Fujitsu's SPARC V Is Real Deal", p. 2. The two integer execution units are designated EXA and EXB.
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Integer-valued functions are ubiquitous in graph theory. They also have similar uses in geometric group theory, where length function represents the concept of norm, and word metric represents the concept of metric. Integer-valued polynomials are important in ring theory.
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Below User RPL is System RPL (SysRPL). Most System RPL commands lack argument checking and are defined only for specific argument types (e.g. short integer vs. long integer), making System RPL programs run dramatically faster than equivalent User RPL ones.
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Here, λ(k) = (−1)Ω(k) is positive if the number of prime factors of the integer k is even, and is negative if it is odd. The big Omega function counts the total number of prime factors of an integer.
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"Light Changed to Matter, Then Stopped and Moved". photonics.com (February 2007). Retrieved on 2015-05-11. The modern view on this is that photons are, by virtue of their integer spin, bosons (as opposed to fermions with half-integer spin).
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In applied mathematics, branch and price is a method of combinatorial optimization for solving integer linear programming (ILP) and mixed integer linear programming (MILP) problems with many variables. The method is a hybrid of branch and bound and column generation methods.
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These two cases are described as integer spin and half-integer spin, respectively. The representations with odd, positive values of m are faithful representations of SU(2), while the representations of SU(2) with non- negative, even m are not faithful.
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In the 1970s Holland developed a compositional technique known as 'Integer Music' that features integer groups in which an integer represents a group of musical tones, each operating independently of other groups. The integer groups generally range from 1 thru 26 tones, which conveniently correlates with the English alphabet, making it possible to construct musical scores that are based on specific non-musical ideas. The player is asked to provide specific tones within each group, while the composer creates the form and structure of the work, typically supplying remarks as to how each group is to be expressed. Many of the composer's scores, including music for solo instruments (with and without signal processing), chamber music, and music for voice, involve the use of integer notation.
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The original Apple II included BASIC interpreter in ROM known originally as Apple BASIC and later as Integer BASIC. Variables in this language can only handle integer numbers ranging from −32,768 to +32,767 (16-bit binary values); floating point numbers are not supported. Apple commissioned Microsoft to develop Applesoft BASIC, capable of handling floating-point numbers. Applesoft BASIC cannot run Integer BASIC programs, causing some users to resist upgrading to it.
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R8000 die photo The R8000 controlled the chip set and executed integer instructions. It contained the integer execution units, integer register file, primary caches and hardware for instruction fetch, branch prediction the translation lookaside buffers (TLBs). In stage one, four instructions are fetched from the instruction cache. The instruction cache is 16 kB large, direct-mapped, virtually tagged and virtually indexed, and has a 32-byte line size.
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This deduction, however, is not very commonly seen in Hashiwokakero puzzles. Determining whether a Hashiwokakero puzzle has a solution is NP-complete, by a reduction from finding Hamiltonian cycles in integer-coordinate unit distance graphs.. There is a solution using integer linear programming in the MathProg examples included in GLPK.. A library of puzzles counting up to 400 islands as well as integer linear programming results are also reported..
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Two to the power of , written as , is the number of ways the bits in a binary word of length can be arranged. A word, interpreted as an unsigned integer, can represent values from 0 () to () inclusively. Corresponding signed integer values can be positive, negative and zero; see signed number representations. Either way, one less than a power of two is often the upper bound of an integer in binary computers.
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The decimal numeral system (also called base-ten positional numeral system, and occasionally called denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system.The History of Arithmetic, Louis Charles Karpinski, 200 pp, Rand McNally & Company, 1925. The way of denoting numbers in the decimal system is often referred to as decimal notation.
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The integer unit consisted of two integer pipelines and the integer register file. The two pipelines, the add pipeline and the multiply pipeline are not identical, each are responsible for executing different instructions, although both are capable of executing common add, logical, load, compare, and conditional move instructions. The multiply pipeline exclusively executes shift, store, and multiply instructions (in a non-pipelined multiplier). The add pipeline exclusively executes branch instructions.
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Let S(x) be the number of non-integer hyperharmonic numbers such that (n,x)\in[0,x]\times[0,x]. Then, assuming the Cramér's conjecture, : S(x)=x^2+O(x\log^3x). Note that the number of integer lattice points in [0,x]\times[0,x] is x^2+O(x^2), which shows that most of the hyperharmonic numbers cannot be integer. The conjecture, however, is still open.
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The holomorph of a polycyclic group is also such a group of integer matrices.
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The exponentiation operation with integer exponents may be defined directly from elementary arithmetic operations.
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A closed trajectory is called a harmonic orbit if k is an integer, i.e.
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Gynoeryx integer is a moth of the family Sphingidae. It is known from Madagascar.
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235 (two hundred [and] thirty-five) is the integer following 234 and preceding 236.
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The Journal of Integer Sequences is a peer-reviewed open-access academic journal in mathematics, specializing in research papers about integer sequences. It was founded in 1998 by Neil Sloane. Sloane had previously published two books on integer sequences, and in 1996 he founded the On-Line Encyclopedia of Integer Sequences (OEIS). Needing an outlet for research papers concerning the sequences he was collecting in OEIS, he founded the journal... Since 2002 the journal has been hosted by the David R. Cheriton School of Computer Science at the University of Waterloo, with Waterloo professor Jeffrey Shallit as its editor-in-chief.
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The original Unix time datatype (`time_t`) stores a date and time as a signed long integer (on 32-bit systems a 32-bit integer) representing the number of seconds since 1 January 1970. During and after 2038, this number will exceed 2 − 1, the largest number representable by a signed long integer on 32-bit systems, causing the Year 2038 problem (also known as the Unix Millennium bug or Y2K38). As a long integer in 64-bit systems uses 64 bits, the problem does not realistically exist on 64-bit systems that use the LP64 model.
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A Heronian triangle is commonly defined as one with integer sides whose area is also an integer, and we shall consider Heronian triangles with distinct integer sides. The lengths of the sides of such a triangle form a Heronian triple (a, b, c) provided a < b < c. Clearly, any Pythagorean triple is a Heronian triple, since in a Pythagorean triple at least one of the legs a, b must be even, so that the area ab/2 is an integer. Not every Heronian triple is a Pythagorean triple, however, as the example (4, 13, 15) with area 24 shows.
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Only 8% of atoms are in the ground state near absolute zero, rather than the 100% of a true condensate. The bosonic behavior of some of these alkaline gases appears odd at first sight, because their nuclei have half-integer total spin. It arises from a subtle interplay of electronic and nuclear spins: at ultra-low temperatures and corresponding excitation energies, the half-integer total spin of the electronic shell and half-integer total spin of the nucleus are coupled by a very weak hyperfine interaction. The total spin of the atom, arising from this coupling, is an integer lower value.
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Several authors have studied algorithms for constructing orthogonal convex hulls: ; ; ; . By the results of these authors, the orthogonal convex hull of points in the plane may be constructed in time , or possibly faster using integer searching data structures for points with integer coordinates.
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There exist polynomial- time randomized reductions from the integer factorization problem to WEAK- PIGEON. Additionally, under the generalized Riemann hypothesis, there also exist deterministic polynomial reductions. However, it is still an open problem to unconditionally show that integer factorization is in PPP.
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Bessel functions for integer are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer are obtained when the Helmholtz equation is solved in spherical coordinates.
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More generally, Hasse principle allows deciding whether a homogeneous Diophantine equation of degree two has an integer solution, and computing a solution if there exist. If a non-trivial integer solution is known, one may produce all other solutions in the following way.
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SPECint is a computer benchmark specification for CPU integer processing power. It is maintained by the Standard Performance Evaluation Corporation (SPEC). SPECint is the integer performance testing component of the SPEC test suite. The first SPEC test suite, CPU92, was announced in 1992.
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Concretely, transform the integer as `(n << 1) ^ (n >> k - 1)` for fixed k-bit integers.
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Integer values are returned in eax, pointers in edx and floating point types in st0.
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In computer programming many functions return values of integer type due to simplicity of implementation.
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An improper fraction is therefore equal either to an integer or to a mixed number.
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In December 2014, it was reported that PSY's music video "Gangnam Style" had exceeded the 32-bit integer limit for YouTube view count, necessitating YouTube to upgrade the counter to a 64-bit integer. In fact, this was an "easter egg" joke shared by the company, as Google had reportedly switched to a 64-bit integer several months prior. The data type time_t, used on operating systems such as Unix, is a signed integer counting the number of seconds since the start of the Unix epoch (midnight UTC of 1 January 1970), and is often implemented as a 32-bit integer. The latest time that can be represented in this form is 03:14:07 UTC on Tuesday, 19 January 2038 (corresponding to 2,147,483,647 seconds since the start of the epoch).
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Even and odd numbers: An integer is even if it is a multiple of two, and is odd otherwise. Prime number: An integer with exactly two positive divisors: itself and 1. The primes form an infinite sequence 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... Composite number: A number that can be factored into a product of smaller integers. Every integer greater than one is either prime or composite.
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A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite. The article is a table of Gaussian Integers followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime. The factorizations take the form of an optional unit multiplied by integer powers of Gaussian primes. Note that there are rational primes which are not Gaussian primes.
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In mathematics, a square number or perfect square is an integer that is the square of an integer;Some authors also call squares of rational numbers perfect squares. in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared".
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In defining the discrete Zak transform, the input function is a function of an integer variable. So, let f(n) be a function of an integer variable n (n taking all positive, zero and negative integers as values). The discrete Zak transform of f(n) is a function of two real variables one of which is the integer variable n. The other variable is a real variable which may be denoted by w.
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De Moivre's formula does not hold for non-integer powers. The derivation of de Moivre's formula above involves a complex number raised to the integer power . If a complex number is raised to a non-integer power, the result is multiple-valued (see failure of power and logarithm identities). For example, when , de Moivre's formula gives the following results: :for the formula gives 1 = 1, and :for the formula gives 1 = −1.
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Autoboxing is the term for getting a reference type out of a value type just through type conversion (either implicit or explicit). The compiler automatically supplies the extra source code that creates the object. For example, in versions of Java prior to J2SE 5.0, the following code did not compile: Integer i = new Integer(9); Integer i = 9; // error in versions prior to 5.0! Compilers prior to 5.0 would not accept the last line.
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A tied score, that is, the players have an equal number of points at the conclusion of the game. With a half integer komi this situation is impossible, but it could happen if the players followed the handicap rules with no komi or integer komi.
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The integer broom space, together with the integer broom topology, is a compact topological space. It is a so-called Kolmogorov space, but it is neither a Fréchet space nor a Hausdorff space. The space is path connected, while neither locally connected nor arc connected.
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LEB128 or Little Endian Base 128 is a form of variable-length code compression used to store an arbitrarily large integer in a small number of bytes. LEB128 is used in the DWARF debug file format and the WebAssembly binary encoding for all integer literals.
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Fermions have half-integer spin while bosons have integer spin. All the particles of the Standard Model have been experimentally observed, recently including the Higgs boson in 2012. Many other hypothetical elementary particles, such as the graviton, have been proposed, but not observed experimentally.
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Even more challenging than their NLP counterparts, deterministically solving an MINLP problem can be very difficult. Techniques such as integer cuts, or branching a problem on its integer variables (hence creating NLP sub-problems which can in turn be solved deterministically), are commonly used.
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The conditions are equality, inequality, less than or equal to, and less than. With the exception of the instructions that specify the former two conditions, there are versions that perform signed and unsigned compares. The integer arithmetic instructions use the integer operate instruction formats.
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Each repeating decimal number satisfies a linear equation with integer coefficients, and its unique solution is a rational number. To illustrate the latter point, the number above satisfies the equation , whose solution is . The process of how to find these integer coefficients is described below.
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Alternative ways to encode character values include specifying an integer value for a code point, such as an ASCII code value or a Unicode code point. This may be done directly via converting an integer literal to a character, or via an escape sequence.
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John Younger calculated the number of Mycenaean wet and dry units (96.1 L and 28.8 L, resp.) in the capacities of 300 pots of known dimensions, including several storage pithoi. He attempted to assign the volumes to integer or half-integer multiples of these units.
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0 is the integer immediately preceding 1. Zero is an even numberLemma B.2.2, The integer 0 is even and is not odd, in because it is divisible by 2 with no remainder. 0 is neither positive nor negative. Many definitions, Extract of pp.
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The longest instructions, divide or square root, took 5.3 milliseconds (ms) for floating point. Integer adds took about 200 μS, but other operations were handled in subroutines as opposed to hardware and took much longer; an integer division/square root required 8.2 ms for instance.
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In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Integer matrices find frequent application in combinatorics.
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For elliptic curves, potential good reduction is equivalent to the j-invariant being an algebraic integer.
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385 = 5 × 7 × 11, sphenic number, square pyramidal number, the number of integer partitions of 18.
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Janus integer, the currant stem girdler, is a species of stem sawfly in the family Cephidae.
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Production planning by mixed integer programming. Springer, 2006. for production, public transport and health care planning.
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For example, is an integer of 19,729 decimal digits (equivalent to 265536−3, or 22222−3).
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In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms.
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For a state of half-integer spin the argument can be circumvented by having fermionic statistics.
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In mathematics, an ergodic sequence is a certain type of integer sequence, having certain equidistribution properties.
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Demonstration, with Cuisenaire rods, of the first four: 1, 2, 4, 6 __FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan (1915). However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato, who set 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it.. Kahane cites Plato's Laws, 771c. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer.
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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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For example, an intensional definition of square number can be "any number that can be expressed as some integer multiplied by itself". The rule—"take an integer and multiply it by itself"—always generates members of the set of square numbers, no matter which integer one chooses, and for any square number, there is an integer that was multiplied by itself to get it. Similarly, an intensional definition of a game, such as chess, would be the rules of the game; any game played by those rules must be a game of chess, and any game properly called a game of chess must have been played by those rules.
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Integer-valued polynomials may be used effectively to solve questions about fixed divisors of polynomials. For example, the polynomials P with integer coefficients that always take on even number values are just those such that P/2 is integer valued. Those in turn are the polynomials that may be expressed as a linear combination with even integer coefficients of the binomial coefficients. In questions of prime number theory, such as Schinzel's hypothesis H and the Bateman–Horn conjecture, it is a matter of basic importance to understand the case when P has no fixed prime divisor (this has been called Bunyakovsky's property, after Viktor Bunyakovsky).
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This is also the probability of that integer winning the game, if all the players are following the distribution to pick their integer. A striking feature is the existence of a maximum integer, above which the strategy recommends a weighting of zero. In the case here p(n)=0 for n>31. Above this number, the chance that the game has not been won by a lower integer falls below 1/(N+1), so that even if one knew with certainty that one would be the only player with such a number, the chance of winning would still be less than one could achieve with a lower number.
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The register files are read during stage seven. The SPARC architecture has separate register files for integer and floating-point instructions. The integer register file has eight register windows. The JWR (Joint Work Register) contains 64 entries and has eight read ports and two write ports.
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Every real number can be essentially uniquely represented as a continued fraction, namely as the sum of its integer part and the reciprocal of its fractional part which is written as the sum of its integer part and the reciprocal of its fractional part, and so on.
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Mathematically, the problem can be formulated as follows: : Given an integer m and a set V of positive integers, find the smallest integer z that cannot be written as the sum v1 \+ v2 \+ ··· + vk of some number k ≤ m of (not necessarily distinct) elements of V.
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The floor function on real numbers. Its discontinuities are pictured with white discs outlines with blue circles. In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member of its domain.
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Most integer quantum Hall experiments are now performed on gallium arsenide heterostructures, although many other semiconductor materials can be used. In 2007, the integer quantum Hall effect was reported in graphene at temperatures as high as room temperature, and in the magnesium zinc oxide ZnO–MgxZn1−xO.
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The TSP can be formulated as an integer linear program., pp.308-309.Tucker, A. W. (1960), "On Directed Graphs and Integer Programs", IBM Mathematical research Project (Princeton University)Dantzig, George B. (1963), Linear Programming and Extensions, Princeton, NJ: PrincetonUP, pp. 545–7, , sixth printing, 1974.
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Zeckendorf's theorem has two parts: #Existence: every positive integer has a Zeckendorf representation. #Uniqueness: no positive integer has two different Zeckendorf representations. The first part of Zeckendorf's theorem (existence) can be proven by induction. For it is clearly true (as these are Fibonacci numbers), for we have .
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The story behind a formula for Pi, groups.google.com; accessed June 23, 2003 He co-authored The Encyclopedia of Integer Sequences, made into the web site On-Line Encyclopedia of Integer Sequences dedicated to integer sequences later in 1995. In 1975, Plouffe broke the world record for memorizing digits of π by reciting 4096 digits, a record which stood until 1977. His "Inverter" is a web site that contains 11.3 billion mathematical constants as of August 15, 2017.
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As long as the parameter is a positive integer, the Lanczos kernel is continuous everywhere, and its derivative is defined and continuous everywhere (even at = , where both sinc functions go to zero). Therefore, the reconstructed signal too will be continuous, with continuous derivative. The Lanczos kernel is zero at every integer argument , except at = 0, where it has value 1. Therefore, the reconstructed signal exactly interpolates the given samples: we will have = for every integer argument = .
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The integer register file contained 136 registers, providing eight register windows, a feature defined in the SPARC ISA. It had two read ports. The integer unit had a four-stage pipeline, of which two stages were added so the pipeline would be equal to all non-floating-point pipelines. Integer multiply and divide, instructions added in the V8 version of the SPARC architecture, had an 18- and 37-cycle latency, respectively, and stalled the pipeline until they were completed.
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In computing, an integer version of this sequence is often used to generate a discrete uniform distribution rather than a continuous one. Instead of using an irrational number, which cannot be calculated on a digital computer, the ratio of two integers is used in its place. An integer k is chosen, relatively prime to an integer modulus m. In the common case that m is a power of 2, this amounts to requiring that k is odd.
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By the law of cosines, every angle of an integer triangle has a rational cosine. If the angles of any triangle form an arithmetic progression then one of its angles must be 60°. For integer triangles the remaining angles must also have rational cosines and a method of generating such triangles is given below. However, apart from the trivial case of an equilateral triangle there are no integer triangles whose angles form either a geometric or harmonic progression.
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A set of incommensurate frequencies is required to perform the transform and most frequencies are irrational. To facilitate computation a set of integer frequencies is selected instead of the irrational frequencies. The integer frequencies are not strictly incommensurate, resulting in an error between the multi-dimensional integral and the transformed one-dimensional integral. However, the integer frequencies can be selected to be incommensurate to any order so that the error can be controlled meeting any precision requirement in theory.
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69 (1962), No. 5, 360–365. However, to provide a more complete geometric interpretation, the integer values can be allowed to be negative and zero (thus allowing Pythagorean triples to be included) with the only condition being that . In this setting, a Pythagorean quadruple defines a cuboid with integer side lengths , , and , whose space diagonal has integer length ; with this interpretation, Pythagorean quadruples are thus also called Pythagorean boxes.R. A. Beauregard and E. R. Suryanarayan, Pythagorean boxes, Math.
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In January 2017 the number 1,999,291,987,030,606,810 was published in OEIS as A281509, and described as "The Largest Known Most Delayed Palindrome", with a delay of 261. Several smaller 261-delay palindromes were published separately as A281508. Every positive integer can be written as the sum of three palindromic numbers in every number system with base 5 or greater. The continued fraction of + ⌊⌋ is a repeating palindrome when n is an integer, where ⌊x⌋ denotes the integer part of x.
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As a more practical example, a language might allow integer values to be used wherever floating point values are expected (`Integer` <: `Float`), or it might define a generic type Number as a common supertype of integers and the reals. In this second case, we only have `Integer` <: `Number` and `Float` <: `Number`, but `Integer` and `Float` are not subtypes of each other. Programmers may take advantage of subtyping to write code in a more abstract manner than would be possible without it. Consider the following example: function max (x as Number, y as Number) is if x < y then return y else return x end If integer and real are both subtypes of `Number`, and an operator of comparison with an arbitrary Number is defined for both types, then values of either type can be passed to this function.
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KENNETH L. CLARKSON. A Las Vegas algorithm for linear and integer programming when the dimension is small.
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If some decision variables are not discrete the problem is known as a mixed-integer programming problem.
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Let n be the number of processes. Each process is identified by an integer in 1, ..., n.
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A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n. Examples are 1/1, 1/2, 1/3, 1/4 ,1/5, etc.
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This piece of code outputs Floyd's Triangle to the console: Imports System.Console Module Program Sub Main() Dim rows As Integer ' Input validation. Do Until Integer.TryParse(ReadLine("Enter a value for how many rows to be displayed: " & vbcrlf), rows) AndAlso rows >= 1 WriteLine("Allowed range is 1 and {0}", Integer.
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The integer DCT is used in Advanced Video Coding (AVC), introduced in 2003, and High Efficiency Video Coding (HEVC), introduced in 2013. The integer DCT is also used in the High Efficiency Image Format (HEIF), which uses a subset of the HEVC video coding format for coding still images.
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Fermat's Last Theorem considers solutions to the Fermat equation: with positive integers , , and and an integer greater than 2. There are several generalizations of the Fermat equation to more general equations that allow the exponent to be a negative integer or rational, or to consider three different exponents.
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Level of Im(f) = 0 is shown with a thick green line. Negative integer values of Im(f) are shown with thick red lines. Positive integer values of Im(f) are shown with thick blue lines. Intermediate levels of Im(f) = constant are shown with thin green lines.
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Because 2 is a prime number, it must also divide p, by Euclid's lemma. So p = 2r, for some integer r. But then, :2q^2 = (2r)^2 = 4r^2, :q^2 = 2r^2, which shows that 2 must divide q as well. So q = 2s for some integer s.
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If `11111111` represents two's complement signed integer −1 (`ADD al,-1`), then the interpretation of the result is `11111110` because `Overflow_Flag` is clear, and `Carry_Flag` is ignored. The sign of the result is negative, because `Sign_Flag` is set. `11111110` is the two's complement form of signed integer −2.
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Pascal structures programs into procedures and functions. : program Printing; var i : integer; procedure Print(j : integer); begin ... end; begin { main program } ... Print(i); end. Procedures and functions can be nested to any depth, and the 'program' construct is the logical outermost block. By default, parameters are passed by value.
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Paul Erdős gave a proof that also relies on the fundamental theorem of arithmetic. Every positive integer has a unique factorization into a square-free number and a square number . For example, . Let be a positive integer, and let be the number of primes less than or equal to .
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The next back is the program version number. The next is the > program serial number (a converted integer). The first word of the trailer > is currently undefined. After the initial integer word count, the rest of > the file consists entirely of 4-byte (32-bit) floating-point words.
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The TurboSPARC had an integer unit and a floating-point unit. Most integer arithmetic instructions except for multiply and divide have a single-cycle latency. Multiply and divide was executed by the FPU. Multiply had a seven cycle latency while divide had an 8- to 33-cycle latency.
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There is another type of statistic, known as braid statistics, which are associated with particles known as plektons. The spin-statistics theorem relates the exchange symmetry of identical particles to their spin. It states that bosons have integer spin, and fermions have half-integer spin. Anyons possess fractional spin.
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One of these is the integer DCT (IntDCT), an integer approximation of the standard DCT, used in several ISO/IEC and ITU-T international standards. DCT compression, also known as block compression, compresses data in sets of discrete DCT blocks. DCT blocks can have a number of sizes, including 8x8 pixels for the standard DCT, and varied integer DCT sizes between 4x4 and 32x32 pixels. The DCT has a strong "energy compaction" property, capable of achieving high quality at high data compression ratios.
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Geometrically, this solution will be a vertex of the convex polytope consisting of all feasible points. If this vertex is not an integer point then the method finds a hyperplane with the vertex on one side and all feasible integer points on the other. This is then added as an additional linear constraint to exclude the vertex found, creating a modified linear program. The new program is then solved and the process is repeated until an integer solution is found.
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However, the number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2). Clearly, an integer cannot be both odd and even at the same time: we have a contradiction. The only assumption we made was that log2 3 is rational (and so expressible as a quotient of integers m/n with n ≠ 0).
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A Diophantine equation is a polynomial equation in two or more unknowns for which only the integer solutions are sought (an integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. An example of linear Diophantine equation is where a, b, and c are constants. An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns.
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Die shot of PA-7200 The PA-7200 was mostly derived from the PA-7100 and was improved adding a second integer unit, enabling it to issue up to two integer instructions per cycle. The second integer unit was not identical to the first, and was only capable of executing simple but more frequently used instructions. The PA-7200 contained 1.3 million transistors and measured 14.0 mm by 15.0 mm (210 mm2)."Design of the HP PA 7200 CPU", p. 2.
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An integer is square- free if and only if q_i=1 for all . An integer greater than one is the th power of another integer if and only if is a divisor of all such that q_i eq 1. The use of the square-free factorization of integers is limited by the fact that its computation is as difficult as the computation of the prime factorization. More precisely every known algorithm for computing a square- free factorization computes also the prime factorization.
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RISC-V segregates math into a minimal set of integer instructions (set I) with add, subtract, shift, bit-wise logic and comparing-branches. These can simulate most of the other RISC-V instruction sets with software. (The atomic instructions are a notable exception.) RISC-V currently lacks the count leading zero and bit-field operations normally used to speed software floating-point in a pure-integer processor. The integer multiplication instructions (set M) includes signed and unsigned multiply and divide.
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The Pythagorean prime 5 and its square root are both hypotenuses of right triangles with integer legs. The formulas show how to transform any right triangle with integer legs into another right triangle with integer legs whose hypotenuse is the square of the first triangle's hypotenuse. A Pythagorean prime is a prime number of the form 4n + 1\. Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares.
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In some PLLs the reference frequency may also be divided by an integer multiple before being input to the phase comparator. This allows the synthesis of frequencies that are N/M times the reference frequency. This can be accomplished in a different manner by periodically changing the integer value of an integer-N frequency divider, effectively resulting in a multiplier with both whole number and fractional component. Such a multiplier is called a fractional-N synthesizer after its fractional component.
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Any integer can be expressed as the rational number , which is its canonical form as a rational number.
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Vertices of this honeycomb can be positioned in 4-space in all integer coordinates (i,j,k,l).
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There is a randomized polynomial-time reduction from the problem of Integer factorization to problems complete for PPA.
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The game becomes unplayable at the 256th level due to an integer overflow that affects the game's memory.
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While specialising in statistics undertaking a Master of Arts degree, she developed a technique for integer linear programming.
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The slow electrons need one electric field oscillation, the faster electrons need an integer multiple of this oscillation.
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Integer programming can globally optimize a scoring function with pairwise contact potential and produce a global optimal alignment.
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279 = 32·31 Every positive integer is the sum of at most 279 eighth powers. See Waring's problem.
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The output signal then contains frequencies such as f1+kf2 and f1−kf2 where k is an integer.
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For example, see Goemans' and Williamson's semidefinite programming-based Max-Cut approximation algorithm.) The challenge in the first step is to choose a suitable integer linear program. Familiarity with linear programming is required, in particular, familiarity with how to model problems using linear programs and integer linear programs. But, for many problems, there is a natural integer linear program that works well, such as in the Set Cover example below. (The integer linear program should have a small integrality gap; indeed randomized rounding is often used to prove bounds on integrality gaps.) In the second step, the optimal fractional solution can typically be computed in polynomial time using any standard linear programming algorithm.
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In MS BASIC, for a statement such as , the line number and would be tokenized, while the was left in its original format and had to be re-parsed into a 16-bit integer every time the line was encountered. In contrast, Integer BASIC would convert the line entirely into tokens that could be immediately read and performed. Where Microsoft BASIC had one token for the keyword , Integer BASIC had three tokens: one if the keyword was followed by no arguments, one if followed by an arithmetic expression, and one if followed by a string literal. Since Integer BASIC processed more of the original source code into tokens, the runtime was faster than versions that required additional runtime parsing.
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Combining restrictions that a shape be equable and that its dimensions be integers is significantly more restrictive than either restriction on its own. For instance, there are infinitely many Pythagorean triples describing integer-sided right triangles, and there are infinitely many equable right triangles with non-integer sides; however, there are only two equable integer right triangles, with side lengths (5,12,13) and (6,8,10). More generally, the problem of finding all equable triangles with integer sides (that is, equable Heronian triangles) was considered by B. Yates in 1858. As W. A. Whitworth and D. Biddle proved in 1904, there are exactly three solutions, beyond the right triangles already listed, with sides (6,25,29), (7,15,20), and (9,10,17).
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The spin–statistics theorem splits particles into two groups: bosons and fermions, where bosons obey Bose-Einstein statistics and fermions obey Fermi-Dirac statistics (and therefore the Pauli Exclusion Principle). Specifically, the theory states that particles with an integer spin are bosons while all other particles have half-integer spins and are fermions. As an example, electrons have half-integer spin and are fermions that obey the Pauli exclusion principle, while photons have integer spin and do not. The theorem relies on both quantum mechanics and the theory of special relativity, and this connection between spin and statistics has been called "one of the most important applications of the special relativity theory".
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Integer relation algorithms have numerous applications. The first application is to determine whether a given real number x is likely to be algebraic, by searching for an integer relation between a set of powers of x {1, x, x2, ..., xn}. The second application is to search for an integer relation between a real number x and a set of mathematical constants such as e, π and ln(2), which will lead to an expression for x as a linear combination of these constants. A typical approach in experimental mathematics is to use numerical methods and arbitrary precision arithmetic to find an approximate value for an infinite series, infinite product or an integral to a high degree of precision (usually at least 100 significant figures), and then use an integer relation algorithm to search for an integer relation between this value and a set of mathematical constants.
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Most file systems that support hard links use reference counting. An integer value is stored with each physical data section. This integer represents the total number of hard links that have been created to point to the data. When a new link is created, this value is increased by one.
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NEC VR5000 die shot. The R5000 is a two-way superscalar design that executes instructions in-order. The R5000 could simultaneously issue an integer and a floating-point instruction. It had one simple pipeline for integer instructions and another for floating-point to save transistors and die area to reduce cost.
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The design of floating-point format allows various optimisations, resulting from the easy generation of a base-2 logarithm approximation from an integer view of the raw bit pattern. Integer arithmetic and bit-shifting can yield an approximation to reciprocal square root (fast inverse square root), commonly required in computer graphics.
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In mathematics, the Mian–Chowla sequence is an integer sequence defined recursively in the following way. The sequence starts with :a_1 = 1. Then for n>1, a_n is the smallest integer such that every pairwise sum :a_i + a_j is distinct, for all i and j less than or equal to n.
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Often, categorical and ordinal data are grouped together; likewise for integer-valued and real-valued data. Furthermore, many algorithms work only in terms of categorical data and require that real-valued or integer-valued data be discretized into groups (e.g., less than 5, between 5 and 10, or greater than 10).
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Two types of labels can be put in a switch statement. A case label consists of the keyword `case`, followed by an expression that evaluates to integer constant. A default label consists of the keyword `default`. Case labels are used to associate an integer value with a statement in the code.
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In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are :1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 .
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In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ħ, all particles that move in 3 dimensions have either integer spin or half-integer spin.
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So, a well-designed open instruction set designed using well-established principles should attract long-term support by many vendors. RISC-V also supports the designers' academic uses. The simplicity of the integer subset permits basic student exercises. The integer subset is a simple ISA enabling software to control research machines.
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Load and store instructions are executed by the integer pipeline, and access the on-chip 8 KB data cache.
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From it further follows that is a positive integer. Finally, implies that and thus , which contradicts the minimality of .
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Conotrachelus integer is a species of true weevil in the beetle family Curculionidae. It is found in North America.
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In this case the notation eventually resolves to being the leftmost number raised to some (usually enormous) integer power.
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Prionus integer is a species of long-horned beetle in the family Cerambycidae. It is found in North America.
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Conversely, Integer BASIC was now removed from ROM and turned into an executable file on the DOS 3.3 disk.
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This defines an integer, as divides both and . Otherwise, the proof is very similar to that for coprime moduli.
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Fermat proved that the only integer solutions of m^2+2=n^3 are n=3,m=\pm 5.
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True nHD frames on the other hand has integer scale factors, for example Nokia 808 PureView with nHD display.
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The first moment method is a simple application of Markov's inequality for integer-valued variables. For a non-negative, integer-valued random variable X, we may want to prove that X = 0 with high probability. To obtain an upper bound for P(X > 0), and thus a lower bound for P(X = 0), we first note that since X takes only integer values, P(X > 0) = P(X ≥ 1). Since X is non-negative we can now apply Markov's inequality to obtain P(X ≥ 1) ≤ E[X].
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Lee is the author of A First Course in Combinatorial Optimization (Cambridge University Press, 2004) and A First Course in Linear Optimization (Reex Press, 2013). He is co-editor of: Trends in Optimization (American Mathematical Society, 2004), Mixed Integer Nonlinear Programming (Springer, 2012), Integer Programming and Combinatorial Optimization (Lecture Notes in Computer Science, Vol. 8494; Springer, 2014), Special Issue: Integer Programming and Combinatorial Optimization, 2014 (Mathematical Programming, Series B. Issue 1-2, December 2015), and Combinatorial Optimization (Lecture Notes in Computer Science, Vol. 10856; Springer, 2018).
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For instance: INTEGER PROCEDURE FACT(N); VALUE N; INTEGER N; Declares a function FACT that takes a value N that is an integer. The indicates that this variable is also the return value for the procedure. Although frowned upon, ALGOL and Pascal allowed code to be labeled using a leading name ending with a colon, which could then be used for the target of loops and statements. One minor difference is that SPL required the label names to be declared in the variable section using the keyword.
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Investigating pairs of distinct integer-valued quadratic forms that represent every integer the same number of times, Schiemann found that such quadratic forms must be in four or more variables, and the least possible discriminant of a four-variable pair is 1729. \- D1 mentions the Hardy–Ramanujan number. 1729 is the lowest number which can be represented by a Loeschian quadratic form a² + ab + b² in four different ways with a and b positive integers. The integer pairs (a,b) are (25,23), (32,15), (37,8) and (40,3).
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The use of cutting planes to solve MILP was introduced by Ralph E. Gomory. Cutting plane methods for MILP work by solving a non-integer linear program, the linear relaxation of the given integer program. The theory of Linear Programming dictates that under mild assumptions (if the linear program has an optimal solution, and if the feasible region does not contain a line), one can always find an extreme point or a corner point that is optimal. The obtained optimum is tested for being an integer solution.
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Gates complained that the original test series "let an integer BASIC be compared against... more powerful BASIC... using floating point." He suggested that the test include a `DEFINT A-Z` at the start, which would make newer versions of Altair BASIC use integer math as well. Instead, Rugg and Feldman took another approach and eliminated all integer-only BASICs from the new test results. This did not eliminate the Apple II, which by this time had introduced the MS-derived Applesoft BASIC on cassette.
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However, there is no integer which has 120 as the sum of its proper divisors, making 120 an untouchable number. The sum of Euler's totient function φ(x) over the first nineteen integers is 120. 120 figures in Pierre de Fermat's modified Diophantine problem as the largest known integer of the sequence 1, 3, 8, 120. Fermat wanted to find another positive integer that multiplied with any of the other numbers in the sequence yields a number that is one less than a square.
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When the matrix A is not totally unimodular, there are a variety of algorithms that can be used to solve integer linear programs exactly. One class of algorithms are cutting plane methods which work by solving the LP relaxation and then adding linear constraints that drive the solution towards being integer without excluding any integer feasible points. Another class of algorithms are variants of the branch and bound method. For example, the branch and cut method that combines both branch and bound and cutting plane methods.
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In general, an underdetermined system of linear equations has an infinite number of solutions, if any. However, in optimization problems that are subject to linear equality constraints, only one of the solutions is relevant, namely the one giving the highest or lowest value of an objective function. Some problems specify that one or more of the variables are constrained to take on integer values. An integer constraint leads to integer programming and Diophantine equations problems, which may have only a finite number of solutions.
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As such, the Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability, to the chemical behavior of atoms. "Half-integer spin" means that the intrinsic angular momentum value of fermions is \hbar = h/2\pi (reduced Planck's constant) times a half-integer (1/2, 3/2, 5/2, etc.). In the theory of quantum mechanics fermions are described by antisymmetric states. In contrast, particles with integer spin (called bosons) have symmetric wave functions; unlike fermions they may share the same quantum states.
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20] An equivalent formula is given in Fibonacci's Liber Abaci (1202, ch. II.12). In modern mathematics, figurate numbers are formalized by the Ehrhart polynomials. The Ehrhart polynomial of a polyhedron is a polynomial that counts the number of integer points in a copy of that is expanded by multiplying all its coordinates by the number . The Ehrhart polynomial of a pyramid whose base is a unit square with integer coordinates, and whose apex is an integer point at height one above the base plane, is ..
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The most basic method of checking the primality of a given integer n is called trial division. This method divides n by each integer from 2 up to the square root of n. Any such integer dividing n evenly establishes n as composite; otherwise it is prime. Integers larger than the square root do not need to be checked because, whenever n = a\cdot b, one of the two factors a and b is less than or equal to the square root of n.
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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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Moreover, it is weakly typed in some of the same ways that Perl is: one can insert a string into an integer column (although SQLite will try to convert the string to an integer first, if the column's preferred type is integer). This adds flexibility to columns, especially when bound to a dynamically typed scripting language. However, the technique is not portable to other SQL products. A common criticism is that SQLite's type system lacks the data integrity mechanism provided by statically typed columns in other products.
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Simplified decay scheme of 60Co, with angular momenta and parities shownA state of a nuclide is described by its energy above the ground state, by its angular momentum J (in units of \hbar), and by its parity, i.e., its behaviour under reflection (positive + or negative −). Since the spin of nucleons is ½ (in units of \hbar), and since orbital angular momentum has integer values, J may be an integer or a half integer number. Electric and magnetic multipole radiations of the same order \ell (i.e.
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The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer.
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The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer.
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Narcissistic numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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In the following two examples, we always assume it is difficult to factorize a large composite number (see Integer factorization).
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Dudeney numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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Methods for breaking modern cryptosystems often involve solving carefully constructed problems in pure mathematics, the best-known being integer factorization.
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Any Pisot number or Salem number is a Perron number, as is the Mahler measure of a monic integer polynomial.
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The Big Integer case study was in use prior to 2000. It was replaced by the Marine Biology case study.
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Thus, to use it efficiently, integer and floating point instructions must be interleaved so as to keep both processors busy.
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Kaprekar numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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Int32` can be used as `Integer` and `Boolean` (`System.Boolean`), `Char` (`System.Char`), `Real` (`System.Double`) join the family of pascal-typenames, too.
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This is in contrast to the overflow flag which provides a mechanism for comparing the values as signed integer values.
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Self numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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Two families of such triangles are the ones with parametrizations given above at #Heronian triangles with integer inradius and exradii.
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MPS (Mathematical Programming System) is a file format for presenting and archiving linear programming (LP) and mixed integer programming problems.
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In particular, even integers are exactly those integers k where This formulation is useful for investigating integer zeroes of polynomials.
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The number 334 is the smallest integer which leaves remainders 15 and 19 when divided by 29 and 45 respectively.
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In computer science, an integer literal is a kind of literal for an integer whose value is directly represented in source code. For example, in the assignment statement `x = 1`, the string `1` is an integer literal indicating the value 1, while in the statement `x = 0x10` the string `0x10` is an integer literal indicating the value 16, which is represented by `10` in hexadecimal (indicated by the `0x` prefix). By contrast, in `x = cos(0)`, the expression `cos(0)` evaluates to 1 (as the cosine of 0), but the value 1 is not literally included in the source code. More simply, in `x = 2 + 2,` the expression `2 + 2` evaluates to 4, but the value 4 is not literally included.
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Tom Jenkyns and Eric Muller, Triangular Triples from Ceilings to Floors, American Mathematical Monthly 107:7 (August 2000) 634–639Ross Honsberger, Mathematical Gems III, pp. 39–37 It also means that the number of integer triangles with even numbered perimeters p = 2n is the same as the number of integer triangles with odd numbered perimeters p = 2n − 3\. Thus there is no integer triangle with perimeter 1, 2 or 4, one with perimeter 3, 5, 6 or 8, and two with perimeter 7 or 10. The sequence of the number of integer triangles with perimeter p, starting at p = 1, is: :0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8 ...
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However, one useful programming function of unions is to map smaller data elements to larger ones for easier manipulation. A data structure consisting, for example, of 4 bytes and a 32-bit integer, can form a union with an unsigned 64-bit integer, and thus be more readily accessed for purposes of comparison etc.
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A variable-length quantity (VLQ) is a universal code that uses an arbitrary number of binary octets (eight-bit bytes) to represent an arbitrarily large integer. A VLQ is essentially a base-128 representation of an unsigned integer with the addition of the eighth bit to mark continuation of bytes. See the example below.
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A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. They are often used in geometry, grouping and time measurement. The first 20 highly composite numbers are: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560.
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Boolean values in Common Lisp are represented by the self-evaluating symbols T and NIL. Common Lisp has namespaces for symbols, called 'packages'. A number of functions are available for rounding scalar numeric values in various ways. The function `round` rounds the argument to the nearest integer, with halfway cases rounded to the even integer.
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Further, in `x = "1"` the `"1"` is a string literal, not an integer literal, because it is in quotes. The value of the string is `1`, which happens to be an integer string, but this is semantic analysis of the string literal – at the syntactic level `"1"` is simply a string, no different from `"foo"`.
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No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UPLance Fortnow. Computational Complexity Blog: Complexity Class of the Week: Factoring. 13 September 2002.).
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We say that and together is a refinement of a tagged partition together with if for each integer with , there is an integer such that and such that for some with . Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.
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124 is the sum of eight consecutive primes (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29). It is a nontotient since there is no integer with 124 coprimes below it. It is an untouchable number since there is no integer whose proper divisors add up to 124. In base 5 it is a repdigit (4445).
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This is found in Pascal and Fortran 66 and Fortran 77, as in this Pascal example: function f(x, y: integer): integer; begin f := x + y; end; This is semantically different in that when called, the function is simply evaluated – it is not passed a variable from the calling scope to store the output in.
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Euphoria has two basic data types: :Atom – A number, implemented as a 31-bit signed integer or a 64-bit IEEE floating-point. Euphoria dynamically changes between integer and floating point representation according to the current value. :Sequence – A vector (array) with zero or more elements. Each element may be an atom or another sequence.
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Algebraic numbers are those that are a solution to a polynomial equation with integer coefficients. Real numbers that are not rational numbers are called irrational numbers. Complex numbers which are not algebraic are called transcendental numbers. The algebraic numbers that are solutions of a monic polynomial equation with integer coefficients are called algebraic integers.
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The class of integer-valued polynomials was described fully by . Inside the polynomial ring \Q[t] of polynomials with rational number coefficients, the subring of integer-valued polynomials is a free abelian group. It has as basis the polynomials :P_k(t) = t(t-1)\cdots (t-k+1)/k! for k = 0,1,2, \dots, i.e.
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The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general. Other fractional arguments can be approximated through efficient infinite products, infinite series, and recurrence relations.
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In nonstandard analysis, a hyperinteger n is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is given by the class of the sequence in the ultrapower construction of the hyperreals.
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The Standard Model of particle physics provides a uniform framework for understanding the electromagnetic, weak, and strong interactions. An interaction occurs when two particles (typically but not necessarily half-integer spin fermions) exchange integer-spin, force-carrying bosons. The fermions involved in such exchanges can be either elementary (e.g. electrons or quarks) or composite (e.g.
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Variable-length representations of integers, such as bignums, can store any integer that fits in the computer's memory. Other integer data types are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10).
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A positive answer to Büchi's problem would imply, using the negative answer to Hilbert's Tenth Problem by Yuri Matiyasevich, that there is no algorithm to decide whether a system of diagonal quadratic forms with integer coefficients represents an integer tuple. Indeed, Büchi observed that squaring, therefore multiplication, would be existentially definable in the integers over the first-order language having two symbols of constant for 0 and 1, a symbol of function for the sum, and a symbol of relation P to express that an integer is a square.
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The file format, which is binary, uses 32-bit little endian format for all integer fields, and 32-bit binary IEEE 754 format for all floating point fields (also little endian). The file begins with a special integer identifier, 0x4D54434F, which, if interpreted as four ASCII characters, forms the string “OCTM”. Following the identifier is an integer value that specifies the file format version (the latest official file format version is 5). The rest of the file, which is described in the file format specification,OpenCTM Format Specification, version 5 contains the triangle mesh information.
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For defining entity/model classes Jakarta EE provides the Jakarta Persistence (JPA), and for expressing constraints on those entities it provides the Bean Validation API. The example below uses both these technologies. @Entity public class User { @Id @GeneratedValue(strategy = IDENTITY) private Integer id; @Size(min = 2, message="First name too short") private String firstName; @Size(min = 2, message="Last name too short") private String lastName; public Integer getId() { return id; } public void setId(Integer id) { this.id = id; } public String getFirstName() { return firstName; } public void setFirstName(String firstName) { this.
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The minimum number of colours k required for which G is (k, d)-colourable is called the d-defective chromatic number, \chi_d (G).. For a graph class G, the defective chromatic number of G is minimum integer k such that for some integer d, every graph in G is (k,d)-colourable. For example, the defective chromatic number of the class of planar graphs equals 4, since every planar graph is (4,0)-colourable and for every integer d there is a planar graph that is not (3,d)-colourable.
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The Cartesian product structure of on Cartesian plane of ordered pairs . Blue lines denote coordinate axes, horizontal green lines are integer , vertical cyan lines are integer , brown-orange lines show half-integer or , magenta and its tint show multiples of one tenth (best seen under magnification) In mathematics, a real coordinate space of dimension , written ( ) or , is a coordinate space over the real numbers. This means that it is the set of the -tuples of real numbers (sequences of real numbers). With component-wise addition and scalar multiplication, it is a real vector space.
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Fermat's little theorem states that if p is prime and a is coprime to p, then ap−1 − 1 is divisible by p. For an integer a > 1, if a composite integer x divides ax−1 − 1, then x is called a Fermat pseudoprime to base a. In other words, a composite integer is a Fermat pseudoprime to base a if it successfully passes the Fermat primality test for the base a. The false statement that all numbers that pass the Fermat primality test for base 2, are prime, is called the Chinese hypothesis.
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Green Integer is an American publishing house of pocket-sized belles-lettres books, based in Los Angeles, California. It was founded in 1997 by Douglas Messerli, whose former publishing house was Sun & Moon, and it is edited by Per Bregne. Green Integer is one of the most active publishers of literary translations in the United States—particularly poetry. Notable authors published by Green Integer include: Djuna Barnes, Paul Auster, Eleanor Antin, Adonis, Ko Un, Tomas Tranströmer, Arthur Schnitzler, Paul Celan, Gertrude Stein, Robert Bresson, Richard Kalich, Charles Bernstein.
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Lagrange had proved that every positive integer is the sum of not more than four squares; Waring suggested that every positive integer is either a cube or the sum of not more than nine cubes. He also advanced the hypothesis that every positive integer is either a biquadrate (fourth power) or the sum of not more than nineteen biquadrates. These hypotheses form what is known as Waring's problem. He also published a theorem, due to his friend John Wilson, concerning prime numbers; it was later proven rigorously by Lagrange.
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If an integer relation is found, this suggests a possible closed- form expression for the original series, product or integral. This conjecture can then be validated by formal algebraic methods. The higher the precision to which the inputs to the algorithm are known, the greater the level of confidence that any integer relation that is found is not just a numerical artifact. A notable success of this approach was the use of the PSLQ algorithm to find the integer relation that led to the Bailey–Borwein–Plouffe formula for the value of .
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The path label of X in this binary tree is 0221, which is the base 3 representation of the integer 25. List-labeling in a universe of size polynomial in the number n of elements in the total order is connected to the problem of maintaining balance in a binary search tree. Note that every node X of a binary search tree is implicitly labeled with an integer that corresponds to its path from the root of the tree. We call this integer the path label of X and define it as follows.
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Once the queue is empty or the value of the counter is insufficient, the scheduler will skip to the next queue. If the queue is empty, the value of the deficit counter is reset to 0. Variables and Constants const integer N // Nb of queues const integer Q[1..N] // Per queue quantum integer DC[1..N] // Per queue deficit counter queue queue[1..N] // The queues Scheduling Loop while true do for i in 1..N do if not queue[i].empty() then DC[i]:= DC[i] + Q[i] while( not queue[i].
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The SQLite web site describes a "strict affinity" mode, but this feature has not yet been added. However, it can be implemented with constraints like `CHECK(typeof(x)='integer')`. Tables normally include a hidden rowid index column, which gives faster access. If a database includes an Integer Primary Key column, SQLite will typically optimize it by treating it as an alias for rowid, causing the contents to be stored as a strictly typed 64-bit signed integer and changing its behavior to be somewhat like an auto-incrementing column.
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For a general positive real number, the binary logarithm may be computed in two parts.. First, one computes the integer part, \lfloor\log_2 x\rfloor (called the characteristic of the logarithm). This reduces the problem to one where the argument of the logarithm is in a restricted range, the interval [1, 2), simplifying the second step of computing the fractional part (the mantissa of the logarithm). For any , there exists a unique integer such that , or equivalently . Now the integer part of the logarithm is simply , and the fractional part is .
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It is usually easy to write brainfuck programs that do not ever cause integer wraparound or overflow, and therefore don't depend on cell size. Generally this means avoiding increment of +255 (unsigned 8-bit wraparound), or avoiding overstepping the boundaries of [-128, +127] (signed 8-bit wraparound) (since there are no comparison operators, a program cannot distinguish between a signed and unsigned two's complement fixed-bit-size cell and negativeness of numbers is a matter of interpretation). For more details on integer wraparound, see the Integer overflow article.
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Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even.
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The UltraSPARC is a four-issue superscalar microprocessor that executes instructions in in-order. It has a nine-stage integer pipeline.
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This relation is indeed an equivalence relation in the set of integer binary quadratic forms and it preserves discriminants and primitivity.
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An eclipse cycle must comprise close to an integer number of anomalistic months in order to perform well in predicting eclipses.
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That is, a repeating decimal with repeat length is equal to the quotient of the repeating part (as an integer) and .
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In number theory, a pernicious number is a positive integer such that the Hamming weight of its binary representation is prime.
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On Unix, Plan 9, and Unix-like computer systems, factor is a utility for factoring an integer into its prime factors.
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Here \alpha(n) denotes the single-valued inverse Ackermann function, for which any realistic input yields an integer less than five.
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This article gives some specific values of the Riemann zeta function, including values at integer arguments, and some series involving them.
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Blennidus integer is a species of ground beetle in the subfamily Pterostichinae. It was described by Henry Walter Bates in 1891.
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The digit sum can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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Dyschirius integer is a species of ground beetle in the subfamily Scaritinae. It was described by John Lawrence LeConte in 1852.
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Since an interval class is the lower number selected among the interval integer and its inversion, interval classes cannot be inverted.
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Relativistic examples of the integer quantum Hall effect and quantum spin Hall effect arise in the context of lattice gauge theory.
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The Whetstone benchmark primarily measures the floating-point arithmetic performance. A similar benchmark for integer and string operations is the Dhrystone.
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In words, this is the integer that has the largest absolute value less than or equal to the absolute value of .
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In mathematics, a Hofstadter sequence is a member of a family of related integer sequences defined by non-linear recurrence relations.
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Glover F., (1986). Future paths for integer programming and links to artificial intelligence, Computers and Operations Research, 13, 533–549 (1986).
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Perfect digital invariants can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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Sum-product numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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In number theory, a sphenic number (from , 'wedge') is a positive integer that is the product of three distinct prime numbers.
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Computer hardware, including virtual machines, nearly always provide a way to represent a processor register or memory address as an integer.
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Integer factorization is the process of determining which prime numbers divide a given positive integer. Doing this quickly has applications in cryptography. The difficulty depends on both the size and form of the number and its prime factors; it is currently very difficult to factorize large semiprimes (and, indeed, most numbers which have no small factors).
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The integer register file has two read ports and one write port, while the floating-point register file has two read ports and two write ports. Execution begins at stage four (EX) for both integer and floating-point instructions; and is written back to the register files when completed in stage eight (WB). Results may be bypassed if possible.
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All known elementary particles, including quarks, have charges that are integer multiples of e. Therefore, one can say that the "quantum of charge" is e. In this case, one says that the "elementary charge" is three times as large as the "quantum of charge". On the other hand, all isolatable particles have charges that are integer multiples of e.
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BARON is a computational system for solving non-convex optimization problems to global optimality. Purely continuous, purely integer, and mixed-integer nonlinear problems can be solved with the software. BARON is available under the AIMMS, AMPL, and GAMS modeling languages on a variety of platforms. The GAMS/BARON solver is also available on the NEOS Server.
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For a positive integer , an angle of measure is trisectible if and only if does not divide .MacHale, Desmond. "Constructing integer angles", Mathematical Gazette 66, June 1982, 144–145. In contrast, is constructible if and only if is a power of or the product of a power of with the product of one or more distinct Fermat primes.
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The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T; Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009. Sloane is president of the OEIS Foundation.
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242 is the smallest integer to start a run of four consecutive integers with the same number of divisors.R. K. Guy Unsolved Problems in Number Theory, section B18.D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986: 147, 176. 242 is a nontotient since there is no integer with 242 coprimes below it.
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Delphi introduced anonymous functions in version 2009. program demo; type TSimpleProcedure = reference to procedure; TSimpleFunction = reference to function(const x: string): Integer; var x1: TSimpleProcedure; y1: TSimpleFunction; begin x1 := procedure begin Writeln('Hello World'); end; x1; //invoke anonymous method just defined y1 := function(const x: string): Integer begin Result := Length(x); end; Writeln(y1('bar')); end.
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This series of targets all have integer sample values and form an open eye- pattern (e.g. PR4 forms a ternary eye). In general, however, the target can just as readily have non-integer values. The classical approach to maximum- likelihood detection on a channel with intersymbol interference (ISI) is to equalize to a minimum-phase, whitened, matched-filter target.
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In general, if the integer n is partitioned into a sum in which "1" appears j1 times, "2" appears j2 times, and so on, then the number of partitions of a set of size n that collapse to that partition of the integer n when the members of the set become indistinguishable is the corresponding coefficient in the polynomial.
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In telecommunication, an n-ary code is a code that has n significant conditions, where n is a positive integer greater than 1. The integer substituted for n indicates the specific number of significant conditions, i.e., quantization states, in the code. For example, an 8-ary code has eight significant conditions and can convey three bits per code symbol.
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Small Eisenstein primes. If and are Eisenstein integers, we say that divides if there is some Eisenstein integer such that . A non-unit Eisenstein integer is said to be an Eisenstein prime if its only non-unit divisors are of the form , where is any of the six units. There are two types of Eisenstein primes.
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C# supports unsigned in addition to the signed integer types. The unsigned types are `byte`, `ushort`, `uint` and `ulong` for 8, 16, 32 and 64 bit widths, respectively. Unsigned arithmetic operating on the types are supported as well. For example, adding two unsigned integers (`uint`s) still yields a `uint` as a result; not a long or signed integer.
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Intel i860 instructions acted on data sizes from 8-bit through 128-bit. The graphics unit was unique for the era. It was essentially a 64-bit integer unit using the FPU registers as eight 128-bit registers. It supported a number of commands for SIMD-like instructions in addition to basic 64-bit integer math.
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A triangle with integer sides and integer area has sides in arithmetic progression if and only if the sides are (b – d, b, b + d), where :b=2(m^2+3n^2)/g, \, :d=(m^2-3n^2)/g, \, and where g is the greatest common divisor of m^2-3n^2, 2mn, and m^2+3n^2.
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300px The APR effect is very similar to EPR: every electron or nucleus, either free or in a solid, has a magnetic moment and an associated with it spin. The spin can take integer or half-integer values, e.g. 1/2, 1, 3/2, etc., and the corresponding magnetic components ms = ±1/2, ±1, ±3/2, etc.
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In C++03, the largest integer type is `long int`. It is guaranteed to have at least as many usable bits as `int`. This resulted in `long int` having size of 64 bits on some popular implementations and 32 bits on others. C++11 adds a new integer type `long long int` to address this issue.
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The standard definition of "even number" can be used to directly prove that zero is even. A number is called "even" if it is an integer multiple of 2. As an example, the reason that 10 is even is that it equals . In the same way, zero is an integer multiple of 2, namely so zero is even.
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A counting process is a stochastic process {N(t), t ≥ 0} with values that are non-negative, integer, and non-decreasing: # N(t) ≥ 0. # N(t) is an integer. # If s ≤ t then N(s) ≤ N(t). If s < t, then N(t) − N(s) is the number of events occurred during the interval (s, t ].
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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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INTERCAL-72 (the original version of INTERCAL) had only four data types: the 16-bit integer (represented with a `.`, called a "spot"), the 32-bit integer (`:`, a "twospot"), the array of 16-bit integers (`,`, a "tail"), and the array of 32-bit integers (`;`, a "hybrid"). There are 65535 available variables of each type, numbered from `.1` to `.
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Chinese retailers tend to avoid decimal values (such as ¥9.99), opting instead for integer values of yuan (such as ¥9 or ¥10).
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384 (three hundred [and] eighty-four) is the natural number following 383 and preceding 385. It is an even composite positive integer.
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A positive integer n is of the above form if and only if its totient φ(n) is a power of 2.
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In some cases, this can be expressed in terms of a linear program, but the variables must be constrained to be integer.
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This specification includes key agreement, signature, and encryption schemes using several mathematical approaches: integer factorization, discrete logarithm, and elliptic curve discrete logarithm.
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In addition there are instructions for moving data between the integer registers and XMM (used in SSE)/FPU (used in MMX) registers.
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It is sponsored by the Contemporary Arts Educational Project, Inc., in loving memory of Anna Fahrni, and by the publisher Green Integer.
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The multiplicative digital root can be extended to the negative integers by use of a signed-digit representation to represent each integer.
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As with other literals, integer literals are generally evaluated at compile time, as part of the semantic analysis phase. In some cases this semantic analysis is done in the lexer, immediately on recognition of an integer literal, while in other cases this is deferred until the parsing stage, or until after the parse tree has been completely constructed. For example, on recognizing the string `0x10` the lexer could immediately evaluate this to 16 and store that (a token of type `integer` and value 16), or defer evaluation and instead record a token of type `integer` and value `0x10`. Once literals have been evaluated, further semantic analysis in the form of constant folding is possible, meaning that literal expressions involving literal values can be evaluated at the compile phase.
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Second edition of the book Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics. The database was at first stored on punched cards. He published selections from the database in book form twice: #A Handbook of Integer Sequences (1973, ), containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. #The Encyclopedia of Integer Sequences with Simon Plouffe (1995, ), containing 5,488 sequences and assigned M-numbers from M0000 to M5487. The Encyclopedia includes the references to the corresponding sequences (which may differ in their few initial terms) in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 (instead of 1 to 2372.) The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not.
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In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. It is named for James Whitbread Lee Glaisher.
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In mathematics, specifically in number theory, a Cunningham number is a certain kind of integer named after English mathematician A. J. C. Cunningham.
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Suffixes are simple strings. These strings are sorted (as in a paper dictionary), before their starting positions (integer indices) are saved in A.
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His conclusion was that Prout's hypothesis as advocated by Thomson (that all atomic weights are integer multiples of that of hydrogen) was untenable.
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The remainder of the stable bosonic nuclides are 5 odd-proton, odd-neutron stable nuclides (, , , and ), all having a non-zero integer spin.
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It is currently unknown why isolatable particles are restricted to integer charges; much of the string theory landscape appears to admit fractional charges.
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Many early synthesizer modules had modules with height in integer inches: 11" (e.g., Roland 700), 10" (e.g., Wavemakers), 9" (e.g., Aries), 8" (e.g.
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It is known to exist whenever r-1 is the power of a prime integer. There are other families of such extremal hypergraphs.
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It can be shown that C is an integer. The form Ax^2 + Bxy + Cy^2 is "the" composition of f_1 and f_2.
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For example, ⌊3.2 gives 3, the largest integer not above the argument, and 3⌊2 gives 2, the lower of the two arguments.
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An integer x that is a Fermat pseudoprime for all values of a that are coprime to x is called a Carmichael number.
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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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An integer x that is a Fermat pseudoprime to all values of a that are coprime to x is called a Carmichael number.
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This system has been extended to represent some non-integer numbers, called decimal fractions or decimal numbers, for forming the decimal numeral system.
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Fortran provides several integer and real types, and the 64-bit type `real64`, accessible via Fortran's intrinsic module `iso_fortran_env`, corresponds to double precision.
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In number theory, a Behrend sequence is an integer sequence whose multiples include almost all integers. The sequences are named after Felix Behrend.
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Young's lattice is a partially ordered set Y formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers diagrams).
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Certain processors, particularly DSPs common in the embedded system industry, have built in support for the fixed- point arithmetic, such as Q and IQ formats. Since the fractional part of a number takes up some bits in the field, the range of values possible in a fixed9point value is less than the same number of bits would provide to an integer. For instance, in an 8 bit field, an unsigned integer can store values from [0, 255], but an unsigned fixed-point with 5 bits allocated to the fractional part only has 3 bits left over for the integer value, and so can only store integer values from [0, 7]. (The number of distinct values that the two fields can store is the same, 28 = 256, because the fixed-point field can also store 32 fractional values for each integer value.) It is therefore common that a scaling factor is used to store real world values that may be larger than the maximum value of the fixed-point format.
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Planes not derived from finite fields also exist (e.g. for n=9), but all known examples have order a prime power. The best general result to date is the Bruck–Ryser theorem of 1949, which states: :If n is a positive integer of the form or and n is not equal to the sum of two integer squares, then n does not occur as the order of a finite plane. The smallest integer that is not a prime power and not covered by the Bruck–Ryser theorem is 10; 10 is of the form , but it is equal to the sum of squares .
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In C and C++, a type is given in a function declaration or variable declaration by giving one or more type specifiers, and optionally type qualifiers. For example, an integer variable can be declared as: int x; where `int` is the type specifier. An unsigned integer variable can be declared as: unsigned int x; where both `unsigned` and `int` are type specifiers. A constant unsigned integer variable can be declared as: const unsigned int x; where `const` is a type qualifier, which the qualified type of `x` is `const unsigned int` and the unqualified type is `unsigned int`.
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Bijective mapping from integer to even numbers To understand what this means, we first examine what it does not mean. For example, there are infinitely many odd integers, infinitely many even integers, and (hence) infinitely many integers overall. However, it turns out that the number of even integers, which is the same as the number of odd integers, is also the same as the number of integers overall. This is because we can arrange things such that for every integer, there is a distinct even integer: ... −2→−4, −1→−2, 0→0, 1→2, 2→4, ...; or, more generally, n→2n (see picture).
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A problem with five linear constraints (in blue, including the non-negativity constraints). In the absence of integer constraints the feasible set is the entire region bounded by blue, but with integer constraints it is the set of red dots. A closed feasible region of a linear programming problem with three variables is a convex polyhedron. In mathematical optimization, a feasible region, feasible set, search space, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potentially including inequalities, equalities, and integer constraints.
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Each word could hold a 48-bit integer or floating-point number, two 24-bit integer or floating-point numbers, six 8-bit instruction syllables, or eight 6-bit characters. There was also provision for efficient handling of double-word (96-bit) numbers in both integer and floating point formats. However, there was no facility for byte or character addressing, so that non-numerical work suffered by comparison. Its standard character set was a version of the Friden Flexowriter paper tape code that was oriented to Algol 60, and included unusual characters such as the Algol subscript 10.
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Algebraic numbers colored by leading coefficient (red signifies 1 for an algebraic integer) An algebraic integer is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are , and . Therefore, the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monic polynomials for all In this sense, algebraic integers are to algebraic numbers what integers are to rational numbers. The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring.
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This means that for large enough , the probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (). In other words, the average gap between consecutive prime numbers among the first integers is roughly .
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A 2D lattice is a regular array of isolated points where if any one point is chosen as the Cartesian origin (0, 0), then all the other points are at (x, y) where x and y range over all positive and negative integers. A lattice triangle is any triangle drawn within a 2D lattice such that all vertices lie on lattice points. By Pick's theorem a lattice triangle has a rational area that either is an integer or has a denominator of 2. If the lattice triangle has integer sides then it is Heronian with integer area.
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The encryption resembles the RSA algorithm. This algorithm encodes data in 3-byte blocks. First, each byte is permuted through an 8-position permutation, then the bits are split into two 12-bit integers (by taking the low 4 bits of the second byte and the 8 bits of the first byte as the first 12-bit integer, and the 8 bits of the third byte and the 4 high bits of the second integer as the second 12-bit integer). Each is individually encrypted by taking them to the 3491st power, mod 4097; the bits are then reassembled into 3 bytes.
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In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific conditions. While there may be probable primes that are composite (called pseudoprimes), the condition is generally chosen in order to make such exceptions rare. Fermat's test for compositeness, which is based on Fermat's little theorem, works as follows: given an integer n, choose some integer a that is not a multiple of n; (typically, we choose a in the range ).
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Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values. For example, the Diophantine equation 3x^2-2xy-y^2z-7=0 has an integer solution: x=1,\ y=2,\ z=-2. By contrast, the Diophantine equation x^2+y^2+1=0 has no such solution.
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While a 32-bit signed integer may be used to hold a 16-bit unsigned value with relative ease, a 32-bit unsigned value would require a 64-bit signed integer. Additionally, a 64-bit unsigned value cannot be stored using any integer type in Java because no type larger than 64 bits exists in the Java language. If abstracted using functions, function calls become necessary for many operations which are native to some other languages. Alternatively, it is possible to use Java's signed integers to emulate unsigned integers of the same size, but this requires detailed knowledge of complex bitwise operations.
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To convert a decimal fraction to octal, multiply by 8; the integer part of the result is the first digit of the octal fraction. Repeat the process with the fractional part of the result, until it is null or within acceptable error bounds. Example: Convert 0.1640625 to octal: :0.1640625 × 8 = 1.3125 = 1 + 0.3125 :0.3125 × 8 = 2.5 = 2 + 0.5 :0.5 × 8 = 4.0 = 4 + 0 Therefore, 0.164062510 = 0.1248. These two methods can be combined to handle decimal numbers with both integer and fractional parts, using the first on the integer part and the second on the fractional part.
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It is possible to use rounded arithmetic to evaluate the exact value of a function with integer domain and range. For example, if we know that an integer is a perfect square, we can compute its square root by converting to a floating-point value , computing the approximate square root of with floating point, and then rounding to the nearest integer . If is not too big, the floating-point round-off error in will be less than 0.5, so the rounded value will be the exact square root of . This is essentially why slide rules could be used for exact arithmetic.
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The type of a reference type can be determined from values of self-describing types. Self-describing types are further split into arrays and class types. The class types are user-defined classes, boxed value types, and delegates. The following example written in Visual Basic .NET shows the difference between reference types and value types: Imports System Class Class1 Public Value As Integer = 0 End Class 'Class1 Class Test Shared Sub Main() Dim val1 As Integer = 0 Dim val2 As Integer = val1 'Coded By val2 = 123 Dim ref1 As New Class1() Dim ref2 As Class1 = ref1 ref2.
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Two joint approaches are ILP formulation and Island Hopping. The ILP formulation listed above can be solved using a traditional ILP solver. This is typically done by temporarily relaxing the integer constraints, solving the problem optimally, and converting the real solution to an integer solution. Additional constraints can be added and the process repeated indefinitely using a branch and bound approach.
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All elementary particles are either bosons or fermions. These classes are distinguished by their quantum statistics: fermions obey Fermi–Dirac statistics and bosons obey Bose–Einstein statistics. Their spin is differentiated via the spin–statistics theorem: it is half-integer for fermions, and integer for bosons. In the Standard Model, elementary particles are represented for predictive utility as point particles.
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The quantization rules are technically true even for macroscopic systems, like the angular momentum L of a spinning tire. However they have no observable effect. For example, if L_z/\hbar is roughly 100000000, it makes essentially no difference whether the precise value is an integer like 100000000 or 100000001, or a non-integer like 100000000.2—the discrete steps are too small to notice.
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In the case of ATI the photoelectron peaks should appear at : E_s = (n + s) \hbar \omega - W, where the integer n represents the minimal number of photons absorbed, and the integer s represents the number of additional photons absorbed. W is the ionization energy, and E_s is the electron kinetic energy of the peak corresponding to s additional photons being absorbed.
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The subtraction gives the conventional sign to the equation of time. For any given value of , (sometimes written as ) has multiple values, differing from each other by integer numbers of half turns. The value generated by a calculator or computer may not be the appropriate one for this calculation. This may cause to be wrong by an integer number of half turns.
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Fürer's algorithm is an integer multiplication algorithm for extremely large integers with very low asymptotic complexity. It was published in 2007 by the Swiss mathematician Martin Fürer of Pennsylvania State UniversityM. Fürer (2007). "Faster Integer Multiplication" Proceedings of the 39th annual ACM Symposium on Theory of Computing (STOC), 55–67, San Diego, CA, June 11–13, 2007, and SIAM Journal on Computing, Vol.
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Signed 32- and 64-bit integers will only hold at most 6 or 13 base-36 digits, respectively (that many base-36 digits can overflow the 32- and 64-bit integers). For example, the 64-bit signed integer maximum value of "9223372036854775807" is "" in base-36. Similarly, the 32-bit signed integer maximum value of "2147483647" is "" in base-36.
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The standard 78-edge network data set for Zachary's karate club is publicly available on the internet.Zachary's Karate Club data set 78 edges The data can be summarized as list of integer pairs. Each integer represents one karate club member and a pair indicates the two members interacted. The data set is summarized below and also in the adjoining image.
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In mathematical logic such concepts as a primitive recursive function and a μ-recursive function represent integer-valued functions of several natural variables or, in other words, functions on . Gödel numbering, defined on well-formed formulae of some formal language, is a natural-valued function. Computability theory is essentially based on natural numbers and natural (or integer) functions on them.
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A convex polyhedron in which all vertices have integer coordinates is called a lattice polyhedron or integral polyhedron. The Ehrhart polynomial of a lattice polyhedron counts how many points with integer coordinates lie within a scaled copy of the polyhedron, as a function of the scale factor. The study of these polynomials lies at the intersection of combinatorics and commutative algebra.
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Starting in GHC 7.4, the Haskell base package has a `popCount` function available on all types that are instances of the `Bits` class (available from the `Data.Bits` module). MySQL version of SQL language provides `BIT_COUNT()` as a standard function. Fortran 2008 has the standard, intrinsic, elemental function `popcnt` returning the number of nonzero bits within an integer (or integer array).
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1988 PolyLib, PPL, isl, the Cloog polyhedral code generator,Cedric Bastoul. Code Generation in the Polyhedral Model Is Easier Than You Think. PACT'13 IEEE International Conference on Parallel Architecture and Compilation Techniques (2004) and the barvinok library for counting integer solutions. Of these libraries, PolyLib and PPL focus mostly on rational values, while the other libraries focus on integer values.
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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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No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co- UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP).
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Angular momentum is a so-called "classical" concept measuring the momentum of a mass in circular motion about a point. The shell numbers start at 1 and increase indefinitely. Each shell of number contains ² orbitals. Each orbital is characterized by its number , where takes integer values from 0 to −1, and its angular momentum number , where takes integer values from + to −.
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The first field is usually the rationals. The second is a higher degree field. The efficiency of the algorithm strongly depends on the norms of certain elements in these fields. When an integer can be represented as a polynomial with small coefficients, the norms that arise are much smaller than those that arise when an integer is represented by a general polynomial.
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Square packing in a square is a packing problem where the objective is to determine how many squares of side one (unit squares) can be packed into a square of side a. If a is an integer, the answer is a^2, but the precise, or even asymptotic, amount of wasted space for non-integer a is an open question.
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One has to show that 1 is the smallest number that is no less than all . For this, it suffices to prove that, if a number is not larger than 1 and no less than all , then . So let such that for every positive integer . Therefore, This implies that the difference between and is less than the inverse of any positive integer.
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The following subsets of N are numerical semigroups. #⟨ 1 ⟩ = {0, 1, 2, 3, ...} #⟨ 1, 2 ⟩ = {0, 1, 2, 3, ...} #⟨ 2, 3 ⟩ = {0, 2, 3, 4, 5, 6, ...} #Let a be a positive integer. ⟨ a, a + 1, a + 2, ... , 2a - 1 ⟩ = {0, a, a + 1, a + 2, a + 3, ...}. #Let b be an odd integer greater than 1. Then ⟨ 2, b ⟩ = {0, 2, 4, . . .
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The following block of Java code illustrates a problem that exists when not using generics. First, it declares an `ArrayList` of type `Object`. Then, it adds a `String` to the `ArrayList`. Finally, it attempts to retrieve the added `String` and cast it to an `Integer`—an error in logic, as it is not generally possible to cast an arbitrary string to an integer.
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A Diophantine equation is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions. Algebraic geometry is the study of the solutions in an algebraically closed field of multivariate polynomial equations. Two equations are equivalent if they have the same set of solutions. In particular the equation P = Q is equivalent to P-Q = 0.
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In general, the transform is applied to the product of the waveform and a window function. Any window (including rectangular) affects the spectral estimate computed by this method. Figure 2: Windowing a sinusoid causes spectral leakage. The same amount of leakage occurs whether there are an integer (blue) or non-integer (red) number of cycles within the window (rows 1 and 2).
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The same proof applies more generally when the set of integer positions is replaced by any discrete group , the space of configurations is replaced by the set of functions from to a finite alphabet, and shift-equivariance is replaced by equivariance under the action of on itself. In particular, it applies to cellular automata defined on an integer grid of any dimension.
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The full statement of Vinogradov's theorem gives asymptotic bounds on the number of representations of an odd integer as a sum of three primes.
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In both of these types of extension, as for Fermat's quadruple, it is possible to add a fifth rational number rather than an integer.
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A. Karatsuba, Fast Evaluation of Riemann zeta-function \zeta(s) for integer values of argument s. Probl. Peredachi Informat., Vol. 31, No. 4 (1995).
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This keyword is followed by an integer to indicate the version number of the font. For more detailed information, consult the version 2.2 specification.
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On a RAM machine with word size and integer inputs , the problem can be solved in operations by means of the fast Fourier transform.
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Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes Pontryagin index.
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77 (seventy-seven) is the natural number following 76 and preceding 78. Seventy-seven is the smallest positive integer requiring five syllables in English.
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Some support for unsigned integer types was provided in JDK 8, but not for unsigned bytes and with no support in the Java language.
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The floor corner brackets and , the ceiling corner brackets and (U+2308, U+2309) are used to denote the integer floor and ceiling functions.
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So this says that , or . Since there is no integer that is both even and odd, we have reached a contradiction and is irrational.
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Thus the standard Young tableaux are precisely the semistandard tableaux of weight (1,1,...,1), which requires every integer up to to occur exactly once.
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After a floating-point operation was started, integer instructions could be run in parallel until the result of the floating-point operation was needed.
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As is true in general for pairs of Beatty sequences, these two sequences are complementary: each positive integer appears exactly once in either sequence.
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MIDI represents pitch bend as a 14-bit integer, allowing for 16,384 possible values. General MIDI implementations default to a range of ±2 semitones.
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In Pascal, a similar end is performed by declaring a subrange of integer (a compiler may then choose to allocate a smaller amount of storage for the declared variable): type a = 1..100; b = -20..20; c = 0..100000; This subrange feature is not supported by C. A major, if subtle, difference between C and Pascal is how they promote integer operations. In Pascal, all operations on integers or integer subranges have the same effect, as if all of the operands were promoted to a full integer. In C, there are defined rules as to how to promote different types of integers, typically with the resultant type of an operation between two integers having a precision that is greater than or equal to the precisions of the operands. This can make machine code generated from C efficient on many processors.
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Exp-Golomb coding is used in the H.264/MPEG-4 AVC and H.265 High Efficiency Video Coding video compression standards, in which there is also a variation for the coding of signed numbers by assigning the value 0 to the binary codeword '0' and assigning subsequent codewords to input values of increasing magnitude (and alternating sign, if the field can contain a negative number): 0 ⇒ 0 ⇒ 1 ⇒ 1 1 ⇒ 1 ⇒ 10 ⇒ 010 −1 ⇒ 2 ⇒ 11 ⇒ 011 2 ⇒ 3 ⇒ 100 ⇒ 00100 −2 ⇒ 4 ⇒ 101 ⇒ 00101 3 ⇒ 5 ⇒ 110 ⇒ 00110 −3 ⇒ 6 ⇒ 111 ⇒ 00111 4 ⇒ 7 ⇒ 1000 ⇒ 0001000 −4 ⇒ 8 ⇒ 1001 ⇒ 0001001 ... In other words, a non-positive integer x≤0 is mapped to an even integer −2x, while a positive integer x>0 is mapped to an odd integer 2x−1. Exp-Golomb coding is also used in the Dirac video codec.
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Integer overflow occurs when an arithmetic operation results in an integer too large to be represented within the available space. A program which does not properly check for integer overflow introduces potential software bugs and exploits. Below is a function in C++ which attempts to confirm that the sum of x and y is less than or equal to a defined value MAX: bool sumIsValid_flawed(unsigned int x, unsigned int y) { unsigned int sum = x + y; return sum <= MAX; } The problem with the code is it does not check for integer overflow on the addition operation. If the sum of x and y is greater than the maximum possible value of an `unsigned int`, the addition operation will overflow and perhaps result in a value less than or equal to MAX, even though the sum of x and y is greater than MAX.
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Finally, the extension adds LDDQU, an alternative misaligned integer vector load that has better performance on NetBurst based platforms for loads that cross cacheline boundaries.
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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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A plane curve cusp (of order two) may be put in the following form by a diffeomorphism of the plane: , where is a positive integer.
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Equivalently, the injective dimension of M is the minimal integer (if there is such, otherwise ∞) n such that Ext(–,M) = 0 for all N > n.
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These considerations occur frequently in practice and so integer linear programming can be used in many applications areas, some of which are briefly described below.
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Abu Kamil (850 - 930 CE) considered non-negative integer solutions of the following equations: :x + y + z = 100 :3x + (/20)y+ (1/3)z = 100.
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Numerical polynomials can be defined over other rings and fields, in which case the integer-valued polynomials above are referred to as classical numerical polynomials.
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In this case the spectrum of the diffracted beam contains multiple frequency shifts, in any case integer multiples of the frequency of the sound wave.
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In particular, a bitwise trie is keyed on the individual bits making up any fixed-length binary datum, such as an integer or memory address.
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A boson, named after S. N. Bose, is a particle with full integer spin, such as photons and W's. This is a force carrying particle.
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In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: : xy + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of : xy + (\alpha+1 - x)y' + ny = 0~. where is still a non-negative integer.
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The basic approach has three steps: # Formulate the problem to be solved as an integer linear program (ILP). # Compute an optimal fractional solution x to the linear programming relaxation (LP) of the ILP. # Round the fractional solution x of the LP to an integer solution x' of the ILP. (Although the approach is most commonly applied with linear programs, other kinds of relaxations are sometimes used.
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A Mandelbrot fractal with Fractint's colour palette editor (version 20.0 in DOSBOX 0.72 on Vista). contains near copies of itself. Fractint originally appeared in 1988 as FRACT386, a computer program for rendering fractals very quickly on the Intel 80386 processor using integer arithmetic. Most 386 processors of the era did not come with floating point units (Intel 80387), so the integer approach was much faster.
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The rank of a partition, shown as its Young diagram In number theory and combinatorics rank of a partition of a positive integer is a certain integer associated with the partition. Dyson introduced the concept in a paper published in the journal Eureka. It was presented in the context of a study of certain congruence properties of the partition function discovered by the mathematician Srinivasa Ramanujan.
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It supported 32-bit addressing, and was word-addressed. It was a load/store architecture--all references to memory used load and store instructions that copied data between the main memory and 32 general-purpose registers (GPRs). All other instructions, such as integer arithmetic, operated on the GPRs. It possessed a basic instruction set consisting of instructions for control flow, integer arithmetic, and logical operations.
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However, this recursion does not permit us to compute the factorial of a negative integer; use of the formula to compute (−1)! would require a division of a nonzero value by zero, and thus blocks us from computing a factorial value for every negative integer. Similarly, the gamma function is not defined for zero or negative integers, though it is defined for all other complex numbers.
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The range of a variable is given as the set of possible values that that variable can hold. In the case of an integer, the variable definition is restricted to whole numbers only, and the range will cover every number within its range (including the maximum and minimum). For example, the range of a signed 16-bit integer variable is all the integers from −32,768 to +32,767.
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Its symmetry group is the dihedral group of order 6 D3. Equilateral triangles are the only triangles whose Steiner inellipse is a circle (specifically, it is the incircle). The integer-sided equilateral triangle is the only triangle with integer sides and three rational angles as measured in degrees.Conway, J. H., and Guy, R. K., "The only rational triangle", in The Book of Numbers, 1996, Springer-Verlag, pp.
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Let r = sin(nθ) be a rose in the polar coordinate system, where n is a positive integer. The rose has n petals if n is odd, and 2n petals if n is even. We then take 361 points on the rose: : (sin(nk), k) (k = 0, d, 2d, 3d, ..., 360d), where d is a positive integer and the angles are in degrees, not radians.
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Sarason, Section IV.9. This construction is analogous to the real logarithm function , which is the inverse of the real exponential function , satisfying for positive real numbers . If a non-zero complex number is given in polar form as ( and real numbers, with ), then is one logarithm of . Since exactly for all integer , adding integer multiples to the argument gives all the numbers that are logarithms of : :.
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Modular exponentiation is a type of exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography. The operation of modular exponentiation calculates the remainder when an integer (the base) raised to the th power (the exponent), , is divided by a positive integer (the modulus). In symbols, given base , exponent , and modulus , the modular exponentiation is: .
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Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned. 1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself.
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Land worked with Helen Makower, Alison Doig and George Morton in the late 1950s on a number of integer programming problems such as the travelling salesman problem and aircraft scheduling. However these were seemingly too complex to solve. British Petroleum commissioned Land and Doig to investigate using discrete variables within linear programming models. Through this investigation they developed the branch and bound algorithm for solving integer problems.
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POSIX requires the UID to be an integer type. Most Unix-like operating systems represent the UID as an unsigned integer. The size of UID values varies amongst different systems; some UNIX OS's used 15-bit values, allowing values up to 32767, while others such as Linux (before version 2.4) supported 16-bit UIDs, making 65536 unique IDs possible. The majority of modern Unix-like systems (e.g.
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Q is a binary fixed point number format where the number of fractional bits (and optionally the number of integer bits) is specified. For example, a Q15 number has 15 fractional bits; a Q1.14 number has 1 integer bit and 14 fractional bits. Q format is often used in hardware that does not have a floating-point unit and in applications that require constant resolution.
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Examples of polycyclic groups include finitely generated abelian groups, finitely generated nilpotent groups, and finite solvable groups. Anatoly Maltsev proved that solvable subgroups of the integer general linear group are polycyclic; and later Louis Auslander (1967) and Swan proved the converse, that any polycyclic group is up to isomorphism a group of integer matrices.Dmitriĭ Alekseevich Suprunenko, K. A. Hirsch, Matrix groups (1976), pp. 174–5; Google Books.
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Allen Forte defines transposition so as to apply to unordered sets of other than twelve pitches: :the addition mod 12 of any integer k in S to every integer p of P. thus giving, "12 transposed forms of P".Forte (1964). "A Theory of Set-Complexes for Music", p. 149, Journal of Music Theory 8/2:136–83. cited in Schuijer (2008), p. 57.
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This is unchanged from Cortex-A76. What did change is the issue queues. As with the integer cluster, the ASIMD cluster now features a unified issue queue for both pipelines, improving efficiency. As with Cortex-A76, the ASIMD on Cortex-A77 are both 128-bit wide capable of 2 double-precision operations, 4 single-precision, 8 half-precision, or 16 8-bit integer operations.
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Conversely, the OS may deny access, and thus neither open the file nor return a handle. In a capability-based system, handles can be passed between processes, with associated access rights. Note that in these cases the handle must be something other than a systemwide- unique small integer, otherwise it is forgeable. Such an integer may nevertheless be used to identify a capability inside a process; e.g.
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The BG cryptosystem is semantically secure based on the assumed intractability of integer factorization; specifically, factoring a composite value N = pq where p, q are large primes. BG has multiple advantages over earlier probabilistic encryption schemes such as the Goldwasser–Micali cryptosystem. First, its semantic security reduces solely to integer factorization, without requiring any additional assumptions (e.g., hardness of the quadratic residuosity problem or the RSA problem).
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The natural numbers are generally written in the way one usually expects to see them, but they can also be written using an explicit quote, an explicit radix point, or redundant zeros on either end. For example, the integer two can be written as 2 or 2. or 0'2 or 0'2. or even 000'02.000, and the integer zero can be written as 0 or 0' or 0.
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In a similar line of research, Random Manhattan Integer Indexing (RMII)Qasemi Zadeh, Behrang. & Handschuh, Siegfried. (2014) random Manhattan integer indexing: Incremental L1 Normed Vector Space Construction, In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 1713–1723, October 25-29, 2014, Doha, Qatar. is proposed for improving the performance of the methods that employ the Manhattan distance between text units.
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Paola Loreti is an Italian mathematician, and a professor of mathematical analysis at Sapienza University of Rome. She is known for her research on Fourier analysis, control theory, and non-integer representations. The Komornik–Loreti constant, the smallest non-integer base for which the representation of 1 is unique, is named after her and Vilmos Komornik. Loreti earned a laurea from Sapienza University in 1984.
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A number is odd if there is an integer such that . One way to prove that zero is not odd is by contradiction: if then , which is not an integer. Since zero is not odd, if an unknown number is proven to be odd, then it cannot be zero. This apparently trivial observation can provide a convenient and revealing proof explaining why an odd number is nonzero.
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For instance, the triangle with sides and is a right triangle, but is not a Pythagorean triple because is not an integer. Moreover, and do not have an integer common multiple because is irrational. Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system.
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An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable number itself). That is, these numbers are not in the image of the aliquot sum function. Their study goes back at least to Abu Mansur al-Baghdadi (circa 1000 AD), who observed that both 2 and 5 are untouchable.
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Sometimes, when a specification becomes too large, it is split into independent modules which can mature at their own pace. Subsequent editions of a module or specification are known as levels and are denoted by the first integer in the title (e.g. CSS3 = Level 3). Subsequent revisions on each level are denoted by an integer following a decimal point (for example, CSS2.1 = Revision 1).
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For example: CAST (NULL AS INTEGER) represents an absent value of type INTEGER. The actual typing of Unknown (distinct or not from NULL itself) varies between SQL implementations. For example, the following SELECT 'ok' WHERE (NULL <> 1) IS NULL; parses and executes successfully in some environments (e.g. SQLite or PostgreSQL) which unify a NULL boolean with Unknown but fails to parse in others (e.g.
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PHP's built-in array type is, in reality, an associative array. Even when using numerical indexes, PHP internally stores arrays as associative arrays.About the implementation of Arrays in PHP So, PHP can have non-consecutively numerically-indexed arrays. The keys have to be of integer (floating point numbers are truncated to integer) or string type, while values can be of arbitrary types, including other arrays and objects.
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Because these graphs are bipartite and have Hamiltonian paths, their maximum independent sets have a number of vertices that is equal to half of the number of vertices in the whole graph, rounded up to the nearest integer., p.6. The diameter of a Fibonacci cube of order n is n, and its radius is n/2 (again, rounded up to the nearest integer)., p.9.
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We can either consider our integer to be the (only) digit of a nonstandard base-φ numeral, and standardize it, or do the following: 1 × 1 = 1, φ × φ = 1 + φ and = −1 + φ. Therefore, we can compute : (a + bφ) + (c + dφ) = ((a + c) + (b + d)φ), : (a + bφ) − (c + dφ) = ((a − c) + (b − d)φ) and : (a + bφ) × (c + dφ) = ((ac + bd) + (ad + bc + bd)φ). So, using integer values only, we can add, subtract and multiply numbers of the form (a + bφ), and even represent positive and negative integer powers of φ. (a + bφ) > (c + dφ) if and only if 2(a − c) − (d − b) > (d − b) × .
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Consider the expression: :"The smallest positive integer not definable in under sixty letters." Since there are only twenty-six letters in the English alphabet, there are finitely many phrases of under sixty letters, and hence finitely many positive integers that are defined by phrases of under sixty letters. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under sixty letters. If there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property "not definable in under sixty letters".
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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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Frequency ratios are used to describe intervals in both Western and non-Western music. They are most often used to describe intervals between notes tuned with tuning systems such as Pythagorean tuning, just intonation, and meantone temperament, the size of which can be expressed by small-integer ratios. When a musical instrument is tuned using a just intonation tuning system, the size of the main intervals can be expressed by small-integer ratios, such as 1:1 (unison), 2:1 (octave), 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third). Intervals with small-integer ratios are often called just intervals, or pure intervals.
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The current now flows in the opposite direction, opposing the difference between the admitted flux \Phi_0 and the external field of just over \Phi_0/2. The current decreases as the external field is increased, is zero when the flux is exactly \Phi_0, and again reverses direction as the external field is further increased. Thus, the current changes direction periodically, every time the flux increases by additional half-integer multiple of \Phi_0, with a change at maximum amperage every half-plus-integer multiple of \Phi_0 and at zero amps every integer multiple. If the input current is more than I_c, then the SQUID always operates in the resistive mode.
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Numerous problems in integer geometry can be solved using parametric equations. A classical such solution is Euclid's parametrization of right triangles such that the lengths of their sides and their hypotenuse are coprime integers. As a and b are not both even (otherwise and would not be coprime), one may exchange them to have even, and the parameterization is then :a = 2mn, \ \ b = m^2 - n^2, \ \ c = m^2 + n^2, where the parameters and are positive coprime integers that are not both odd. By multiplying and by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths.
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There are no primitive Pythagorean triangles with integer altitude from the hypotenuse. This is because twice the area equals any base times the corresponding height: 2 times the area thus equals both ab and cd where d is the height from the hypotenuse c. The three side lengths of a primitive triangle are coprime, so d = is in fully reduced form; since c cannot equal 1 for any primitive Pythagorean triangle, d cannot be an integer. However, any Pythagorean triangle with legs x, y and hypotenuse z can generate a Pythagorean triangle with an integer altitude, by scaling up the sides by the length of the hypotenuse z.
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The symbols (# and #) introduce patterns. The colon is used to declare patterns and variables. The @ sign before the integer type in the field definitions specifies that these are integer fields, and not, by contrast, references, arrays or other patterns. As another comparison, a procedure in C++ could have the form int max(int x, int y) { if(x >= y) { return x; } else { return y; } } In BETA, such a function could be written using a pattern max: (# x, y, z: @integer enter (x, y) do (if x >= y // True then x -> z else y -> z if) exit z #) The x, y and z are local variables.
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SU(2) is the universal covering group of SO(3), and so its representation theory includes that of the latter, by dint of a surjective homomorphism to it. This underlies the significance of SU(2) for the description of non-relativistic spin in theoretical physics; see below for other physical and historical context. As shown below, the finite-dimensional irreducible representations of SU(2) are indexed by a non-negative integer m and have dimension m+1. In the physics literature, the representations are labeled by the quantity l=m/2, where l is then either an integer or a half- integer, and the dimension is 2l+1.
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On December 1, 2014, the YouTube video garnered over 231 − 1 views, overflowing the YouTube counter to a negative number, resulting in a public comment from Google/YouTube saying "We never thought a video would be watched in numbers greater than a 32-bit integer (=2,147,483,647 views), but that was before we met PSY. "Gangnam Style" has been viewed so many times we had to upgrade to a 64-bit integer (9,223,372,036,854,775,808)!" Hovering over the counter of the YouTube video triggered an easter egg. A YouTube representative later revealed that the comment was a joke and that the company had already updated to a 64-bit integer months ago.
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The function rounds to the nearest integer. Applying this yields: :. The parameter arranges here to set to the zero nearest value which is the desired one.
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XBLite also has 64 bit integer data type, User Defined Types and the ability to have multiple modules in order to create GUI applications or games.
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By typing "FP", Integer BASIC is switched out and Applesoft switched back in. The machine language monitor at $F800 may also be banked out for RAM.
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19］ Hence ［VII. 20, 21］ b＝nc, where n is some integer. Therefore c measures b. Similarly, if c does not measure b, c measures a.
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The LLL algorithm for computing a nearly-orthogonal basis was used to show that integer programming in any fixed dimension can be done in polynomial time.
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A modified Granger causality test based on the GARCH (generalized auto-regressive conditional heteroscedasticity) type of integer- valued time series models is available in many areas.
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Given an integer k and an election, it is NP- complete to determine whether a candidate can become a Condorcet winner with fewer than k swaps.
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In recreational mathematics, van Eck's sequence is an integer sequence defined recursively as follows. Let a0 = 0\. Then, for n ≥ 0, if there exists an m < n such that am = an, take the largest such m and set an+1 = n − m; otherwise an+1 = 0. Thus the first occurrence of an integer in the sequence is followed by a 0, and the second and subsequent occurrences are followed by the size of the gap between the two most-recent occurrences. The first few terms of the sequence are (OEIS: A181391): :0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5 ... van Eck's sequence (A181391) at the On-Line Encyclopedia of Integer Sequences The sequence was named by Neil Sloane after Jan Ritsema van Eck, who contributed it to the On-Line Encyclopedia of Integer Sequences in 2010.
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Matiyasevich proved that there is no algorithm that, given a multivariate polynomial p(x1, x2,...,xk) with integer coefficients, determines whether there is an integer solution to the equation p = 0. Because polynomials with integer coefficients, and integers themselves, are directly expressible in the language of arithmetic, if a multivariate integer polynomial equation p = 0 does have a solution in the integers then any sufficiently strong system of arithmetic T will prove this. Moreover, if the system T is ω-consistent, then it will never prove that a particular polynomial equation has a solution when in fact there is no solution in the integers. Thus, if T were complete and ω-consistent, it would be possible to determine algorithmically whether a polynomial equation has a solution by merely enumerating proofs of T until either "p has a solution" or "p has no solution" is found, in contradiction to Matiyasevich's theorem.
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Although there can be no infinite non-collinear set of points with integer distances, there are infinite non-collinear sets of points whose distances are rational numbers. The (still unsolved) Erdős–Ulam problem asks whether there can exist a dense set of points in the plane at rational distances from each other. For any finite set S of points at rational distances from each other, it is possible to find a similar set of points at integer distances from each other, by expanding S by a factor of the least common denominator of the distances in S. Therefore, there exist arbitrarily large finite sets of non-collinear points with integer distances from each other. However, including more points into S may cause the expansion factor to increase, so this construction does not allow infinite sets of points at rational distances to be transformed into infinite sets of points at integer distances.
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The following is a FORTRAN 66 hello world program using Hollerith constants. It assumes that at least four characters per word are supported by the implementation: PROGRAM HELLO1 C INTEGER IHWSTR(3) DATA IHWSTR/4HHELL,4HO WO,3HRLD/ C WRITE (6,100) IHWSTR STOP 100 FORMAT (3A4) END Besides `DATA` statements, Hollerith constants were also allowed as actual arguments in subroutine calls. However, there was no way that the callee could know how many characters were passed in. The programmer had to pass the information explicitly. The hello world program could be written as follows on a machine where four characters are stored in a word: PROGRAM HELLO2 CALL WRTOUT (11HHELLO WORLD, 11) STOP END C SUBROUTINE WRTOUT (IARRAY, NCHRS) C INTEGER IARRAY(1)FORTRAN 66 did not have a way to indicate a variable-sized array. So a '1' was typically used to indicate that the size is unknown. INTEGER NCHRS C INTEGER ICPW DATA ICPW/4/Four characters per word. INTEGER I, NWRDS C NWRDS = (NCHRS + ICPW - 1) /ICPW WRITE (6,100) (IARRAY(I), I=1,NWRDS) RETURN 100 FORMAT (100A4)A count of 100 is a 'large enough' value that any reasonable number of characters can be written.
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Since orientation assigns an ordering, we must label the tail (v) with a smaller color (or integer from our set) than the head (w) of our arc.
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Production planning by mixed integer programming. As a result, manufacturers are forced to rely on an outdated routine for planning that is largely unchanged since the 1960s.
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In LTspice, numeric values can be expressed in four different ways: integer (i.e. 1000), real (i.e. 1000.0), scientific e-notation (i.e. 1e3, 1.0e3), scale factor notation (i.e.
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Every m by n matrix A with integer entries has a unique m by n matrix H, such that H=UA for some square unimodular matrix U.
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More precisely, the conjecture predicts the leading coefficient of the L-function at an integer point in terms of regulators and a height pairing on motivic cohomology.
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ANTIGONE is an evolution of GloMIQO, a global Mixed-Integer Quadratic Programming solver written by Ruth Misener. ANTIGONE extends the functionality of GloMIQO to general MINLP problems.
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The Baer–Specker group is the group B = ZN of all integer sequences with componentwise addition, that is, the direct product of countably many copies of Z.
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However, these definitions are equivalent by using a different unimodular matrix U. A unimodular matrix is a square invertible integer matrix whose determinant is 1 or -1.
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Cryptophilus integer is a species of pleasing fungus beetle in the family Erotylidae. It is found in Australia, Europe & Northern Asia (excluding China), North America, and Oceania.
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In either case, for every positive integer n, there is at least one prime bigger than n. The conclusion is that the number of primes is infinite.
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Results of interest occur when the size of the coefficients in an integer relation is small compared to the precision with which the real numbers are specified.
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It is also conjectured that when b is neither a perfect power nor −4k4 with k positive integer, then there are infinity many base-b repunit primes.
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Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of four squares.
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Data types may not only be primitive types, such as integer and floating-point types, but also pointer types, and compound datatypes (unions, structs, and C++ classes).
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Suppose, to the contrary, there is an integer that has two distinct prime factorizations. Let n be the least such integer and write n = p1 p2 ... pj = q1 q2 ... qk, where each pi and qi is prime. (Note j and k are both at least 2.) We see p1 divides q1 q2 ... qk, so p1 divides some qi by Euclid's lemma. Without loss of generality, say p1 divides q1.
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Fermions differ from bosons, which obey Bose–Einstein statistics. Some fermions are elementary particles, such as the electrons, and some are composite particles, such as the protons. According to the spin-statistics theorem in relativistic quantum field theory, particles with integer spin are bosons, while particles with half-integer spin are fermions. In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers.
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In applied mathematics, Graver bases enable iterative solutions of linear and various nonlinear integer programming problems in polynomial time. They were introduced by Jack E. Graver.Jack E. Graver: On the foundations of linear and linear integer programming, Mathematical Programming 9:207–226, 1975 Their connection to the theory of Gröbner bases was discussed by Bernd Sturmfels.Bernd Sturmfels, Gröbner Bases and Convex Polytopes, American Mathematical Society, xii+162 pp.
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A (k, d)-coloring of a graph G is a coloring of its vertices with k colours such that each vertex v has at most d neighbours having the same colour as the vertex v. We consider k to be a positive integer (it is inconsequential to consider the case when k = 0) and d to be a non-negative integer. Hence, (k, 0)-coloring is equivalent to proper vertex coloring.
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If problem A is hard, there exists a formal security reduction from a problem which is widely considered unsolvable in polynomial time, such as integer factorization problem or discrete logarithm problem. However, non-existence of a polynomial time algorithm does not automatically ensure that the system is secure. The difficulty of a problem also depends on its size. For example, RSA public key cryptography relies on the difficulty of integer factorization.
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In quantum mechanics, the spin of a particle or collection of particles is quantized. In particular, all particles have either half integer or integer spin. In the most general case, the eigenspinors for a system can be quite complicated. If you have a collection of the Avogadro number of particles, each one with two (or more) possible spin states, writing down a complete set of eigenspinors would not be practically possible.
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HEVC specifies four transform units (TUs) sizes of 4x4, 8x8, 16x16, and 32x32 to code the prediction residual. A CTB may be recursively partitioned into 4 or more TUs. TUs use integer basis functions based on the discrete cosine transform (DCT). In addition, 4x4 luma transform blocks that belong to an intra coded region are transformed using an integer transform that is derived from discrete sine transform (DST).
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In 1817 Degen got printed his large work on the fundamental solutions (x, y) of Pell's equation x2 – ny2 = 1 where n is a positive integer. Euler had earlier shown that these could be systematically calculated with the use of continued fractions. Degen used this method and presented integer solutions for all C.F. Degen, Canon Pellianus Sive Tabula simplicissimam Aequationis Celebratissimae, Bonnier, København (1817). Electronic version from Göttinger Digitalisierungszentrum.
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In most ALGOL-like languages, such as Pascal, Modula-2, Ada and Delphi, conversion and casting are distinctly different concepts. In these languages, conversion refers to either implicitly or explicitly changing a value from one data type storage format to another, e.g. a 16-bit integer to a 32-bit integer. The storage needs may change as a result of the conversion, including a possible loss of precision or truncation.
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Adding BCD numbers using these opcodes is a complex task, and requires many instructions to add even modest numbers. It can also require a large amount of memory. If only doing integer calculations, then all integer calculations are exact, so the radix of the number representation is not important for accuracy. On an x86 processor, calculations with binary numbers are usually a lot faster than the same calculations with BCD numbers.
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In Python, the `int` type has a `bit_count()` method to count the number of bits set. This functionality is new in Python 3.10, scheduled for release in 2021. In Common Lisp, the function logcount, given a non-negative integer, returns the number of 1 bits. (For negative integers it returns the number of 0 bits in 2's complement notation.) In either case the integer can be a BIGNUM.
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Dirac was also noted for his personal modesty. He called the equation for the time evolution of a quantum-mechanical operator, which he was the first to write down, the "Heisenberg equation of motion". Most physicists speak of Fermi–Dirac statistics for half-integer-spin particles and Bose–Einstein statistics for integer-spin particles. While lecturing later in life, Dirac always insisted on calling the former "Fermi statistics".
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The sequence 0, 3, 8, 15, ... is formed according to the formula n2 − 1 for the nth term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, even though we do not have a formula for the nth perfect number.
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SMT permits multiple independent threads of execution to better utilize the resources provided by modern processor architectures. Superscalar processors differ from multi-core processors in that the several execution units are not entire processors. A single processor is composed of finer-grained execution units such as the ALU, integer multiplier, integer shifter, FPU, etc. There may be multiple versions of each execution unit to enable execution of many instructions in parallel.
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Fermat's little theorem states that if p is a prime number, then for any integer b, the number b − b is an integer multiple of p. Carmichael numbers are composite numbers which have this property. Carmichael numbers are also called Fermat pseudoprimes or absolute Fermat pseudoprimes. A Carmichael number will pass a Fermat primality test to every base b relatively prime to the number, even though it is not actually prime.
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Several variations on Euclid's proof exist, including the following: The factorial n! of a positive integer n is divisible by every integer from 2 to n, as it is the product of all of them. Hence, is not divisible by any of the integers from 2 to n, inclusive (it gives a remainder of 1 when divided by each). Hence is either prime or divisible by a prime larger than n.
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In modern positional numbers systems, such as the decimal system, the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in the radix or base, in this case, . As another example, in radix 5, a string of digits such as 132 denotes the (decimal) number = 42. This representation is unique. Let b be a positive integer greater than 1.
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For every positive integer n, the set of all positive divisors of n becomes a partially ordered set if we use divisibility as the order relation. This partially ordered set is always a distributive lattice. It is a Boolean algebra if and only if n is square-free. A positive integer n is square-free if and only if μ(n) ≠ 0, where μ denotes the Möbius function.
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Let p ≥ 3 be an integer. A group presentation (∗) as above is said to satisfy the C(p) small cancellation condition if whenever r ∈ R and :r=u_1\dots u_m where ui are pieces and where the above product is freely reduced as written, then m ≥ p. That is, no defining relator can be written as a reduced product of fewer than p pieces. Let q ≥ 3 be an integer.
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A variation on that is MI for free negative, giving an upper bound of 0 but no lower bound. Bound type PL is for a free positive from zero to plus infinity, but as this is the normal default, it is seldom used. There are also bound types for use in MIP models – BV for binary, being 0 or 1. UI for upper integer and LI for lower integer.
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If the endpoints of intervals are within a small integer range (e.g., in the range [1, \ldots, O(n)]), faster and in fact optimal data structures existJens M. Schmidt. Interval Stabbing Problems in Small Integer Ranges. DOI. ISAAC'09, 2009Range Queries#Semigroup operators with preprocessing time O(n) and query time O(1+ m) for reporting m intervals containing a given query point (see for a very simple one).
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The address unit, also known as the "A-box", executed load and store instructions. To enable the address unit and integer unit to operate in parallel, the address unit has its own displacement adder, which it uses to calculate virtual addresses, instead of using the adder in the integer unit.McLellan 1993, p. 43 A 32-entry fully associative translation lookaside buffer (TLB) is used to translate virtual addresses into physical addresses.
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The offset is encoded as a "motion vector". Frequently, the offset is zero, but if something in the picture is moving, the offset might be something like 23 pixels to the right and 4-and-a-half pixels up. In MPEG-1 and MPEG-2, motion vector values can either represent integer offsets or half-integer offsets. The match between the two regions will often not be perfect.
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This conjecture was justified for a class of parameters by R. Amrane and H. Belbachir. Especially, these authors proved that H_n^{(4)} is not integer for all r<26 and n=2,3,... Extension to high orders was made by Göral and Sertbaş. These authors have also shown that H_n^{(r)} is never integer when n is even or a prime power, or r is odd. Another result is the following.
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For instance, by April 2001, 12 two-bedroom flats with INTEGER-inspired features had been built in Harlow, Essex. Core INTEGER-style technologies that were used in the flats included enhanced insulation and a high-efficiency communal boiler system that reduced energy demand by 50%, as well as reduced servicing by replacing 12 small boilers with two large ones, and a grey water recycling system that reduced water consumption by 30%.
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Quantum mechanical particles are classified as bosons or fermions. The spin–statistics theorem of quantum field theory demands that all particles with half-integer spin behave as fermions and all particles with integer spin behave as bosons. Multiple bosons may occupy the same quantum state; however, by the Pauli exclusion principle, no two fermions can occupy the same state. Since electrons have spin 1/2, they are fermions.
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The weak Collatz conjecture asserts the following: "The 3x + 1 semigroup contains every positive integer." This was formulated by Farkas and it has been proved to be true as a consequence of the following property of the 3x + 1 semigroup: :The 3x + 1 semigroup S equals the set of all positive rationals in lowest terms having the property that b ≠ 0 (mod 3). In particular, S contains every positive integer.
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If R possesses no non-zero zero divisors, it is called an integral domain (or domain). An element a satisfying for some positive integer n is called nilpotent.
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It is a divisibility sequence, and there exists an integer so that the subsequence ( ± ) ≥ 1 (with an appropriate choice of signs) is an EDS in the earlier sense.
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2011 07 12 07 18 47 The 349 bytes, resulting from the UTF-8 encoded XML, correlates to a size overhead of 8625% over the original integer representation.
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2011 07 12 07 18 47 The 349 bytes resulting from the UTF-8 encoded XML correspond to a size overhead of 8625% over the original integer representation.
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Let is a finite group where for all element of . Then has the order for some non negative integer . Let is . In the case of is 1, then .
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This expansion converges to the meromorphic function defined by the ratio of the two convergent series (provided, of course, that a is neither zero nor a negative integer).
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Similarly, the set of integers in W is itself a multiplicative semigroup. It is called the wild integer semigroup and members of this semigroup are called wild numbers.
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It is used in a special case of integer programming, in which all the decision variables are integers. It can assume the values either as zero or one.
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In fact is solution of the equation :x^2-2ax+a^2+b^2, and this equation has integer coefficients if and only if and are both integers.
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Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers. The notation of integer intervals is considered in the special section below.
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The system included vector floating-point hardware which was not rich in integer operations. The floating-point values were floated to allow manipulation to create the initial approximation.
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There are a \epsilon > 0 and an integer d_0 such that any graph G with \delta(G) \ge d_0 satisfies \chi'_g(G) \ge (1+\epsilon)\delta(G).
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42 thus multiply instructions have a variable latency of 19 to 23 cycles depending on the operands. In stage seven, integer instructions write their results to the IRF.
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Octeract Engine is a proprietary massively parallel deterministic global optimization solver for general Mixed-Integer Nonlinear Programs (MINLP). It uses MPI as a means of accelerating solution times.
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Hence, for this choice, the difference between (2n-1)!e^{-1} and (2n-1)!s_{2n-1} would be an integer. But from the above inequality, that's impossible.
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Storage of integers is usually done with dedicated 32 or 64 bits per integer. For small integers, packing multiple integers into the same space makes storage more efficient.
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It is possible to choose between bilinear interpolation and bicubic interpolation for the grey level evaluation at non integer coordinates. The bi-cubic interpolation is the recommended one.
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A representation of a value using binary scaling is more precise than a floating-point representation occupying the same number of bits, but typically represents values of a more limited range, therefore more easily leading to arithmetic overflow during computation. Implementation of operations using integer arithmetic instructions is often (but not always) faster than the corresponding floating-point instructions. A position for the 'binary point' is chosen for each variable to be represented, and binary shifts associated with arithmetic operations are adjusted accordingly. The binary scaling corresponds in Q (number format) to the first digit, i.e. Q1.15 is a 16 bit integer scaled with one bit as integer and fifteen as fractional. A Bscal 1 or Q1.15 number would represent approximately 1.999 to −2.0. To give an example, a common way to use integer arithmetic to simulate floating point, using 32-bit numbers, is to multiply the coefficients by 65536. Using binary scientific notation, this will place the binary point at B16.
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A traditional solution to such a problem is locking. For instance, locks can be placed around modifications to an account to ensure that credits and debits occur atomically. In Haskell, locking is accomplished with MVars: type Account = MVar Integer credit :: Integer -> Account -> IO () credit amount account = do current <\- takeMVar account putMVar account (current + amount) debit :: Integer -> Account -> IO () debit amount account = do current <\- takeMVar account putMVar account (current - amount) Using such procedures will ensure that money will never be lost or gained due to improper interleaving of reads and writes to any individual account. However, if one tries to compose them together to create a procedure like transfer: transfer :: Integer -> Account -> Account -> IO () transfer amount from to = do debit amount from credit amount to a race condition still exists: the first account may be debited, then execution of the thread may be suspended, leaving the accounts as a whole in an inconsistent state.
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To avoid this, one can use the STM monad, which allows one to write atomic transactions. This means that all operations inside the transaction fully complete, without any other threads modifying the variables that our transaction is using, or it fails, and the state is rolled back to where it was before the transaction was begun. In short, atomic transactions either complete fully, or it is as if they were never run at all. The lock-based code above translates in a relatively straightforward way: type Account = TVar Integer credit :: Integer -> Account -> STM () credit amount account = do current <\- readTVar account writeTVar account (current + amount) debit :: Integer -> Account -> STM () debit amount account = do current <\- readTVar account writeTVar account (current - amount) transfer :: Integer -> Account -> Account -> STM () transfer amount from to = do debit amount from credit amount to The return types of `STM ()` may be taken to indicate that we are composing scripts for transactions.
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The necessity of introducing half- integer spin goes back experimentally to the results of the Stern–Gerlach experiment. A beam of atoms is run through a strong heterogeneous magnetic field, which then splits into N parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two—the ground state therefore could not be an integer, because even if the intrinsic angular momentum of the atoms were the smallest (non-zero) integer possible, 1, the beam would be split into 3 parts, corresponding to atoms with Lz = −1, +1, and 0, with 0 simply being the value known to come between -1 and +1 while also being a whole-integer itself, and thus a valid quantized spin number in this case. The existence of this hypothetical "extra step" between the two polarized quantum states would necessitate a third quantum state; a third beam, which is not observed in the experiment.
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In other words, an integer GCD computation reduces the factorization of a polynomial over the rationals to the factorization of a primitive polynomial with integer coefficients, and the factorization over the integers to the factorization of an integer and a primitive polynomial. Everything that precedes remains true if Z is replaced by a polynomial ring over a field F and Q is replaced by a field of rational functions over F in the same variables, with the only difference that "up to a sign" must be replaced by "up to the multiplication by an invertible constant in F". This reduces the factorization over a purely transcendental field extension of F to the factorization of multivariate polynomials over F.
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A solution (a, b, c) for a given n leads to a solution for all the factors of n: if h is a factor of n then there is an integer g such that n = gh. Then (ag, bg, cg) is a solution for the exponent h: : (ag)h \+ (bg)h = (cg)h. Therefore, to prove that Fermat's equation has no solutions for n > 2, it suffices to prove that it has no solutions for n = 4 and for all odd primes p. For any such odd exponent p, every positive-integer solution of the equation ap + bp = cp corresponds to a general integer solution to the equation ap + bp + cp = 0\.
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Example JPA Classes, getters and setters omitted for simplicity. @Entity public class Author { @Id private Integer id; private String firstName; private String lastName; @ManyToMany private List books; } @Entity public class Book { @Id private Integer id; private String title; private String isbn; @ManyToOne private Publisher publisher; @ManyToMany private List authors; } @Entity public class Publisher { @Id private Integer id; private String name; private String address; @OneToMany(mappedBy = "publisher") private List books; } Then a simple query to retrieve the list of all authors, ordered alphabetically, would be: SELECT a FROM Author a ORDER BY a.firstName, a.lastName To retrieve the list of authors that have ever been published by XYZ Press: SELECT DISTINCT a FROM Author a INNER JOIN a.
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Each positive integer n has 2n−1 distinct compositions. Bijection between 3 bit binary numbers and compositions of 4 A weak composition of an integer n is similar to a composition of n, but allowing terms of the sequence to be zero: it is a way of writing n as the sum of a sequence of non-negative integers. As a consequence every positive integer admits infinitely many weak compositions (if their length is not bounded). Adding a number of terms 0 to the end of a weak composition is usually not considered to define a different weak composition; in other words, weak compositions are assumed to be implicitly extended indefinitely by terms 0\.
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Pentium II processor with MMX technology MMX defines eight registers, called MM0 through MM7, and operations that operate on them. Each register is 64 bits wide and can be used to hold either 64-bit integers, or multiple smaller integers in a "packed" format: a single instruction can then be applied to two 32-bit integers, four 16-bit integers, or eight 8-bit integers at once. MMX provides only integer operations. When originally developed, for the Intel i860, the use of integer math made sense (both 2D and 3D calculations required it), but as graphics cards that did much of this became common, integer SIMD in the CPU became somewhat redundant for graphical applications.
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The great disadvantage of Euler's factorization method is that it cannot be applied to factoring an integer with any prime factor of the form 4k + 3 occurring to an odd power in its prime factorization, as such a number can never be the sum of two squares. Even odd composite numbers of the form 4k + 1 are often the product of two primes of the form 4k + 3 (e.g. 3053 = 43 × 71) and again cannot be factored by Euler's method. This restricted applicability has made Euler's factorization method disfavoured for computer factoring algorithms, since any user attempting to factor a random integer is unlikely to know whether Euler's method can actually be applied to the integer in question.
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Superscalar execution was improved by adding the extra integer unit and modifying the control logic so that two integer instructions, two load-store units, or an integer and a load- store can be issued in one cycle in addition to the existing instruction combinations supported by the PA-7100. A number of modifications were made to circuits derived from the PA-7100LC. Prominently, the floating-point unit multiplier was modified to take up less area by halving the tree of carry-save adders that summed the partial products of the mantissa. This simplification left the latency of single precision multiplies unchanged (two cycles), but increased the latency of double precision multiplies to three cycles.
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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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10 is square-free, as its divisors greater than 1 are 2, 5, and 10, none of which is a perfect square (the first few perfect squares being 1, 4, 9, and 16) In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no perfect square other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-free, but is not, because 18 is divisible by . The smallest positive square-free numbers are :1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, ...
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The weighing pans of this balance scale contain zero objects, divided into two equal groups. Zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. This can be easily verified based on the definition of "even": it is an integer multiple of 2, specifically . As a result, zero shares all the properties that characterize even numbers: for example, 0 is neighbored on both sides by odd numbers, any decimal integer has the same parity as its last digit—so, since 10 is even 0 will be even, and if is even then has the same parity as —and and always have the same parity.
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An array can be used to implement a (bounded) stack, as follows. The first element, usually at the zero offset, is the bottom, resulting in `array[0]` being the first element pushed onto the stack and the last element popped off. The program must keep track of the size (length) of the stack, using a variable top that records the number of items pushed so far, therefore pointing to the place in the array where the next element is to be inserted (assuming a zero-based index convention). Thus, the stack itself can be effectively implemented as a three-element structure: structure stack: maxsize : integer top : integer items : array of item procedure initialize(stk : stack, size : integer): stk.
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A linear DDA starts by calculating the smaller of dy or dx for a unit increment of the other. A line is then sampled at unit intervals in one coordinate and corresponding integer values nearest the line path are determined for the other coordinate. Considering a line with positive slope, if the slope is less than or equal to 1, we sample at unit x intervals (dx=1) and compute successive y values as : y_{k+1} = y_k + m : x_{k+1} = x_k + 1 Subscript k takes integer values starting from 0, for the 1st point and increases by 1 until endpoint is reached. y value is rounded off to nearest integer to correspond to a screen pixel.
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133 is a semiprime: a product of two prime numbers, namely 7 and 19. Since those prime factors are Gaussian primes, this means that 133 is a Blum integer.
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TC0 contains several important problems, such as sorting n n-bit numbers, multiplying two n-bit numbers, integer division or recognizing the Dyck language with two types of parentheses.
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Conversely, the restriction to the elements of order coprime to p of each ordinary irreducible character is uniquely expressible as a non-negative integer combination of irreducible Brauer characters.
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Wolsey has made seminal contributions in duality theory for integer programming, submodular optimization, the group-theoretic approach and polyhedral analysis of fixed-charge network flow and production planning models.
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In mathematics, especially the field of group theory, the Parker vector is an integer vector that describes a permutation group in terms of the cycle structure of its elements.
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The Fibonacci code word for a particular integer is exactly the integer's Zeckendorf representation with the order of its digits reversed and an additional "1" appended to the end.
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The fundamental theorem of arithmetic establishes that the factorization of any positive integer into prime numbers is essentially unique, i.e., unique up to the ordering of the prime factors.
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Computational number theory, also known as algorithmic number theory, is the study of algorithms for performing number theoretic computations. The best known problem in the field is integer factorization.
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Other notable examples include the travelling salesman problem, which asks for the route taken by the salesman, and the integer factorization problem, which asks for the list of factors.
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Destroy; begin FreeAndNil(FObservers); inherited; end; procedure TEdijsObserverManager.NotifyObservers(const AValue: string); var i: Integer; begin for i := 0 to FObservers.Count - 1 do FObservers[i].Update(AValue); end; procedure TEdijsObserverManager.
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Zerogram Press is an imprint of Green Integer dedicated to publishing contemporary literary fiction written in English. It also publishes literary criticism and essays. It was founded in 2016.
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The gamma function interpolates the factorial function to non-integer values. The main clue is the recurrence relation generalized to a continuous domain. Besides nonnegative integers, the factorial can also be defined for non-integer values, but this requires more advanced tools from mathematical analysis. One function that fills in the values of the factorial (but with a shift of 1 in the argument), that is often used, is called the gamma function, denoted .
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The JOSS system used a hard disk to store user programs in an allocated space. Within that space were a number of files that were referred to using a multi-part filename consisting of an integer and a five-letter name in parentheses, for instance, . The integer part is the equivalent of a directory, and the name is the sub-file within it, maintained by JOSS itself. Files are written with and loaded with .
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In mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed to be exact (there is no remainder). The method can also be used to compute the determinant of matrices with (approximated) real entries, avoiding the introduction any round-off errors beyond those already present in the input.
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A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of −5 and −2 as well.
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For every unitary irreducible representations there is an equivalent one, . All infinite-dimensional irreducible representations must be non-unitary, since the group is compact. In quantum mechanics, the Casimir invariant is the "angular-momentum-squared" operator; integer values of spin characterize bosonic representations, while half-integer values fermionic representations. The antihermitian matrices used above are utilized as spin operators, after they are multiplied by , so they are now hermitian (like the Pauli matrices).
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The Microwatt is a tiny 64-bit little endian scalar integer processor core, implementing a subset of the Power ISA 3.0 instruction set. It has 64× 64-bit general purpose registers. It uses Wishbone for the memory interface. The initial development was done in a couple of months, included the entire integer processing functionality of the instruction set; the bare minimum to make it compliant, with no memory management unit and no floating-point unit.
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Integer range can also affect the number of memory locations the CPU can directly address (an address is an integer value representing a specific memory location). For example, if a binary CPU uses 32 bits to represent a memory address then it can directly address 232 memory locations. To circumvent this limitation and for various other reasons, some CPUs use mechanisms (such as bank switching) that allow additional memory to be addressed.
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SAL supports integer and string data types. An integer in SAL is a 32-bit base-2 signed integral numeric; a string is a (declared) fixed-size array of characters, with a maximum definable length of 255 characters. SAL APIs support creation of arbitrary buffers, including non-interactive buffers not visible to the user. Many functions are available to access and manipulate buffers and their contents, effectively acting as very large text containers.
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There are a few disadvantages with integer notation. First, theorists have traditionally used the same integers to indicate elements of different tuning systems. Thus, the numbers 0, 1, 2, ... 5, are used to notate pitch classes in 6-tone equal temperament. This means that the meaning of a given integer changes with the underlying tuning system: "1" can refer to C in 12-tone equal temperament, but D in 6-tone equal temperament.
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The cutting-plane method is an umbrella term for optimization methods which iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts. Such procedures are popularly used to find integer solutions to mixed integer linear programming (MILP) problems, as well as to solve general, not necessarily differentiable convex optimization problems. The use of cutting planes to solve MILP was introduced by Ralph E. Gomory and Václav Chvátal.
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This version of IPOPT is generally known as IPOPT-C (with the 'C' standing for 'complementarity'). While in theory any mixed-integer program can be recast as an MPEC, it may or may not be solvable with IPOPT-C. Solution of MINLPs (Mixed-Integer Nonlinear Programs) using IPOPT is still being explored . Carl Laird and Andreas Wächter are the developers of IPOPT 3.0, which is a re-implementation of IPOPT in C++.
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All instructions begin execution during stage six in the ten functional units. Integer instructions except for multiply are executed in two arithmetic logic units (ALUs) and two shift/merge units. All instructions executed in these units have a single-cycle latency and their results are written to the destination register in stage seven. Floating-point instructions and integer multiply instructions are executed in two fused multiply–accumulate (FMAC) units and two divide/square-root units.
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Integer solutions to the optic equation + = for . The number in the circle is c. In [ the SVG file,] hover over a circle to see its solution. In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers a and b to equal the reciprocal of a third positive integer c:Dickson, L. E., History of the Theory of Numbers, Volume II: Diophantine Analysis, Chelsea Publ.
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In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of (strictly) positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same partition of that number. Every integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence.
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Some real numbers have decimal expansions that eventually get into loops, endlessly repeating a sequence of one or more digits: :1/3 = 0.33333... :1/7 = 0.142857142857... :1318/185 = 7.1243243243... Every time this happens the number is still a rational number (i.e. can alternatively be represented as a ratio of an integer and a positive integer). Also the converse is true: The decimal expansion of a rational number is either finite, or endlessly repeating.
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Most fixed-size integer formats cannot explicitly indicate invalid data. In such a case, when converting NaN to an integer type, the IEEE 754 standard requires that an invalid operation exception be signaled. For example in Java, such operations throw instances of `java.lang.ArithmeticException`. In C, they lead to undefined behavior, but if annex F is supported, the operation yields an "invalid" floating-point exception (as required by the IEEE standard) and an unspecified value.
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In C++20, a new header `` was added, containing functions `std::popcount` and `std::has_single_bit`, taking arguments of unsigned integer types. In Java, the growable bit-array data structure has a method that counts the number of bits that are set. In addition, there are and functions to count bits in primitive 32-bit and 64-bit integers, respectively. Also, the arbitrary-precision integer class also has a method that counts bits.
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Compiled code runs at 1/3 to 1/2 the speed of a pure C application on a Palm platform, and on a Windows API (Win32) platform is approximately three to four times faster than a JIT-compiled Java due partly to its integer-only implementation. Additional mathematics libraries for non-integer arithmetic are provided with the SDK. The SuperWaba VM and API are both extensible. They are published under the GPL free software license.
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The largest remainder method requires the numbers of votes for each party to be divided by a quota representing the number of votes required for a seat (i.e. usually the total number of votes cast divided by the number of seats, or some similar formula). The result for each party will usually consist of an integer part plus a fractional remainder. Each party is first allocated a number of seats equal to their integer.
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In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular numbers, and as the sum of four or fewer square numbers, and as the sum of five or fewer pentagonal numbers, and so on. That is, the -gonal numbers form an additive basis of order .
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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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An example of an infinite diophantine equation is: :, which can be expressed as "How many ways can a given integer be written as the sum of a square plus twice a square plus thrice a square and so on?" The number of ways this can be done for each forms an integer sequence. Infinite Diophantine equations are related to theta functions and infinite dimensional lattices. This equation always has a solution for any positive .
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Furthermore, it has been proved that all Heronian triangles can be drawn as lattice triangles.P. Yiu, "Heronian triangles are lattice triangles", American Mathematical Monthly 108 (2001), 261–263. Consequently, an integer triangle is Heronian if and only if it can be drawn as a lattice triangle. There are infinitely many primitive Heronian (non-Pythagorean) triangles which can be placed on an integer lattice with all vertices, the incenter, and all three excenters at lattice points.
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Some integer triangles with one angle at vertex A having given rational cosine h/k (h<0 or >0; k>0) are given bySastry, K. R. S., "Integer-sided triangles containing a given rational cosine", Mathematical Gazette 68, December 1984, 289−290. :a=p^2-2pqh+q^2k^2, :b=p^2-q^2k^2, :c=2qk(p-qh), where p and q are any coprime positive integers such that p>qk.
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The out-of-order window size has been increased to 160 entries. The backend is 12 execution ports with a 50% increase over Cortex-A76. It has a pipeline depth of 13 stages and the execution latencies of 10 stages. There are six pipelines in the integer cluster - an increase of two additional integer pipelines from Cortex-A76. One of the changes from Cortex-A76 is the unification of the issue queues.
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These variables can be pushed to or popped from the stack. In addition, the current font f is held as an integer value, but is not pushed and popped with the rest of the state variables when the opcodes `push` or `pop` are encountered. Font spacing information is loaded from TFM files. The fonts themselves are not embedded in the DVI file, only referenced by an integer value defined in the relevant `fnt_defi` op.
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In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent. Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself. Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done.
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An enum type cannot be instantiated directly. Internally, each enum value contains an integer, corresponding to the order in which they are declared in the source code, starting from 0. The programmer cannot set a custom integer for an enum value directly, but one can define overloaded constructors that can then assign arbitrary values to self-defined members of the enum class. Defining getters allows then access to those self-defined members.
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The internal integer can be obtained from an enum value using the method, and the list of enum values of an enumeration type can be obtained in order using the `values()` method. It is generally discouraged for programmers to convert enums to integers and vice versa. Enumerated types are `Comparable`, using the internal integer; as a result, they can be sorted. The Java standard library provides utility classes to use with enumerations.
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1719–1736; LBNL-44481.I. S. Kotsireas, and K. Karamanos, "Exact Computation of the bifurcation Point B4 of the logistic Map and the Bailey–Broadhurst Conjectures", I. J. Bifurcation and Chaos 14(7):2417–2423 (2004) Integer relation algorithms are combined with tables of high precision mathematical constants and heuristic search methods in applications such as the Inverse Symbolic Calculator or Plouffe's Inverter. Integer relation finding can be used to factor polynomials of high degree.
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Dividing integers in a computer program requires special care. Some programming languages, such as C, treat integer division as in case 5 above, so the answer is an integer. Other languages, such as MATLAB and every computer algebra system return a rational number as the answer, as in case 3 above. These languages also provide functions to get the results of the other cases, either directly or from the result of case 3.
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The notion of the integral part or integer part of x was first introduced in 1798 by Adrien- Marie Legendre, under the name partie entière ("entière" being the feminine form of the French adjective "entier", the French word for "integer"), when he needed the concept for his proof of the Legendre's formula. Carl Friedrich Gauss introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808).Lemmermeyer, pp. 10, 23.
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The Kleitman–Wang algorithms are two different algorithms in graph theory solving the digraph realization problem, i.e. the question if there exists for a finite list of nonnegative integer pairs a simple directed graph such that its degree sequence is exactly this list. For a positive answer the list of integer pairs is called digraphic. Both algorithms construct a special solution if one exists or prove that one cannot find a positive answer.
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Create tables using the CREATE TABLE statement: CREATE TABLE name (colname datatype [, colname datatype...]) Data types currently "supported" are integer, float, and varchar. They are ignored by the implementation, anything that is hashable and marshallable can currently go in any column (but that is likely to change). For example: CREATE TABLE frequents (drinker VARCHAR, bar VARCHAR, perweek INTEGER) At present tuples, complexes, or anything else can be put into a column specified as "VARCHAR".
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This fundamental difference in 1H and 2H manifests in many physical properties. Integer spin particles like deuterium follow Bose-Einstein statistics while fermions with half integer spins follow Fermi-Dirac statistics. Wave functions that describe multiple fermions must be antisymmetric with respect to swapping particles, while boson wave functions are symmetric. Because bosons are indistinguishable and can occupy the same state, collections of bosons behave very differently than fermions at colder temperatures.
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Branch and cut is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some or all the unknowns are restricted to integer values. Branch and cut involves running a branch and bound algorithm and using cutting planes to tighten the linear programming relaxations. Note that if cuts are only used to tighten the initial LP relaxation, the algorithm is called cut and branch.
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The Archimedean property is a property of certain ordered fields such as the rational numbers or the real numbers, stating that every two elements are within an integer multiple of each other. If a field contains two positive elements for which this is not true, then must be an infinitesimal, greater than zero but smaller than any integer unit fraction. Therefore, the negation of the Archimedean property is equivalent to the existence of infinitesimals.
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Locks-and-keys is a solution to dangling pointers in computer programming languages. The locks-and-keys approach represents pointers as ordered pairs (key, address) where the key is an integer value. Heap-dynamic variables are represented as the storage for the variable plus a cell for an integer lock value. When a variable is allocated, a lock value is created and placed both into the variable's cell and into the pointer's key cell.
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This type of logic error can be detected during compile time by using generics and is the primary motivation for using them. The above code fragment can be rewritten using generics as follows: List v = new ArrayList(); v.add("test"); Integer i = (Integer)v.get(0); // (type error) compilation-time error The type parameter `String` within the angle brackets declares the `ArrayList` to be constituted of `String` (a descendant of the `ArrayList`'s generic `Object` constituents).
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Register names are preceded with a % character and constants are literal, e.g.: shr.u64 %rd14, %rd12, 32; // shift right an unsigned 64-bit integer from %rd12 by 32 positions, result in %rd14 cvt.u64.u32 %rd142, %r112; // convert an unsigned 32-bit integer to 64-bit There are predicate registers, but compiled code in shader model 1.0 uses these only in conjunction with branch commands; the conditional branch is @%p14 bra $label; // branch to $label The `setp.cc.
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The table is usually constructed so that the value follows the name, to save time during the final lookup. In contrast, in I-code the address of the variable is determined in advance and the reference in code is replaced by that address. This avoids a runtime search through the variable table. Other optimizations include a separate `FOR/NEXT` routine used when the index variable is an INTEGER, and separate INTEGER and REAL math libraries.
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The output frequency of a quartz oscillator can be either that of the fundamental resonance or of a multiple of that resonance, called a harmonic frequency. Harmonics are an exact integer multiple of the fundamental frequency. But, like many other mechanical resonators, crystals exhibit several modes of oscillation, usually at approximately odd integer multiples of the fundamental frequency. These are termed "overtone modes", and oscillator circuits can be designed to excite them.
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In computer science, an integer is a datum of integral data type, a data type that represents some range of mathematical integers. Integral data types may be of different sizes and may or may not be allowed to contain negative values. Integers are commonly represented in a computer as a group of binary digits (bits). The size of the grouping varies so the set of integer sizes available varies between different types of computers.
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The value of an item with an integral type is the mathematical integer that it corresponds to. Integral types may be unsigned (capable of representing only non-negative integers) or signed (capable of representing negative integers as well). An integer value is typically specified in the source code of a program as a sequence of digits optionally prefixed with + or −. Some programming languages allow other notations, such as hexadecimal (base 16) or octal (base 8).
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Copelatus integer is a species of diving beetle. It is part of the genus Copelatus in the subfamily Copelatinae of the family Dytiscidae. It was described by Sharp in 1882.
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Kirkpatrick-Reisch sorting is a fast sorting algorithm for items with limited- size integer keys. It is notable for having an asymptotic time complexity that is better than radix sort.
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The numbers occur frequently in the study of the zeta function and are significant because is a -integer for primes where does not divide . The are called divided Bernoulli numbers.
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If all the phase contributions in a closed loop sum to an integer multiple of 2π at a certain frequency, a frequency mode is allowed to exist at that frequency.
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If p is a polynomial with integer coefficients then the functions z1,...,zn,ep(z1,...,zn) are all algebraic at a dense set of points of the hypersurface p=0.
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Another application of this Fourier series is to solve the Basel problem by using Parseval's theorem. The example generalizes and one may compute ζ(2n), for any positive integer n.
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"Page proportion is 2:3, type area proportioned in the Golden Section." Tschichold's "golden canon of page construction" is based on simple integer ratios, equivalent to Rosarivo's "typographical divine proportion".
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For example, for smoking cessation, there is an adult protocol and adolescent protocol. PhenX also provides a datatype (e.g., integer, string, date, enumerated, or encoded values) for each PhenX variable.
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Astronomical years do not have an integer number of days or lunar months. Any calendar that follows an astronomical year must have a system of intercalation such as leap years.
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Speranza is the co-author, with Renata Mansini and Włodzimierz Ogryczak, of the book Linear and Mixed Integer Programming for Portfolio Optimization (EURO Advanced Tutorials on Operational Research, Springer, 2015).
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If the boundary is a sphere in three space dimensions, the angular components of the eigenfunctions are spherical harmonics, and the radial components are Bessel functions of half-integer order.
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The stipulated leaves vary from small and entire (Artocarpus integer) to large and lobed (Artocarpus altilis), with the cordate leaves of the species A. altilis ending in long, sharp tips.
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In certain approaches to loop optimization, the set of the executions of the loop body is viewed as the set of integer points in a polyhedron defined by loop constraints.
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Many optimization problems are formulated as constrained or unconstrained mathematical programs, either linear programs (LP) or mixed integer programs (MIP), for which many solution techniques and commercial solvers are available.
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The values in an integer circuit are sets of integers and the gates compute set union, set intersection, and set complement, as well as the arithmetic operations addition and multiplication.
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Constants can also be used, and are represented by a `#` ("mesh") followed by the constant itself, written as a decimal number; only integer constants from 0 to 65535 are supported.
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In mathematics, a Hughes plane is one of the non-Desarguesian projective planes found by . There are examples of order p2n for every odd prime p and every positive integer n.
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This is this method that is used for proving that, if P ≠ NP (an unsolved conjecture), the complexity of every NP-complete problem is \Omega(n^k), for every positive integer .
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Many later CPU designs use similar mixed bit width, especially when the processor is meant for general-purpose usage where a reasonable balance of integer and floating point capability is required.
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It is just one particularly interesting aspect of the general theory about values of L-functions L(s) at integer values of s, and there is much empirical evidence supporting it.
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For contact process on all integer lattices, a major breakthrough came in 1990 when Bezuidenhout and Grimmett showed that the contact process also dies out almost surely at the critical value.
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There are five numbers (15, 16, 20, 24, and 30) whose totient number is 8. No positive integer smaller than 8 has as many such numbers, so 8 is highly totient.
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A prefix that indicates an integer, e.g., "bin", "tern," or "quatern", may be used in lieu of a numeral, to produce "binary", "ternary", or "quaternary" (2, 3, and 4 states respectively).
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A sequence of positive integers is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.
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David Singmaster read the article, analysed the structure and named the object the Ulam-Warburton cellular automaton in his 2003 article. Since then it has given rise to numerous integer sequences.
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Major improvements were improved superscalar execution and an extra integer unit. The PA-7100LC also implemented architectural improvements including the MAX-1 multimedia instructions, uncacheable memory pages, and bi-endian support.
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Then the function n \mapsto \dim_k M_n is called the Hilbert function of M. The function coincides with the integer- valued polynomial for large n called the Hilbert polynomial of M.
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Mansini is the co-author, with Włodzimierz Ogryczak and M. Grazia Speranza, of the book Linear and Mixed Integer Programming for Portfolio Optimization (EURO Advanced Tutorials on Operational Research, Springer, 2015).
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An easter egg in most of the games in the series references a supposed integer overflow bug in Civilization that causes a computer- controlled Gandhi, normally a highly peaceful leader, to become a nuclear warmonger. The game is said to start Gandhi's "aggression value" at 1 out of a maximum 255 possible for a 8-bit unsigned integer, making a computer- controlled Gandhi tend to avoid armed conflict. However, once a civilization achieves Democracy as its form of government, its leader's aggression value falls by 2. Under normal arithmetic principles, Gandhi's "1" would be reduced to "-1", but because the value is an 8-bit unsigned integer, it wraps around to "255," causing Gandhi to suddenly become the most aggressive opponent in the game.
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That is, polynomials with integer or rational coefficients have the unique factorization property. More precisely, every polynomial with rational coefficients may be factorized in a product :P(x)=q\,P_1(x)\cdots P_k(x), where is a rational number and P_1, \ldots, P_k are non-constant polynomials with integer coefficients that are irreducible and primitive; this means that none of the P_i may be written as the product two polynomials (with integer coefficients) that are neither 1 nor –1 (integers are considered as polynomials of degree zero). Moreover, this factorization is unique up to the order of the factors and the multiplication by –1 of an even number of factors. There are efficient algorithms for computing this factorization, which are implemented in most computer algebra systems.
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This revision changes the coding scheme to make it compatible with the AVX scheme, but with a differing prefix byte in order to avoid overlap between instructions introduced by AMD and instructions introduced by Intel. The revised instruction set no longer carries the name SSE5, which has been criticized for being misleading, but most of the instructions in the new revision are functionally identical to the original SSE5 specification—only the way the instructions are coded differs. The planned additions to the AMD instruction set consists of three subsets: #XOP: Integer vector multiply–accumulate instructions, integer vector horizontal addition, integer vector compare, shift and rotate instructions, byte permutation and conditional move instructions, floating point fraction extraction. #FMA4: Floating-point vector multiply–accumulate.
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To convert a number from a fixed point type with scaling factor R to another type with scaling factor S, the underlying integer must be multiplied by R and divided by S; that is, multiplied by the ratio R/S. Thus, for example, to convert the value 1.23 = 123/100 from a type with scaling factor R=1/100 to one with scaling factor S=1/1000, the underlying integer 123 must be multiplied by (1/100)/(1/1000) = 10, yielding the representation 1230/1000. If S does not divide R (in particular, if the new scaling factor S is greater than the original R), the new integer will have to be rounded. The rounding rules and methods are usually part of the language's specification.
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Equivalently, the vertices correspond to variables, and two variables form an edge if they share an inequality. The sparsity measure d of A is the minimum between the tree-depth of the graph of A and the tree-depth of the graph of the transpose of A. Let a be the numeric measure of A defined as the maximum absolute value of any entry of A. Let n be the number of variables of the integer program. Then it was shown in 2018 that integer programming can be solved in strongly polynomial and fixed-parameter tractable time parameterized by a and d. That is, for some computable function f and some constant k, integer programming can be solved in time f(a,d)n^k.
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If is not an integer, then round up to the next integer to get the appropriate index; the corresponding data value is the -th -quantile. On the other hand, if is an integer then any number from the data value at that index to the data value of the next can be taken as the quantile, and it is conventional (though arbitrary) to take the average of those two values (see Estimating quantiles from a sample). If, instead of using integers and , the “-quantile” is based on a real number with then replaces in the above formulas. Some software programs (including Microsoft Excel) regard the minimum and maximum as the 0th and 100th percentile, respectively; however, such terminology is an extension beyond traditional statistics definitions.
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Operator syntax makes it possible to use either form by declaring the feature (for example in `INTEGER`, but this applies to other basic classes and can be used in any other for which such an operator is appropriate): plus alias "+" (other: INTEGER): INTEGER \-- ... Normal function declaration... end The range of operators that can be used as "alias" is quite broad; they include predefined operators such as "+" but also "free operators" made of non-alphanumeric symbols. This makes it possible to design special infix and prefix notations, for example in mathematics and physics applications. Every class may in addition have one function aliased to "[]", the "bracket" operator, allowing the notation `a [i, ...]` as a synonym for `a.f (i, ...)` where `f` is the chosen function.
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In some cases, such as variable elimination ("projection"), PolyLib and PPL primarily use algorithms for the rational domain, and thus produce an approximation of the result for integer variables. It may be the case that this reduces the common experience with the Omega Library in which a minor change to one coefficient can cause a dramatic shift in the response of the library's algorithms. Polylib has some operations to produce exact results for Z-polyhedra (integer points bounded by polyhedra), but at the time of this writing, significant bugs have been reported. Note that bugs also exist in the Omega Library, including reliance on hardware-supplied integer types and cases of the full Presburger Arithmetic algorithms that were not implemented in the library.
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As only an integer number of flux quanta are allowed to penetrate the superconducting ring, clockwise or counter-clockwise mesoscopic supercurrents (typically 300 nA) are developed in the loop to compensate (screen or enhance) a non-integer external flux bias. When the applied flux through the loop area is close to a half integer number of flux quanta, the two lowest energy eigenstates of the loop will be a quantum superposition of the clockwise and counter-clockwise currents. The two lowest energy eigenstates differ only by the relative quantum phase between the composing current-direction states. Higher energy eigenstates correspond to much larger (macroscopic) persistent currents, that induce an additional flux quantum to the qubit loop, thus are well separated energetically from the lowest two eigenstates.
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The most relevant rules concern addition, subtraction, and multiplication: :even ± even = even :odd ± odd = even :even × integer = even Inserting appropriate values into the left sides of these rules, one can produce 0 on the right sides: :2 − 2 = 0 :−3 + 3 = 0 :4 × 0 = 0 The above rules would therefore be incorrect if zero were not even. At best they would have to be modified. For example, one test study guide asserts that even numbers are characterized as integer multiples of two, but zero is "neither even nor odd". Accordingly, the guide's rules for even and odd numbers contain exceptions: :even ± even = even (or zero) :odd ± odd = even (or zero) :even × nonzero integer = even These rules are given, but they are not quoted verbatim.
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A variable or value of that type is usually represented as a fraction m/n where m and n are two integer numbers, either with a fixed or arbitrary precision. Depending on the language, the denominator n may be constrained to be non-zero, and the two numbers may be kept in reduced form (without any common divisors except 1). Languages that support a rational data type usually provide special syntax for building such values, and also extend the basic arithmetic operations ('+', '−', '×', '/', integer powers) and comparisons ('=', '<', '>', '≤') to act on them — either natively or through operator overloading facilities provided by the language. These operations may be translated by the compiler into a sequence of integer machine instructions, or into library calls.
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Fractint is a freeware computer program to render and display many kinds of fractals. The program originated on MS-DOS, then ported to the Atari ST, Linux, and Macintosh. During the early 1990s, Fractint was the definitive fractal generating program for personal computers. The name is a portmanteau of fractal and integer, since the first versions of Fractint used only integer arithmetic (also known as fixed-point arithmetic), for faster rendering on computers without math coprocessors.
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A counter machine consists of: # Labeled unbounded integer-valued registers: a finite (or infinite in some models) set of registers r0 ... rn each of which can hold any single non-negative integer (0, 1, 2, ... - i.e. unbounded). The registers do their own arithmetic; there may or may not be one or more special registers e.g. "accumulator" (See Random-access machine for more on this). # A state register that stores/identifies the current instruction to be executed.
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The original implementation of the Unix operating system stored system time as a 32-bit signed integer representing the number of seconds past the Unix epoch: midnight UTC, 1 January 1970. This value will roll over on 19 January 2038. This problem has been addressed in most modern Unix and Unix-like operating systems by storing system time as a 64-bit signed integer, although individual applications, protocols, and file formats will still need to be changed as well.
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It is often found on algae, and is most closely associated with the seaweed Fucus vesiculosus. It lives in shallow water, and is often found around artificial constructions, such as docks. It is an omnivore, feeding on debris and preying on small crustaceans, especially harpacticoid copepods, but consumes a greater proportion of macrozooplankton than other common littoral mysids, such as Neomysis integer and Praunus inermis. P. fleuosus is less gregarious than species such as N. integer.
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Therefore, the absolute value of e z can be 1 only if e x is 1; since x is real, that happens only if x = 0. Therefore z is pure imaginary and cos(y) + i sin(y) = 1. Since y is real, that happens only if cos(y) = 1 and sin(y) = 0, so that y is an integer multiple of 2\. Consequently the singular points of this function occur at : z = a nonzero integer multiple of 2i.
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There are only 242 different values (if +0 and −0 are regarded as different), because 14 of the bit patterns represent NaNs. The values between 0 and 16 have the same bit pattern as minifloat or two's-complement integer. The first pattern with a different value is 00010001, which is 18 as a minifloat and 17 as a two's-complement integer. This coincidence does not occur at all with negative values, because this minifloat is a signed-magnitude format.
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Integer precision (bit-size) must be kept during vector instruction execution. The correct vector instruction must be chosen based on the size and behavior of the internal integers. Also, with mixed integer types, extra care must be taken to promote/demote them correctly without losing precision. Special care must be taken with sign extension (because multiple integers are packed inside the same register) and during shift operations, or operations with carry bits that would otherwise be taken into account.
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The mass defect used in nuclear physics is different from its use in mass spectrometry. In nuclear physics, the mass defect is the difference in the mass of a composite particle and the sum of the masses of its component parts. In mass spectrometry the mass defect is defined as the difference between the exact mass and the nearest integer mass. The Kendrick mass defect is the exact Kendrick mass subtracted from the nearest integer Kendrick mass.
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The norm of an algebraic integer is again an integer, because it is equal (up to sign) to the constant term of the characteristic polynomial. In algebraic number theory one defines also norms for ideals. This is done in such a way that if I is a nonzero ideal of OK, the ring of integers of the number field K, N(I) is the number of residue classes in O_K / I – i.e. the cardinality of this finite ring.
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Let n ≥ 0 be a non-negative integer. The graph Γ is said to satisfy e(Γ) ≤ n if for every finite collection F of edges of Γ the graph Γ − F has at most n infinite connected components. By definition, e(Γ) = m if e(Γ) ≤ m and if for every 0 ≤ n < m the statement e(Γ) ≤ n is false. Thus e(Γ) = m if m is the smallest nonnegative integer n such that e(Γ) ≤ n.
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The wave nature of light becomes important at small scales such as those in ASPs, meaning a pure-moire model of ASP function is insufficient. However, at half-integer multiples of the Talbot depth, the periodicity of the diffraction grating is recapitulated, and the moire effect is rescued. By building ASPs where the vertical separation between the gratings is approximately equal to a half-integer multiple of the Talbot depth, the sinusoidal sensitivity with incident angle is observed.
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Secondly, in Java an extra cast is needed whenever one tries to directly dereference a primitive value, as it will not be boxed automatically. The expression `((Integer)42).toString()` will convert an integer literal to string in Java while `42.ToString()` performs the same operation in C#. This is because the latter one is an instance call on the primitive value `42`, while the former one is an instance call on an object of type `java.lang.Integer`.
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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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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 expression is an effective initial guess for computing the square root of a 32-bit integer using Newton's method. CLZ can efficiently implement null suppression, a fast data compression technique that encodes an integer as the number of leading zero bytes together with the nonzero bytes. It can also efficiently generate exponentially distributed integers by taking the clz of uniformly random integers. The log base 2 can be used to anticipate whether a multiplication will overflow, since .
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In March 2019, David Harvey and Joris van der Hoeven submitted an article presenting an integer multiplication algorithm with a claimed complexity of O(n\log n).David Harvey, Joris Van Der Hoeven (2019). Integer multiplication in time O(n log n) The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal. The algorithm is not considered practically useful, as its advantages only appear when multiplying extremely large numbers (having more than bits).
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This means the following: take a point (α,β) in the plane, and then consider the sequence of points :(2α,2β), (3α,3β), ... . For each of these, multiply the distance to the closest line with integer x-coordinate by the distance to the closest line with integer y-coordinate. This product will certainly be at most 1/4. The conjecture makes no statement about whether this sequence of values will converge; it typically does not, in fact.
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The question of whether integer multiplication or table lookup operations should be permitted goes back to ; see also . Other more specialized models of computation such as the parallel random access machine have also been considered.; comment in ; ; ; . showed that in some cases the multiplications or table lookups required by some integer sorting algorithms could be replaced by customized operations that would be more easily implemented in hardware but that are not typically available on general- purpose computers.
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The Hausdorff dimension is useful for studying structurally complicated sets, especially fractals. The Hausdorff dimension is defined for all metric spaces and, unlike the dimensions considered above, can also have non-integer real values.Fractal Dimension , Boston University Department of Mathematics and Statistics The box dimension or Minkowski dimension is a variant of the same idea. In general, there exist more definitions of fractal dimensions that work for highly irregular sets and attain non-integer positive real values.
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Every cyclic group is abelian. That is, its group operation is commutative: (for all g and h in G). This is clear for the groups of integer and modular addition since , and it follows for all cyclic groups since they are all isomorphic to these standard groups. For a finite cyclic group of order n, gn is the identity element for any element g. This again follows by using the isomorphism to modular addition, since for every integer k.
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For the purpose of dimensional analysis, the units of angular velocity and angular acceleration are s−1 and s−2 respectively. Likewise, the phase difference of two waves can also be measured in radians. For example, if the phase difference of two waves is (k⋅2) radians, where k is an integer, they are considered in phase, whilst if the phase difference of two waves is (), where k is an integer, they are considered in antiphase.
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A domain describes the set of possible values for a given attribute, and can be considered a constraint on the value of the attribute. Mathematically, attaching a domain to an attribute means that any value for the attribute must be an element of the specified set. The character string "ABC", for instance, is not in the integer domain, but the integer value 123 is. Another example of domain describes the possible values for the field "CoinFace" as ("Heads","Tails").
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The main difference between mesons and baryons is that mesons have integer spin (thus are bosons) while baryons are fermions (half-integer spin). Because mesons are bosons, the Pauli exclusion principle does not apply to them. Because of this, they can act as force mediating particles on short distances, and thus play a part in processes such as the nuclear interaction. Since mesons are composed of quarks, they participate in both the weak and strong interactions.
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The latter can be proved by contradiction. The previous example employed the contrapositive of a definition to prove a theorem. One can also prove a theorem by proving the contrapositive of the theorem's statement. To prove that if a positive integer N is a non-square number, its square root is irrational, we can equivalently prove its contrapositive, that if a positive integer N has a square root that is rational, then N is a square number.
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The inner product on the Leech lattice is defined as 1/8 the sum of the products of respective co-ordinates of the two multiplicand vectors; it is an integer. The square norm of a vector is its inner product with itself, always an even integer. It is common to speak of the type of a Leech lattice vector: half the square norm. Subgroups are often named in reference to the types of relevant fixed points.
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Let k be a positive integer. In number theory, Jordan's totient function J_k(n) of a positive integer n is the number of k-tuples of positive integers all less than or equal to n that form a coprime (k+1)-tuple together with n. (A tuple is coprime if and only if it is coprime as a set.) This is a generalisation of Euler's totient function, which is J_1. The function is named after Camille Jordan.
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The Alpha 21064 is a superpipelined dual-issue superscalar microprocessor that executes instructions in-order. It is capable of issuing up to two instructions every clock cycle to four functional units: an integer unit, a floating-point unit (FPU), an address unit, and a branch unit. The integer pipeline is seven stages long, and the floating-point pipeline ten stages. The first four stages of both pipelines are identical and are implemented by the I-Box.
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The Alpha 21164 is a four-issue superscalar microprocessor capable of issuing a maximum of four instructions per clock cycle to four execution units: two integer and two floating-point. The integer pipeline is seven stages long, and the floating- point pipeline is ten stages long. The 21164 implemented a 43-bit virtual address and a 40-bit physical address. It was therefore capable of addressing 8 TB of virtual memory and 1 TB of physical memory.
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CCSDS 122.0 makes use of a three-level two-dimensional discrete wavelet transform (DWT) using a biorthogonal 9/7 tap filters, followed by a bit-plane encoder. It has some design commonalities with ICER and JPEG 2000 that use similar wavelet coding schemes. The transform (DWT) can be computed using either floating-point or integer arithmetic. The integer transform uses non-linear approximation of the 9/7 wavelet and it is used in the lossless coding scheme.
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The spin–statistics theorem implies that half-integer–spin particles are subject to the Pauli exclusion principle, while integer-spin particles are not. Only one fermion can occupy a given quantum state at any time, while the number of bosons that can occupy a quantum state is not restricted. The basic building blocks of matter such as protons, neutrons, and electrons are fermions. Particles such as the photon, which mediate forces between matter particles, are bosons.
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Double-precision integer multiplies and divides are included, as multiplies and divides that produce the high word of the result. The ISA document recommends that implementors of CPUs and compilers fuse a standardized sequence of high and low multiply and divide instructions to one operation if possible. The floating-point instructions (set F) includes single-precision arithmetic and also comparison-branches similar to the integer arithmetic. It requires an additional set of 32 floating-point registers.
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The neutrino was first confirmed experimentally in 1956 by Frederick Reines and Clyde Cowan, two and a half years before Pauli's death. On receiving the news, he replied by telegram: "Thanks for message. Everything comes to him who knows how to wait. Pauli." In 1940, he re-derived the spin-statistics theorem, a critical result of quantum field theory which states that particles with half-integer spin are fermions, while particles with integer spin are bosons.
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In mathematics, and especially general topology, the prime integer topology and the relatively prime integer topology are examples of topologies on the set of positive whole numbers, i.e. the set }. To give the set Z+ a topology means to say which subsets of Z+ are "open", and to do so in a way that the following axioms are met: # The union of open sets is an open set. # The finite intersection of open sets is an open set.
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The number 4,294,967,295, equivalent to the hexadecimal value , is the maximum value for a 32-bit unsigned integer in computing. It is therefore the maximum value for a variable declared as an unsigned integer (usually indicated by the codeword) in many programming languages running on modern computers. The presence of the value may reflect an error, overflow condition, or missing value. This value is also the largest memory address for CPUs using a 32-bit address bus.
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Mathematical programs involving binary variables and disjunction definitions for modeling discrete choices are called disjunctive programs. Disjunctive programs have many applications, including ordering of tasks in a production process, organizing complex projects in a time saving manner and choosing the optimal route in a circuit. Procedures for linear and nonlinear disjunctive programming extensions are implemented within EMP. Linear disjunctive programs are reformulated as mixed integer programs (MIPs) and nonlinear disjunctive programs are reformulated as mixed integer nonlinear programs (MINLPs).
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Geometrical proof that an irrational number exists: If the isosceles right triangle ABC had integer side lengths, so had the strictly smaller triangle A'B'C. Repeating this construction would obtain an infinitely descending sequence of integer side lengths. In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase "there exist(s)", or it might a universal statement whose last quantifier is existential (e.g.
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This generates the impression of sound at integer multiple frequencies of the fundamental known as harmonics, or more precisely, harmonic partials. For most string instruments and other long and thin instruments such as a bassoon, the first few overtones are quite close to integer multiples of the fundamental frequency, producing an approximation to a harmonic series. Thus, in music, overtones are often called harmonics. Depending upon how the string is plucked or bowed, different overtones can be emphasized.
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His concept "Dyson's transform" led to one of the most important lemmas of Olivier Ramaré's theorem: that every even integer can be written as a sum of no more than six primes.
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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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Every non- zero vector of II25,1 can be written uniquely as a positive integer multiple of a primitive vector, so to classify all vectors it is sufficient to classify the primitive vectors.
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In computational number theory, a factor base is a small set of prime numbers commonly used as a mathematical tool in algorithms involving extensive sieving for potential factors of a given integer.
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So we can drop the ± from A2n± notation. The cusps are then given by the zero-level-sets of the representatives of the A2n equivalence classes, where n ≥ 1 is an integer.
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Consider the case where , where is a positive integer. We proceed by mathematical induction. In the base case, , so . We have already shown that the inequality holds when , so we are done.
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Integer Arithmetic Logic Unit (ALU): Supports full 32-bit precision for all instructions, consistent with standard programming language requirements. It is also optimized to efficiently support 64-bit and extended precision operations.
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In mathematics, specifically in number theory, a binomial number is an integer which can be obtained by evaluating a homogeneous polynomial containing two terms. It is a generalization of a Cunningham number.
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The respective equalities follow because χ is a class-function of G and N is a normal subgroup. The integer e appearing in the statement of the theorem is this common multiplicity.
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If A is an involutory matrix then every integer power of A is involutory. In fact, An will be equal to A if n is odd and I if n is even.
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Chryseobacterium artocarpi is a Gram-negative and non-spore-forming bacteria from the genus of Chryseobacterium which has been isolated from rhizosphere soil from the tree Artocarpus integer. Chryseobacterium artocarpi produces flexirubin.
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Klaus von Klitzing (, born 28 June 1943, Schroda) is a German physicist, known for discovery of the integer quantum Hall effect, for which he was awarded the 1985 Nobel Prize in Physics.
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The TomSym symbolic source transformation makes it possible to define any the set of decision variables (both continuous and integer) and any type of constraint as well as scalars and constant parameters.
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The set of HSASMs is a superset of the ASMs. The extreme points of the convex hull of the set of r-spin HSASMs are themselves integer multiples of the usual ASMs.
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Algebraic geometry now finds applications in statistics, control theory, robotics, error-correcting codes, phylogenetics and geometric modelling. There are also connections to string theory, game theory, graph matchings, solitons and integer programming.
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Neutrinos have half-integer spin (); therefore they are fermions. Neutrinos are leptons. They have only been observed to interact through the weak force, although it is assumed that they also interact gravitationally.
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A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime. The first Fibonacci primes are : :2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, ....
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On average, x32 is 5–8% faster on the SPEC CPU integer benchmarks compared to x86-64. There is no speed advantage over x86-64 in the SPEC CPU floating-point benchmarks.
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If both are odd or even, the resulting integer will be even, and 2 is the only even prime. The first few quartan primes are :2, 17, 97, 257, 337, 641, 881, … .
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There are quantitative forms of this, stating more precisely bounds (depending on degree) on the largest absolute value of a conjugate that imply that an algebraic integer is a root of unity.
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Denoting this integer by v, the problem reduces to solving the following linear Diophantine equation: : Kuttaka may be applied to solve this equation. The smallest solution is :u = 346,688,814 and v = 32,202.
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Denoting this integer by v, the problem reduces to solving the following linear Diophantine equation: :. Kuttaka may be applied to solve this equation. The smallest solution is :u = 118,076,020 and v = 171,872.
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The three functions `Low()`, `High()` and `Length()` would still retrieve lower bound, upper bound and array length correctly. Example: type TIntArray = array of Integer; T2DimIntArray = array of array of Integer; var a : TIntArray; a2 : T2DimIntArray; i,j: Integer; begin SetLength(a,10); // allocate 10 storage for i := Low(a) to High(a) do ... // do something with a[i] SetLength(a2,10,10); // allocate 10 x 10 storage for i := Low(a2) to High(a2) do for j := Low(a2[i]) to High(a2[i]) do ... // do something with a[i,j] end; Assignment between dynamic arrays copies the reference of the source array to the destination. If a real copy is required, one can use the `Copy` function. Example: type TIntegerArray = array of Integer; var a,b: TIntegerArray; begin ... // initialize a and b a := b; // a now points to the same array pointed by b a[1] := 0; // b[1] should be 0 as well after this a := Copy(b,3,5); // Copy 5 elements from b starting from index 3 // a would access it from 0 to 4 however end.
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The classical integer sorting algorithms of pigeonhole sort, counting sort, and radix sort are widely used and practical.; . Much of the subsequent research on integer sorting algorithms has focused less on practicality and more on theoretical improvements in their worst case analysis, and the algorithms that come from this line of research are not believed to be practical for current 64-bit computer architectures, although experiments have shown that some of these methods may be an improvement on radix sorting for data with 128 or more bits per key.. Additionally, for large data sets, the near-random memory access patterns of many integer sorting algorithms can handicap them compared to comparison sorting algorithms that have been designed with the memory hierarchy in mind.. Integer sorting provides one of the six benchmarks in the DARPA High Productivity Computing Systems Discrete Mathematics benchmark suite,DARPA HPCS Discrete Mathematics Benchmarks, Duncan A. Buell, University of South Carolina, retrieved 2011-04-20. and one of eleven benchmarks in the NAS Parallel Benchmarks suite.
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Every node in this cluster has a unique alias, an 8 bit (1 byte) integer, which can be assigned to the node a priori or set at any time via the configuration interface.
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This section describes textbook methods that can be convenient when computing by hand. These methods are not used for computer computations because they use integer factorization, which is currently slower than polynomial factorization.
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The `Byte`, `Short`, `Integer`, `Long`, `Float`, and `Double` wrapper classes are all subclasses of the class. The wrapper classes `BigDecimal` and `BigInteger ` are not one of the primitive wrapper classes but are immutable.
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The Brunn-Minkowski inequality continues to be relevant to modern geometry and algebra. For instance, there are connections to algebraic geometry, and combinatorial versions about counting sets of points inside the integer lattice.
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In Wolfram Mathematica the backslash is used this way for integer divide. In MATLAB and GNU Octave the backslash is used for left matrix divide, while the slash is for right matrix divide.
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Its magnitude is A, and it completes one cycle every 2/ω seconds. θ is the angle it forms with the real axis at t = n•2/ω, for integer values of n.
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This definition of Euclidean division may be interpreted geometrically in the complex plane (see the figure), by remarking that the distance from a complex number to the closest Gaussian integer is at most .
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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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A male's singing skills do not guarantee him instant success: other, silent, males may be waiting nearby to intercept the females he attracts.Huber et al., p. 46, describe this behaviour in Gryllus integer.
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The shift instructions perform arithmetic right shift, and logical left and right shifts. The shift amount is given by a register or literal. Logical and shift instructions use the integer operate instruction formats.
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The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5×4×3×2×1 = 120.
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For specifying attributes and actions we use user-defined data types (nat,int(integer), real,bool(boolean), string,date etc.). Using these types we construct new data types like lists, records, enumeration etc.
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This article collects together a variety of proofs of Fermat's little theorem, which states that :a^p \equiv a \pmod p for every prime number p and every integer a (see modular arithmetic).
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Java has classes that correspond to scalar values, such as Integer, Boolean and Float. Combined with autoboxing (automatic usage-driven conversion between object and value), this effectively allows nullable variables for scalar values.
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The details are outlined in these notes. The Finite Difference Coefficients Calculator constructs finite difference approximations for non-standard (and even non-integer) stencils given an arbitrary stencil and a desired derivative order.
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A delta sigma synthesizer adds a randomization to programmable-N frequency divider of the fractional-N synthesizer. This is done to shrink sidebands created by periodic changes of an integer-N frequency divider.
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FIX messages are formed from a number of fields; each field is a tag value pairing that is separated from the next field by a delimiter SOH (0x01). The tag is an integer that indicates the meaning of the field. The value is an array of bytes that hold a specific meaning for the particular tag (e.g. tag 48 is securityID, a string that identifies the security; tag 22 is IDSource, an integer that indicates the identifier class being used).
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Polygonal numbers: These are numbers that can be represented as dots that are arranged in the shape of a regular polygon, including Triangular numbers, Square numbers, Pentagonal numbers, Hexagonal numbers, Heptagonal numbers, Octagonal numbers, Nonagonal numbers, Decagonal numbers, Hendecagonal numbers, and Dodecagonal numbers. There are many other famous integer sequences, such as the sequence of Fibonacci numbers, the sequence of factorials, the sequence of perfect numbers, and so forth, many of which are enumerated in the On-Line Encyclopedia of Integer Sequences.
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In this section, we show that factoring over Q (the rational numbers) and over Z (the integers) is essentially the same problem. The content of a polynomial p ∈ Z[X], denoted "cont(p)", is, up to its sign, the greatest common divisor of its coefficients. The primitive part of p is primpart(p)=p/cont(p), which is a primitive polynomial with integer coefficients. This defines a factorization of p into the product of an integer and a primitive polynomial.
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Although every concrete Boolean algebra is a Boolean algebra, not every Boolean algebra need be concrete. Let n be a square-free positive integer, one not divisible by the square of an integer, for example 30 but not 12. The operations of greatest common divisor, least common multiple, and division into n (that is, ¬x = n/x), can be shown to satisfy all the Boolean laws when their arguments range over the positive divisors of n. Hence those divisors form a Boolean algebra.
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FLOPS and MIPS are units of measure for the numerical computing performance of a computer. Floating-point operations are typically used in fields such as scientific computational research. The unit MIPS measures integer performance of a computer. Examples of integer operation include data movement (A to B) or value testing (If A = B, then C). MIPS as a performance benchmark is adequate when a computer is used in database queries, word processing, spreadsheets, or to run multiple virtual operating systems.
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Early digital audio gear and video games used 8-bit integer samples or less. Roland's classic TR-909 drum machine used 6-bit integer samples. The number of bits used in each sample directly affects the signal-to-noise ratio and dynamic range of the digital signal, specifically by determining the amplitude of a kind of noise called quantization noise that is similar to low-pass- filtered white noise. Resolution reduction intentionally reduces the number of bits used for audio samples.
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The genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n handles (i.e. an oriented surface of genus n). Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing. The non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps (i.e.
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A 128-bit type provided by a C compiler can be available in Perl via the `Math::Int128` module. A 128-bit register can store 2128 (over 3.40 × 1038) different values. The range of integer values that can be stored in 128 bits depends on the integer representation used. With the two most common representations, the range is 0 through 340,282,366,920,938,463,463,374,607,431,768,211,455 (2128 − 1) for representation as an (unsigned) binary number, and −170,141,183,460,469,231,731,687,303,715,884,105,728 (−2127) through 170,141,183,460,469,231,731,687,303,715,884,105,727 (2127 − 1) for representation as two's complement.
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The data structure representing the mesh provides support for two basic operations: inserting triangles and removing triangles. It also supports an edge collapse operation that is useful in triangle decimation schemes. The structure provides no support for the vertex positions, but it does assume that each vertex is assigned a unique integer identifier, typically the index of that vertex in an array of contiguous vertex positions. A mesh vertex is defined by a single integer and is denoted by hvi.
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In number theory, a Thabit number, Thâbit ibn Kurrah number, or 321 number is an integer of the form 3 \cdot 2^n - 1 for a non-negative integer n. The first few Thabit numbers are: :2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, ... The 9th Century mathematician, physician, astronomer and translator Thābit ibn Qurra is credited as the first to study these numbers and their relation to amicable numbers.
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The R4600 was a simple design; it was a scalar processor, issuing up to one instruction per cycle to its integer pipeline or floating-point unit (FPU). Most integer instructions have a single cycle latency and throughput, except for multiplies and divides. Multiplies, 32-bit and 64-bit, have an eight-cycle latency and six-cycle throughput. Divides have a 32-cycle latency and throughput for 32-bit integers and a 61-cycle latency and throughput for 64-bit integers.
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The Euler genus of a graph is the minimal integer n such that the graph can be embedded in an orientable surface of (orientable) genus n/2 or in a non-orientable surface of (non-orientable) genus n. A graph is orientably simple if its Euler genus is smaller than its non-orientable genus. The maximum genus of a graph is the maximal integer n such that the graph can be 2-cell embedded in an orientable surface of genus n.
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Some authors classify tensor densities into the two types called (authentic) tensor densities and pseudotensor densities in this article. Other authors classify them differently, into the types called even tensor densities and odd tensor densities. When a tensor density weight is an integer there is an equivalence between these approaches that depends upon whether the integer is even or odd. Note that these classifications elucidate the different ways that tensor densities may transform somewhat pathologically under orientation-reversing coordinate transformations.
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Fermat's little theorem states that if p is prime and a is coprime to p, then ap−1 − 1 is divisible by p. For an integer a > 1, if a composite integer x divides ax−1 − 1, then x is called a Fermat pseudoprime to base a. It follows that if x is a Fermat pseudoprime to base a, then x is coprime to a. Some sources use variations of this definition, for example to allow only odd numbers to be pseudoprimes.
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This equation can be viewed as a Diophantine equation, that is, an equation for which only integer solutions are sought. In this case, the solution set is the empty set, since 2 is not the square of an integer. However, if one searches for real solutions, there are two solutions, and ; in other words, the solution set is . When an equation contains several unknowns, and when one has several equations with more unknowns than equations, the solution set is often infinite.
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In the 1760s, Johann Heinrich Lambert proved that the number (pi) is irrational: that is, it cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven, and Nicolas Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich.
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If F(X) = a_0 + a_1 X + \dots + a_n X^n is a polynomial with integer coefficients, then F is called primitive if the greatest common divisor of all the coefficients a_0, a_1, \dots, a_n is 1; in other words, no prime number divides all the coefficients. Proof: Clearly the product f(x).g(x) of two primitive polynomials has integer coefficients. Therefore, if it is not primitive, there must be a prime p which is a common divisor of all its coefficients.
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Fibonacci sequence on a building in Gothenburg In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description.
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The original proof of the irrationality of the non-square natural numbers depends on Euclid's lemma. Many proofs of the irrationality of the square roots of non-square natural numbers implicitly assume the fundamental theorem of arithmetic, which was first proven by Carl Friedrich Gauss in his Disquisitiones Arithmeticae. This asserts that every integer has a unique factorization into primes. For any rational non-integer in lowest terms there must be a prime in the denominator which does not divide into the numerator.
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Decimal may also refer specifically to the digits after the decimal separator, such as in " is the approximation of to two decimals". The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form , where is an integer, and is a non- negative integer. The decimal system has been extended to infinite decimals for representing any real number, by using an infinite sequence of digits after the decimal separator (see Decimal representation).
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The quotients are generally found by rounding the real and complex parts of the exact ratio (such as the complex number ) to the nearest integers. The second difference lies in the necessity of defining how one complex remainder can be "smaller" than another. To do this, a norm function is defined, which converts every Gaussian integer into an ordinary integer. After each step of the Euclidean algorithm, the norm of the remainder is smaller than the norm of the preceding remainder, .
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A locally cyclic group is a group in which each finitely generated subgroup is cyclic. An example is the additive group of the rational numbers: every finite set of rational numbers is a set of integer multiples of a single unit fraction, the inverse of their lowest common denominator, and generates as a subgroup a cyclic group of integer multiples of this unit fraction. A group is locally cyclic if and only if its lattice of subgroups is a distributive lattice..
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There are two representations of the rational numbers in common use. One uses a sign ( + or – ), followed by a nonnegative integer (numerator), followed by a division symbol, followed by a positive integer (denominator). For example, –58/2975 . (If no sign is written, the sign is + .) The other is a sign followed by a sequence of digits, with a radix point (called a decimal point in base ten) somewhere in the sequence, and an overscore over one or more of the rightmost digits.
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The integer programming approach to RAPTOR produces higher quality models than other protein threading methods. Most threading software use dynamic programming to optimize their scoring functions when aligning a sequence with a template. Dynamic programming is much easier to implement than integer programming; however if a scoring function has pairwise contact potential included, dynamic programming cannot globally optimize such a scoring function and instead just generates a local optimal alignment. Pairwise contacts are very conserved in protein structure and crucial for prediction accuracy.
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In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function f(x) = x^n is an even function if n is an even integer, and it is an odd function if n is an odd integer.
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A major and peculiar feature of the language is the concept of patterns. In another programming language, such as C++, one would have several classes and procedures. BETA expresses both of these concepts using patterns. For example, a simple class in C++ would have the form class point { int x, y; }; In BETA, the same class could be represented by the pattern point: (# x, y: @integer #) That is, a class called point will have two fields, x and y, of type integer.
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Recamán's sequence was named after its inventor, Colombian mathematician Bernardo Recamán Santos, by Neil Sloane, creator of the On-Line Encyclopedia of Integer Sequences (OEIS). The OEIS entry for this sequence is . Even when Neil Sloane has collected more than 325,000 sequences since 1964, the Recamán's sequence was referenced in his paper My favorite integer sequences.N. J. A. Sloane, Sequences and their Applications (Proceedings of SETA '98), C. Ding, T. Helleseth and H. Niederreiter (editors), Springer-Verlag, London, 1999, pp. 103–130.
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The precise definition of a mathematical term, such as "even" meaning "integer multiple of two", is ultimately a convention. Unlike "even", some mathematical terms are purposefully constructed to exclude trivial or degenerate cases. Prime numbers are a famous example. Before the 20th century, definitions of primality were inconsistent, and significant mathematicians such as Goldbach, Lambert, Legendre, Cayley, and Kronecker wrote that 1 was prime. The modern definition of "prime number" is "positive integer with exactly 2 factors", so 1 is not prime.
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Heath-Brown (1983), Theorem. More generally, Birch's theorem says that for any odd positive integer d, there is an integer N such that for all n ≥ N, every hypersurface of degree d in Pn over Q has a rational point. For hypersurfaces of smaller dimension (in terms of their degree), things can be more complicated. For example, the Hasse principle fails for the smooth cubic surface 5x3 \+ 9y3 \+ 10z3 \+ 12w3 = 0 in P3 over Q, by Ian Cassels and Richard Guy.
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In the Dirac equation, a four- > component spinor (known as a "Dirac spinor") was used for the electron wave- > function. Relativistic spin explained gyromagnetic anomaly, which was (in > retrospect) first observed by Samuel Jackson Barnett in 1914 (see > Einstein–de Haas effect). In 1940, Pauli proved the spin-statistics theorem, > which states that fermions have half-integer spin and bosons have integer > spin. In retrospect, the first direct experimental evidence of the electron > spin was the Stern–Gerlach experiment of 1922.
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For progress on the general problem of finding planar drawings with rational edge lengths, see . It is possible in polynomial time to find a drawing of a planar 3-tree with integer coordinates minimizing the area of the bounding box of the drawing, and to test whether a given planar 3-tree may be drawn with its vertices on a given set of points.For the drawings with integer coordinates, see , and for the drawings on a given vertex set, see .
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The E8 root system also contains a copy of A8 (which has 72 roots) as well as E6 and E7 (in fact, the latter two are usually defined as subsets of E8). In the odd coordinate system, E8 is given by taking the roots in the even coordinate system and changing the sign of any one coordinate. The roots with integer entries are the same while those with half-integer entries have an odd number of minus signs rather than an even number.
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Neither `List` nor `List` is a subtype of the other; even though `Integer` is a subtype of `Number`. So, any method that takes `List` as a parameter does not accept an argument of `List`. If it did, it would be possible to insert a `Number` that is not an `Integer` into it; which violates type safety. Here is an example that demonstrates how type safety would be violated if `List` were a subtype of `List`: List ints = new ArrayList(); ints.
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In a linear laser, an integer multiple of the wavelength fits the length of the laser cavity. This means that in traveling back and forth the laser light goes through an integer number of cycles of its frequency. In the case of a ring laser the same applies: the number of cycles of the laser light's frequency is the same in both directions. This quality of the same number of cycles in both directions is preserved when the ring laser setup is rotating.
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In computer science, they count the number of steps needed for binary search and related algorithms. Other areas in which the binary logarithm is frequently used include combinatorics, bioinformatics, the design of sports tournaments, and photography. Binary logarithms are included in the standard C mathematical functions and other mathematical software packages. The integer part of a binary logarithm can be found using the find first set operation on an integer value, or by looking up the exponent of a floating point value.
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The first example of SPT order is the Haldane phase of odd-integer-spin chain. It is a SPT phase protected by SO(3) spin rotation symmetry. (Note that Haldane phases of even-integer-spin chain do not have SPT order.) A more well known example of SPT order is the topological insulator of non-interacting fermions, a SPT phase protected by U(1) and time reversal symmetry. On the other hand, fractional quantum Hall states are not SPT states.
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Algorithms and computer programs have been developed for determining nonisomorphic finite semigroups of a given order. These have been applied to determine the nonisomorphic semigroups of small order. (Accessed on 9 May 2009) The number of nonisomorphic semigroups with n elements, for n a nonnegative integer, is listed under in the On-Line Encyclopedia of Integer Sequences. lists the number of non-equivalent semigroups, and the number of associative binary operations, out of a total of nn2, determining a semigroup.
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The algorithm described in: "A Method for the Construction of Minimum-Redundancy Codes" David A. Huffman, Proceedings of the I.R.E. is: algorithm compute huffman code is input: message ensemble (set of (message, probability)). base D. output: code ensemble (set of (message, code)). 1- sort the message ensemble by decreasing probability. 2- N is the cardinal of the message ensemble (number of different messages). 3- compute the integer n_0 such as 2≤n_0≤D and (N−n_0)/(D−1) is integer.
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From this paper André Neveu and John Schwarz developed a string theory with both fermions and bosons.] According to quantum mechanics, particles can be divided into two types: bosons and fermions. The distinction between bosons and fermions is basic. Fermions are particles which have half integer spin (1/2, 3/2, 5/2 and so on), measured in units of Planck's constant and bosons are particles which have integer spin (0, 1, 2 and so on), measured in units of Planck's constant.
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However, a mutant bamboo plant flowering at an integer multiple of its population's flowering interval would release its seeds only during collective flowering events, and would release more seeds than the average plant in the population. It could, therefore, take over the population, establishing a flowering interval that is an integer multiple of the previous flowering interval. The hypothesis predicts that observed bamboo flowering intervals should factorize into small prime numbers. The mass fruiting also has direct economic and ecological consequences, however.
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The first term is not needed when is finite, that is when is not a non-positive integer and the real part of goes to negative infinity, whereas the second term is not needed when is finite, that is, when is a not a non-positive integer and the real part of goes to positive infinity. There is always some solution to Kummer's equation asymptotic to as . Usually this will be a combination of both and but can also be expressed as .
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These reference tones are chosen such that all chord tones have integer relations to these reference tones. For each chord there exists a lower and an upper reference tone, with all chord tones being integer multiples of the frequency of the lower reference tone and integer fractions of the frequency of the upper reference tone. In quadruple notation there are only positive (or zero) values if the chord is related to the lower reference tone, and only negative (or zero) values if the chord is related to the higher reference tone. To get a single numerical value describing the complexity of a chord, Vogel builds a weighted sum of the quadruples describing the notes of the chord. He suggests the weights 1, 3, 5, and 7 for the prime numbers 2, 3, 5, and 7.
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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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This method of generating primitive Pythagorean triples also provides integer solutions to Descartes' Circle Equation, : \left( k_1 + k_2 + k_3 + k_4 \right)^2 = 2\left( k_1^2 + k_2^2 + k_3^2 + k_4^2 \right), where integer curvatures ki are obtained by multiplying the reciprocal of each radius by the area A. The result is k1 = pp', k2 = qp', k3 = q'p, k4 = qq'. Here, the largest circle is taken as having negative curvature with respect to the other three. The largest circle (curvature k4) may also be replaced by a smaller circle with positive curvature ( k0 = 4pp' − qq' ). EXAMPLE: Using the area and four radii obtained above for primitive triple [44, 117, 125] we obtain the following integer solutions to Descartes' Equation: k1 = 143, k2 = 99, k3 = 26, k4 = (−18), and k0 = 554.
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In computational complexity theory, and more specifically in the analysis of algorithms with integer data, the transdichotomous model is a variation of the random access machine in which the machine word size is assumed to match the problem size. The model was proposed by Michael Fredman and Dan Willard,. who chose its name "because the dichotomy between the machine model and the problem size is crossed in a reasonable manner.". In a problem such as integer sorting in which there are integers to be sorted, the transdichotomous model assumes that each integer may be stored in a single word of computer memory, that operations on single words take constant time per operation, and that the number of bits that can be stored in a single word is at least .
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Pascal has had a number of type safety requirements, some of which are kept in some compilers. Where a Pascal compiler dictates "strict typing", two variables cannot be assigned to each other unless they are either compatible (such as conversion of integer to real) or assigned to the identical subtype. For example, if you have the following code fragment: type TwoTypes = record I: Integer; Q: Real; end; DualTypes = record I: Integer; Q: Real; end; var T1, T2: TwoTypes; D1, D2: DualTypes; Under strict typing, a variable defined as `TwoTypes` is not compatible with `DualTypes` (because they are not identical, even though the components of that user defined type are identical) and an assignment of ` T1 := D2; ` is illegal. An assignment of `T1 := T2; ` would be legal because the subtypes they are defined to are identical.
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A consequence of this theorem is that if there is a number system in which it is true that for any positive integer n there is a positive number x such that 0 < x < 1/n, then there exists an extension of that number system in which it is true that there exists a positive number x such that for any positive integer n we have 0 < x < 1/n. The possibility to switch "for any" and "there exists" is crucial. The first statement is true in the real numbers as given in ZFC set theory : for any positive integer n it is possible to find a real number between 1/n and zero, but this real number depends on n. Here, one chooses n first, then one finds the corresponding x.
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While a 32-bit signed integer may be used to hold a 16-bit unsigned value losslessly and a 32-bit unsigned value would require a 64-bit signed integer, a 64-bit unsigned value cannot be stored easily using any integer type because no type larger than 64 bits exists in the Java language. In all cases, the memory consumed may increase by a factor of up to two, and any logic that depends on the rules of two's complement overflow must typically be rewritten. If abstracted using functions, function calls become necessary for many operations which are native to some other languages. Alternatively, it is possible to use Java's signed integers to emulate unsigned integers of the same size, but this requires detailed knowledge of bitwise operations.
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Matiyasevich's theorem, also called the Matiyasevich–Robinson–Davis–Putnam or MRDP theorem, says: :Every computably enumerable set is Diophantine. A set S of integers is computably enumerable if there is an algorithm such that: For each integer input n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever. That is equivalent to saying there is an algorithm that runs forever and lists the members of S. A set S is Diophantine precisely if there is some polynomial with integer coefficients f(n, x1, ..., xk) such that an integer n is in S if and only if there exist some integers x1, ..., xk such that f(n, x1, ..., xk) = 0. Conversely, every Diophantine set is computably enumerable: consider a Diophantine equation f(n, x1, ..., xk) = 0.
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If p is a prime other than 2 or 5, the decimal representation of the fraction repeats: : = 0.. The period (repetend length) must be a factor of λ(49) = 42, where λ(n) is known as the Carmichael function. This follows from Carmichael's theorem which states that if n is a positive integer then λ(n) is the smallest integer m such that :a^m \equiv 1 \pmod n for every integer a that is coprime to n. The period of is usually pTp, where Tp is the period of . There are three known primes for which this is not true, and for those the period of is the same as the period of because p2 divides 10p−1−1\. These three primes are 3, 487, and 56598313 .
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This implies that a trigonometric number is an algebraic number, and twice a trigonometric number is an algebraic integer. Ivan Niven gave proofs of theorems regarding these numbers.Niven, Ivan. Numbers: Rational and Irrational, 1961.
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It originated in the following, unexpected observation. Define ΔMm = Mmono \- Mnom, where Mmono is the monoisotopic mass (e.g. O = 15.994915 Da) and Mnom is the nominal (integer) mass, i.e., the number of nucleons (e.g.
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The only non- singleton set with this property is the empty set. The Bell number integer sequence counts the number of partitions of a set (), if singletons are excluded then the numbers are smaller ().
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The distinction based on z being an integer or not accounts for the fact that the Jacobi theta function converges to the periodic delta function, or Dirac comb in z as t\rightarrow 0.
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Therefore, the output has the correct residue class. Third, m is in , and therefore is between 0 and . Hence t is less than , and because it's an integer, this puts t in the range .
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Stand Firm in Faith website. Retrieved January 6, 2009. He banned the LGBT group Integrity from the Cathedral in 1983.Louie Crew, "Where Are We Going?" First appeared in Integer 42 (1983): 6-7.
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In mathematics, the sieve of Sundaram is a simple deterministic algorithm for finding all the prime numbers up to a specified integer. It was discovered by Indian mathematician Mr. S. P. Sundaram in 1934.
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In research mathematics, figurate numbers are studied by way of the Ehrhart polynomials, polynomials that count the number of integer points in a polygon or polyhedron when it is expanded by a given factor..
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The 2012 Conference will take place during June 25–30 at the Institute of Mathematics and Informatics, Eszerházy Károly College, Eger, Hungary, with keynote speaker Neil Sloane, founder of the Encyclopedia of Integer Sequences.
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Nevertheless, even in NTSC regions, film productions are often shot at exactly 24 frame/s (this is called integer frame rate), especially for DCI. This can be a source of confusion and technical difficulties.
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Finally, given two solutions such that , one deduces that . As and are coprime, Euclid's lemma shows that divides , and thus that there exists an integer such that and . Therefore, and , which completes the proof.
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Carl Pomerance, Analysis and Comparison of Some Integer Factoring Algorithms, in Computational Methods in Number Theory, Part I, H.W. Lenstra, Jr. and R. Tijdeman, eds., Math. Centre Tract 154, Amsterdam, 1982, pp 89-139.
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For "conventional" cyclic compounds, aromaticity is conferred by the presence of 4n + 2 delocalized pi electrons, where n is an integer. Particular instability (antiaromaticity) is conferred by the presence of 4n conjugated pi electrons.
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When the list S cannot be reduced to a list S' of nonnegative integer pairs in any step of this approach, the theorem proves that the list S from the beginning is not digraphic.
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When the list S cannot be reduced to a list S' of nonnegative integer pairs in any step of this approach, the theorem proves that the list S from the beginning is not digraphic.
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In mathematics, the Meyer signature cocycle, introduced by . is an integer- valued 2-cocyle on a symplectic group that describes the signature of a fiber bundle whose base and fiber are both Riemann surfaces.
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The early versions are also backward-compatible with newer x86 software by means of an on-chip emulator mode. Integer performance was disappointing and despite improvements, sales in volume markets continue to be low.
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Eiffel includes inline agents defining closures. An inline agent is an object representing a routine, defined by giving the code of the routine in-line. For example, in ok_button.click_event.subscribe ( agent (x, y: INTEGER) do map.
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Xylotrechus integer is a species of beetle in the family Cerambycidae. It was described by Haldeman in 1847.Bezark, Larry G. A Photographic Catalog of the Cerambycidae of the World. Retrieved on 22 May 2012.
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Bag structure is constructed from the pennant data structure. A pennant is a tree of 2k nodex, where k is a nonnegative integer. Each root x in this tree contains two pointers x.left and x.
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Unconditional jumps use Goto and Lbl to operate. When Goto n (where n is an integer in 0-9) is executed, the program will jump to Lbl n. Loops can be created with unconditional jumps.
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In mathematics, a Størmer number or arc-cotangent irreducible number, named after Carl Størmer, is a positive integer n for which the greatest prime factor of n2 + 1 is greater than or equal to 2n.
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Later work has shown that the question of solvability of a Diophantine equation is undecidable even if the equation only has 9 natural number variables (Matiyasevich, 1977) or 11 integer variables (Zhi Wei Sun, 1992).
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This extension can be defined by for , for , and for with a positive integer. The sequence of intervals within which this function is positive or negative follows the same pattern as the Thue–Morse sequence.
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The basic idea behind LP decoding is to first represent the maximum likelihood decoding of a linear code as an integer linear program, and then relax the integrality constraints on the variables into linear inequalities.
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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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In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects of each size...
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The density is at most \Gamma(4/3)^3/6\approx 0.119. Every integer can be represented as a sum of three cubes of rational numbers (rather than as a sum of cubes of integers).
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These include improper fractions as well as mixed numbers. Continued fraction: An expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. Scientific notation: A method for writing very small and very large numbers using powers of 10. When used in science, such a number also conveys the precision of measurement using significant figures.
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Consider the integer multi-commodity flow problem of routing k types of integer commodities from m suppliers to n consumers subject to supply, consumption, and capacity constraints, so as to minimize the sum of linear or congestion-dependent convex costs on the arcs. Then for every fixed k and m, the Graver basis of the defining system can be computed and the resulting separable-convex objective function minimized in time which is polynomial in the variable number n of consumers and in the rest of the data.
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Advanced process monitor (APMonitor) is a modeling language for differential algebraic (DAE) equations. It is a free web-service or local server for solving representations of physical systems in the form of implicit DAE models. APMonitor is suited for large-scale problems and solves linear programming, integer programming, nonlinear programming, nonlinear mixed integer programming, dynamic simulation, moving horizon estimation, and nonlinear model predictive control. APMonitor does not solve the problems directly, but calls nonlinear programming solvers such as APOPT, BPOPT, IPOPT, MINOS, and SNOPT.
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Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to László Babai and Eugene Luks, has run time 2O() for graphs with n vertices. The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k.
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455: "It is very likely that q x and p x cannot be rational at the same time except if x is an integer. … At present we can not show this. Professor Siegel has communicated to us the result that q x, r x and s x can not be simultaneously rational except if x is an integer." An equivalent statement was first mentioned in print by Theodor Schneider who set it as the first of eight important, open problems in transcendental number theory in 1957.
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A range check is a check to make sure a number is within a certain range; for example, to ensure that a value about to be assigned to a 16-bit integer is within the capacity of a 16-bit integer (i.e. checking against wrap-around). This is not quite the same as type checking. Other range checks may be more restrictive; for example, a variable to hold the number of a calendar month may be declared to accept only the range 1 to 12.
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When we allow the exponent to be the reciprocal of an integer, i.e. for some integer , we have the inverse Fermat equation a^{1/m} + b^{1/m} = c^{1/m}. All solutions of this equation were computed by Hendrik Lenstra in 1992. In the case in which the mth roots are required to be real and positive, all solutions are given by :a=rs^m :b=rt^m :c=r(s+t)^m for positive integers r, s, t with s and t coprime.
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The SPARC64 V fetches up to eight instructions from the instruction cache during the first stage and places them into a 48-entry instruction buffer. In the next stage, four instructions are taken from this buffer, decoded and issued to the appropriate reserve stations. The SPARC64 V has six reserve stations, two that serve the integer units, one for the address generators, two for the floating-point units, and one for branch instructions. Each integer, address generator and floating-point unit has an eight-entry reserve station.
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This argument generalizes. The compact dimension imposes specific boundary conditions on all fields, for example periodic boundary conditions in the case of a periodic dimension, and typically Neumann or Dirichlet boundary conditions in other cases. Now suppose the size of the compact dimension is L; then the possible eigenvalues under gradient along this dimension are integer or half-integer multiples of 1/L (depending on the precise boundary conditions). In quantum mechanics this eigenvalue is the momentum of the field, and is therefore related to its energy.
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In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proven from ZFC. In 1970, Russian mathematician Yuri Matiyasevich showed that Hilbert's Tenth Problem, posed in 1900 as a challenge to the next century of mathematicians, cannot be solved. Hilbert's challenge sought an algorithm which finds all solutions of a Diophantine equation. A Diophantine equation is a more general case of Fermat's Last Theorem; we seek the integer roots of a polynomial in any number of variables with integer coefficients.
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A mesh edge is defined by a pair of integers hv0,v1i, each integer corresponding to an end point of the edge. To support edge maps, the edges are stored so that v0 = min(v0,v1). A triangle component is defined by a triple of integers hv0,v1,v2i, each integer corresponding to a vertex of the triangle. To support triangle maps, the triangles are stored so that v0 = min(v0,v1,v2). Observe that hv0,v1,v2i and hv0,v2,v1i are treated as different triangles.
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It is known that no non-constant polynomial function P(n) with integer coefficients exists that evaluates to a prime number for all integers n. The proof is as follows: suppose such a polynomial existed. Then P(1) would evaluate to a prime p, so P(1) \equiv 0 \pmod p. But for any integer k, P(1+kp) \equiv 0 \pmod p also, so P(1+kp) cannot also be prime (as it would be divisible by p) unless it were p itself.
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Ford also proved that if there exists a counterexample to the Conjecture, then a positive proportion (in the sense of asymptotic density) of the integers are likewise counterexamples. Although the conjecture is widely believed, Carl Pomerance gave a sufficient condition for an integer n to be a counterexample to the conjecture . According to this condition, n is a counterexample if for every prime p such that p − 1 divides φ(n), p2 divides n. However Pomerance showed that the existence of such an integer is highly improbable.
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Multi-Architecture Binary files are in a special archive format, in which a single file stores one or more Mach-O subfiles for each architecture supported by the Multi-Architecture Binary. Every Multi-Architecture Binary starts with a structure (struct fat_header) containing two unsigned integers. The first integer ("magic") is used as a magic number to identify this file as a Fat Binary. The second integer ("nfat_arch") defines how many Mach-O Files the archive contains (how many instances of the same program for different architectures).
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In many cases, the task or the programmer can guarantee that the integer values in a specific application will not grow large enough to cause an overflow. Such guarantees may be based on pragmatic limits: a school attendance program may have a task limit of 4,000 students. A programmer may design the computation so that intermediate results stay within specified precision boundaries. Some programming languages such as Lisp, Python, Perl, Haskell and Ruby use, or have an option to use, arbitrary-precision numbers for all integer arithmetic.
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The FICO Xpress optimizer is a commercial optimization solver for linear programming (LP), mixed integer linear programming (MILP), convex quadratic programming (QP), convex quadratically constrained quadratic programming (QCQP), second-order cone programming (SOCP) and their mixed integer counterparts. Xpress includes a general purpose non-linear solver, Xpress NonLinear, including a successive linear programming algorithm (SLP, first- order method), and Artelys Knitro (second-order methods). Xpress was originally developed by Dash Optimization, and was acquired by FICO in 2008. "Dash Optimization acquired by FICO" Jan 22, 2008.
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The goal of these problems is to design a network of lines to install so that a predefined set of communication requirements are met and the total cost of the network is minimal. This requires optimizing both the topology of the network along with the setting the capacities of the various lines. In many cases, the capacities are constrained to be integer quantities. Usually there are, depending on the technology used, additional restrictions that can be modeled as linear inequalities with integer or binary variables.
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The CSX700 processor consists of two processing arrays, each with 96 processing elements. The processing elements each contain a 32/64-bit floating point multiplier, a 32/64-bit floating point adder, 6 KB of SRAM, an integer arithmetic logic unit, and a 16-bit integer multiply–accumulate unit.CSX700 specification It currently sells its CSX700 processor on a PCI Express expansion card with 2 GB of memory, called the Advance e710. The card is supplied with the ClearSpeed Software Development Kit and application libraries.
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In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z. Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in Rn, the Hermite normal form can solve problems about the solution to the linear system Ax=b where this time x is restricted to have integer coordinates only. Other applications of the Hermite normal form include integer programming, cryptography, and abstract algebra.
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In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring (e.g. measurements which can only yield integer values may have a non-integer mean). It is a fundamental concept in all areas of quantum physics.
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Analog signals can use voltage or current that is proportional to the size of the monitored variable and can take any value within their scale. Pressure, temperature, flow, and weight are often represented by analog signals. These are typically interpreted as integer values with various ranges of accuracy depending on the device and the number of bits available to store the data. For example, an analog 0 to 10 V or 4-20 mA input would be converted into an integer value of 0 to 32,767.
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Quaternions are also used in one of the proofs of Lagrange's four- square theorem in number theory, which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant theorem in its own right, Lagrange's four square theorem has useful applications in areas of mathematics outside number theory, such as combinatorial design theory. The quaternion-based proof uses Hurwitz quaternions, a subring of the ring of all quaternions for which there is an analog of the Euclidean algorithm.
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Let us look at a simple multiplication: 5×7 = 35, (3 + 5 = 8). Now consider (7 + 9)×5 = 16×5 = 80, (8 + 0 = 8) or 7×(9 + 5) = 7×14 = 98, (9 + 8 = 17, (1 + 7 = 8). Any non-negative integer can be written as 9×n + a, where 'a' is a single digit from 0 to 8, and 'n' is some non-negative integer. Thus, using the distributive rule, (9×n + a)×(9×m + b)= 9×9×n×m + 9(am + bn) + ab.
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If, for example, the definition "not divisible by any integer other than 1 and itself" happened to be 43rd, then this would be true. Since 43 is itself not divisible by any integer other than 1 and itself, then the number of this definition has the property of the definition itself. However, this may not always be the case. If the definition: "divisible by 3" were assigned to the number 58, then the number of the definition does not have the property of the definition itself.
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The mass number should also not be confused with the standard atomic weight (also called atomic weight) of an element, which is the ratio of the average atomic mass of the different isotopes of that element (weighted by abundance) to the unified atomic mass unit. The atomic weight is an actual mass (made relative, i.e., a ratio), while the mass number is a counted number (and so an integer). This weighted average can be quite different from the near-integer values for individual isotopic masses.
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It had an in-order, five-stage pipeline with a single integer unit, a double precision floating point unit (FPU) and separate 16 kB instruction and 8 kB data caches. While the integer unit was a brand new design, the FPU was derived from the R4600 to save time. It was 69 mm2 small using a 0.5 μm fabrication process and drew just 1.2 W at 120 MHz. The 603q was designed for Motorola, but they withdrew from the contract before the 603q went into full production.
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Phase synchronization is the process by which two or more cyclic signals tend to oscillate with a repeating sequence of relative phase angles. Phase synchronisation is usually applied to two waveforms of the same frequency with identical phase angles with each cycle. However it can be applied if there is an integer relationship of frequency, such that the cyclic signals share a repeating sequence of phase angles over consecutive cycles. These integer relationships are called Arnold tongues which follow from bifurcation of the circle map.
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It had been conjectured that 6 cannot be so represented, and Golomb conjectured that there are infinitely many integers which cannot be represented as a difference between two powerful numbers. However, Narkiewicz showed that 6 can be so represented in infinitely many ways such as :6 = 5473 − 4632, and McDaniel showed that every integer has infinitely many such representations (McDaniel, 1982). Erdős conjectured that every sufficiently large integer is a sum of at most three powerful numbers; this was proved by Roger Heath-Brown (1987).
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There exist tetrahedra having integer-valued edge lengths, face areas and volume. These are called Heronian tetrahedra. One example has one edge of 896, the opposite edge of 990 and the other four edges of 1073; two faces are isosceles triangles with areas of and the other two are isosceles with areas of , while the volume is . A tetrahedron can have integer volume and consecutive integers as edges, an example being the one with edges 6, 7, 8, 9, 10, and 11 and volume 48.
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Translated into modern notation, Euler stated Lemmermeyer, p. 5, Ireland & Rosen, pp. 54, 61 that for distinct odd primes p and q: # If q ≡ 1 (mod 4) then q is a quadratic residue (mod p) if and only if there exists some integer b such that p ≡ b2 (mod q). # If q ≡ 3 (mod 4) then q is a quadratic residue (mod p) if and only if there exists some integer b which is odd and not divisible by q such that p ≡ ±b2 (mod 4q).
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Each type itself is compatible with or a signed or unsigned integer type, but each implementation defines its own rules for choosing a type. Some compilers warn if an object with enumerated type is assigned a value that is not one of its constants. However, such an object can be assigned any values in the range of their compatible type, and constants can be used anywhere an integer is expected. For this reason, values are often used in place of preprocessor directives to create named constants.
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A primitive Pythagorean triple is one in which , and are coprime (that is, they have no common divisor larger than 1). A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle. The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula ; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples.
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Solving the general case is an NP-hard problem. To see this, note that the two constraints x1(x1 − 1) ≤ 0 and x1(x1 − 1) ≥ 0 are equivalent to the constraint x1(x1 − 1) = 0, which is in turn equivalent to the constraint x1 ∈ {0, 1}. Hence, any 0–1 integer program (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained quadratic program. Since 0–1 integer programming is NP-hard in general, QCQP is also NP-hard.
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The Ehrhart polynomial of the order polytope is a polynomial whose values at integer values x give the number of integer points in a copy of the polytope scaled by a factor of x. For the order polytope, the Ehrhart polynomial equals (after a minor change of variables) the order polynomial of the corresponding partially ordered set. This polynomial encodes several pieces of information about the polytope including its volume (the leading coefficient of the polynomial and its number of vertices (the sum of coefficients).
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The \lceil r\cdot n\rceil is the smallest possible when n is an integer. An r-partite hypergraph is an r-uniform hypergraph in which the vertices are partitioned into r disjoint sets and each hyperedge contains exactly one vertex of each set (so a 2-partite hypergraph is a just bipartite graph). Let n be any positive integer. Any family of rn-r+1 fractional-matchings (=colors) of size at least n in an r-partite hypergraph has a rainbow-fractional-matching of size n.
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In computer science, the AF-heap is a type of priority queue for integer data, an extension of the fusion tree using an atomic heap proposed by M. L. Fredman and D. E. Willard.M. L. Fredman and D. E. Willard. Trans-dichotomous algorithms for minimum spanning trees and shortest paths. Journal of Computer and System Sciences 48, 533-551 (1994) Using an AF-heap, it is possible to perform insert or decrease-key operations and delete-min operations on machine-integer keys in time .
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Kuṭṭaka is an algorithm for finding integer solutions of linear Diophantine equations. A linear Diophantine equation is an equation of the form ax + by = c where x and y are unknown quantities and a, b, and c are known quantities with integer values. The algorithm was originally invented by the Indian astronomer-mathematician Āryabhaṭa (476–550 CE) and is described very briefly in his Āryabhaṭīya. Āryabhaṭa did not give the algorithm the name Kuṭṭaka, and his description of the method was mostly obscure and incomprehensible.
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In relativistic quantum mechanics and quantum field theory, the Joos–Weinberg equation is a relativistic wave equations applicable to free particles of arbitrary spin , an integer for bosons () or half-integer for fermions (). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields. The spin quantum number is usually denoted by in quantum mechanics, however in this context is more typical in the literature (see references). It is named after H. Joos and Steven Weinberg, found in the early 1960s.
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The primary defining characteristic of AMD64 is the availability of 64-bit general-purpose processor registers (for example, rax and rbx), 64-bit integer arithmetic and logical operations, and 64-bit virtual addresses. The designers took the opportunity to make other improvements as well. Some of the most significant changes are described below. ; 64-bit integer capability : All general-purpose registers (GPRs) are expanded from 32 bits to 64 bits, and all arithmetic and logical operations, memory-to-register and register-to-memory operations, etc.
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A cycle k-cover of a graph is a family of cycles which cover every edge of G exactly k times. It has been proven that every bridgeless graph has cycle k-cover for any integer even integer k≥4. For k=2, it is the well-known cycle double cover conjecture is an open problem in graph theory. The cycle double cover conjecture states that in every bridgeless graph there exists a set of cycles that together cover every edge of the graph twice.
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To detect interest points, SURF uses an integer approximation of the determinant of Hessian blob detector, which can be computed with 3 integer operations using a precomputed integral image. Its feature descriptor is based on the sum of the Haar wavelet response around the point of interest. These can also be computed with the aid of the integral image. SURF descriptors have been used to locate and recognize objects, people or faces, to reconstruct 3D scenes, to track objects and to extract points of interest.
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In contrast to JPEG 2000, PGF uses only powers of two, therefore the parameter value i represents a quantization step of 2i. Just using powers of two makes no need of integer multiplication and division operations.
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The ARM Cortex-A76 is a microarchitecture implementing the ARMv8.2-A 64-bit instruction set designed by ARM Holdings' Austin design centre. ARM states a 25% and 35% increase in integer and floating point performance, respectively.
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E.A. Karatsuba "Fast Algorithms and the FEE Method". For arguments that are integer multiples of , the gamma function can also be evaluated quickly using arithmetic–geometric mean iterations (see particular values of the gamma function and ).
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So, the term geocoding (used for addresses) sometimes is generalized for geocodes. In spatial indexing applications the geocode can also be translated between human-readable (e.g. hexadecimal) and internal (e.g. binary 64-bit unsigned integer) representations.
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A counter-based random number generation (CBRNG, also known as a counter-based pseudo-random number generator, or CBPRNG) is a kind of pseudorandom number generator that uses only an integer counter as its internal state.
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Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.
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The length is an integer number (e.g. uint8, or int64) encoding the size of the data payload in bytes. It is used for strings, high-precision numbers and optionally containers. They are omitted for other types.
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Lagaria's wild numbers are connected with the Collatz conjecture and the concept of the 3x + 1 semigroup. The original fictional sequence of wild numbers has found a place in the On- Line Encyclopedia of Integer Sequences.
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To further generalize, an A-restricted composition of an integer n, for a subset A of the (nonnegative or positive) integers, is an ordered collection of one or more elements in A whose sum is n.
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For instance, for every positive integer , it is possible to combine copies of the L-tromino, L-tetromino, or P-pentomino into a single larger shape similar to the smaller polyomino from which it was formed.
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Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation.
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Contains are array of 4 byte integer value, one for each (tagged) point of the TIN. The values are stored in little-endian byte order. Superpoints have the value `0` (zero). Points without a tag (e.g.
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The number 4,294,967,295 is an integer equal to 2 − 1\. It is a perfect totient number. It follows 4,294,967,294 and precedes 4,294,967,296. It has a factorization of 3 \cdot 5 \cdot 17 \cdot 257 \cdot 65537.
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V. Blagouchine, A Note on Some Recent Results for the Bernoulli Numbers of the Second Kind, Journal of Integer Sequences, Vol. 20, No. 3 (2017), Article 17.3.8 arXiv:1612.03292Ernst Schröder, Zeitschrift fur Mathematik und Physik, vol.
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The algorithmL. E. Doggett, Ch. 12, "Calendars", p. 606, in Seidelmann 1992 is valid for all (possibly proleptic) Julian calendar years ≥ −4712, that is, for all JDN ≥ 0. Divisions are integer divisions, fractional parts are ignored.
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The following is a Pascal program by Niklaus Wirth in 1976.Wirth, 1976, p. 145 It finds one solution to the eight queens problem. program eightqueen1(output); var i : integer; q : boolean; a : array[ 1 .. 8] of boolean; b : array[ 2 .. 16] of boolean; c : array[ −7 .. 7] of boolean; x : array[ 1 .. 8] of integer; procedure try( i : integer; var q : boolean); var j : integer; begin j := 0; repeat j := j + 1; q := false; if a[ j] and b[ i + j] and c[ i − j] then begin x[ i ] := j; a[ j ] := false; b[ i + j] := false; c[ i − j] := false; if i < 8 then begin try( i + 1, q); if not q then begin a[ j] := true; b[ i + j] := true; c[ i − j] := true; end end else q := true end until q or (j = 8); end; begin for i := 1 to 8 do a[ i] := true; for i := 2 to 16 do b[ i] := true; for i := −7 to 7 do c[ i] := true; try( 1, q); if q then for i := 1 to 8 do write( x[ i]:4); writeln end.
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The following example uses isort to define a procedure vecsort that takes an integer n and an integer vector v with elements v[0] through v[n−1] and sorts them in either increasing or decreasing order, depending on whether a third parameter incr is true or false, respectively: procedure vecsort(n, v, incr): procedure vprec(r, s): if incr then return v[r] < v[s]; else return v[r] > v[s]; procedure vswap(r, s): integer t; t ← v[r]; v[r] ← v[s]; v[s] ← t isort(0, n−1, vprec, vswap); Note the use of nested function definitions to get a function vprec whose effect depends on the parameter incr passed to vecsort. In languages that do not allow nested function definitions, like standard C, obtaining this effect would require rather complicated and/or thread-unsafe code.
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In 1962, Paul Erdős and Lajos Pósa proved that for every positive integer k there exists a positive integer k' such that for every graph G, either (i) G has k vertex-disjoint (long and/or even) cycles or (ii) there exists a subset X of less than k' vertices of G such that G \ X has no (long and/or even) cycles. This result, known today as the Erdős–Pósa theorem, cannot be extended to odd cycles. In fact, in 1987 Dejter and Víctor Neumann-LaraDejter I. J.; Neumann-Lara V. "Unboundedness for odd cyclic transversality", Coll. Math. Soc. J. Bolyai, 52 (1987), 195–203 showed that given an integer k > 0, there exists a graph G not possessing disjoint odd cycles such that the number of vertices of G whose removal destroys all odd cycles of G is higher than k.
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Starting state of the standard puzzle Water pouring puzzles (also called water jug problems, decanting problems or measuring puzzles) are a class of puzzle involving a finite collection of water jugs of known integer capacities (in terms of a liquid measure such as liters or gallons). Initially each jug contains a known integer volume of liquid, not necessarily equal to its capacity. Puzzles of this type ask how many steps of pouring water from one jug to another (until either one jug becomes empty or the other becomes full) are needed to reach a goal state, specified in terms of the volume of liquid that must be present in some jug or jugs. By Bézout's identity, such puzzles have solution if and only if the desired volume is a multiple of the greatest common divisor of all the integer volume capacities of jugs.
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That is, the maximum ratio of areas on either side of a diameter, another measure of asymmetry, is bigger for the Reuleaux triangle than for other curves of constant width.. Among all shapes of constant width that avoid all points of an integer lattice, the one with the largest width is a Reuleaux triangle. It has one of its axes of symmetry parallel to the coordinate axes on a half-integer line. Its width, approximately 1.545, is the root of a degree-6 polynomial with integer coefficients.. Just as it is possible for a circle to be surrounded by six congruent circles that touch it, it is also possible to arrange seven congruent Reuleaux triangles so that they all make contact with a central Reuleaux triangle of the same size. This is the maximum number possible for any curve of constant width.
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In mathematical terms the problem can be stated: :Given positive integers a1, a2, ..., an such that gcd(a1, a2, ..., an) = 1, find the largest integer that cannot be expressed as an integer conical combination of these numbers, i.e., as a sum :: k1a1 + k2a2 + ··· + knan, :where k1, k2, ..., kn are non-negative integers. This largest integer is called the Frobenius number of the set { a1, a2, ..., an }, and is usually denoted by g(a1, a2, ..., an). The requirement that the greatest common divisor (GCD) equal 1 is necessary in order for the Frobenius number to exist. If the GCD were not 1, then starting at some number m the only sums possible are multiples of the GCD; every number past m that is not divisible by the GCD cannot be represented by any linear combination of numbers from the set.
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The origins of the INTEGER Millennium House date to a May 1996 seminar about green building, which catalysed the formation of the INTEGER building design and consultancy team to "evaluate available design and technical solutions to improve housing performance". The efforts of this design team resulted in the creation of a scale model of a house, which has subsequently been widely exhibited at institutions such as the Science Museum, London as well as profiled in the DK Eyewitness Book Future. Damian Bree, Tim Day, Paul Hodgkins, and Nicholas Thompson began designing the INTEGER Millennium House itself in October 1997, a process that proceeded for 10 weeks. Built on the Building Research Establishment (BRE)'s Innovation Park property in Garston, Watford, the actual construction of the house took 13 weeks before it opened to the public in 1998.
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These are the only possible integer polynomial factors of f(x). Testing them exhaustively reveals that :p(x) = x^2 + x + 1 constructed from p(0)=1, p(1)=3 and p(-1)=1, factors f(x).
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A slight generalization of the last example. Once again consider the group of integers Z under addition. Let n be any positive integer. We will consider the subgroup nZ of Z consisting of all multiples of n.
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It is also equal to 32 (see tetration). Sixteen is the base of the hexadecimal number system, which is used extensively in computer science. Sixteen is the largest known integer n for which 2n+1 is prime.
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The integer unit had 32, 64-bit-wide registers. The floating point unit is discussed below. The XIO had the capacity of 30 MB/s throughput to I/O devices. It included 64 control and data registers.
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11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name.
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McGraw-Hill, New York, 1960. Von Neumann's results have been viewed as a special case of linear programming, where von Neumann's model uses only nonnegative matrices.Alexander Schrijver, Theory of Linear and Integer Programming. John Wiley & sons, 1998, .
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The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial :x^2 - x - 1. Having degree 2, this polynomial actually has two roots, the other being the golden ratio conjugate.
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"A", "B", "AB" or "O", for blood type), ordinal (e.g. "large", "medium" or "small"), integer-valued (e.g. the number of occurrences of a particular word in an email) or real-valued (e.g. a measurement of blood pressure).
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"A", "B", "AB" or "O", for blood type); ordinal (e.g. "large", "medium" or "small"); integer-valued (e.g. the number of occurrences of a particular word in an email); or real-valued (e.g. a measurement of blood pressure).
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Oberwolfach 2011 Gérard Pierre Cornuéjols (born 1950) is the IBM University Professor of Operations Research in the Carnegie Mellon University Tepper School of Business.. His research interests include facility location, integer programming, balanced matrices, and perfect graphs.
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All solutions in integers a, b, c are given in terms of positive integer parameters m, n, k by :a=km(m+n) , \quad b=kn(m+n), \quad c=kmn where m and n are coprime.
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The effective magnetic field of composite fermions has been confirmed by the similarity of the fractional and the integer quantum Hall effects, observation of Fermi sea at half filled Landau level, and measurements of the cyclotron radius.
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When a is a non-positive integer, −n, {}_1F_1(-n;b;z) is a polynomial. Up to constant factors, these are the Laguerre polynomials. This implies Hermite polynomials can be expressed in terms of 1F1 as well.
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Lower bounds for multiplication are also known for some classes of branching programs.Farid Ablayev and Marek Karpinski, A lower bound for integer multiplication on randomized ordered read-once branching programs, Information and Computation 186 (2003), 78–89.
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Then each of these primes divides all but one of the numerator terms and hence does not divide the numerator itself; but each prime does divide the denominator. Thus the expression is irreducible and is non-integer.
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Gou Gu mutual seeking discusses the algorithm of finding the length of a side of the right triangle while knowing the other two. Gou Gu integer is precisely the finding of some significant integer Pythagorean numbers, including famously the triple 3,4,5. Gou Gu dual capacity discusses algorithms for calculating the areas of the inscribed rectangles and other polygons in the circle, which also serves an algorithm to calculate the value of pi. Lastly, Gou Gu similars provide algorithms of calculating heights and lengths of buildings on the mathematical basis of similar right triangles.
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A commutative ring is a set, equipped with an addition and multiplication operation, satisfying all the axioms of a field, except for the existence of multiplicative inverses . For example, the integers form a commutative ring, but not a field: the reciprocal of an integer is not itself an integer, unless . In the hierarchy of algebraic structures fields can be characterized as the commutative rings in which every nonzero element is a unit (which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinct ideals, and .
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Mathematical programming and in particular Mixed integer programming models are another approach to solve MSA problems. The advantage of such optimization models is that they can be used to find the optimal MSA solution more efficiently compared to the traditional DP approach. This is due in part, to the applicability of decomposition techniques for mathematical programs, where the MSA model is decomposed into smaller parts and iteratively solved until the optimal solution is found. Example algorithms used to solve mixed integer programming models of MSA include branch and price and Benders decomposition .
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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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The gamma function interpolates the factorial function to non-integer values. The gamma function can be seen as a solution to the following interpolation problem: : "Find a smooth curve that connects the points (x, y) given by y = (x - 1)! at the positive integer values for x." A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of operations does not depend on the size of x.
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If k2 is greater than one, F2−F1 is a negative number; thus, the added inverse-cube force is attractive. Conversely, if k2 is less than one, F2−F1 is a positive number; the added inverse-cube force is repulsive. If k is an integer such as 3, the orbit of the second particle is said to be a harmonic of the first particle's orbit; by contrast, if k is the inverse of an integer, such as , the second orbit is said to be a subharmonic of the first orbit.
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Bit fields can be used to reduce memory consumption when a program requires a number of integer variables which always will have low values. For example, in many systems storing an integer value requires two bytes (16-bits) of memory; sometimes the values to be stored actually need only one or two bits. Having a number of these tiny variables share a bit field allows efficient packaging of data in the memory. In C and C++, native implementation-defined bit fields can be created using unsigned int, signed int, or (in C99:) _Bool.
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In the C, C++, and D programming languages, a type qualifier is a keyword that is applied to a type, resulting in a qualified type. For example, `const int` is a qualified type representing a constant integer, while `int` is the corresponding unqualified type, simply an integer. In D these are known as type constructors, by analogy with constructors in object-oriented programming. Type qualifiers are a way of expressing additional information about a value through the type system, and ensuring correctness in the use of the data.
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For certain classes of problems, algorithms running on quantum computers are naturally capable of achieving lower time complexity than on classical computers. That is, quantum algorithms can solve certain problems faster than the most efficient algorithm running on a traditional computer. For example, Shor's algorithm can factor an integer N in polynomial time, while the best-known factoring classic algorithm, the general number field sieve, operates in sub-exponential time. This is significant to public key cryptography because the security of RSA is dependent on the infeasibility of factoring integers, the integer factorization problem.
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R8010 die photo The R8010 executed floating-point instructions provided by an instruction queue on the R8000. The queue decoupled the floating-point pipeline from the integer pipeline, implementing a limited form of out-of-order execution by allowing floating-point instructions to execute when possible after or before the integer instructions from the same group are issued. The pipelines were decoupled to help mitigate some of the streaming cache latency. It contained the floating-point register file, a load queue, a store queue, and two identical floating-point units.
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A 32-bit register can store 232 different values. The range of integer values that can be stored in 32 bits depends on the integer representation used. With the two most common representations, the range is 0 through 4,294,967,295 (232 − 1) for representation as an (unsigned) binary number, and −2,147,483,648 (−231) through 2,147,483,647 (231 − 1) for representation as two's complement. One important consequence is that a processor with 32-bit memory addresses can directly access at most 4 GiB of byte-addressable memory (though in practice the limit may be lower).
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CLN uses class inheritance to model the natural subsets of the available number types: E.g. the integer class is a subtype of the rational class, just as the integer numbers are a subset of the rational numbers. The complex numbers and all its subtypes behave exactly like the types of numbers known to the Common Lisp language, giving CLN another meaning: it becomes an abbreviation of Common Lisp Numbers. Due to this, CLN can be and is used for implementations of Common Lisp, other interpreted languages, or computer algebra systems.
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Nemhauser's research concerns large mixed integer programming problems and their applications.. He is one of the co-inventors of the branch and price method for solving integer linear programs.. He also contributed important early studies of approximation algorithms for facility location problems. and for submodular optimization.. Nemhauser, together with Leslie Trotter, showed in 1975 that the optimal solution to the weighted vertex cover problem contains all the nodes that have a value of 1 in the linear programming relaxation as well as some of the nodes that have a value of 0.5.
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Normalized frequency is a unit of measurement of frequency equivalent to cycles/sample. In digital signal processing (DSP), the continuous time variable, t, with units of seconds, is replaced by the discrete integer variable, n, with units of samples. More precisely, the time variable, in seconds, has been normalized (divided) by the sampling interval, T (seconds/sample), which causes time to have convenient integer values at the moments of sampling. This practice is analogous to the concept of natural units, meaning that the natural unit of time in a DSP system is samples.
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The custom instruction set 107 MMI (Multimedia Extensions) was implemented by grouping the two 64-bit integer ALUs. Both the integer and floating-point pipelines are six stages long. To feed the execution units with instructions and data, there is a 16 KB two-way set associative instruction cache, an 8 KB two-way set associative non blocking data cache and a 16 KB scratchpad RAM. Both the instruction and data caches are virtually indexed and physically tagged while the scratchpad RAM exists in a separate memory space.
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If M is an arbitrary set containing zero, the concept of support is immediately generalizable to functions f : X→M. Support may also be defined for any algebraic structure with identity (such as a group, monoid, or composition algebra), in which the identity element assumes the role of zero. For instance, the family ZN of functions from the natural numbers to the integers is the uncountable set of integer sequences. The subfamily { f in ZN :f has finite support } is the countable set of all integer sequences that have only finitely many nonzero entries.
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The Pythagorean equation, , has an infinite number of positive integer solutions for x, y, and z; these solutions are known as Pythagorean triples (with the simplest example 3,4,5). Around 1637, Fermat wrote in the margin of a book that the more general equation had no solutions in positive integers if n is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, and no proof by him has ever been found. His claim was discovered some 30 years later, after his death.
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