Its standard orientation goes from 0 to 1. The unit interval is a totally ordered set and a complete lattice (every subset of the unit interval has a supremum and an infimum).
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The unit interval as a subset of the real line In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter `I`). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: , , and .
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The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1.
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Proper interval graphs are interval graphs that have an interval representation in which no interval properly contains any other interval; unit interval graphs are the interval graphs that have an interval representation in which each interval has unit length. A unit interval representation without repeated intervals is necessarily a proper interval representation. Not every proper interval representation is a unit interval representation, but every proper interval graph is a unit interval graph, and vice versa.; Every proper interval graph is a claw-free graph; conversely, the proper interval graphs are exactly the claw-free interval graphs.
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The dimension of the unit interval is not an obstacle, as was clear already to Norbert Wiener. He constructed the Wiener process (also called Brownian motion) in the form of a measurable map from the unit interval to the space of continuous functions.
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An exponential distribution restricted to the unit interval is equivalent to a continuous Bernoulli distribution with appropriate parameter.
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The elements of the van der Corput sequence (in any base) form a dense set in the unit interval; that is, for any real number in [0, 1], there exists a subsequence of the van der Corput sequence that converges to that number. They are also equidistributed over the unit interval.
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In telecommunication, an isochronous signal is a signal in which the time interval separating any two significant instants is equal to the unit interval or a multiple of the unit interval. Variations in the time intervals are constrained within specified limits. "Isochronous" is a characteristic of one signal, while "synchronous" indicates a relationship between two or more signals.
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A less combinatorial example is the operad of little intervals: The space A(n) consists of all embeddings of n disjoint intervals into the unit interval.
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Every real, separable Banach space is isometrically isomorphic to a closed subspace of , the space of all continuous functions from the unit interval into the real line.
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Any finite space is trivially compact. A non-trivial example of a compact space is the (closed) unit interval of real numbers. If one chooses an infinite number of distinct points in the unit interval, then there must be some accumulation point in that interval. For instance, the odd-numbered terms of the sequence get arbitrarily close to 0, while the even-numbered ones get arbitrarily close to 1\.
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The Hahn–Mazurkiewicz theorem is the following characterization of spaces that are the continuous image of curves: :A non- empty Hausdorff topological space is a continuous image of the unit interval if and only if it is a compact, connected, locally connected, second-countable space. Spaces that are the continuous image of a unit interval are sometimes called Peano spaces. In many formulations of the Hahn–Mazurkiewicz theorem, second-countable is replaced by metrizable. These two formulations are equivalent.
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Pompeiu's construction is described here. Let denote the real cube root of the real number . Let be an enumeration of the rational numbers in the unit interval . Let be positive real numbers with .
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There is an exponential increase in volume associated with adding extra dimensions to a mathematical space. For example, 102=100 evenly spaced sample points suffice to sample a unit interval (a "1-dimensional cube") with no more than 10−2=0.01 distance between points; an equivalent sampling of a 10-dimensional unit hypercube with a lattice that has a spacing of 10−2=0.01 between adjacent points would require 1020=[(102)10] sample points. In general, with a spacing distance of 10−n the 10-dimensional hypercube appears to be a factor of 10n(10-1)=[(10n)10/(10n)] "larger" than the 1-dimensional hypercube, which is the unit interval. In the above example n=2: when using a sampling distance of 0.01 the 10-dimensional hypercube appears to be 1018 "larger" than the unit interval.
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As noted in the previous paragraph, second- order comprehension axioms easily generalize to the higher-order framework. However, theorems expressing the compactness of basic spaces behave quite differently in second- and higher-order arithmetic: on one hand, when restricted to countable covers/the language of second-order arithmetic, the compactness of the unit interval is provable in WKL0 from the next section. On the other hand, given uncountable covers/the language of higher-order arithmetic, the compactness of the unit interval is only provable from (full) second-order arithmetic (). Other covering lemmas (e.g.
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In 1890, Peano discovered a continuous curve, now called the Peano curve, that passes through every point of the unit square (). His purpose was to construct a continuous mapping from the unit interval onto the unit square. Peano was motivated by Georg Cantor's earlier counterintuitive result that the infinite number of points in a unit interval is the same cardinality as the infinite number of points in any finite-dimensional manifold, such as the unit square. The problem Peano solved was whether such a mapping could be continuous; i.e.
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A continuum structure function is defined by Baxter as a nondecreasing mapping from the unit hypercube to the unit interval in Continuum structures I., Baxter, L A, Journal of Applied Probability. Vol. 21, no. 4, pp. 802–815. 1984 .
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The union of two fuzzy sets A and B is specified in general by a binary operation on the unit interval function of the form :u:[0,1]×[0,1] → [0,1]. :For all x ∈ U: μA ∪ B(x) = u[μA(x), μB(x)].
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The intersection of two fuzzy sets A and B is specified in general by a binary operation on the unit interval, a function of the form :i:[0,1]×[0,1] → [0,1]. :For all x ∈ U: μA ∩ B(x) = i[μA(x), μB(x)].
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The Bernstein polynomials of a fixed degree m are a family of m+1 linearly independent polynomials that are a partition of unity for the unit interval [0,1]. Partition of unity is used to establish global smooth approximations for Sobolev functions in bounded domains.
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If is a cover and γ is a path in X (i.e. a continuous map from the unit interval into X) and is a point "lying over" γ(0) (i.e. , then there exists a unique path Γ in C lying over γ (i.e. ) such that .
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If X is some topological space, such as the unit interval [0,1], we can consider the space of all continuous functions from X to R. This is a vector subspace of RX since the sum of any two continuous functions is continuous and scalar multiplication is continuous.
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Multi-valued logics (such as fuzzy logic and relevance logic) allow for more than two truth values, possibly containing some internal structure. For example, on the unit interval such structure is a total order; this may be expressed as the existence of various degrees of truth.
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There is an essentially unique measure that is translation-invariant, strictly positive and locally finite on the real line. In fact, any such measure must be a constant multiple of Lebesgue measure, specifying that the measure of the unit interval should be 1—before determining the solution uniquely.
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The same set of points would not accumulate to any point of the open unit interval ; so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded. In particular, the sequence of points , which is not bounded, has no subsequence that converges to any real number. Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces are encountered in mathematical analysis, where the property of compactness of some topological spaces arises in the hypotheses or in the conclusions of many fundamental theorems, such as the Bolzano–Weierstrass theorem, the extreme value theorem, the Arzelà–Ascoli theorem, and the Peano existence theorem.
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Cographs are exactly the graphs with clique-width at most 2. Every distance-hereditary graph has clique-width at most 3. However, the clique-width of unit interval graphs is unbounded (based on their grid structure). Similarly, the clique-width of bipartite permutation graphs is unbounded (based on similar grid structure).
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Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one. This paper displeased Kronecker and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Karl Weierstrass supported its publication.Dauben 1979, pp. 69, 324 63n.
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The name is due to E. C. Zeeman, who observed that any contractible 2-complex (such as the dunce hat) after taking the Cartesian product with the closed unit interval seemed to be collapsible. This observation became known as the Zeeman conjecture and was shown by Zeeman to imply the Poincaré conjecture.
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Suppose that X is the unit interval with the Lebesgue measurable sets and Lebesgue measure, and Y is the unit interval with all subsets measurable and the counting measure, so that Y is not σ-finite. If f is the characteristic function of the diagonal of X×Y, then integrating f along X gives the 0 function on Y, but integrating f along Y gives the function 1 on X. So the two iterated integrals are different. This shows that Tonelli's theorem can fail for spaces that are not σ-finite no matter what product measure is chosen. The measures are both decomposable, showing that Tonelli's theorem fails for decomposable measures (which are slightly more general than σ-finite measures).
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One might be tempted to think that the meaning of curves intersecting is that they necessarily cross each other, like the intersection point of two non-parallel lines, from one side to the other. However, two curves (or two subcurves of one curve) may contact one another without crossing, as, for example, a line tangent to a circle does. A non-self-intersecting continuous curve cannot fill the unit square because that will make the curve a homeomorphism from the unit interval onto the unit square (any continuous bijection from a compact space onto a Hausdorff space is a homeomorphism). But a unit square has no cut- point, and so cannot be homeomorphic to the unit interval, in which all points except the endpoints are cut-points.
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In telecommunication, the term bias distortion has the following meanings: #Signal distortion resulting from a shift in the bias. #In digital signaling, distortion of the signal in which all the significant intervals have uniformly longer or shorter durations than their theoretical durations. Bias distortion is expressed in percent of the system-specified unit interval.
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Other scales have been in use historically. The Rankine scale, using the degree Fahrenheit as its unit interval, is still in use as part of the English Engineering Units in the United States in some engineering fields. ITS-90 gives a practical means of estimating the thermodynamic temperature to a very high degree of accuracy.
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In mathematics, there are different results that share the common name of the Ky Fan inequality. The Ky Fan inequality presented here is used in game theory to investigate the existence of an equilibrium. Another Ky Fan inequality is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval.
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A totally disconnected Julia set. By the Denjoy–Riesz theorem, there exists an arc passing through all the points in this set. In topology, the Denjoy–Riesz theorem states that every compact set of totally disconnected points in the Euclidean plane can be covered by a continuous image of the unit interval, without self-intersections (a Jordan arc).
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It maps quadratic irrationals to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. In addition, it maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree.
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For example, in the theory of quivers, the (analogue of the) unit interval is the graph whose vertex set is {0,1} and which contains a single edge e whose source is 0 and whose target is 1. One can then define a notion of homotopy between quiver homomorphisms analogous to the notion of homotopy between continuous maps.
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361 in . Haar used these functions to give an example of an orthonormal system for the space of square-integrable functions on the unit interval [0, 1]. The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the Daubechies wavelet, the Haar wavelet is also known as Db1.
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In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named for who introduced it for continuous functions on the unit interval, who extended the result to some non-compact spaces, and who extended the result to compact Hausdorff spaces. There are many closely related variations of the theorem, as the linear functionals can be complex, real, or positive, the space they are defined on may be the unit interval or a compact space or a locally compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or regular Borel measures or Radon measures or signed measures or complex measures.
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In that year he published the Peano axioms, a formal foundation for the collection of natural numbers. The next year, the University of Turin also granted him his full professorship. The Peano curve was published in 1890 as the first example of a space-filling curve which demonstrated that the unit interval and the unit square have the same cardinality.
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In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space has dimension 2 + 1 = 3. In general, fiber bundles over the circle are a special case of mapping tori. Here is the construction: take the Cartesian product of a surface with the unit interval.
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Using the inverse hyperbolic function , the rapidity corresponding to velocity is where c is the velocity of light. For low speeds, is approximately . Since in relativity any velocity is constrained to the interval the ratio satisfies . The inverse hyperbolic tangent has the unit interval for its domain and the whole real line for its range, and so the interval maps onto .
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In telecommunication, a start signal is a signal that prepares a device to receive data or to perform a function. In asynchronous serial communication, start signals are used at the beginning of a character that prepares the receiving device for the reception of the code elements. A start signal is limited to one signal element usually having the duration of a unit interval.
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The Thompson group F is generated by operations like this on binary trees. Here L and T are nodes, but A B and R can be replaced by more general trees. The group F also has realizations in terms of operations on ordered rooted binary trees, and as a subgroup of the piecewise linear homeomorphisms of the unit interval that preserve orientation and whose non-differentiable points are dyadic rationals and whose slopes are all powers of 2. The group F can also be considered as acting on the unit circle by identifying the two endpoints of the unit interval, and the group T is then the group of automorphisms of the unit circle obtained by adding the homeomorphism x→x+1/2 mod 1 to F. On binary trees this corresponds to exchanging the two trees below the root.
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Fuzzy relations, which are now used throughout fuzzy mathematics and has applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1]. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition -- an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1.
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If two spaces have different number of cut-points, they are not homeomorphic. A classic example is using cut-points to show that lines and circles are not homeomorphic. Cut-points are also useful in the characterization of topological continua, a class of spaces which combine the properties of compactness and connectedness and include many familiar spaces such as the unit interval, the circle, and the torus.
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400px In geometric topology, a wild arc is an embedding of the unit interval into 3-dimensional space not equivalent to the usual one in the sense that there does not exist an ambient isotopy taking the arc to a straight line segment. found the first example of a wild arc, and found another example called the Fox-Artin arc whose complement is not simply connected.
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In 1960 Donald Samuel Ornstein constructed an example of a non-singular transformation on the Lebesgue space of the unit interval, which does not admit a \sigma–finite invariant measure equivalent to Lebesgue measure, thus solving a long-standing problem in ergodic theory. A few years later, Rafael V. Chacón gave an example of a positive (linear) isometry of L_1 for which the individual ergodic theorem fails in L_1.
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The Möbius strip corresponds to a double cover of the circle (the θ → 2θ mapping) and by changing the fiber, can also be viewed as having a two-point fiber, the unit interval as a fiber, or the real line. Complex line bundles are closely related to circle bundles. There are some celebrated ones, for example the Hopf fibrations of spheres to spheres. In algebraic geometry, an invertible sheaf (i.e.
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Each of these methods begins with three independent random scalars uniformly distributed on the unit interval. takes advantage of the odd dimension to change a Householder reflection to a rotation by negation, and uses that to aim the axis of a uniform planar rotation. Another method uses unit quaternions. Multiplication of rotation matrices is homomorphic to multiplication of quaternions, and multiplication by a unit quaternion rotates the unit sphere.
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There are numerous characterizations that tell when a second-countable topological space is metrizable, such as Urysohn's metrization theorem. The problem of determining whether a metrizable space is completely metrizable is more difficult. Topological spaces such as the open unit interval (0,1) can be given both complete metrics and incomplete metrics generating their topology. There is a characterization of complete separable metric spaces in terms of a game known as the strong Choquet game.
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In mathematics, a continuum structure function (CSF) is defined by Laurence Baxter as a nondecreasing mapping from the unit hypercube to the unit interval. It is used by Baxter to help in the Mathematical modelling of the level of performance of a system in terms of the performance levels of its components.Baxter, L A (1984) Continuum structures I., Journal of Applied Probability, 21 (4), pp. 802–815 Baxter, L A, (1986), Continuum structures.
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As a result, any surface is nonorientable if and only if it contains a Möbius band as a subspace. The Möbius strip is also a standard example used to illustrate the mathematical concept of a fiber bundle. Specifically, it is a nontrivial bundle over the circle S1 with its fiber equal to the unit interval, . Looking only at the edge of the Möbius strip gives a nontrivial two point (or Z2) bundle over S1.
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The degree of isochronous distortion, in data transmission, is the ratio of the absolute value of the maximum measured difference between the actual and the theoretical intervals separating any two significant instants of modulation (or demodulation), to the unit interval. These instants are not necessarily consecutive. This value is usually expressed as a percentage. The result of the measurement should be qualified by an indication if the period, usually limited, of the observation.
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Cf. p. 765 f. of the Handbook The Heraclitean operator ranges over the domain of all imaginable entities, including existent, non-existent, and fictitious ones, and over all possible frames of reference, a frame of reference being a particular language and a particular logic. As a result, an entity exists, does not exist, or is fictitious only to a particular extent in the unit interval [0, 1] with respect to a particular language and a particular logic.
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In applied statistics, fractional models are, to some extent, related to binary response models. However, instead of estimating the probability of being in one bin of a dichotomous variable, the fractional model typically deals with variables that take on all possible values in the unit interval. One can easily generalize this model to take on values on any other interval by appropriate transformations.Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.
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In mathematics, Maharam's theorem is a deep result about the decomposability of measure spaces, which plays an important role in the theory of Banach spaces. In brief, it states that every complete measure space is decomposable into "non-atomic parts" (copies of products of the unit interval [0,1] on the reals), and "purely atomic parts", using the counting measure on some discrete space. The theorem is due to Dorothy Maharam. It was extended to localizable measure spaces by Irving Segal.
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Also, every polynomial of odd degree admits at least one real root: these two properties make R the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra. The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalized such that the unit interval [0;1] has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g.
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In set theory, the random algebra or random real algebra is the Boolean algebra of Borel sets of the unit interval modulo the ideal of measure zero sets. It is used in random forcing to add random reals to a model of set theory. The random algebra was studied by John von Neumann in 1935 (in work later published as ) who showed that it is not isomorphic to the Cantor algebra of Borel sets modulo meager sets. Random forcing was introduced by .
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In mathematics, continuous geometry is an analogue of complex projective geometry introduced by , where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.
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The value of the game on the unit interval (a graph with two nodes with a link in-between) has been estimated approximatively. The game appears simple but is quite complicated. The obvious search strategy of starting at a random end and "sweeping" the whole interval as fast as possible guarantees a 0.75 expected capture time, and is not optimal. By utilising a more sophisticated mixed searcher and hider strategy, one can reduce the expected capture time by about 8.6%.
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The normal deviate mapping (or normal quantile function, or inverse normal cumulative distribution) is given by the probit function, so that the horizontal axis is x = probit(Pfa) and the vertical is y = probit(Pfr), where Pfa and Pfr are the false-accept and false-reject rates. The probit mapping maps probabilities from the unit interval [0,1], to the extended real line [−∞, +∞]. Since this makes the axes infinitely long, one has to confine the plot to some finite rectangle of interest.
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Hence, synchronous circuitry benefits from minimizing period jitter, so that the shortest clock period approaches the average clock period. ;Cycle-to-cycle jitter :The difference in duration of any two adjacent clock periods. It can be important for some types of clock generation circuitry used in microprocessors and RAM interfaces. In telecommunications, the unit used for the above types of jitter is usually the unit interval (UI) which quantifies the jitter in terms of a fraction of the transmission unit period.
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In probability theory and computer science, a log probability is simply a logarithm of a probability. The use of log probabilities means representing probabilities on a logarithmic scale, instead of the standard [0, 1] unit interval. Since the probability of independent events multiply, and logarithms convert multiplication to addition, log probabilities of independent events add. Log probabilities are thus practical for computations, and have an intuitive interpretation in terms of information theory: the negative of the log probability is the information content of an event.
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An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent", strictly positive and locally finite measure on an infinite-dimensional vector space. Wiener's original construction only applied to the space of real-valued continuous paths on the unit interval, known as classical Wiener space. Leonard Gross provided the generalization to the case of a general separable Banach space. The notion of a Banach space itself was discovered independently by both Wiener and Stefan Banach at around the same time.
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A nowhere dense set is not necessarily negligible in every sense. For example, if is the unit interval , not only is it possible to have a dense set of Lebesgue measure zero (such as the set of rationals), but it is also possible to have a nowhere dense set with positive measure. For one example (a variant of the Cantor set), remove from [0,1] all dyadic fractions, i.e. fractions of the form in lowest terms for positive integers and , and the intervals around them: , .
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In mathematical psychology, indifference graphs arise from utility functions, by scaling the function so that one unit represents a difference in utilities small enough that individuals can be assumed to be indifferent to it. In this application, pairs of items whose utilities have a large difference may be partially ordered by the relative order of their utilities, giving a semiorder.. In bioinformatics, the problem of augmenting a colored graph to a properly colored unit interval graph can be used to model the detection of false negatives in DNA sequence assembly from complete digests..
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While this theorem was proved in 1909 and 1910 separately by Hermann Weyl, Wacław Sierpiński and Piers Bohl, variants of this theorem continue to be studied to this day. In 1916, Weyl proved that the sequence a, 22a, 32a, ... mod 1 is uniformly distributed on the unit interval. In 1935, Ivan Vinogradov proved that the sequence pn a mod 1 is uniformly distributed, where pn is the nth prime. Vinogradov's proof was a byproduct of the odd Goldbach conjecture, that every sufficiently large odd number is the sum of three primes.
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Thus, by choosing the priorities either to be a set of independent random real numbers in the unit interval, or by choosing them to be a random permutation of the numbers from to (where is the number of nodes in the tree), and by maintaining the heap ordering property using tree rotations after any insertion or deletion of a node, it is possible to maintain a data structure that behaves like a random binary search tree. Such a data structure is known as a treap or a randomized binary search tree.; .
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Three iterations of a Peano curve construction, whose limit is a space-filling curve. In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890.. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve..
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In classical mathematics, characteristic functions of sets only take values 1 (members) or 0 (non-members). In fuzzy set theory, characteristic functions are generalized to take value in the real unit interval [0, 1], or more generally, in some algebra or structure (usually required to be at least a poset or lattice). Such generalized characteristic functions are more usually called membership functions, and the corresponding "sets" are called fuzzy sets. Fuzzy sets model the gradual change in the membership degree seen in many real-world predicates like "tall", "warm", etc.
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Let S be a metric space. A curve A in S is a continuous map from the unit interval into S, i.e., A : [0,1] \rightarrow S. A reparameterization \alpha of [0,1] is a continuous, non-decreasing, surjection \alpha: [0,1] \rightarrow [0,1]. Let A and B be two given curves in S. Then, the Fréchet distance between A and B is defined as the infimum over all reparameterizations \alpha and \beta of [0,1] of the maximum over all t \in [0,1] of the distance in S between A(\alpha(t)) and B(\beta(t)).
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In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is transcribed Tychonoff), who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1937 paper of Eduard Čech. Several texts identify Tychonoff's theorem as the single most important result in general topology [e.g.
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Two loops a, b in a torus. In mathematics, a loop in a topological space X is a continuous function f from the unit interval I = [0,1] to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its terminal point.. A loop may also be seen as a continuous map f from the pointed unit circle S1 into X, because S1 may be regarded as a quotient of I under the identification of 0 with 1. The set of all loops in X forms a space called the loop space of X.
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In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be used to represent any continuous function in Fourier analysis.. They can thus be viewed as a discrete, digital counterpart of the continuous, analog system of trigonometric functions on the unit interval. But unlike the sine and cosine functions, which are continuous, Walsh functions are piecewise constant. They take the values −1 and +1 only, on sub-intervals defined by dyadic fractions. The system of Walsh functions is known as the Walsh system.
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The interval [−1,1], with length two, demarcated by the positive and negative units, occurs frequently, such as in the range of the trigonometric functions sine and cosine and the hyperbolic function tanh. This interval may be used for the domain of inverse functions. For instance, when θ is restricted to [−π/2, π/2] then sin(θ) is in this interval and arcsine is defined there. Sometimes, the term "unit interval" is used to refer to objects that play a role in various branches of mathematics analogous to the role that [0,1] plays in homotopy theory.
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In differential geometry, the Lie–Palais theorem states that an action of a finite-dimensional Lie algebra on a smooth compact manifold can be lifted to an action of a finite-dimensional Lie group. For manifolds with boundary the action must preserve the boundary, in other words the vector fields on the boundary must be tangent to the boundary. proved it as a global form of an earlier local theorem due to Sophus Lie. The example of the vector field d/dx on the open unit interval shows that the result is false for non-compact manifolds.
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In particular, if the space Z is assumed to be simply connected (so that is trivial), condition is automatically satisfied, and every continuous map from Z to X can be lifted. Since the unit interval is simply connected, the lifting property for paths is a special case of the lifting property for maps stated above. If is a covering and and are such that , then p# is injective at the level of fundamental groups, and the induced homomorphisms are isomorphisms for all . Both of these statements can be deduced from the lifting property for continuous maps.
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The density of a topological space (the least of the cardinalities of its dense subsets) is a topological invariant. A topological space with a connected dense subset is necessarily connected itself. Continuous functions into Hausdorff spaces are determined by their values on dense subsets: if two continuous functions into a Hausdorff space Y agree on a dense subset of X then they agree on all of X. For metric spaces there are universal spaces, into which all spaces of given density can be embedded: a metric space of density is isometric to a subspace of , the space of real continuous functions on the product of copies of the unit interval.
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One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval , some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence accumulate to 0 (while others accumulate to 1).
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To wit, there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivity axioms; their morphisms are those functions which map connected sets to connected sets . Topological spaces and graphs are special cases of connective spaces; indeed, the finite connective spaces are precisely the finite graphs. However, every graph can be canonically made into a topological space, by treating vertices as points and edges as copies of the unit interval (see topological graph theory#Graphs as topological spaces). Then one can show that the graph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.
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However, the product of continuum many copies of the closed unit interval (with its usual topology) fails to be sequentially compact with respect to the product topology, even though it is compact by Tychonoff's theorem (e.g., see ). This is a critical failure: if X is a completely regular Hausdorff space, there is a natural embedding from X into [0,1]C(X,[0,1]), where C(X,[0,1]) is the set of continuous maps from X to [0,1]. The compactness of [0,1]C(X,[0,1]) thus shows that every completely regular Hausdorff space embeds in a compact Hausdorff space (or, can be "compactified".) This construction is the Stone–Čech compactification.
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Originally, Grigorchuk's group G was constructed as a group of Lebesgue-measure-preserving transformations on the unit interval, but subsequently simpler descriptions of G were found and it is now usually presented as a group of automorphisms of the infinite regular binary rooted tree. The study of Grigorchuk's group informed in large part the development of the theory of branch groups, automata groups and self-similar groups in the 1990s-2000s and Grigorchuk's group remains a central object in this theory. Recently important connections between this theory and complex dynamics, particularly the notion of iterated monodromy groups, have been uncovered in the work of Volodymyr Nekrashevych.
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In mathematics, there are two different results that share the common name of the Ky Fan inequality. One is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval. The result was published on page 5 of the book Inequalities by Edwin F. Beckenbach and Richard E. Bellman (1961), who refer to an unpublished result of Ky Fan. They mention the result in connection with the inequality of arithmetic and geometric means and Augustin Louis Cauchy's proof of this inequality by forward-backward-induction; a method which can also be used to prove the Ky Fan inequality.
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In probability theory, a standard probability space, also called Lebesgue- Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms. The theory of standard probability spaces was started by von Neumann in 1932 and shaped by Vladimir Rokhlin in 1940. Rokhlin showed that the unit interval endowed with the Lebesgue measure has important advantages over general probability spaces, yet can be effectively substituted for many of these in probability theory.
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Bit reversal is most important for radix-2 Cooley–Tukey FFT algorithms, where the recursive stages of the algorithm, operating in-place, imply a bit reversal of the inputs or outputs. Similarly, mixed-radix digit reversals arise in mixed-radix Cooley–Tukey FFTs.B. Gold and C. M. Rader, Digital Processing of Signals (New York: McGraw–Hill, 1969). The bit reversal permutation has also been used to devise lower bounds in distributed computation.. The Van der Corput sequence, a low-discrepancy sequence of numbers in the unit interval, is formed by reinterpreting the indexes of the bit-reversal permutation as the fixed-point binary representations of dyadic rational numbers.
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A/D and D/A conversion A beta encoder is an analog-to-digital conversion (A/D) system in which a real number in the unit interval is represented by a finite representation of a sequence in base beta, with beta being a real number between 1 and 2. Beta encoders are an alternative to traditional approaches to pulse-code modulation. As a form of non-integer representation, beta encoding contrasts with traditional approaches to binary quantization, in which each value is mapped to the first N bits of its base-2 expansion. Rather than using base 2, beta encoders use base beta as a beta-expansion.
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It is important to disambiguate algorithmic randomness with stochastic randomness. Unlike algorithmic randomness, which is defined for computable (and thus deterministic) processes, stochastic randomness is usually said to be a property of a sequence that is a priori known to be generated (or is the outcome of) by an independent identically distributed equiprobable stochastic process. Because infinite sequences of binary digits can be identified with real numbers in the unit interval, random binary sequences are often called (algorithmically) random real numbers. Additionally, infinite binary sequences correspond to characteristic functions of sets of natural numbers; therefore those sequences might be seen as sets of natural numbers.
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For this reason, the Lebesgue definition makes it possible to calculate integrals for a broader class of functions. For example, the Dirichlet function, which is 0 where its argument is irrational and 1 otherwise, has a Lebesgue integral, but does not have a Riemann integral. Furthermore, the Lebesgue integral of this function is zero, which agrees with the intuition that when picking a real number uniformly at random from the unit interval, the probability of picking a rational number should be zero. Lebesgue summarized his approach to integration in a letter to Paul Montel: The insight is that one should be able to rearrange the values of a function freely, while preserving the value of the integral.
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An indifference graph, formed from a set of points on the real line by connecting pairs of points whose distance is at most one In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting two vertices by an edge when their numbers are within one unit of each other.. Indifference graphs are also the intersection graphs of sets of unit intervals, or of properly nested intervals (intervals none of which contains any other one). Based on these two types of interval representations, these graphs are also called unit interval graphs or proper interval graphs; they form a subclass of the interval graphs.
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Basic fuzzy logic is the logic of continuous t-norms (binary operations on the real unit interval [0, 1]). Applications involving fuzzy logic include facial recognition systems, home appliances, anti-lock braking systems, automatic transmissions, controllers for rapid transit systems and unmanned aerial vehicles, knowledge-based and engineering optimization systems, weather forecasting, pricing, and risk assessment modeling systems, medical diagnosis and treatment planning and commodities trading systems, and more. Fuzzy logic is used to optimize efficiency in thermostats for control of heating and cooling, for industrial automation and process control, computer animation, signal processing, and data analysis. Fuzzy logic has made significant contributions in the fields of machine learning and data mining.
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In a letter to Wacław Sierpiński, motivated by some results of Giuseppe Vitali, Tibor Radó observed that for every covering of a unit interval, one can select a subcovering consisting of pairwise disjoint intervals with total length at least 1/2 and that this number cannot be improved. He then asked for an analogous statement in the plane. : If the area of the union of a finite set of squares in the plane with parallel sides is one, what is the guaranteed maximum total area of a pairwise disjoint subset? Radó proved that this number is at least 1/9 and conjectured that it is at least 1/4 a constant which cannot be further improved.
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The Boolean domain {0, 1} can be replaced by the unit interval , in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1-x, conjunction (AND) is replaced with multiplication (xy), and disjunction (OR) is defined via De Morgan's law to be 1-(1-x)(1-y). Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.
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Logics that restrict the t-norm semantics to a subset of the real unit interval (for example, finitely valued Łukasiewicz logics) are usually included in the class as well. Important examples of t-norm fuzzy logics are monoidal t-norm logic MTL of all left-continuous t-norms, basic logic BL of all continuous t-norms, product fuzzy logic of the product t-norm, or the nilpotent minimum logic of the nilpotent minimum t-norm. Some independently motivated logics belong among t-norm fuzzy logics, too, for example Łukasiewicz logic (which is the logic of the Łukasiewicz t-norm) or Gödel–Dummett logic (which is the logic of the minimum t-norm).
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In logic, the unit interval [0,1] can be interpreted as a generalization of the Boolean domain {0,1}, in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1-x ; conjunction (AND) is replaced with multiplication (xy); and disjunction (OR) is defined, per De Morgan's laws, as 1-(1-x)(1-y) . Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.
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In algebraic geometry and algebraic topology, branches of mathematics, homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval , which is not an algebraic variety, with the affine line , which is. The theory requires a substantial amount of technique to set up, but has spectacular applications such as Voevodsky's construction of the derived category of mixed motives and the proof of the Milnor and Bloch-Kato conjectures.
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In telecommunication, the term degree of start-stop distortion has the following meanings: # In asynchronous serial communication data transmission, the ratio of (a) the absolute value of the maximum measured difference between the actual and theoretical intervals separating any significant instant of modulation (or demodulation) from the significant instant of the start element immediately preceding it to (b) the unit interval. # The highest absolute value of individual distortion affecting the significant instants of a start- stop modulation. The degree of distortion of a start-stop modulation (or demodulation) is usually expressed as a percentage. Distinction can be made between the degree of late (positive) distortion and the degree of early (negative) distortion.
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Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine to make the algebra of two values of fundamental importance to computer hardware, mathematical logic, and set theory. Two-valued logic can be extended to multi- valued logic, notably by replacing the Boolean domain {0, 1} with the unit interval [0,1], in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1 − x, conjunction (AND) is replaced with multiplication (xy), and disjunction (OR) is defined via De Morgan's law. Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic.
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Since the relation of two functions f, g\colon X\to Y being homotopic relative to a subspace is an equivalence relation, we can look at the equivalence classes of maps between a fixed X and Y. If we fix X = [0,1]^n, the unit interval [0, 1] crossed with itself n times, and we take its boundary \partial([0,1]^n) as a subspace, then the equivalence classes form a group, denoted \pi_n(Y,y_0), where y_0 is in the image of the subspace \partial([0,1]^n). We can define the action of one equivalence class on another, and so we get a group. These groups are called the homotopy groups. In the case n = 1, it is also called the fundamental group.
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Since for each this removes intervals adding up to at most , the nowhere dense set remaining after all such intervals have been removed has measure of at least (in fact just over 0.535... because of overlaps) and so in a sense represents the majority of the ambient space . This set is nowhere dense, as it is closed and has an empty interior: any interval is not contained in the set since the dyadic fractions in have been removed. Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than 1, although the measure cannot be exactly 1 (else the complement of its closure would be a nonempty open set with measure zero, which is impossible).
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A perfectly normal space is a topological space X in which every two disjoint closed sets E and F can be precisely separated by a continuous function f from X to the real line R: the preimages of {0} and {1} under f are, respectively, E and F. (In this definition, the real line can be replaced with the unit interval [0,1].) It turns out that X is perfectly normal if and only if X is normal and every closed set is a Gδ set. Equivalently, X is perfectly normal if and only if every closed set is a zero set. Every perfectly normal space is automatically completely normal. A Hausdorff perfectly normal space X is a T6 space, or perfectly T4 space.
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Two subsets A and B of a topological space X are said to be separated by neighbourhoods if there are neighbourhoods U of A and V of B that are disjoint. In particular A and B are necessarily disjoint. Two plain subsets A and B are said to be separated by a function if there exists a continuous function f from X into the unit interval [0,1] such that f(a) = 0 for all a in A and f(b) = 1 for all b in B. Any such function is called a Urysohn function for A and B. In particular A and B are necessarily disjoint. It follows that if two subsets A and B are separated by a function then so are their closures.
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In mathematics, the order polytope of a finite partially ordered set is a convex polytope defined from the set. The points of the order polytope are the monotonic functions from the given set to the unit interval, its vertices correspond to the upper sets of the partial order, and its dimension is the number of elements in the partial order. The order polytope is a distributive polytope, meaning that coordinatewise minima and maxima of pairs of its points remain within the polytope. The order polytope of a partial order should be distinguished from the linear ordering polytope, a polytope defined from a number n as the convex hull of indicator vectors of the sets of edges of n-vertex transitive tournaments.
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Using the probability integral transform, if X is any random variable, and F is the cumulative distribution function of X, then as long as F is invertible, the random variable U = F(X) follows a uniform distribution on the unit interval [0,1]. From a uniform distribution, we can transform to any distribution with an invertible cumulative distribution function. If G is an invertible cumulative distribution function, and U is a uniformly distributed random variable, then the random variable G−1(U) has G as its cumulative distribution function. Putting the two together, if X is any random variable, F is the invertible cumulative distribution function of X, and G is an invertible cumulative distribution function then the random variable G−1(F(X)) has G as its cumulative distribution function.
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The points traced by a path from A to B in R². However, different paths can trace the same set of points. In mathematics, a path in a topological space X is a continuous function f from the unit interval I = [0,1] to X :f : I -> X. The initial point of the path is f(0) and the terminal point is f(1). One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path. Note that a path is not just a subset of X which "looks like" a curve, it also includes a parameterization. For example, the maps f(x) = x and g(x) = x2 represent two different paths from 0 to 1 on the real line.
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It is dual to the mapping cone in the sense that the product above is essentially the fibered product or pullback X\times_f Y which is dual to the pushout X\sqcup_f Y used to construct the mapping cone. In this particular case, the duality is essentially that of currying, in that the mapping cone (X\times I)\sqcup_f Y has the curried form X \times_f (I\to Y) where I\to Y is simply an alternate notation for the space Y^I of all continuous maps from the unit interval to Y. The two variants are related by an adjoint functor. Observe that the currying preserves the reduced nature of the maps: in the one case, to the tip of the cone, and in the other case, paths to the basepoint.
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Closed-form expressions are an important sub-class of analytic expressions, which contain a bounded or an unbounded number of applications of well-known functions. Unlike the broader analytic expressions, the closed-form expressions do not include infinite series or continued fractions; neither includes integrals or limits. Indeed, by the Stone–Weierstrass theorem, any continuous function on the unit interval can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include all continuous functions. Similarly, an equation or system of equations is said to have a closed-form solution if, and only if, at least one solution can be expressed as a closed-form expression; and it is said to have an analytic solution if and only if at least one solution can be expressed as an analytic expression.
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In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence be the sequence of moments :m_n = \int_0^1 x^n\,d\mu(x) of some Borel measure supported on the closed unit interval . In the case , this is equivalent to the existence of a random variable supported on , such that . The essential difference between this and other well-known moment problems is that this is on a bounded interval, whereas in the Stieltjes moment problem one considers a half-line , and in the Hamburger moment problem one considers the whole line . The Stieltjes moment problems and the Hamburger moment problems, if they are solvable, may have infinitely many solutions (indeterminate moment problem) whereas a Hausdorff moment problem always has a unique solution if it is solvable (determinate moment problem).
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The maximal product measure can be constructed by applying Carathéodory's extension theorem to the additive function μ such that μ(A×B)=μ1(A)μ2(B) on the ring of sets generated by products of measurable sets. (Carathéodory's extension theorem gives a measure on a measure space that in general contains more measurable sets than the measure space X×Y, so strictly speaking the measure should be restricted to the σ-algebra generated by the products A×B of measurable subsets of X and Y.) The product of two complete measure spaces is not usually complete. For example, the product of the Lebesgue measure on the unit interval I with itself is not the Lebesgue measure on the square I×I. There is a variation of Fubini's theorem for complete measures, which uses the completion of the product of measures rather than the uncompleted product.
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T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics that takes the real unit interval [0, 1] for the system of truth values and functions called t-norms for permissible interpretations of conjunction. They are mainly used in applied fuzzy logic and fuzzy set theory as a theoretical basis for approximate reasoning. T-norm fuzzy logics belong in broader classes of fuzzy logics and many-valued logics. In order to generate a well-behaved implication, the t-norms are usually required to be left-continuous; logics of left-continuous t-norms further belong in the class of substructural logics, among which they are marked with the validity of the law of prelinearity, (A -> B) ∨ (B -> A). Both propositional and first-order (or higher-order) t-norm fuzzy logics, as well as their expansions by modal and other operators, are studied.
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We do this first for K = [0, 1], where the desired extension of f : X → [0, 1] is just the projection onto the f coordinate in [0, 1]C. In order to then get this for general compact Hausdorff K we use the above to note that K can be embedded in some cube, extend each of the coordinate functions and then take the product of these extensions. The special property of the unit interval needed for this construction to work is that it is a cogenerator of the category of compact Hausdorff spaces: this means that if A and B are compact Hausdorff spaces, and f and g are distinct maps from A to B, then there is a map h : B → [0, 1] such that hf and hg are distinct. Any other cogenerator (or cogenerating set) can be used in this construction.
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Intuitively, a curve in two or three (or higher) dimensions can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a curve: In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane (a planar curve) or the 3-dimensional space (space curve). Sometimes, the curve is identified with the image of the function (the set of all possible values of the function), instead of the function itself. It is also possible to define curves without endpoints to be a continuous function on the real line (or on the open unit interval ).
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Near a Rajchman measure, particularly important notion invented by Rajchman is a Rajchman algebra associated with a locally compact group which is defined to be the set of all elements of the Fourier-Stieltjes algebra which vanish at infinity, a closed and complemented ideal in the Fourier-Stieltjes algebra that contains the Fourier algebra. His first doctoral student a noted Polish mathematician Antoni Zygmund created the Chicago school of mathematical analysis with the emphasis onto harmonic analysis, which produced the 1966 Fields Medal winner Paul Cohen. His second doctoral student Zygmunt Zalcwasser, co-advised by Wacław Sierpiński, introduced the Zalcwasser rank to measure the uniform convergence of sequences of continuous functions on the unit interval. In October 2000, the Stefan Banach International Mathematical Center at the Institute of Mathematics of the Polish Academy of Sciences honoured Rajchman's achievements by the Rajchman-Zygmund-Marcinkiewicz Symposium.
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The girth is always at least four, because the shortest path on the unit sphere between two opposite points cannot be shorter than the length-two line segment connecting them through the origin of the space. A Banach space for which it is exactly four is said to be flat. There exist flat Banach spaces of infinite dimension in which the girth is achieved by a minimum-length curve; an example is the space C[0,1] of continuous functions from the unit interval to the real numbers, with the sup norm. The unit sphere of such a space has the counterintuitive property that certain pairs of opposite points have the same distance within the sphere that they do in the whole space.. The girth is a continuous function on the Banach–Mazur compactum, a space whose points correspond to the normed vector spaces of a given dimension.
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In mathematics, a random minimum spanning tree may be formed by assigning random weights from some distribution to the edges of an undirected graph, and then constructing the minimum spanning tree of the graph. When the given graph is a complete graph on vertices, and the edge weights have a continuous distribution function whose derivative at zero is , then the expected weight of its random minimum spanning trees is bounded by a constant, rather than growing as a function of . More precisely, this constant tends in the limit (as goes to infinity) to , where is the Riemann zeta function and is Apéry's constant. For instance, for edge weights that are uniformly distributed on the unit interval, the derivative is , and the limit is just .. In contrast to uniformly random spanning trees of complete graphs, for which the typical diameter is proportional to the square root of the number of vertices, random minimum spanning trees of complete graphs have typical diameter proportional to the cube root.
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Given a map f\colon X \to Y, the mapping cone C_f is defined to be the quotient space of the mapping cylinder (X \times I) \sqcup_f Y with respect to the equivalence relation (x, 0) \sim (x',0)\,, (x,1) \sim f(x) on X. Here I denotes the unit interval [0, 1] with its standard topology. Note that some authors (like J. Peter May) use the opposite convention, switching 0 and 1. Visually, one takes the cone on X (the cylinder X \times I with one end (the 0 end) identified to a point), and glues the other end onto Y via the map f (the identification of the 1 end). Coarsely, one is taking the quotient space by the image of X, so C_f = Y/f(X); this is not precisely correct because of point-set issues, but is the philosophy, and is made precise by such results as the homology of a pair and the notion of an n-connected map.
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The following example in Fortran77 was published in 2008 which includes a discussion of how to initialise : DOUBLE PRECISION FUNCTION ACORNJ(XDUMMY) C C Fortran implementation of ACORN random number generator C of order less than or equal to 120 (higher orders can be C obtained by increasing the parameter value MAXORD) and C modulus less than or equal to 2^60. C C After appropriate initialization of the common block /IACO2/ C each call to ACORNJ generates a single variate drawn from C a uniform distribution over the unit interval. C IMPLICIT DOUBLE PRECISION (A-H,O-Z) PARAMETER (MAXORD=120,MAXOP1=MAXORD+1) COMMON /IACO2/ KORDEJ,MAXJNT,IXV1(MAXOP1),IXV2(MAXOP1) DO 7 I=1,KORDEJ IXV1(I+1)=(IXV1(I+1)+IXV1(I)) IXV2(I+1)=(IXV2(I+1)+IXV2(I)) IF (IXV2(I+1).GE.MAXJNT) THEN IXV2(I+1)=IXV2(I+1)-MAXJNT IXV1(I+1)=IXV1(I+1)+1 ENDIF IF (IXV1(I+1).
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The Klein bottle can be seen as a fiber bundle over the circle S1, with fibre S1, as follows: one takes the square (modulo the edge identifying equivalence relation) from above to be E, the total space, while the base space B is given by the unit interval in y, modulo 1~0. The projection π:E→B is then given by . The Klein bottle can be constructed (in a four dimensional space, because in three dimensional space it cannot be done without allowing the surface to intersect itself) by joining the edges of two Möbius strips together, as described in the following limerick by Leo Moser: The initial construction of the Klein bottle by identifying opposite edges of a square shows that the Klein bottle can be given a CW complex structure with one 0-cell P, two 1-cells C1, C2 and one 2-cell D. Its Euler characteristic is therefore . The boundary homomorphism is given by and , yielding the homology groups of the Klein bottle K to be , and for .
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A homotopy between two embeddings of the torus into R3: as "the surface of a doughnut" and as "the surface of a coffee mug". This is also an example of an isotopy. Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function H: X \times [0,1] \to Y from the product of the space X with the unit interval [0, 1] to Y such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second parameter of H as time then H describes a continuous deformation of f into g: at time 0 we have the function f and at time 1 we have the function g. We can also think of the second parameter as a "slider control" that allows us to smoothly transition from f to g as the slider moves from 0 to 1, and vice versa.
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Because of the simple nature of the dynamics when the iterates are viewed in binary notation, it is easy to categorize the dynamics based on the initial condition: If the initial condition is irrational (as almost all points in the unit interval are), then the dynamics are non-periodic—this follows directly from the definition of an irrational number as one with a non-repeating binary expansion. This is the chaotic case. If x0 is rational the image of x0 contains a finite number of distinct values within [0, 1) and the forward orbit of x0 is eventually periodic, with period equal to the period of the binary expansion of x0. Specifically, if the initial condition is a rational number with a finite binary expansion of k bits, then after k iterations the iterates reach the fixed point 0; if the initial condition is a rational number with a k-bit transient (k ≥ 0) followed by a q-bit sequence (q > 1) that repeats itself infinitely, then after k iterations the iterates reach a cycle of length q.
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An interval exchange transformation is a dynamical system defined from a partition of the unit interval into finitely many smaller intervals, and a permutation on those intervals. Veech and Howard Masur independently discovered that, for almost every partition and every irreducible permutation, these systems are uniquely ergodic, and also made contributions to the theory of weak mixing for these systems.. See in particular p. 51. The Rauzy–Veech–Zorich induction map, a function from and to the space of interval exchange transformations is named in part after Veech: Rauzy defined the map, Veech constructed an infinite invariant measure for it, and Zorich strengthened Veech's result by making the measure finite.. The Veech surface and the related Veech group are named after Veech, as is the Veech dichotomy according to which geodesic flow on the Veech surface is either periodic or ergodic.. Veech played a role in the Nobel-prize-winning discovery of buckminsterfullerene in 1985 by a team of Rice University chemists including Richard Smalley. At that time, Veech was chair of the Rice mathematics department, and was asked by Smalley to identify the shape that the chemists had determined for this molecule.
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In a paper published in 1924, Stefan Banach and Alfred Tarski gave a construction of such a paradoxical decomposition, based on earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical decompositions of the sphere by Felix Hausdorff, and discussed a number of related questions concerning decompositions of subsets of Euclidean spaces in various dimensions. They proved the following more general statement, the strong form of the Banach–Tarski paradox: : Given any two bounded subsets and of a Euclidean space in at least three dimensions, both of which have a nonempty interior, there are partitions of and into a finite number of disjoint subsets, A=A_1 \cup \cdots\cup A_k, B=B_1 \cup \cdots\cup B_k (for some integer k), such that for each (integer) between and , the sets and are congruent. Now let be the original ball and be the union of two translated copies of the original ball. Then the proposition means that you can divide the original ball into a certain number of pieces and then rotate and translate these pieces in such a way that the result is the whole set , which contains two copies of .
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There are 8 possible geometric structures in 3 dimensions, described in the next section. There is a unique minimal way of cutting an irreducible oriented 3-manifold along tori into pieces that are Seifert manifolds or atoroidal called the JSJ decomposition, which is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. (For example, the mapping torus of an Anosov map of a torus has a finite volume solv structure, but its JSJ decomposition cuts it open along one torus to produce a product of a torus and a unit interval, and the interior of this has no finite volume geometric structure.) For non-oriented manifolds the easiest way to state a geometrization conjecture is to first take the oriented double cover. It is also possible to work directly with non-orientable manifolds, but this gives some extra complications: it may be necessary to cut along projective planes and Klein bottles as well as spheres and tori, and manifolds with a projective plane boundary component usually have no geometric structure.
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