Knowing whether a polynomial is always nonnegative might seem like a mathematical triviality.
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When an expression is a sum of squares, you know that it's always nonnegative.
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As you repeat this procedure, you're interested in knowing at each step whether the polynomial is still nonnegative.
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In answering this question you might test different values—can I subtract 3 from the polynomial such that it's still nonnegative?
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If you're handed a complicated polynomial—one with dozens of variables raised to high powers—it's not easy to determine straightaway whether it's always nonnegative.
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One way to find the minimum value is to ask yourself: What's the most I can subtract from a nonnegative polynomial before it turns negative somewhere?
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Over the past 10 months, the back-bench candidate Marianne Williamson, the early exiter Beto O'Rourke, and the out-of-nowhere surprise success story Pete Buttigieg have all flirted with some form of a nonnegative campaign.
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In practice, you want to minimize a value—the distance between the wall and the booth—and so you shift the graph of the polynomial around to see how far you can push it before it ceases to be nonnegative.
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The nonnegative rank of a matrix can be determined algorithmically.J. Cohen and U. Rothblum. "Nonnegative ranks, decompositions and factorizations of nonnegative matrices". Linear Algebra and its Applications, 190:149–168, 1993.
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In linear algebra, the nonnegative rank of a nonnegative matrix is a concept similar to the usual linear rank of a real matrix, but adding the requirement that certain coefficients and entries of vectors/matrices have to be nonnegative. For example, the linear rank of a matrix is the smallest number of vectors, such that every column of the matrix can be written as a linear combination of those vectors. For the nonnegative rank, it is required that the vectors must have nonnegative entries, and also that the coefficients in the linear combinations are nonnegative.
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The exponential of a Metzler (or quasipositive) matrix is a nonnegative matrix because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices of continuous-time finite-state Markov processes are always Metzler matrices, and that probability distributions are always non-negative. A Metzler matrix has an eigenvector in the nonnegative orthant because of the corresponding property for nonnegative matrices.
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It is interesting to study the curvature of polyhedral spaces (the curvature in the sense of Alexandrov spaces), specifically polyhedral spaces of nonnegative and nonpositive curvature. Nonnegative curvature on singularities of codimension 2 implies nonnegative curvature overall. However, this is false for nonpositive curvature. For example, consider R^3 with one octant removed.
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A square of a real number is nonnegative, and a sum of nonnegative numbers is zero iff both numbers are 0. So ƒ(x)2 = 0 for all and is the only solution.
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In case the nonnegative rank of is equal to its actual rank, is called a nonnegative rank factorization. The problem of finding the NRF of , if it exists, is known to be NP-hard.
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It has been proved that determining whether {{rank}_+}(A)= rank(A) is NP-hard.Stephen Vavasis, On the complexity of nonnegative matrix factorization, SIAM Journal on Optimization 20 (3) 1364-1377, 2009. Obvious questions concerning the complexity of nonnegative rank computation remain unanswered to date. For example, the complexity of determining the nonnegative rank of matrices of fixed rank k is unknown for k > 2.
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In linear algebra, a square nonnegative matrix A of order n is said to be productive, or to be a Leontief matrix, if there exists a n \times 1 nonnegative column matrix P such as P-AP is a positive matrix.
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Historically, questions about extensions first surfaced in combinatorial optimization, where extensions arise naturally from extended formulations. A seminal work by Yannakakis connected extension complexity to various other notions in mathematics, in particular nonnegative rank of nonnegative matrices and communication complexity.
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Current research (since 2010) in nonnegative matrix factorization includes, but is not limited to, # Algorithmic: searching for global minima of the factors and factor initialization. # Scalability: how to factorize million-by-billion matrices, which are commonplace in Web-scale data mining, e.g., see Distributed Nonnegative Matrix Factorization (DNMF), Scalable Nonnegative Matrix Factorization (ScalableNMF), Distributed Stochastic Singular Value Decomposition. # Online: how to update the factorization when new data comes in without recomputing from scratch, e.g.
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Yao showed that every nonempty set of nonnegative integers is the score set for some tournament.
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Haddad's treatise on Nonnegative and Compartmental Dynamical Systems, Princeton, NJ: Princeton University Press, 2010, presents a complete analysis and design framework for modeling and feedback control of nonnegative and compartmental dynamical systems. This work is rigorously theoretical in nature yet vitally practical in impact. The concepts are illustrated by examples from biology, chemistry, ecology, economics, genetics, medicine, sociology, and engineering. This book develops a unified stability and dissipativity analysis and control design framework for nonnegative and compartmental dynamical systems in order to foster the understanding of these systems as well as advancing the state-of-the-art in active control of nonnegative and compartmental systems.
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This may be confusing, as sometimes nonnegative matrices (respectively, nonpositive matrices) are also denoted in this way.
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The rank of the matrix A is the largest number of columns which are linearly independent, i.e., none of the selected columns can be written as a linear combination of the other selected columns. It is not true that adding nonnegativity to this characterization gives the nonnegative rank: The nonnegative rank is in general less than or equal to the largest number of columns such that no selected column can be written as a nonnegative linear combination of the other selected columns.
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Nonnegative rank has important applications in Combinatorial optimization:Mihalis Yannakakis. Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci., 43(3):441–466, 1991.
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The older terms were misleading, in view of the examples below. Every line bundle L on a proper curve C over k which has a global section that is not identically zero has nonnegative degree. As a result, a basepoint-free line bundle on a proper scheme X over k has nonnegative degree on every curve in X; that is, it is nef.Lazarsfeld (2004), Example 1.4.5.
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"On the Equivalence of Nonnegative Matrix Factorization and Spectral Clustering". Proc. SIAM Int'l Conf. Data Mining, pp. 606-610. May 2005Ron Zass and Amnon Shashua (2005).
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Note that has the following properties: #It is subadditive: . #It is homogeneous: for all scalars . #It is nonnegative: . Therefore, is a seminorm on , with an induced topology.
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Matsushima and Tanno showed that, on a Kähler manifold, any Codazzi tensor which is hermitian is parallel. Berger showed that, on a compact manifold of nonnegative sectional curvature, any Codazzi tensor h with trgh constant must be parallel. Furthermore, on a compact manifold of nonnegative sectional curvature, if the sectional curvature is strictly positive at least one point, then every symmetric parallel 2-tensor is a constant multiple of the metric.
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With a matrix approach of the input-output model, the consumption matrix is productive if it is economically viable and if the latter and the demand vector are nonnegative.
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An n × n matrix P is doubly stochastic precisely if both P and its transpose PT are stochastic matrices. A stochastic matrix is a square matrix of nonnegative real entries in which the sum of the entries in each column is 1. Thus, a doubly stochastic matrix is a square matrix of nonnegative real entries in which the sum of the entries in each row and the sum of the entries in each column is 1.
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The above example of adding and then becomes: . On the 35s, the number of functions able to handle complex numbers is limited and somewhat arbitrary. For example, directly taking the square root of a negative real number results in an error message instead of a complex number. This is strictly correct given that a nonnegative real number a has a unique nonnegative square root and this is called the principal square root which is denoted by .
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In full generality, polyhedral spaces were first defined by Milka Milka, A. D. Multidimensional spaces with polyhedral metric of nonnegative curvature. I. (Russian) Ukrain. Geometr. Sb. Vyp. 5--6 1968 103–114.
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NMath Stats can be used for descriptive statistics, linear regression, ANOVA, probability distributions, hypothesis tests, principal component analysis, cluster analysis, nonnegative matrix factorization (NMF), partial least squares (PLS) and Savitzky-Golay smoothing.
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The proof of the theorem may be most easily understood as an application of the Perron-Frobenius theorem. This latter theorem comes from a branch of linear algebra known as the theory of nonnegative matrices. A good source text for the basic theory is Seneta (1973). The statement of Okishio's theorem, and the controversies surrounding it, may however be understood intuitively without reference to, or in-depth knowledge of, the Perron-Frobenius theorem or the general theory of nonnegative matrices.
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The unique solution λ represents the rate of growth of the economy, which equals the interest rate. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann.For this problem to have a unique solution, it suffices that the nonnegative matrices A and B satisfy an irreducibility condition, generalizing that of the Perron–Frobenius theorem of nonnegative matrices, which considers the (simplified) eigenvalue problem : A - λ I q = 0, where the nonnegative matrix A must be square and where the diagonal matrix I is the identity matrix. Von Neumann's irreducibility condition was called the "whales and wranglers" hypothesis by David Champernowne, who provided a verbal and economic commentary on the English translation of von Neumann's article.
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The unique solution λ represents the growth factor which is 1 plus the rate of growth of the economy; the rate of growth equals the interest rate.For this problem to have a unique solution, it suffices that the nonnegative matrices A and B satisfy an irreducibility condition, generalizing that of the Perron–Frobenius theorem of nonnegative matrices, which considers the (simplified) eigenvalue problem : A − λ I q = 0, where the nonnegative matrix A must be square and where the diagonal matrix I is the identity matrix. Von Neumann's irreducibility condition was called the "whales and wranglers" hypothesis by D. G. Champernowne, who provided a verbal and economic commentary on the English translation of von Neumann's article. Von Neumann's hypothesis implied that every economic process used a positive amount of every economic good.
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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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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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Once the pivot column has been selected, the choice of pivot row is largely determined by the requirement that the resulting solution be feasible. First, only positive entries in the pivot column are considered since this guarantees that the value of the entering variable will be nonnegative. If there are no positive entries in the pivot column then the entering variable can take any nonnegative value with the solution remaining feasible. In this case the objective function is unbounded below and there is no minimum.
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There are several different definitions and types of stochastic matrices: :A right stochastic matrix is a real square matrix, with each row summing to 1. :A left stochastic matrix is a real square matrix, with each column summing to 1. :A doubly stochastic matrix is a square matrix of nonnegative real numbers with each row and column summing to 1. In the same vein, one may define a stochastic vector (also called probability vector) as a vector whose elements are nonnegative real numbers which sum to 1.
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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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In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only.
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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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A convergent series of numbers can often be reordered in such a way that the new series diverges. This is not possible for series of nonnegative numbers, however, so the notion of absolute-convergence precludes this phenomenon. When dealing with uniformly convergent series of functions, the same phenomenon occurs: the series can potentially be reordered into a non-uniformly convergent series, or a series which does not even converge pointwise. This is impossible for series of nonnegative functions, so the notion of uniform absolute-convergence can be used to rule out these possibilities.
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Karsten Grove is a Danish-American mathematician working in metric and differential geometry, differential topology and global analysis, mainly in topics related to global Riemannian geometry, Alexandrov geometry, isometric group actions and manifolds with positive or nonnegative sectional curvature.
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A hyperbolic Coxeter group is compact if all subgroups are finite (i.e. have positive determinants), and paracompact if all its subgroups are finite or affine (i.e. have nonnegative determinants). Finite and affine groups are also called elliptical and parabolic respectively.
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Primaries of some color spaces are complete (that is, all visible colors are described in terms of their weighted sums with nonnegative weights) but necessarily imaginaryBruce MacEvoy. "Do 'Primary' Colors Exist?" (imaginary or imperfect primaries section ). Handprint. Accessed 10 August 2007.
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In quantum information and quantum computation, an entanglement monotone is a function that quantifies the amount of entanglement present in a quantum state. Any entanglement monotone is a nonnegative function whose value does not increase under local operations and classical communication.
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The degree of a line bundle L on a proper curve C over k is defined as the degree of the divisor (s) of any nonzero rational section s of L. The coefficients of this divisor are positive at points where s vanishes and negative where s has a pole. Therefore, any line bundle L on a curve C such that H^0(C,L) eq 0 has nonnegative degree (because sections of L over C, as opposed to rational sections, have no poles).Hartshorne (1977), Lemma IV.1.2. In particular, every basepoint-free line bundle on a curve has nonnegative degree.
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The following property holds: after h2-sorting of any h1-sorted array, the array remains h1-sorted. Every h1-sorted and h2-sorted array is also (a1h1+a2h2)-sorted, for any nonnegative integers a1 and a2. The worst-case complexity of Shellsort is therefore connected with the Frobenius problem: for given integers h1,..., hn with gcd = 1, the Frobenius number g(h1,..., hn) is the greatest integer that cannot be represented as a1h1\+ ... +anhn with nonnegative integer a1,..., an. Using known formulae for Frobenius numbers, we can determine the worst-case complexity of Shellsort for several classes of gap sequences.
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This was, in turn, an important piece of one of their resolutions of the Calabi–Jörgens problem on rigidity of affine hyperspheres with nonnegative mean curvature.Shiu Yuen Cheng and Shing-Tung Yau. Complete affine hypersurfaces. I. The completeness of affine metrics. Comm.
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Some examples are: monotone circuits (in which all the field elements are nonnegative real numbers), constant depth circuits, and multilinear circuits (in which every gate computes a multilinear polynomial). These restricted models have been studied extensively and some understanding and results were obtained.
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The digraph realization problem is a decision problem in graph theory. Given pairs of nonnegative integers ((a_1,b_1),\ldots,(a_n,b_n)), the problem asks whether there is a labeled simple directed graph such that each vertex v_i has indegree a_i and outdegree b_i.
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When the list S cannot be reduced to a list S' of nonnegative integers in any step of this approach, the theorem proves that the list S from the beginning is not graphic. The time complexity of the algorithm is O(n^2).
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The secretary problem can be generalized to the case where there are multiple different jobs. Again, there are n applicants coming in random order. When a candidate arrives, she reveals a set of nonnegative numbers. Each value specifies her qualification for one of the jobs.
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Because zero is neither positive nor negative, the term nonnegative is sometimes used to refer to a number that is either positive or zero, while nonpositive is used to refer to a number that is either negative or zero. Zero is a neutral number.
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Let ω be a nonnegative parameter with units of radians/second. Then the angular function (angle vs. time) , has slope −ω, which is called a negative frequency. But when the function is used as the argument of a cosine operator, the result is indistinguishable from .
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A topological dynamical system consists of a Hausdorff topological space X (usually assumed to be compact) and a continuous self-map f. Its topological entropy is a nonnegative extended real number that can be defined in various ways, which are known to be equivalent.
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For example, f(x)=x^2 is a superadditive function for nonnegative real numbers because the square of (x+y) is always greater than or equal to the square of x plus the square of y, for nonnegative real numbers x and y. The analogue of Fekete's lemma holds for subadditive functions as well. There are extensions of Fekete's lemma that do not require the definition of superadditivity above to hold for all m and n. There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and subadditivity is present.
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It may happen that it is not. A densely defined operator T is called positive (or nonnegative) if its quadratic form is nonnegative, that is, \langle Tx \mid x \rangle \ge 0 for all x in the domain of T. Such operator is necessarily symmetric. The operator T∗T is self-adjoint and positive for every densely defined, closed T. The spectral theorem applies to self-adjoint operators and moreover, to normal operators, but not to densely defined, closed operators in general, since in this case the spectrum can be empty. A symmetric operator defined everywhere is closed, therefore bounded, which is the Hellinger–Toeplitz theorem.
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A simple iterative method to approach the double stochastic matrix is to alternately rescale all rows and all columns of A to sum to 1. Sinkhorn and Knopp presented this algorithm and analyzed its convergence. Sinkhorn, Richard, & Knopp, Paul. (1967). "Concerning nonnegative matrices and doubly stochastic matrices".
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On the other hand, if is odd, f=p^{(i)} changes of sign at , while p^{(i-1)} does not. There are thus sign variations. Thus, when pass through , the number of sign variation decrease either of or , which are nonnegative even numbers in each case.
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The set of evil numbers (numbers n with t_n=0) forms a subspace of the nonnegative integers under nim-addition (bitwise exclusive or). For the game of Kayles, evil nim-values occur for few (finitely many) positions in the game, with all remaining positions having odious nim- values.
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The general VLQ encoding is simple, but in basic form is only defined for unsigned integers (nonnegative, positive or zero), and is somewhat redundant, since prepending 0x80 octets corresponds to zero padding. There are various signed number representations to handle negative numbers, and techniques to remove the redundancy.
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Robert Plemmons in 2007 Robert James Plemmons (born December 18, 1938) is an American mathematician specializing in computational mathematics. He is the Emeritus Z. Smith Reynolds Professor of Mathematics and Computer Science at Wake Forest University. In 1979, Plemmons co-authored the book Nonnegative Matrices in the Mathematical Sciences.
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A second proof comes from the infinite product for Ep(x): each exponent -μ(n)/n for n not divisible by p is a p-integral, and when a rational number a is p-integral all coefficients in the binomial expansion of (1 - xn)a are p-integral by p-adic continuity of the binomial coefficient polynomials t(t-1)...(t-k+1)/k! in t together with their obvious integrality when t is a nonnegative integer (a is a p-adic limit of nonnegative integers) . Thus each factor in the product of Ep(x) has p-integral coefficients, so Ep(x) itself has p-integral coefficients. The (p-integral) series expansion has radius of convergence 1.
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Dr. Andrzej Cichocki participated in the development of software packages: Independent Component Analysis LAB, Nonnegative Tensor Factorization LAB, Nonnegative Matrix Factorizations LAB and others. He also is the author of 6 patents. Doctor Andrzej Cichocki developed novel multi-way (tensor) and machine learning (ML) technologies for massive brain (and generally multi-modal biomedical) data analysis and for computational (neuro)science, that is modeling and simulations of complex mechanisms and phenomena. He developed novel algorithms and software for multiway component analysis, including multilinear Independent Component Analysis (ICA), non- negative matrix/tensor factorization (NMF/NTF), Smooth Component Analysis and Sparse Component Analysis (SCA), for tensor decomposition and tensor networks to simulate complex systems and process massive large-scale multidimensional data sets.
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This is an exercise. If all are symmetric then if and only if from which it follows that and . If all are balanced then the inequality for all unit scalars is proved similarly. Since is a nonnegative subadditive function satisfying , is uniformly continuous on if and only if is continuous at .
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In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics, as well as applied mathematics.
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The graph of the square function is a parabola. The squaring operation defines a real function called the ' or the '. Its domain is the whole real line, and its image is the set of nonnegative real numbers. The square function preserves the order of positive numbers: larger numbers have larger squares.
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All of the difficulties of the "problem of cumulants" are absent when one works formally. The simplest example is that the second cumulant of a probability distribution must always be nonnegative, and is zero only if all of the higher cumulants are zero. Formal cumulants are subject to no such constraints.
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The event horizon is defined as the boundary of the causal past of null infinity. Such boundaries are generated by null geodesics. The affine parameter goes to infinity as we approach null infinity, and no caustics form until then. So, the expansion of the event horizon has to be nonnegative.
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25, 1967, p.5 allows rank functions to have arbitrary (rather than only nonnegative) integer values. In this variant, the integers can be graded (by the identity function) in his setting, and the compatibility of ranks with the ordering is not redundant. As a third variant, Brightwell and WestSee reference [2], p.722.
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For surfaces, the theorem was proved by Stefan Cohn-Vossen. Victor Andreevich Toponogov generalized it to manifolds with non-negative sectional curvature. Jeff Cheeger and Detlef Gromoll proved that non-negative Ricci curvature is sufficient. Later the splitting theorem was extended to Lorentzian manifolds with nonnegative Ricci curvature in the time-like directions.
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Then for all nonnegative rationals, we have :S(p/q) = [p+q; p+2q, p+3q, p+4q, \dots], with similar formulas for negative rationals; in particular we have :S(0) = S(0/1) = [1; 2, 3, 4, 5, 6, 7, \dots]. Many of the formulas can be proved using Gauss's continued fraction.
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In Hilbert's axioms, congruence is an equivalence relation on segments. One can thus define the length of a segment as its equivalence class. One must thus prove that this length satisfies properties that characterize nonnegative real numbers. It is what did Artin, with axioms that are not Hilbert's ones, but are equivalent.
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In the one-dimensional case, the inequality is first proved when the functions f, g and h are characteristic functions of a finite unions of intervals. Then the inequality can be extended to characteristic functions of measurable sets, to measurable functions taking a finite number of values and finally to nonnegative measurable functions.
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Suppose f : [a, b] → R is continuous and g is a nonnegative integrable function on [a, b]. By the extreme value theorem, there exists m and M such that for each x in [a, b], m\leqslant f(x) \leqslant M and f[a,b] = [m, M]. Since g is nonnegative, :m \int_a^b g(x) \, dx \leqslant \int^b_a f(x)g(x) \, dx \leqslant M \int_a^b g(x) \, dx. Now let :I = \int_a^b g(x) \, dx. If I = 0, we're done since :0 \leqslant \int_a^b f(x) g(x)\, dx \leqslant 0 means :\int_a^b f(x)g(x)\, dx=0, so for any c in (a, b), :\int_a^b f(x)g(x)\, dx = f(c) I = 0.
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The lexicographical order is used not only in dictionaries, but also commonly for numbers and dates. One of the drawbacks of the Roman numeral system is that it is not always immediately obvious which of two numbers is the smaller. On the other hand, with the positional notation of the Hindu–Arabic numeral system, comparing numbers is easy, because the natural order on nonnegative integers is the same as the variant shortlex of the lexicographic order. In fact, with positional notation, a nonnegative integer is represented by a sequence of numerical digits, and an integer is larger than another one if either it has more digits (ignoring leading zeroes) or the number of digits is the same and the first (most significant) digit which differs is larger.
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Sylver coinage is a mathematical game for two players, invented by John H. Conway. It is discussed in chapter 18 of Winning Ways for Your Mathematical Plays. This article summarizes that chapter. The two players take turns naming positive integers greater than 1 that are not the sum of nonnegative multiples of previously named integers.
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The ring Z and its quotients Z/nZ have no subrings (with multiplicative identity) other than the full ring. Every ring has a unique smallest subring, isomorphic to some ring Z/nZ with n a nonnegative integer (see characteristic). The integers Z correspond to in this statement, since Z is isomorphic to Z/0Z.
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In mathematics, specifically linear algebra, a real matrix A is copositive if :x^TAx\geq 0 for every nonnegative vector x\geq 0. The collection of all copositive matrices is a proper cone; it includes as a subset the collection of real positive-definite matrices. Copositive matrices find applications in economics, operations research, and statistics.
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Let be an instance of the travelling salesman problem. That is, is a complete graph on the set of vertices, and the function assigns a nonnegative real weight to every edge of . According to the triangle inequality, for every three vertices , , and , it should be the case that . Then the algorithm can be described in pseudocode as follows.
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The algorithm can be further simplified for linear feasibility problems, that is for linear systems with nonnegative variables; these problems can be formulated for oriented matroids. The criss-cross algorithm has been adapted for problems that are more complicated than linear programming: There are oriented-matroid variants also for the quadratic-programming problem and for the linear-complementarity problem.
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For this reason, Gauss's result is sometimes known as the Eureka theorem.. The full polygonal number theorem was not resolved until it was finally proven by Cauchy in 1813. The proof of is based on the following lemma due to Cauchy: For odd positive integers and such that and we can find nonnegative integers , , , and such that and .
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Hamilton extended the maximum principle for parabolic partial differential equations to the setting of symmetric 2-tensors which satisfy a parabolic partial differential equations. He also put this into the general setting of a parameter-dependent section of a vector bundle over a closed manifold which satisfies a heat equation, giving both strong and weak formulations. Partly due to these foundational technical developments, Hamilton was able to give an essentially complete understanding of how Ricci flow behaves on three- dimensional closed Riemannian manifolds of positive Ricci curvature and nonnegative Ricci curvature, four-dimensional closed Riemannian manifolds of positive or nonnegative curvature operator, and two-dimensional closed Riemannian manifolds of nonpositive Euler characteristic or of positive curvature. In each case, after appropriate normalizations, the Ricci flow deforms the given Riemannian metric to one of constant curvature.
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In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called (Gromov-)hyperbolic groups.
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Plemmons has focused his work on applied computational mathematics. At Auburn in the early 1960s, Plemmons' work with Ph.D. advisors Richard Ball and Emily Haynesworth was focused on finite semigroups theory. He continued this research until the early 1980s at the University of Tennessee Knoxville. In 1979, he co-authored the book Nonnegative Matrices in the Mathematical Sciences along with Abraham Berman.
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Then on the edges of this octant (singularities of codimension 2) the curvature is nonpositive (because of branching geodesics), yet it is not the case at the origin (singularity of codimension 3), where a triangle such as (0,0,e), (0,e,0), (e,0,0) has a median longer than would be in the Euclidean plane, which is characteristic of nonnegative curvature.
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Lyapunov optimization refers to the use of a Lyapunov function to optimally control a dynamical system. Lyapunov functions are used extensively in control theory to ensure different forms of system stability. The state of a system at a particular time is often described by a multi- dimensional vector. A Lyapunov function is a nonnegative scalar measure of this multi-dimensional state.
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The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals. Every nonnegative real number has a square root in R, although no negative number does. This shows that the order on R is determined by its algebraic structure.
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A similar theorem can be proven for signed and complex measures: namely, that if is a nonnegative σ-finite measure, and is a finite-valued signed or complex measure such that , i.e. is absolutely continuous with respect to , then there is a -integrable real- or complex-valued function on such that for every measurable set , : u(A) = \int_A g \, d\mu.
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Since the remainders decrease with every step but can never be negative, a remainder rN must eventually equal zero, at which point the algorithm stops. The final nonzero remainder rN−1 is the greatest common divisor of a and b. The number N cannot be infinite because there are only a finite number of nonnegative integers between the initial remainder r0 and zero.
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Iterating the aliquot sum function produces the aliquot sequence n, s(n), s(s(n)), ... of a nonnegative integer n (in this sequence, we define s(0) = 0). It remains unknown whether these sequences always converge (the limit of the sequence must be 0 or a perfect number), or whether they can diverge (i.e. the limit of the sequence does not exist).
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Let N be the set of nonnegative integers. A subset S of N is called a numerical semigroup if the following conditions are satisfied. #0 is an element of S #N − S, the complement of S in N, is finite. #If x and y are in S then x + y is also in S. There is a simple method to construct numerical semigroups.
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In standard NMF, matrix factor ， i.e., can be anything in that space. Convex NMFC Ding, T Li, MI Jordan, Convex and semi-nonnegative matrix factorizations, IEEE Transactions on Pattern Analysis and Machine Intelligence, 32, 45-55, 2010 restricts the columns of to convex combinations of the input data vectors (v_1, \cdots, v_n) . This greatly improves the quality of data representation of .
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If all are symmetric then if and only if from which it follows that and . If all are balanced then the inequality for all unit scalars is proved similarly. Since is a nonnegative subadditive function satisfying , is uniformly continuous on if and only if is continuous at . If all are neighborhoods of the origin then for any real , pick an integer such that so that implies .
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This can be viewed as a sharpening or quantification of the positive energy theorem, which provides the weaker statement that the energy is nonnegative. In the 1990s, Yun Gang Chen, Yoshikazu Giga, and Shun'ichi Goto, and independently Lawrence Evans and Joel Spruck, developed a theory of weak solutions for mean curvature flow by considering level sets of solutions of a certain elliptic partial differential equation.
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A generalization of PageRank for the case of ranking two interacting groups of objects was described in. In applications it may be necessary to model systems having objects of two kinds where a weighted relation is defined on object pairs. This leads to considering bipartite graphs. For such graphs two related positive or nonnegative irreducible matrices corresponding to vertex partition sets can be defined.
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It does not imply that the ensemble averaged entropy production is non-negative at all times. This is untrue, as consideration of the entropy production in a viscoelastic fluid subject to a sinusoidal time dependent shear rate shows. In this example the ensemble average of the time integral of the entropy production over one cycle is however nonnegative - as expected from the Second Law Inequality.
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A fractional program in which f is nonnegative and concave, g is positive and convex, and S is a convex set is called a concave fractional program. If g is affine, f does not have to be restricted in sign. The linear fractional program is a special case of a concave fractional program where all functions f, g, h_j, j=1, \ldots, m are affine.
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More generally, Penrose conjectured that an inequality as above should hold for spacelike submanifolds of spacetimes that are not necessarily time-symmetric. In this case, nonnegative scalar curvature is replaced with the dominant energy condition, and one possibility is to replace the minimal surface condition with an apparent horizon condition. Proving such an inequality remains an open problem in general relativity, called the Penrose conjecture.
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Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include commutative semigroups, commutative monoids and semirings. Richman considers two such systems, designed so that 0.999... < 1\. First, Richman defines a nonnegative decimal number to be a literal decimal expansion.
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In mathematics, the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of the system. Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. Their definition was modelled after the definition of the Kolmogorov–Sinai, or metric entropy. Later, Dinaburg and Rufus Bowen gave a different, weaker definition reminiscent of the Hausdorff dimension.
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Visualisation of using the binary GCD algorithm to find the greatest common divisor (GCD) of 36 and 24. Thus, the GCD is 22 × 3 = 12. The binary GCD algorithm, also known as Stein's algorithm, is an algorithm that computes the greatest common divisor of two nonnegative integers. Stein's algorithm uses simpler arithmetic operations than the conventional Euclidean algorithm; it replaces division with arithmetic shifts, comparisons, and subtraction.
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In a monoid, one can define positive integer powers of an element x : x1 = x, and xn = x • ... • x (n times) for n > 1 . The rule of powers xn + p = xn • xp is obvious. From the definition of a monoid, one can show that the identity element e is unique. Then, for any x, one can set x0 = e and the rule of powers is still true with nonnegative exponents.
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Many circuit complexity classes are defined in terms of class hierarchies. For each nonnegative integer i, there is a class NCi, consisting of polynomial-size circuits of depth O(\log^i(n)), using bounded fan-in AND, OR, and NOT gates. We can talk about the union NC of all of these classes. By considering unbounded fan-in gates, we construct the classes ACi and AC (which is equal to NC).
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Doris Fischer-Colbrie is a ceramic artist and former mathematician. She received her Ph.D. in mathematics in 1978 from University of California at Berkeley, where her advisor was H. Blaine Lawson. Many of her contributions to the theory of minimal surfaces are now considered foundational to the field. In particular, her collaboration with Richard Schoen is a landmark contribution to the interaction of stable minimal surfaces with nonnegative scalar curvature.
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The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math. 6 (1953), 337–394. Hence Shi and Tam's result gives a striking sense in which, given a compact and smooth three-dimensional Riemannian manifold-with-boundary of nonnegative scalar curvature, whose boundary components have positive intrinsic curvature and positive mean curvature, the extrinsic geometry of the boundary components are controlled by their intrinsic geometry.
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The symbol √ is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by = 3, because and 3 is nonnegative. However raising x to the power of 0.5 using the key works if the number is entered as a real number with a complex part equal to zero. Inverse and hyperbolic trigonometry functions cannot be used with complex numbers.
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Construction of a regular pentagon using straightedge and compass. This is only possible because 5 is a alt=Construction of a regular pentagon using straightedge and compass Fermat primes are primes of the form :F_k = 2^{2^k}+1, with k a nonnegative integer. also include 2^0+1=2, which is not of this form. They are named after Pierre de Fermat, who conjectured that all such numbers are prime.
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All results described in this article are based on Descartes' rule of signs. If is a univariate polynomial with real coefficients, let us denote by the number of its positive real roots, counted with their multiplicity,This means that a root of multiplicity is counted as roots. and by the number of sign variations in the sequence of its coefficients. Descartes's rule of signs asserts that : is a nonnegative even integer.
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In a product multicommodity flow problem, there is a nonnegative weight for each node v \in V in graph G=(V,E). The demand for the commodity between nodes and is the product of the weights of node and node . The uniform multicommodity flow problem is a special case of the product multicommodity flow problem for which the weight is set to 1 for all nodes u \in V.
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The Euclidean algorithm proceeds in a series of steps such that the output of each step is used as an input for the next one. Let k be an integer that counts the steps of the algorithm, starting with zero. Thus, the initial step corresponds to k = 0, the next step corresponds to k = 1, and so on. Each step begins with two nonnegative remainders rk−1 and rk−2.
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The theorem has particular use in algebraic graph theory. The "underlying graph" of a nonnegative n-square matrix is the graph with vertices numbered 1, ..., n and arc ij if and only if Aij ≠ 0. If the underlying graph of such a matrix is strongly connected, then the matrix is irreducible, and thus the theorem applies. In particular, the adjacency matrix of a strongly connected graph is irreducible.
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Given a vector space over a field of the real numbers or complex numbers , a norm on V is a nonnegative-valued function with the following properties: For all and all , # (being subadditive or satisfying the triangle inequality). # (being absolutely homogeneous or absolutely scalable). # If then is the zero vector (being positive definite or being point-separating). A seminorm on V is a function with the properties 1 and 2 above.
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An object is weakly labelled if each of its atoms has a nonnegative integer label, and each of these labels is distinct. An object is (strongly or well) labelled, if furthermore, these labels comprise the consecutive integers [1 \ldots n]. Note: some combinatorial classes are best specified as labelled structures or unlabelled structures, but some readily admit both specifications. A good example of labelled structures is the class of labelled graphs.
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By proving extension of Eells and Sampson's vanishing theorem in various geometric settings, they were able to draw striking geometric conclusions, such as that if is a complete Riemannian manifold of nonnegative Ricci curvature, then for any precompact open set with smooth and simply-connected boundary, there cannot exist a nontrivial homomorphism from the fundamental group of into any group which is the fundamental group of a closed Riemannian manifold of nonpositive curvature.
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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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Suppose such an individual, say "Fred", must walk exactly k blocks to get to a point B that is exactly k blocks from A. It is convenient to regard Fred's starting point A as the origin, (0,0), of a rectangular array of lattice points and B as some lattice point (e,n), e units "East" and n units "North" of A, where e+n=k and both e and n are nonnegative.
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In analogy with the real numbers, we call an element c of an ordered ring R positive if 0 < c, and negative if c < 0\. 0 is considered to be neither positive nor negative. The set of positive elements of an ordered ring R is often denoted by R+. An alternative notation, favored in some disciplines, is to use R+ for the set of nonnegative elements, and R++ for the set of positive elements.
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Bessel himself originally proved that for nonnegative integers , the equation has an infinite number of solutions in .Bessel, F. (1824) "Untersuchung des Theils der planetarischen Störungen", Berlin Abhandlungen, article 14. When the functions are plotted on the same graph, though, none of the zeros seem to coincide for different values of except for the zero at . This phenomenon is known as Bourget's hypothesis after the 19th-century French mathematician who studied Bessel functions.
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In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of a nef divisor.
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An odds ratio of 1 indicates that the condition or event under study is equally likely to occur in both groups. An odds ratio greater than 1 indicates that the condition or event is more likely to occur in the first group. And an odds ratio less than 1 indicates that the condition or event is less likely to occur in the first group. The odds ratio must be nonnegative if it is defined.
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In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number p (which is suppressed in the notation), it consists of theories K(n) for each nonnegative integer n, each a ring spectrum in the sense of homotopy theory. published the first account of the theories.
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The result is a partial order. If and are each totally ordered, then the result is a total order as well. The lexicographical order of two totally ordered sets is thus a linear extension of their product order. One can define similarly the lexicographic order on the Cartesian product of an infinite family of ordered sets, if the family is indexed by the nonnegative integers, or more generally by a well-ordered set.
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Other methods flag observations based on measures such as the interquartile range. For example, if Q_1 and Q_3 are the lower and upper quartiles respectively, then one could define an outlier to be any observation outside the range: : \big[ Q_1 - k (Q_3 - Q_1 ) , Q_3 + k (Q_3 - Q_1 ) \big] for some nonnegative constant k. John Tukey proposed this test, where k=1.5 indicates an "outlier", and k=3 indicates data that is "far out".
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Hamiltonian systems can be generalized in various ways. Instead of simply looking at the algebra of smooth functions over a symplectic manifold, Hamiltonian mechanics can be formulated on general commutative unital real Poisson algebras. A state is a continuous linear functional on the Poisson algebra (equipped with some suitable topology) such that for any element of the algebra, maps to a nonnegative real number. A further generalization is given by Nambu dynamics.
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The Fulkerson–Chen–Anstee theorem is a result in graph theory, a branch of combinatorics. It provides one of two known approaches solving the digraph realization problem, i.e. it gives a necessary and sufficient condition for pairs of nonnegative integers ((a_1,b_1),\ldots,(a_n,b_n)) to be the indegree- outdegree pairs of a simple directed graph; a sequence obeying these conditions is called "digraphic". D. R. Fulkerson D.R. Fulkerson: Zero-one matrices with zero trace.
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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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In 0 + 1, 2 + 1, 3 + 1, 4 + 1, 6 + 1, 7 + 1, 8 + 1, 10 + 1 dimensions, etc., a SUSY algebra is classified by a positive integer N. In 1 + 1, 5 + 1, 9 + 1 dimensions, etc., a SUSY algebra is classified by two nonnegative integers (M, N), at least one of which is nonzero. M represents the number of left-handed SUSYs and N represents the number of right-handed SUSYs.
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In mathematics, many sequences of numbers or of polynomials are indexed by nonnegative integers, for example the Bernoulli numbers and the Bell numbers. In both mechanics and statistics, the zeroth moment is defined, representing total mass in the case of physical density, or total probability, i.e. one, for a probability distribution. The zeroth law of thermodynamics was formulated after the first, second, and third laws, but considered more fundamental, thus its name.
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The Havel–Hakimi algorithm is an algorithm in graph theory solving the graph realization problem. That is, it answers the following question: Given a finite list of nonnegative integers, is there a simple graph such that its degree sequence is exactly this list? The degree sequence is a list of numbers that for each vertex of the graph states how many neighbors it has. For a positive answer, the list of integers is called graphic.
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In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If f is a complex- valued holomorphic function germ then the Milnor number of f, denoted μ(f), is either a nonnegative integer, or is infinite. It can be considered both a geometric invariant and an algebraic invariant. This is why it plays an important role in algebraic geometry and singularity theory.
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A Fibonacci sequence of order is an integer sequence in which each sequence element is the sum of the previous n elements (with the exception of the first n elements in the sequence). The usual Fibonacci numbers are a Fibonacci sequence of order 2. The cases n = 3 and n = 4 have been thoroughly investigated. The number of compositions of nonnegative integers into parts that are at most n is a Fibonacci sequence of order n.
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As a consequence, one can find a formula for the Laplacian of the norm-squared of the second fundamental form. This "Simons formula" simplifies significantly when the mean curvature of the submanifold is zero and when the Riemannian manifold has constant curvature. In this setting, Shiing-Shen Chern, Manfredo do Carmo, and Kobayashi studied the algebraic structure of the zeroth order terms, showing that they are nonnegative provided that the norm of the second fundamental form is sufficiently small.
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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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In mathematics, particularly matrix theory, a Stieltjes matrix, named after Thomas Joannes Stieltjes, is a real symmetric positive definite matrix with nonpositive off-diagonal entries. A Stieltjes matrix is necessarily an M-matrix. Every n×n Stieltjes matrix is invertible to a nonsingular symmetric nonnegative matrix, though the converse of this statement is not true in general for n > 2\. From the above definition, a Stieltjes matrix is a symmetric invertible Z-matrix whose eigenvalues have positive real parts.
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In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, x^2yz^3=xxyzzz is a monomial. The constant 1 is a monomial, being equal to the empty product and to 0 for any variable .
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The term originated as an abstraction of the sequence: single, couple/double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., ‑tuple, ..., where the prefixes are taken from the Latin names of the numerals. The unique 0‑tuple is called the null tuple or empty tuple. A 1‑tuple is called a single (or singleton), a 2‑tuple is called an ordered pair or couple, and a 3‑tuple is called a triple (or triplet). The number can be any nonnegative integer.
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There are two equivalent common definitions of an ordered field. The definition of total order appeared first historically and is a first-order axiomatization of the ordering ≤ as a binary predicate. Artin and Schreier gave the definition in terms of positive cone in 1926, which axiomatizes the subcollection of nonnegative elements. Although the latter is higher-order, viewing positive cones as maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings.
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One finds that this equation is a Pöschl-Teller equation (i.e. a second order differential equation like the Schrödinger equation with Pöschl-Teller potential) with nonnegative eigenvalues. The nonnegativity of the eigenvalues is indicative of the stability of the instanton. As stated above, the instanton is the pseudoparticle configuration defined on an infinite line of Euclidean time that communicates between the two wells of the potential and is responsible for the ground state of the system.
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If the leading digit equals the first digit after the quote, then either the number is 0!0 = 0, or the representation can be shortened by rolling the repetition to the right. For example, 23'25 = 32'5 which is positive because 3 is less than 5. In binary, if it starts with 1 it is negative, and if it starts with 0 it is nonnegative, assuming the repetition has been rolled to the right as far as possible.
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The definition of a cone may be extended to higher dimensions (see convex cones). In this case, one says that a convex set C in the real vector space Rn is a cone (with apex at the origin) if for every vector x in C and every nonnegative real number a, the vector ax is in C. In this context, the analogues of circular cones are not usually special; in fact one is often interested in polyhedral cones.
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Form an arbelos with the two inner semicircles tangent at point C. Let m denote any nonnegative real number. Draw two circles, with radii m times the radii of the smaller two arbelos semicircles, centered on the arbelos ground line, also tangent to each other at point C and with radius m times the radius of the corresponding small arbelos arc. Any circle centered on the Schoch line and externally tangent to the circles is a Woo circle.
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Descartes' rule of signs asserts that the difference between the number of sign variations in the sequence of the coefficients of a polynomial and the number of its positive real roots is a nonnegative even integer. It results that if this number of sign variations is zero, then the polynomial does not have any positive real roots, and, if this number is one, then the polynomial has a unique positive real root, which is a single root. Unfortunately the converse is not true, that is, a polynomial which has either no positive real root or as a single positive simple root may have a number of sign variations greater than 1. This has been generalized by Budan's theorem (1807), into a similar result for the real roots in a half-open interval : If is a polynomial, and is the difference between of the numbers of sign variations of the sequences of the coefficients of and , then minus the number of real roots in the interval, counted with their multiplicities, is a nonnegative even integer.
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A bit array (also known as bit map, bit set, bit string, or bit vector) is an array data structure that compactly stores bits. It can be used to implement a simple set data structure. A bit array is effective at exploiting bit-level parallelism in hardware to perform operations quickly. A typical bit array stores kw bits, where w is the number of bits in the unit of storage, such as a byte or word, and k is some nonnegative integer.
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Given a probability density function w (meaning that w is nonnegative and integrates to 1), the function : f(x) = \int_A \, w(a) \, p(x;a) \, da is again a probability density function for x. A similar integral can be written for the cumulative distribution function. Note that the formulae here reduce to the case of a finite or infinite mixture if the density w is allowed to be a generalized function representing the "derivative" of the cumulative distribution function of a discrete distribution.
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Their result deals with noncompact three-dimensional Riemannian manifolds-with-boundary of nonnegative scalar curvature whose boundary is minimal, relating the geometry near infinity to the surface area of the largest boundary component. Hubert Bray, by making use of the positive mass theorem instead of the inverse mean curvature flow, was able to improve Huisken and Ilmanen's inequality to involve the total surface area of the boundary.Hubert L. Bray. Proof of the Riemannian Penrose inequality using the positive mass theorem.
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Her dissertation produced fundamental new examples of manifolds with positive Ricci curvature and was published in the Bulletin of the American Mathematical Society. These examples were later expanded upon by Burkard Wilking. In addition to her work on the topology of manifolds with nonnegative Ricci curvature, she has completed work on the isometry groups of manifolds with negative Ricci curvature with coauthors Xianzhe Dai and Zhongmin Shen. She also has major work with Peter Petersen on manifolds with integral Ricci curvature bounds.
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The phrase "single postulate" is just used in comparison with the original "two postulate" formulation. The real question here is whether universal lightspeed can be deduced rather than assumed. The Lorentz transformations, up to a nonnegative free parameter, can be derived without first postulating the universal lightspeed. Experiment rules out the validity of the Galilean transformations and this means the parameter in the Lorentz transformations is nonzero hence there is a finite maximum speed before anything has been said about light.
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In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every nonzero vector of . According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "always nonnegative" and "always nonpositive", respectively. In other words, it may take on zero values.
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Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of (M, g) having nonnegative scalar curvature is equivalent to the spacetime obeying the dominant energy condition. This inequality was first proved by Gerhard Huisken and Tom Ilmanen in 1997 in the case where A is the area of the largest component of the outermost minimal surface. Their proof relied on the machinery of weakly defined inverse mean curvature flow, which they developed.
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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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Further information on the gaps comes from applying Clifford's theorem. Multiplication of functions gives the non-gaps a numerical semigroup structure, and an old question of Adolf Hurwitz asked for a characterization of the semigroups occurring. A new necessary condition was found by R.-O. Buchweitz in 1980 and he gave an example of a subsemigroup of the nonnegative integers with 16 gaps that does not occur as the semigroup of non-gaps at a point on a curve of genus 16 (see ).
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An n-ary quasigroup is a set with an n-ary operation, with , such that the equation has a unique solution for any one variable if all the other n variables are specified arbitrarily. Polyadic or multiary means n-ary for some nonnegative integer n. A 0-ary, or nullary, quasigroup is just a constant element of Q. A 1-ary, or unary, quasigroup is a bijection of Q to itself. A binary, or 2-ary, quasigroup is an ordinary quasigroup.
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In 2000, Andreas Hirsch and coworkers in Erlangen, Germany, formulated a rule to determine when a fullerene would be aromatic. They found that if there were 2(n + 1)2 π-electrons, then the fullerene would display aromatic properties. This follows from the fact that an aromatic fullerene must have full icosahedral (or other appropriate) symmetry, so the molecular orbitals must be entirely filled. This is possible only if there are exactly 2(n + 1)2 electrons, where n is a nonnegative integer.
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The Atkinson index is for no value of \varepsilon highly sensitive to top incomes because of the common restriction that \varepsilon is nonnegative. The Atkinson index is related to the generalized entropy (GE) class of inequality indexes by \epsilon=1-\alpha \- i.e an Atkinson index with high inequality aversion is derived from a GE index with small \alpha. GE indexes with large \alpha are sensitive to the existence of large top incomes but the corresponding Atkinson index would have negative \varepsilon.
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Exact solutions for the variants of NMF can be expected (in polynomial time) when additional constraints hold for matrix . A polynomial time algorithm for solving nonnegative rank factorization if contains a monomial sub matrix of rank equal to its rank was given by Campbell and Poole in 1981. Kalofolias and Gallopoulos (2012) solved the symmetric counterpart of this problem, where is symmetric and contains a diagonal principal sub matrix of rank r. Their algorithm runs in time in the dense case.
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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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An important consequence is that, in three dimensions, a limiting Ricci flow as produced by the compactness theory automatically has nonnegative curvature. As such, Hamilton's Harnack inequality is applicable to the limiting Ricci flow. These methods were extended by Grigori Perelman, who due to his "noncollapsing theorem" was able to apply Hamilton's compactness theory in a number of extended contexts. In 1997, Hamilton was able to combine the methods he had developed to define "Ricci flow with surgery" for four-dimensional Riemannian manifolds of positive isotropic curvature.
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To work with inequalities, it is convenient to consider R-divisors, meaning finite linear combinations of Cartier divisors with real coefficients. The R-divisors modulo numerical equivalence form a real vector space N^1(X) of finite dimension, the Néron–Severi group tensored with the real numbers.Lazarsfeld (2004), Example 1.3.10. (Explicitly: two R-divisors are said to be numerically equivalent if they have the same intersection number with all curves in X.) An R-divisor is called nef if it has nonnegative degree on every curve.
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The nef R-divisors form a closed convex cone in N^1(X), the nef cone Nef(X). The cone of curves is defined to be the convex cone of linear combinations of curves with nonnegative real coefficients in the real vector space N_1(X) of 1-cycles modulo numerical equivalence. The vector spaces N^1(X) and N_1(X) are dual to each other by the intersection pairing, and the nef cone is (by definition) the dual cone of the cone of curves.Lazarsfeld (2004), Definition 1.4.25.
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In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree in variables for each nonnegative integer , and it is formed by adding together all distinct products of distinct variables.
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Recall that it follows from Lusin's theorem that a Lebesgue-measurable function is approximately continuous almost everywhere (and conversely). The key theorem in constructing the Khinchin integral is this: a function f that is generalized absolutely continuous (or even of "generalized bounded variation", a weaker notion) has an approximate derivative almost everywhere. Furthermore, if f is generalized absolutely continuous and its approximate derivative is nonnegative almost everywhere, then f is nondecreasing, and consequently, if this approximate derivative is zero almost everywhere, then f is constant.
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In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p. Abelian p-groups are also called p-primary or simply primary.
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A lattice is unimodular if and only if its dual lattice is integral. Unimodular lattices are equal to their dual lattices, and for this reason, unimodular lattices are also known as self- dual. Given a pair (m,n) of nonnegative integers, an even unimodular lattice of signature (m,n) exists if and only if m-n is divisible by 8, but an odd unimodular lattice of signature (m,n) always exists. In particular, even unimodular definite lattices only exist in dimension divisible by 8.
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The primitive recursive functions are among the number-theoretic functions, which are functions from the natural numbers (nonnegative integers) {0, 1, 2, ...} to the natural numbers. These functions take n arguments for some natural number n and are called n-ary. The basic primitive recursive functions are given by these axioms: More complex primitive recursive functions can be obtained by applying the operations given by these axioms: Example. We take f(x) as the S(x) defined above. This f is a 1-ary primitive recursive function.
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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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Google's Protocol Buffers "zig-zag encoding" is a system similar to sign-and-magnitude, but uses the least significant bit to represent the sign and has a single representation of zero. This allows a variable-length quantity encoding intended for nonnegative (unsigned) integers to be used efficiently for signed integers.Protocol Buffers: Signed Integers Another approach is to give each digit a sign, yielding the signed-digit representation. For instance, in 1726, John Colson advocated reducing expressions to "small numbers", numerals 1, 2, 3, 4, and 5.
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For avoidance of doubt a non-zero non-negative square matrix A such that 1 + A is primitive is sometimes said to be connected. Then irreducible non-negative square matrices and connected matrices are synonymous.For surveys of results on irreducibility, see Olga Taussky-Todd and Richard A. Brualdi. The nonnegative eigenvector is often normalized so that the sum of its components is equal to unity; in this case, the eigenvector is the vector of a probability distribution and is sometimes called a stochastic eigenvector.
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In mathematics, a numerical semigroup is a special kind of a semigroup. Its underlying set is the set of all nonnegative integers except a finite number and the binary operation is the operation of addition of integers. Also, the integer 0 must be an element of the semigroup. For example, while the set {0, 2, 3, 4, 5, 6, ...} is a numerical semigroup, the set {0, 1, 3, 5, 6, ...} is not because 1 is in the set and 1 + 1 = 2 is not in the set.
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If is the adjacency matrix of the directed or undirected graph , then the matrix (i.e., the matrix product of copies of ) has an interesting interpretation: the element gives the number of (directed or undirected) walks of length from vertex to vertex . If is the smallest nonnegative integer, such that for some , , the element of is positive, then is the distance between vertex and vertex . This implies, for example, that the number of triangles in an undirected graph is exactly the trace of divided by 6.
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In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers :n = x^2 + y^2 + z^2 if and only if is not of the form n = 4^a(8b + 7) for nonnegative integers and . The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as n = 4^a(8b + 7)) are :7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71 ... .
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One notable variant of a Markov random field is a conditional random field, in which each random variable may also be conditioned upon a set of global observations o. In this model, each function \phi_k is a mapping from all assignments to both the clique k and the observations o to the nonnegative real numbers. This form of the Markov network may be more appropriate for producing discriminative classifiers, which do not model the distribution over the observations. CRFs were proposed by John D. Lafferty, Andrew McCallum and Fernando C.N. Pereira in 2001.
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Relaxation methods are important especially in the solution of linear systems used to model elliptic partial differential equations, such as Laplace's equation and its generalization, Poisson's equation. These equations describe boundary-value problems, in which the solution-function's values are specified on boundary of a domain; the problem is to compute a solution also on its interior. Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences.Abraham Berman, Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, 1994, SIAM. .
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An alternative way to encode negative numbers is to use the least-significant bit for sign. This is notably done for Google Protocol Buffers, and is known as a zigzag encoding for signed integers.Protocol Buffers: Encoding: Signed Integers One can encode the numbers so that encoded 0 corresponds to 0, 1 to −1, 10 to 1, 11 to −2, 100 to 2, etc.: counting up alternates between nonnegative (starting at 0) and negative (since each step changes the least-significant bit, hence the sign), whence the name "zigzag encoding".
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Born in Driehuizen, Texel, Zijm received both his BSc in mathematics, physics and astronomy in 1977, and his MSc cum laude in applied mathematics at the University of Amsterdam. In 1982 he received his Phd in operations research at the Eindhoven University of Technology under supervision of Jaap Wessels and Gerhard Willem Veltkamp with a thesis entitled "Nonnegative Matrices in Dynamic Programming."Henk Zijm at Mathematics Genealogy Project. Zijm had started his academic career in 1981 as assistant professor at the Department of Actuarial Sciences and Econometrics of the University of Amsterdam.
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The player who cannot name such a number loses. For instance, if player A opens with 2, B can win by naming 3. Sylver coinage is named after James Joseph Sylvester, who proved that if a and b are relatively prime positive integers, then (a − 1)(b − 1) − 1 is the largest number that is not a sum of nonnegative multiples of a and b. Thus, if a and b are the first two moves in a game of sylver coinage, this formula gives the largest number that can still be played.
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In mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices).. The name M-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski, who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive..
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Didier G. Leibovici and Christian Beckmann, An introduction to Multiway Methods for Multi-Subject fMRI experiment, FMRIB Technical Report 2001, Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB), Department of Clinical Neurology, University of Oxford, John Radcliffe Hospital, Headley Way, Headington, Oxford, UK. The negentropy of a distribution is equal to the Kullback–Leibler divergence between p_x and a Gaussian distribution with the same mean and variance as p_x (see Differential entropy#Maximization in the normal distribution for a proof). In particular, it is always nonnegative.
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Since the norm is a nonnegative integer and decreases with every step, the Euclidean algorithm for Gaussian integers ends in a finite number of steps. The final nonzero remainder is , the Gaussian integer of largest norm that divides both and ; it is unique up to multiplication by a unit, or . Many of the other applications of the Euclidean algorithm carry over to Gaussian integers. For example, it can be used to solve linear Diophantine equations and Chinese remainder problems for Gaussian integers; continued fractions of Gaussian integers can also be defined.
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Diagram of a counter automaton. Each cell of its stack either contains an "A" or a space symbol. In computer science, more particular in the theory of formal languages, a counter automaton, or counter machine, is a pushdown automaton with only two symbols, A and the initial symbol in \Gamma, the finite set of stack symbols. Equivalently, a counter automaton is a nondeterministic finite automaton with an additional memory cell that can hold one nonnegative integer number (of unlimited size), which can be incremented, decremented, and tested for being zero.
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In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix. The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics.
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Z is quasi-equal to blue, or the S cone response, and X is a mix of response curves chosen to be nonnegative. The XYZ tristimulus values are thus analogous to, but different from, the LMS cone responses of the human eye. Setting Y as luminance has the useful result that for any given Y value, the XZ plane will contain all possible chromaticities at that luminance. The unit of the tristimulus values , , and is often arbitrarily chosen so that or is the brightest white that a color display supports.
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From the mathematical point of view, the object corresponds to a function and the problem posed is to reconstruct this function from its integrals or sums over subsets of its domain. In general, the tomographic inversion problem may be continuous or discrete. In continuous tomography both the domain and the range of the function are continuous and line integrals are used. In discrete tomography the domain of the function may be either discrete or continuous, and the range of the function is a finite set of real, usually nonnegative numbers.
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The nonnegative orthant of Rn and the space of all positive semidefinite matrices are self-dual, as are the cones with ellipsoidal base (often called "spherical cones", "Lorentz cones", or sometimes "ice-cream cones"). So are all cones in R3 whose base is the convex hull of a regular polygon with an odd number of vertices. A less regular example is the cone in R3 whose base is the "house": the convex hull of a square and a point outside the square forming an equilateral triangle (of the appropriate height) with one of the sides of the square.
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The Shortest Path Faster Algorithm (SPFA) is an improvement of the Bellman–Ford algorithm which computes single-source shortest paths in a weighted directed graph. The algorithm is believed to work well on random sparse graphs and is particularly suitable for graphs that contain negative- weight edges.About the so-called SPFA algorithm However, the worst-case complexity of SPFA is the same as that of Bellman–Ford, so for graphs with nonnegative edge weights Dijkstra's algorithm is preferred. The SPFA algorithm was first published by Edward F. Moore in 1959, as a generalization of breadth first search; SPFA is Moore's “Algorithm D”.
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In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some nonnegative real number 0 \leq k < 1 such that for all x and y in M, :d(f(x),f(y)) \leq k\,d(x,y). The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for k ≤ 1, then the mapping is said to be a non-expansive map.
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Weaker "irreducibility" conditions were given by David Gale and by John Kemeny, Morgenstern, and Gerald L. Thompson in the 1950s and then by Stephen M. Robinson in the 1970s. Von Neumann's results have been viewed as a special case of linear programming, where his model uses only nonnegative matrices. The study of his model of an expanding economy continues to interest mathematical economists with interests in computational economics. This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, linear inequalities, complementary slackness, and saddlepoint duality.
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An exponential-Golomb code (or just Exp-Golomb code) is a type of universal code. To encode any nonnegative integer x using the exp-Golomb code: # Write down x+1 in binary # Count the bits written, subtract one, and write that number of starting zero bits preceding the previous bit string. The first few values of the code are: 0 ⇒ 1 ⇒ 1 1 ⇒ 10 ⇒ 010 2 ⇒ 11 ⇒ 011 3 ⇒ 100 ⇒ 00100 4 ⇒ 101 ⇒ 00101 5 ⇒ 110 ⇒ 00110 6 ⇒ 111 ⇒ 00111 7 ⇒ 1000 ⇒ 0001000 8 ⇒ 1001 ⇒ 0001001 ... This is identical to the Elias gamma code of x+1, allowing it to encode 0.
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Dickerson et al use simulations to check under what conditions an envy-free assignment of discrete items is likely to exist. They generate instances by sampling the value of each item to each agent from two probability distributions: uniform and correlated. In the correlated sampling, they first sample an intrinsic value for each good, and then assign a random value to each agent drawn from a truncated nonnegative normal distribution around that intrinsic value. Their simulations show that, when the number of goods is larger than the number of agents by a logarithmic factor, envy-free allocations exist with high probability.
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Press, Cambridge, MA, 1995. By modifying the integral methods he developed in 1984, Huisken and Carlo Sinestrari carried out an elaborate inductive argument on the elementary symmetric polynomials of the second fundamental form to show that any singularity model resulting from such rescalings must be a mean curvature flow which moves by translating a single convex hypersurface in some direction. This passage from mean-convexity to full convexity is comparable with the much easier Hamilton- Ivey estimate for Ricci flow, which says that any singularity model of a Ricci flow on a closed 3-manifold must have nonnegative sectional curvature.
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PCA of the multivariate Gaussian distribution centered at (1, 3) with a standard deviation of 3 in roughly the (0.878, 0.478) direction and of 1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite) covariance matrix scaled by the square root of the corresponding eigenvalue. (Just as in the one-dimensional case, the square root is taken because the standard deviation is more readily visualized than the variance. The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue.
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For several parameters, the covariance matrices and information matrices are elements of the convex cone of nonnegative-definite symmetric matrices in a partially ordered vector space, under the Loewner (Löwner) order. This cone is closed under matrix addition and inversion, as well as under the multiplication of positive real numbers and matrices. An exposition of matrix theory and Loewner order appears in Pukelsheim. The traditional optimality criteria are the information matrix's invariants, in the sense of invariant theory; algebraically, the traditional optimality criteria are functionals of the eigenvalues of the (Fisher) information matrix (see optimal design).
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Because primitive recursive functions use natural numbers rather than integers, and the natural numbers are not closed under subtraction, a truncated subtraction function (also called "proper subtraction") is studied in this context. This limited subtraction function sub(a, b) [or b ∸ a] returns b - a if this is nonnegative and returns 0 otherwise. The predecessor function acts as the opposite of the successor function and is recursively defined by the rules: :pred(0) = 0, :pred(n + 1) = n. These rules can be converted into a more formal definition by primitive recursion: :pred(0) = 0, :pred(S(n)) = P12(n, pred(n)).
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The positive integer powers give the number of possible values for an -bit integer binary number; for example, a byte may take different values. The binary number system expresses any number as a sum of powers of , and denotes it as a sequence of and , separated by a binary point, where indicates a power of that appears in the sum; the exponent is determined by the place of this : the nonnegative exponents are the rank of the on the left of the point (starting from ), and the negative exponents are determined by the rank on the right of the point.
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A matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. In the system of linear equations () above, with A non-singular, the matrix A can be split, that is, written as a difference so that () can be re-written as () above. The expression () is a regular splitting of A if and only if B−1 ≥ 0 and C ≥ 0, that is, and C have only nonnegative entries. If the splitting () is a regular splitting of the matrix A and A−1 ≥ 0, then ρ(T) < 1 and T is a convergent matrix.
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Thus this difference must be zero, and, thus ; that is This proof relies on the fact that zero is the only nonnegative number that is less than all inverses of integers, or equivalently that there is no number that is larger than every integer. This is the Archimedean property, that is verified for rational numbers and real numbers. Real numbers may be enlarged into number systems, such as hyperreal numbers, with infinitely small numbers (infinitesimals) and infinitely large numbers (infinite numbers). When using such systems, notation 0.999... is generally not used, as there is no smallest number that is no less than all .
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Generalizing this theorem to the continuous-variable case, it can be shown that, likewise, a class of continuous-variable quantum computations can be simulated using only classical analog computations. This class includes, in fact, some computational tasks that use quantum entanglement. When the Wigner quasiprobability representations of all the quantities—states, time evolutions and measurements—involved in a computation are nonnegative, then they can be interpreted as ordinary probability distributions, indicating that the computation can be modeled as an essentially classical one. This type of construction can be thought of as a continuum generalization of the Spekkens Toy Model.
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Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include the squaring function x^2 and the exponential function e^x. In simple terms, a convex function refers to a function that is in the shape of a cup \cup, and a concave function is in the shape of a cap \cap.
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In mathematics, a norm is a function from a vector space over the real or complex numbers to the nonnegative real numbers, that satisfies certain properties pertaining to scalability and additivity and takes the value zero only if the input vector is zero. A pseudonorm or seminorm satisfies the same properties, except that it may have a zero value for some nonzero vectors. The Euclidean norm, or 2-norm, is a specific norm on a Euclidean vector space that is strongly related to the Euclidean distance. It is also equal to the square root of the inner product of a vector with itself.
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The function given by is not injective, since each possible result y (except 0) corresponds to two different starting points in – one positive and one negative, and so this function is not invertible. With this type of function, it is impossible to deduce a (unique) input from its output. Such a function is called non-injective or, in some applications, information-losing. If the domain of the function is restricted to the nonnegative reals, that is, the function is redefined to be with the same rule as before, then the function is bijective and so, invertible.
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Finally, sequent calculus generalizes the form of a natural deduction judgment to : A_1, \ldots, A_n \vdash B_1, \ldots, B_k, a syntactic object called a sequent. The formulas on left-hand side of the turnstile are called the antecedent, and the formulas on right-hand side are called the succedent or consequent; together they are called cedents or sequents. Again, A_i and B_i are formulae, and n and k are nonnegative integers, that is, the left-hand-side or the right-hand-side (or neither or both) may be empty. As in natural deduction, theorems are those B where \vdash B is the conclusion of a valid proof.
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The book has been cited over 7,500 times. In 1994, it was revised and republished by the Society for Industrial and Applied Mathematics (SIAM). In the mid-to-late 1980s until mid 1990s, his research focused on numerical linear algebra, specifically in Matrix Theory with applications in Markov chains and nonnegative matrices. Plemmons has been recognized internationally for his significant contributions to the field, celebrated at the Linear Algebra: Theory, Applications, and Computations Conference held at Wake Forest University in 1999 in honor of Plemmons' 60th birthday, and the International Workshop on Numerical Linear Algebra with Applications held in Hong Kong in 2013 in honor of his 75th birthday.
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More generally, a line bundle L on a proper scheme X over a field k is said to be nef if it has nonnegative degree on every (closed irreducible) curve in X.Lazarsfeld (2004), Definition 1.4.1. (The degree of a line bundle L on a proper curve C over k is the degree of the divisor (s) of any nonzero rational section s of L.) A line bundle may also be called an invertible sheaf. The term "nef" was introduced by Miles Reid as a replacement for the older terms "arithmetically effective" and "numerically effective", as well as for the phrase "numerically eventually free".Reid (1983), section 0.12f.
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The purpose of the compression achieved by sketching is to allow all of the keys to be stored in one w-bit word. Let the node sketch of a node be the bit string :1`sketch`(x1)1`sketch`(x2)...1`sketch`(xk) We can assume that the sketch function uses exactly b ≤ r4 bits. Then each block uses 1 + b ≤ w4/5 bits, and since k ≤ w1/5, the total number of bits in the node sketch is at most w. A brief notational aside: for a bit string s and nonnegative integer m, let sm denote the concatenation of s to itself m times.
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This does not change the set of feasible solutions or the optimal solution, and it ensures that the slack variables will constitute an initial feasible solution. The new tableau is in canonical form but it is not equivalent to the original problem. So a new objective function, equal to the sum of the artificial variables, is introduced and the simplex algorithm is applied to find the minimum; the modified linear program is called the Phase I problem. The simplex algorithm applied to the Phase I problem must terminate with a minimum value for the new objective function since, being the sum of nonnegative variables, its value is bounded below by 0.
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Hilbert proved that, for every integer k > 1, every non-negative integer is the sum of a bounded number of k-th powers. In general, a set A of nonnegative integers is called a basis of order h if hA contains all positive integers, and it is called an asymptotic basis if hA contains all sufficiently large integers. Much current research in this area concerns properties of general asymptotic bases of finite order. For example, a set A is called a minimal asymptotic basis of order h if A is an asymptotic basis of order h but no proper subset of A is an asymptotic basis of order h.
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Unlike earlier versions, the models of von Neumann had inequality constraints. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of Brouwer's fixed point theorem. Von Neumann's model of an expanding economy considered the matrix pencil A - λ B with nonnegative matrices A and B; von Neumann sought probability vectors p and q and a positive number λ that would solve the complementarity equation : pT (A - λ B) q = 0, along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run.
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In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to 1. The terms "probability distribution function"Probability distribution function PlanetMath and "probability function"Probability Function at MathWorld have also sometimes been used to denote the probability density function.
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Fisher's factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic. If the probability density function is ƒθ(x), then T is sufficient for θ if and only if nonnegative functions g and h can be found such that : f_\theta(x)=h(x) \, g_\theta(T(x)), i.e. the density ƒ can be factored into a product such that one factor, h, does not depend on θ and the other factor, which does depend on θ, depends on x only through T(x). It is easy to see that if F(t) is a one-to- one function and T is a sufficient statistic, then F(T) is a sufficient statistic.
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A complementarity problem is a type of mathematical optimization problem. It is the problem of optimizing (minimizing or maximizing) a function of two vector variables subject to certain requirements (constraints) which include: that the inner product of the two vectors must equal zero, i.e. they are orthogonal. In particular for finite-dimensional real vector spaces this means that, if one has vectors X and Y with all nonnegative components (xi ≥ 0 and yi ≥ 0 for all i: in the first quadrant if 2-dimensional, in the first octant if 3-dimensional), then for each pair of components xi and yi one of the pair must be zero, hence the name complementarity. e.g.
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As in calculus, the derivative detects multiple roots. If R is a field then R[x] is a Euclidean domain, and in this situation we can define multiplicity of roots; for every polynomial f(x) in R[x] and every element r of R, there exists a nonnegative integer mr and a polynomial g(x) such that :f(x) = (x - r)^{m_r} g(x) where g(r)≠0. mr is the multiplicity of r as a root of f. It follows from the Leibniz rule that in this situation, mr is also the number of differentiations that must be performed on f(x) before r is no longer a root of the resulting polynomial.
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Assuming that \Sigma has nonnegative self intersection number this was generalized to Kähler manifolds (an example being the complex projective plane) by John Morgan, Zoltán Szabó, and Clifford Taubes, also using the Seiberg–Witten invariants. There is at least one generalization of this conjecture, known as the symplectic Thom conjecture (which is now a theorem, as proved for example by Peter Ozsváth and Szabó in 2000). It states that a symplectic surface of a symplectic 4-manifold is genus minimizing within its homology class. This would imply the previous result because algebraic curves (complex dimension 1, real dimension 2) are symplectic surfaces within the complex projective plane, which is a symplectic 4-manifold.
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Execution time (upper bounds, lower bounds, exact time) can be proven the same way, just by introducing a time variable. To prove termination, prove the execution time is finite. To prove nontermination, prove the execution time is infinite. For example, if the time variable is t, and time is measured by counting iterations, then to prove that execution of the previous while-loop takes time x when x is initially nonnegative, and takes forever when x is initially negative, prove if x>0 then x:= x–1; t:= t+1; (x≥0 ⇒ t'=t+x) ∧ (x<0 ⇒ t'=∞) else ok fi ⇒ (x≥0 ⇒ t'=t+x) ∧ (x<0 ⇒ t'=∞) where ok = (x'=x ∧ t'=t).
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More precisely, the extrinsic geometry is controlled by the extrinsic geometry of the isometric embedding uniquely determined by the intrinsic geometry. Shi and Tam's proof adopts a method, due to Robert Bartnik, of using parabolic partial differential equations to construct noncompact Riemannian manifolds-with-boundary of nonnegative scalar curvature and prescribed boundary behavior. By combining Bartnik's construction with the given compact manifold-with-boundary, one obtains a complete Riemannian manifold which is non-differentiable along a closed and smooth hypersurface. By using Bartnik's method to relate the geometry near infinity to the geometry of the hypersurface, and by proving a positive energy theorem in which certain singularities are allowed, Shi and Tam's result follows.
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The following program (written in pseudocode) emulates the execution of a -based OISC: int memory[], program_counter, a, b, c program_counter = 0 while (program_counter >= 0): a = memory[program_counter] b = memory[program_counter+1] c = memory[program_counter+2] if (a < 0 or b < 0): program_counter = -1 else: memory[b] = memory[b] - memory[a] if (memory[b] > 0): program_counter += 3 else: program_counter = c This program assumes that is indexed by nonnegative integers. Consequently, for a instruction (, , ), the program interprets , , or an executed branch to as a halting condition. Similar interpreters written in a -based language (i.e., self-interpreters, which may use self-modifying code as allowed by the nature of the instruction) can be found in the external links below.
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Given a positive (or more generally irreducible non-negative matrix) A, one defines the function f on the set of all non- negative non-zero vectors x such that f(x) is the minimum value of [Ax]i / xi taken over all those i such that xi ≠ 0. Then f is a real valued function, whose maximum is the Perron–Frobenius eigenvalue r. For the proof we denote the maximum of f by the value R. The proof requires to show R = r. Inserting the Perron-Frobenius eigenvector v into f, we obtain f(v) = r and conclude r ≤ R. For the opposite inequality, we consider an arbitrary nonnegative vector x and let ξ=f(x).
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Examples of motifs are the AID motif WRCY/RGYW (W = A or T, R = purine and Y = pyrimidine) with C to T/G/A mutations, and error-prone DNA pol η attributed AID-related mutations (A to G/C/G) in WA/TW motifs. Another (agnostic) way to analyze the observed mutational spectra and DNA sequence context of mutations in tumors involves pooling all mutations of different types and contexts from cancer samples into a discrete distribution. If multiple cancer samples are available, their context-dependent mutations can be represented in the form of a nonnegative matrix. This matrix can be further decomposed into components (mutational signatures) which ideally should describe individual mutagenic factors.
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This is possible only if there are exactly 2(n+1)2 electrons, where n is a nonnegative integer. In particular, for example, buckminsterfullerene, with 60 π-electrons, is non-aromatic, since 60/2 = 30, which is not a perfect square.. In 2011, Jordi Poater and Miquel Solà, expanded the rule to determine when an open-shell fullerene species would be aromatic. They found that if there were 2n2+2n+1 π-electrons, then the fullerene would display aromatic properties. This follows from the fact that a spherical species having a same-spin half-filled last energy level with the whole inner levels being fully filled is also aromatic.. It is similar to Baird's rule.
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Dedekind, before 1886 While teaching calculus for the first time at the Polytechnic school, Dedekind developed the notion now known as a Dedekind cut (German: Schnitt), now a standard definition of the real numbers. The idea of a cut is that an irrational number divides the rational numbers into two classes (sets), with all the numbers of one class (greater) being strictly greater than all the numbers of the other (lesser) class. For example, the square root of 2 defines all the nonnegative numbers whose squares are less than 2 and the negative numbers into the lesser class, and the positive numbers whose squares are greater than 2 into the greater class. Every location on the number line continuum contains either a rational or an irrational number.
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A 1775 color triangle by Tobias Mayer. The sRGB color triangle, shown as a subset of x,y space, a chromaticity space based on CIE 1931 colorimetry A colour triangle is an arrangement of colours within a triangle, based on the additive combination of three primary colors at its corners. An additive colour space defined by three primary colors has a chromaticity gamut that is a color triangle, when the amounts of the primaries are constrained to be nonnegative. Before the theory of additive color was proposed by Thomas Young and further developed by James Clerk Maxwell and Hermann von Helmholtz, triangles were also used to organize colors, for example around a system of red, yellow, and blue primary colors.
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Von Neumann raised the intellectual and mathematical level of economics in several influential publications. For his model of an expanding economy, he proved the existence and uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem. Von Neumann's model of an expanding economy considered the matrix pencil A − λB with nonnegative matrices A and B; von Neumann sought probability vectors p and q and a positive number λ that would solve the complementarity equation :p^T (A - \lambda B) q = 0 along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run.
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In mathematics, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S. An ordered monoid and an ordered group are, respectively, a monoid or a group that are endowed with a partial order that makes them ordered semigroups. The terms posemigroup, pogroup and pomonoid are sometimes used, where "po" is an abbreviation for "partially ordered". The positive integers, the nonnegative integers and the integers form respectively a posemigroup, a pomonoid, and a pogroup under addition and the natural ordering. Every semigroup can be considered as a posemigroup endowed with the trivial (discrete) partial order "=".
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Because the variance of the estimator of a parameter vector is a matrix, the problem of "minimizing the variance" is complicated. Using statistical theory, statisticians compress the information-matrix using real- valued summary statistics; being real-valued functions, these "information criteria" can be maximized. Traditionally, statisticians have evaluated estimators and designs by considering some summary statistic of the covariance matrix (of an unbiased estimator), usually with positive real values (like the determinant or matrix trace). Working with positive real numbers brings several advantages: If the estimator of a single parameter has a positive variance, then the variance and the Fisher information are both positive real numbers; hence they are members of the convex cone of nonnegative real numbers (whose nonzero members have reciprocals in this same cone).
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The fourth central moment is a measure of the heaviness of the tail of the distribution, compared to the normal distribution of the same variance. Since it is the expectation of a fourth power, the fourth central moment, where defined, is always nonnegative; and except for a point distribution, it is always strictly positive. The fourth central moment of a normal distribution is . The kurtosis is defined to be the standardized fourth central moment (Equivalently, as in the next section, excess kurtosis is the fourth cumulant divided by the square of the second cumulant.) If a distribution has heavy tails, the kurtosis will be high (sometimes called leptokurtic); conversely, light-tailed distributions (for example, bounded distributions such as the uniform) have low kurtosis (sometimes called platykurtic).
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Other examples of intervals are the set of numbers such that , the set of all real numbers \R, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "size" (or "measure" or "length") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations, and arithmetic roundoff.
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Any nonzero polynomial may be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees, which are called the homogeneous components of the polynomial. Given a polynomial ring R=K[x_1, \ldots,x_n] over a field (or, more generally, a ring) K, the homogeneous polynomials of degree d form a vector space (or a module), commonly denoted R_d. The above unique decomposition means that R is the direct sum of the R_d (sum over all nonnegative integers). The dimension of the vector space (or free module) R_d is the number of different monomials of degree d in n variables (that is the maximal number of nonzero terms in a homogeneous polynomial of degree d in n variables).
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In linear algebra, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type); to economics (Okishio's theorem, Hawkins–Simon condition); to demography (Leslie population age distribution model); to social networks (DeGroot learning process), to Internet search engines and even to ranking of football teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau.
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Rind, Bauserman and Tromovitch stated that research findings can be skewed by an investigator's personal biases, and in Rind et al. claimed that "[r]eviewers who are convinced that CSA is a major cause of adult psychopathology may fall prey to confirmation bias by noting and describing study findings indicating harmful effects but ignoring or paying less attention to findings indicating nonnegative outcomes". They defended their deliberate choice of non-legal and non-clinical samples, accordingly avoiding individuals who received psychological treatment or were engaged in legal proceedings as a way of correcting this bias through the use of a sample of college students. Dallam and Anna Salter have stated that Rind and Bauserman have associated with age of consent reform organizations in the past.
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Lasso was introduced in order to improve the prediction accuracy and interpretability of regression models by altering the model fitting process to select only a subset of the provided covariates for use in the final model rather than using all of them. It was developed independently in geophysics, based on prior work that used the \ell^1 penalty for both fitting and penalization of the coefficients, and by the statistician, Robert Tibshirani based on Breiman’s nonnegative garrote. Prior to lasso, the most widely used method for choosing which covariates to include was stepwise selection, which only improves prediction accuracy in certain cases, such as when only a few covariates have a strong relationship with the outcome. However, in other cases, it can make prediction error worse.
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A weighted matroid is a matroid together with a function from its elements to the nonnegative real numbers. The weight of a subset of elements is defined to be the sum of the weights of the elements in the subset. The greedy algorithm can be used to find a maximum-weight basis of the matroid, by starting from the empty set and repeatedly adding one element at a time, at each step choosing a maximum- weight element among the elements whose addition would preserve the independence of the augmented set. This algorithm does not need to know anything about the details of the matroid's definition, as long as it has access to the matroid through an independence oracle, a subroutine for testing whether a set is independent.
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When the entries of A are nonnegative, the permanent can be computed approximately in probabilistic polynomial time, up to an error of εM, where M is the value of the permanent and ε > 0 is arbitrary. In other words, there exists a fully polynomial-time randomized approximation scheme (FPRAS) (). The most difficult step in the computation is the construction of an algorithm to sample almost uniformly from the set of all perfect matchings in a given bipartite graph: in other words, a fully polynomial almost uniform sampler (FPAUS). This can be done using a Markov chain Monte Carlo algorithm that uses a Metropolis rule to define and run a Markov chain whose distribution is close to uniform, and whose mixing time is polynomial.
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Fix an abelian category, such as a category of modules over a ring. A spectral sequence is a choice of a nonnegative integer r0 and a collection of three sequences: # For all integers r ≥ r0, an object Er, called a sheet (as in a sheet of paper), or sometimes a page or a term, # Endomorphisms dr : Er → Er satisfying dr o dr = 0, called boundary maps or differentials, # Isomorphisms of Er+1 with H(Er), the homology of Er with respect to dr. The E2 sheet of a cohomological spectral sequence A doubly graded spectral sequence has a tremendous amount of data to keep track of, but there is a common visualization technique which makes the structure of the spectral sequence clearer. We have three indices, r, p, and q.
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Number of sides of known constructible polygons having up to 1000 sides (bold) or odd side count (red) Carl Friedrich Gauss developed the theory of Gaussian periods in his Disquisitiones Arithmeticae and formulated a sufficient condition for the constructibility of regular polygons. Gauss stated that this condition was also necessary, but never published a proof. Pierre Wantzel gave a full proof of necessity in 1837. The result is known as the Gauss–Wantzel theorem: : An n-sided regular polygon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and distinct Fermat primes: in other words, if and only if n is of the form n = 2kp1p2…ps, where k is a nonnegative integer and the pi are distinct Fermat primes.
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Based on his monotonicity formula, Huisken showed that many of these regions, specifically those known as type I singularities, are modeled in a precise way by self-shrinking solutions of the mean curvature flow. There is now a reasonably complete understanding of the rescaling process in the setting of mean curvature flows which only involve hypersurfaces whose mean curvature is strictly positive. Following provisional work by Huisken, Tobias Colding and William Minicozzi have shown that (with some technical conditions) the only self-shrinking solutions of mean curvature flow which have nonnegative mean curvature are the round cylinders, hence giving a complete local picture of the type I singularities in the "mean-convex" setting.Tobias H. Colding and William P. Minicozzi, II. Generic mean curvature flow I: generic singularities. Ann.
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An m by n matrix A with integer entries has a (row) Hermite normal form H if there is a square unimodular matrix U where H=UA and H has the following restrictions: # H is upper triangular (that is, hij = 0 for i > j), and any rows of zeros are located below any other row. # The leading coefficient (the first nonzero entry from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it; moreover, it is positive. # The elements below pivots are zero and elements above pivots are nonnegative and strictly smaller than the pivot. The third condition is not standard among authors, for example some sources force non-pivots to be nonpositive or place no sign restriction on them.
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In information theory and statistics, negentropy is used as a measure of distance to normality.Aapo Hyvärinen, Survey on Independent Component Analysis, node32: Negentropy, Helsinki University of Technology Laboratory of Computer and Information ScienceAapo Hyvärinen and Erkki Oja, Independent Component Analysis: A Tutorial, node14: Negentropy, Helsinki University of Technology Laboratory of Computer and Information ScienceRuye Wang, Independent Component Analysis, node4: Measures of Non-Gaussianity Out of all distributions with a given mean and variance, the normal or Gaussian distribution is the one with the highest entropy. Negentropy measures the difference in entropy between a given distribution and the Gaussian distribution with the same mean and variance. Thus, negentropy is always nonnegative, is invariant by any linear invertible change of coordinates, and vanishes if and only if the signal is Gaussian.
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Building on his results on matrix games and on his model of an expanding economy, von Neumann invented the theory of duality in linear programming when George Dantzig described his work in a few minutes, and an impatient von Neumann asked him to get to the point. Dantzig then listened dumbfounded while von Neumann provided an hourlong lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming. Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Paul Gordan (1873), which was later popularized by Karmarkar's algorithm. Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squares subproblem with a convexity constraint (projecting the zero-vector onto the convex hull of the active simplex).
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Shi is well-known for his foundational work with Luen-Fai Tam on compact and smooth Riemannian manifolds-with-boundary whose scalar curvature is nonnegative and whose boundary is mean-convex. In particular, if the manifold has a spin structure, and if each connected component of the boundary can be isometrically embedded as a strictly convex hypersurface in Euclidean space, then the average value of the mean curvature of each boundary component is less than or equal to the average value of the mean curvature of the corresponding hypersurface in Euclidean space. This is particularly simple in three dimensions, where every manifold has a spin structure and a result of Louis Nirenberg shows that any positively-curved Riemannian metric on the two-dimensional sphere can be isometrically embedded in three-dimensional Euclidean space in a geometrically unique way.Louis Nirenberg.
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Given two compact oriented Riemannian manifolds, Mi, possibly with boundary: : dSWIF(M1, M2) = 0 iff there is an orientation preserving isometry from M1 to M2. If Mi converge in the Gromov–Hausdorff sense to a metric space Y then a subsequence of the Mi converge SWIF-ly to an integral current space contained in Y but not necessarily equal to Y. For example, the GH limit of a sequence of spheres with a long thin neck pinch is a pair of spheres with a line segment running between them while the SWIF limit is just the pair of spheres. The GH limit of a sequence of thinner and thinner tori is a circle but the flat limit is the 0 space. In the setting with nonnegative Ricci curvature and a uniform lower bound on volume, the GH and SWIF limits agree.
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The primordial example is the classifying space of a discrete group G. We regard G as a category with one object whose endomorphisms are the elements of G. Then the k-simplices of N(G) are just k-tuples of elements of G. The face maps act by multiplication, and the degeneracy maps act by insertion of the identity element. If G is the group with two elements, then there is exactly one nondegenerate k-simplex for each nonnegative integer k, corresponding to the unique k-tuple of elements of G containing no identities. After passing to the geometric realization, this k-tuple can be identified with the unique k-cell in the usual CW structure on infinite-dimensional real projective space. The latter is the most popular model for the classifying space of the group with two elements.
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A polynomial expression is an expression built with scalars (elements of ), indeterminates, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers. As all these operations are defined in K[X_1,\dots, X_n] a polynomial expression represents a polynomial, that is an element of K[X_1,\dots, X_n]. The definition of a polynomial as a linear combination of monomials is a particular polynomial expression, which is often called the canonical form, normal form, or expanded form of the polynomial. Given a polynomial expression, one can compute the expanded form of the represented polynomial by expanding with the distributive law all the products that have a sum among their factors, and then using commutativity (except for the product of two scalars), and associativity for transforming the terms of the resulting sum into products of a scalar and a monomial; then one gets the canonical form by regrouping the like terms.
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Let X be a variety of nonnegative Kodaira dimension over a field of characteristic zero, and let B be the canonical model of X, B = Proj R(X, KX); the dimension of B is equal to the Kodaira dimension of X. There is a natural rational map X – → B; any morphism obtained from it by blowing up X and B is called the Iitaka fibration. The minimal model and abundance conjectures would imply that the general fiber of the Iitaka fibration can be arranged to be a Calabi–Yau variety, which in particular has Kodaira dimension zero. Moreover, there is an effective Q-divisor Δ on B (not unique) such that the pair (B, Δ) is klt, KB \+ Δ is ample, and the canonical ring of X is the same as the canonical ring of (B, Δ) in degrees a multiple of some d > 0.O. Fujino and S. Mori, J. Diff. Geom.
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In computer-aided geometric design a control point is a member of a set of points used to determine the shape of a spline curve or, more generally, a surface or higher-dimensional object.. For Bézier curves, it has become customary to refer to the d-vectors p_i in a parametric representation \sum_i p_i \phi_i of a curve or surface in d-space as control points, while the scalar-valued functions \phi_i, defined over the relevant parameter domain, are the corresponding weight or blending functions. Some would reasonably insist, in order to give intuitive geometric meaning to the word "control", that the blending functions form a partition of unity, i.e., that the \phi_i are nonnegative and sum to one. This property implies that the curve lies within the convex hull of its control points.. This is the case for Bézier's representation of a polynomial curve as well as for the B-spline representation of a spline curve or tensor-product spline surface.
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A m by n matrix A with integer entries has a (column) Hermite normal form H if there is a square unimodular matrix U where H=AU and H has the following restrictions: # H is lower triangular, hij = 0 for i < j, and any columns of zeros are located on the right. # The leading coefficient (the first nonzero entry from the top, also called the pivot) of a nonzero column is always strictly below of the leading coefficient of the column before it; moreover, it is positive. # The elements to the right of pivots are zero and elements to the left of pivots are nonnegative and strictly smaller than the pivot. Note that the row-style definition has a unimodular matrix U multiplying A on the left (meaning U is acting on the rows of A), while the column-style definition has the unimodular matrix action on the columns of A. The two definitions of Hermite normal forms are simply transposes of each other.
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85 of: :"In accordance with this, already in the plane there is no nonnegative additive measure (for which the unit square has a measure of 1), which is invariant with respect to all transformations belonging to A2 [the group of area- preserving affine transformations]." To explain this a bit more, the question of whether a finitely additive measure exists, that is preserved under certain transformations, depends on what transformations are allowed. The Banach measure of sets in the plane, which is preserved by translations and rotations, is not preserved by non-isometric transformations even when they do preserve the area of polygons. As explained above, the points of the plane (other than the origin) can be divided into two dense sets which we may call A and B. If the A points of a given polygon are transformed by a certain area- preserving transformation and the B points by another, both sets can become subsets of the B points in two new polygons.
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The nonnegative integer r is called the free rank or Betti number of the module M. The module is determined up to isomorphism by specifying its free rank , and for class of associated irreducible elements and each positive integer the number of times that occurs among the elementary divisors. The elementary divisors can be obtained from the list of invariant factors of the module by decomposing each of them as far as possible into pairwise relatively prime (non-unit) factors, which will be powers of irreducible elements. This decomposition corresponds to maximally decomposing each submodule corresponding to an invariant factor by using the Chinese remainder theorem for R. Conversely, knowing the multiset of elementary divisors, the invariant factors can be found, starting from the final one (which is a multiple of all others), as follows. For each irreducible element such that some power occurs in , take the highest such power, removing it from , and multiply these powers together for all (classes of associated) to give the final invariant factor; as long as is non-empty, repeat to find the invariant factors before it.
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This agrees with the previous definitions when X is an affine or projective variety (viewed as a scheme over k). When X is a variety over an algebraically closed field k, much of the structure of X is determined by its set X(k) of k-rational points. For a general field k, however, X(k) gives only partial information about X. In particular, for a variety X over a field k and any field extension E of k, X also determines the set X(E) of E-rational points of X, meaning the set of solutions of the equations defining X with values in E. Example: Let X be the conic curve x2 \+ y2 = −1 in the affine plane A2 over the real numbers R. Then the set of real points X(R) is empty, because the square of any real number is nonnegative. On the other hand, in the terminology of algebraic geometry, the algebraic variety X over R is not empty, because the set of complex points X(C) is not empty.
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G(3) is at least four (since cubes are congruent to 0, 1 or −1 mod 9); for numbers less than 1.3, 1290740 is the last to require six cubes, and the number of numbers between N and 2N requiring five cubes drops off with increasing N at sufficient speed to have people believe that ; the largest number now known not to be a sum of four cubes is 7373170279850, and the authors give reasonable arguments there that this may be the largest possible. The upper bound is due to Linnik in 1943.U.V. Linnik. Mat. Sb. N.S. 12(54), 218–224 (1943) On the representation of large numbers as sums of seven cubes. (All nonnegative integers require at most 9 cubes, and the largest integers requiring 9, 8, 7, 6 and 5 cubes are conjectured to be 239, 454, 8042, 1290740 and 7373170279850, respectively.) 13792 is the largest number to require seventeen fourth powers (Deshouillers, Hennecart and Landreau showed in 2000 that every number between 13793 and 10245 required at most sixteen, and Kawada, Wooley and Deshouillers extended Davenport's 1939 result to show that every number above 10220 required no more than sixteen).
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Half-coins have infinitely many sides numbered with 0,1,2,... and the positive even numbers are taken with negative probabilities. Two half-coins make a complete coin in the sense that if we flip two half-coins then the sum of the outcomes is 0 or 1 with probability 1/2 as if we simply flipped a fair coin. In Convolution quotients of nonnegative definite functions and Algebraic Probability Theory Imre Z. Ruzsa and Gábor J. Székely proved that if a random variable X has a signed or quasi distribution where some of the probabilities are negative then one can always find two random variables, Y and Z, with ordinary (not signed / not quasi) distributions such that X, Y are independent and X + Y = Z in distribution. Thus X can always be interpreted as the "difference" of two ordinary random variables, Z and Y. If Y is interpreted as a measurement error of X and the observed value is Z then the negative regions of the distribution of X are masked / shielded by the error Y. Another example known as the Wigner distribution in phase space, introduced by Eugene Wigner in 1932 to study quantum corrections, often leads to negative probabilities.
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