pinecone/core-2020-05-10-deduplication huggingface dataset:
The Algebra of Flows in Graphs

AbstractWe define a contravariant functorKfrom the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graphX, an abelian groupB, and a nonnegative integerj, an element of Hom(Kj(X),B) is a coherent family ofB-valued flows on the set of all graphs obtained by contracting some (j−1)-set of edges ofX; in particular, Hom(K1(X),R) is the familiar (real) “cycle-space” ofX. We show thatK·(X) is torsion-free and that its Poincaré polynomial is the specializationtn−kTX(1/t, 1+t) of the Tutte polynomial ofX(hereXhasnvertices andkcomponents). Functoriality ofK·induces a functorial coalgebra structure onK·(X); dualizing, for any ringBwe obtain a functorialB-algebra structure on Hom(K·(X),B). WhenBis commutative we present this algebra as a quotient of a divided power algebra, leading to some interesting inequalities on the coefficients of the above Poincaré polynomial. We also provide a formula for the theta function of the lattice of integer-valued flows inX, and conclude with 10 open problems
The Fundamental Group of the Complement of the Complexification of a Real Arrangement of Hyperplanes

AbstractLet A be an arrangement of hyperplanes (i.e., a finite set of 1-codimension vector subspaces) in Rd. We say that the linear ordering of the hyperplanes A,H1≺H2≺···≺Hn, is ashellabilityorder of A, if there is an oriented affine linelcrossing the hyperplanes of A on the given linear order. The intersection latticeL(A) is the set of all intersections of the hyperplanes of A partially ordered by reversed inclusion. Set M(Ac)≔Cd\⋃{H⊗C:H∈A}. We will prove:Suppose that there are shellability orders H1≺H2≺···≺Hnand H′1≺′H′2≺′···≺′H′n, respectively, ofAandA′,such that the bijective map Hi→H′i, i∈[n]determines an isomorphism of the intersection lattices L(A)and L(A′).Then the fundamental groupsπ1(M(Ac))andπ1(M(A′c)) are isomorphic
Generating Functions of Circular Codes

AbstractWe describe in terms of different parameters the generating series of the star of a circular code. We extend the characterization of length distributions of circular codes established for a finite alphabet by Schützenberger to an arbitrary “weighted” alphabet. In this framework, we give a new characterization of these length distributions. This one directly concerns the coefficients of the generating series of the code instead of the number of primitive conjugacy classes. This result shows that we can decide whether a finite sequence is the length distribution of a circular code. We also establish a necessary and sufficient condition for a series to be the length distribution of a maximal circular code over a finite alphabet
Cubical Species and Nonassociative Algebras

AbstractWe lay down the foundations of a theory ofcubical species, as a variant of Joyal's classical theory of species (A. Joyal,Adv. Math.42(1981), 1–82). Informally, a cubical species associates in a functorial way a set of structures to each hypercube. In this context, the hyperoctahedral groups replace the symmetric groups. We analyze cubical species, molecular cubical species, and basic operations among them, along with explicit examples. We show, in particular, that the cubical product gives rise, in a natural way, to a commutative nonassociative ring of formal power series. We conclude with a detailed analysis of this nonassociative ring
Two-Best Solutions under Distance Constraints: The Model and Exemplary Results for Matroids

AbstractIn discrete maximization problems one typically wants to find an optimal solution. However, in topics like “alignment of DNA-strings,” “automatic speech recognition,” and “computer chess” people have been asking for finding not only the best, but thekbest solutions. Sometimes it becomes a problem when thesekbest solutions are too similar. This similarity problem may be overcome by demanding that theksolutions obey certain distance conditions. We investigate the simplest casek=2 and we look at valuated matroids. We present several exemplary results, concerning also time complexity. These results are interesting in themselves. But they are also good references for similar studies in other fields with less smooth structures
On the Ramsey Problem for Multicolor Bipartite Graphs

AbstractGiveni,jpositive integers, letKi,jdenote a bipartite complete graph and letRr(m,n) be the smallest integerasuch that for anyr-coloring of the edges ofKa,aone can always find a monochromatic subgraph isomorphic toKm,n. In other words, ifa≥Rr(m,n) then every matrixa×awith entries in {0,1,…,r−1} always contains a submatrixm×norn×mwhose entries arei, 0≤i≤r−1. We shall prove thatR2(m,n)≤2m(n−1)+2m−1−1, which generalizes the previous resultsR2(2,n)≤4n−3 andR2(3,n)≤8n−5 due to Beineke and Schwenk. Moreover, we find a class of lower bounds based on properties of orthogonal Latin squares which establishes that limr→∞Rr(2,2)r−2=1
Solution of the Boolean Markus–Yamabe Problem

AbstractThe main point in the design of content addressable memory would be under what conditions the state possessing the total information can attract all other states in the phase of the system. The problem can be formulated as a global asymptotic stability problem of Boolean dynamical systems. In this article we give a complete answer to this global asymptotic stability problem. The conditions employed involve the Hamming distance on the phase space {0,1}nas well as the spectral condition on the Jacobian Boolean matrix of F: {0,1}n→{0,1}n evaluated at each point of {0,1}n. This article furnishes a complete solution of the Boolean Markus–Yamabe problem
Tensor Products and the Loomis–Sikorski Theorem for MV-Algebras

AbstractMV-algebras are the models of the time-honored equational theory of magnitudes with unit. Introduced by Chang as a counterpart of the infinite-valued sentential calculus of Łukasiewicz, they are currently investigated for their relations with AFC*-algebras, toric desingularizations, and lattice-ordered abelian groups. Using tensor products, in this paper we shall characterize multiplicatively closed MV-algebras. Generalizing work of Loomis and Sikorski, we shall investigate the relationships between σ-complete multiplicatively closed MV-algebras, and pointwise σ-complete MV-algebras of [0,1]-valued functions
A Sparse Effective Nullstellensatz

AbstractWe present bounds for the sparseness in the Nullstellensatz. These bounds can give a much sharper characterization than degree bounds of the monomial structure of the polynomials in the Nullstellensatz in case that the input system is sparse. As a consequence we derive a degree bound which can substantially improve the known ones in case of a sparse system.In addition we introduce the notion of algebraic degree associated to a polynomial system of equations. We obtain a new degree bound which is sharper than the known ones when this parameter is small. We also improve the previous effective Nullstellensätze in case the input polynomials are quadratic.Our approach is completely algebraic, and the obtained results are independent of the characteristic of the base field
Symmetrization Inequalities for Difference Equations on Graphs

AbstractWe prove symmetrization inequalities for positive solutions of (not necessarily linear) difference equations of the form−Δu=φu−c·u+λ,where Δ is a discrete Laplacian, φ is a convex decreasing function,cis a positive function and λ is a real function, on subsets ofX×Y, whereXis a graph andYis the line Z, the circle graph Zm, them-regular treeTm, the line graphL(Tm) ofTm, or the edge graph of the octahedron. Instead of Δ, we may also have some other operators, including the heat operator. The typical result says that we use a discrete version of Steiner symmetrization to symmetrize the domain on which the equation is defined, and if all the functions and boundary values involved are appropriately symmetrized as well, then the increasing convex means ofu(x,·) are increased for each fixedx∈X
A Unified Treatment of the Geometric Algebra of Matroids and Even Δ-Matroids

AbstractThe concept of acombinatorial(W;P;U)-geometryfor a Coxeter groupW, a subsetPof its generating involutions and a subgroupUofWwithP⊆Uyields the combinatorial foundation for a unified treatment of the representation theories of matroids and of even Δ-matroids. The concept of a (W,P)-matroid as introduced by I. M. Gelfand and V. V. Serganova is slightly different, although for many important classes ofWandPone gets the same structures. In the present paper, we extend the concept of the Tutte group of an ordinary matroid to combinatorial (W;P;U)-geometries and suggest two equivalent definitions of a (W;P;U)-matroid with coefficients in a fuzzy ringK. While the first one is more appropriate for many theoretical considerations, the second one has already been used to show that (W;P;U)-matroids with coefficients encompass matroids with coefficients and Δ-matroids with coefficients
Testing Orientability for Matroids is NP-Complete

AbstractMatroids and oriented matroids are fundamental objects in combinatorial geometry. While matroids model the behavior of vector configurations over general fields, oriented matroids model the behavior of vector configurations over ordered fields. For every oriented matroid there is a corresponding underlying matroid. This article addresses the question how complex it is to algorithmically decide whether, on the other hand, one can assign an orientation to a given (rank 3) matroid. We will prove that this problem is NP-complete
Block Toeplitz and Hurwitz Matrices: A Recursive Approach

AbstractRecursive matrices—bi-infinite matrices such that each row can be recursively computed from the previous one—have been revealed to be a useful tool in the study of signal analysis and filter theory. In particular, recursive Toeplitz and Hurwitz matrices are used in [3] to give an algebraic interpretation of signal processing. In this work we introduce the notion of block recursive matrix, and we show that an important property of scalar recursive matrices, namely, the product rule, also holds in the case of block matrices. We also give conditions under which a block Hurwitz matrix of step k can be seen as a scalar Hurwitz matrix of the same step
Cubature Formulae and Polynomial Ideals

AbstractThe structure of cubature formulae of degree 2n−1 is studied from a polynomial ideal point of view. The main result states that if I is a polynomial ideal generated by a proper set of (2n−1)-orthogonal polynomials and if the cardinality of the variety V(I) is equal to the codimension of I, then there exists a cubature formula of degree 2n−1 based on the points in the variety. The result covers a number of cubature formulae in the literature, including Gaussian cubature formulae on one end and the usual product formulae on the classical domains on the other end. The result also offers a new method for constructing cubature formulae
The Frequency Interpretation in Probability

AbstractWe formulate and discuss the notion of generic sequence (and random sequence) associated with a sequence of random variables. These are substantial generalizations of the notion of “collective” introduced by R. von Mises as an attempt to give an operational meaning to various probabilistic ideas. The definition of collectives lacked a precise formulation, and A. Church and others attempted to give a rigorous meaning to the theory by utilizing ideas from the theory of computability. Nevertheless, the theory had major flaws. Even though some important contributions to special cases were made by P. Martin-Löf, the whole circle of ideas has languished due to lack of generality. The theory presented in the present article resolves many of these problems, and provides a coherent framework for the relevant ideas
L∞ Metric Criteria for Convergence in Bayesian Recursive Inference Systems

AbstractMotivated by applications to probabilistic inference, we consider a sequence of probability measures, called “conclusion measures,” on a fixed space X. The sequence is generated recursively via conditional probability, driven by a sequence of input measures (rather than by a sequence of punctual data, as in Bayesian statistical inference). The general problem is to give conditions on the input measures such that the sequence of conclusion measures converges weakly. We develop L∞-metric criteria defined recursively on the input measures, which are sufficient (but not necessary) for the sequence of conclusion measures to converge at a given rate. We discuss the applications of this to the “directed convergence strategy” introduced in [1]. Finally, we show that if the input measures satisfy the criteria, then the input sequence also converges at a comparable rate
The Nullstellensatz for Systems of PDE

AbstractGiven an ideal I in A, the polynomial ring in n-indeterminates, the affine variety of I is the set of common zeros in Cn of all the polynomials that belong to I, and the Hilbert Nullstellensatz states that there is a bijective correspondence between these affine varieties and radical ideals of A. If, on the other hand, one thinks of a polynomial as a (constant coefficient) partial differential operator, then instead of its zeros in Cn, one can consider its zeros, i.e., its homogeneous solutions, in various function and distribution spaces. An advantage of this point of view is that one can then consider not only the zeros of ideals of A, but also the zeros of submodules of free modules over A (i.e., of systems of PDEs). The question then arises as to what is the analogue here of the Hilbert Nullstellensatz. The answer clearly depends on the function–distribution space in which solutions of PDEs are being located, and this paper considers the case of the classical spaces. This question is related to the more general question of embedding a partial differential system in a (two-sided) complex with minimal homology. This paper also explains how these questions are related to some questions in control theory
Variants of the Rogers–Ramanujan Identities

AbstractWe evaluate several integrals involving generating functions of continuous q-Hermite polynomials in two different ways. The resulting identities give new proofs and generalizations of the Rogers–Ramanujan identities. Two quintic transformations are given, one of which immediately proves the Rogers–Ramanujan identities without the Jacobi triple product identity. Similar techniques lead to new transformations for unilateral and bilateral series. The quintic transformations lead to curious identities involving primitive fifth roots of unity which are then extended to primitive pth roots of unity for odd p
The Strong Independence Theorem for Automorphism Groups and Congruence Lattices of Arbitrary Lattices

AbstractIn the book, General Lattice Theory (Birkhäuser, Basel, 1978), the first author raised the following problem (Problem II.18):Let L be a nontrivial lattice and let G be a group. Does there exist a lattice K such that K and L have isomorphic congruence lattices and the automorphism group of K is isomorphic to G?The finite case was solved, independently, in the affirmative, by V. A. Baranskiĭ (1979, in “Abstracts of Lectures of the 15th All-Soviet Algebraic Conference, Krasnojarsk, July 1979,” Vol. 1, p. 11) and A. Urquhart (1978, Algebra Universalis8, 45–58). In 1995, the first author and E. T. Schmidt (Beitrage Algebra Geom.36, 97–108) proved a much stronger result, the strong independence of the automorphism group and the congruence lattice in the finite case. In this paper, we provide a full affirmative solution of the above problem. In fact, we prove much stronger results, verifying strong independence for general lattices and also for lattices with zero
Tree Representations of Non-symmetric Group-Valued Proximities

AbstractLet X be a finite set and let d be a function from X×X into an arbitrary group G. An example of such a function arises by taking a tree T whose vertices include X, assigning two elements of G to each edge of T (one for each orientation of the edge), and setting d(i,j) equal to the product of the elements along the directed path from i to j. We characterize conditions when an arbitrary function d can be represented in this way, and show how such a representation may be explicitly constructed. We also describe the extent to which the underlying tree and the edge weightings are unique in such a representation. These results generalize a recent theorem involving undirected edge assignments by an Abelian group. The non-Abelian bi-directed case is of particular relevance to phylogeny reconstruction in molecular biology
Enumeration of m-Ary Cacti

AbstractThe purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both labelled and unlabelled m-ary cacti, according to (i) the number of polygons, (ii) the vertex-color distribution, (iii) the vertex-degree distribution of each color. We also enumerate m-ary cacti according to the order of their automorphism group. Using a generalization of Otter's formula, we express the species of m-ary cacti in terms of rooted and of pointed cacti. A variant of the m-dimensional Lagrange inversion is then used to enumerate these structures. The method of Liskovets for the enumeration of unrooted planar maps can also be adapted to m-ary cacti
Large Deviations of Random Vector Fields with Applications to Economics

AbstractLet Zn(p), n=1,2,…, be a sequence of random vector fields on Rl, and let π*n={p∈Rl;Zn(p)=0} denote the random set of zeros of Zn. We study the asymptotics of probabilities of the type P(π*n∩B≠∅) for subsets B⊃Rl. Under suitable regularity conditions these probabilities are of the order e−nI(B) where I(B) is a large deviation rate function obtained as a minimum of an associated entropy function I(p) over B. We study also the large deviations of random graphs of the form {(p,n−1Xn(p));p∈π*n}⊂Rl+d, where Xn is an auxiliary sequence of random maps Xn: Rl→Rd. In economic applications Zn(p) refers to the total excess demand in a random economy of size n and hence π*n becomes the random set of equilibrium prices for the economy. The auxiliary map Xn(p) may denote any price–depending total characteristic like total demand, supply, etc. The conditional laws of large numbers which ensue on the large deviation estimates admit an interpretation in terms of a principle of entropy minimization in analogy with the classical maximum entropy principles in statistics and statistical mechanics
The Excedance Set of a Permutation

AbstractThe excedance set of a permutation π=π1π2···πn is the set of indices i for which πi>i. We give a formula for the number of permutations with a given excedance set and recursive formulas satisfied by these numbers. We prove log-concavity of certain sequences of these numbers and we show that the most common excedance set among permutations in the symmetric group Sn is {1,2,…,⌊n/2⌋}. We also relate certain excedance set numbers to Stirling numbers of the second kind, and others to the Genocchi numbers
Enumeration of Planar Constellations

AbstractThe enumeration of transitive ordered factorizations of a given permutation is a combinatorial problem related to singularity theory. Let n≥1, and let σ0 be a permutation of Sn having di cycles of length i, for i≥1. Let m≥2. We prove that the number of m-tuples (σ1,…,σm) of permutation of Sn such that•σσ···σ=σ,•the group generated by σ,…,σ acts transitively on {1,2,…,},•∑(σ)=(−1)+2, where (σ) denotes the number of cycles of σA one-to-one correspondence relates these m-tuples to some rooted planar maps, which we call constellations and enumerate via a bijection with some bicolored trees. For m=2, we recover a formula of Tutte for the number of Eulerian maps. The proof relies on the idea that maps are conjugacy classes of trees and extends the method previously applied to Eulerian maps by the second author. Our result might remind the reader of an old theorem of Hurwitz, giving the number of m-tuples of transpositions satisfying the above conditions. Indeed, we show that our result implies Hurwitz' theorem. We also briefly discuss its implications for the enumeration of nonequivalent coverings of the sphere
Inferring Three-Dimensional Crack Statistics from Two-Dimensional Data

AbstractIn rock mechanics it is often assumed that the number of cracks whose size exceeds c is given by the exponential N0e−c/c̄. It is difficult, however, to examine the cracks in a three-dimensional body to verify this exponential variation, and one is normally limited to observations on an outcropping, a cut, or a plane obtained by sectioning a sample. In this paper, we consider two mathematical problems. The direct problem is to find the distribution of the line segments in a plane section when the three-dimensional distribution of cracks is homogeneous, isotropic, and exponential. It will be shown that this distribution can be expressed by means of a Hankel function and that the distribution in a plane section is qualitatively different from the three-dimensional exponential distribution in having a peak at a finite value of segment length. It is found that the mean segment size in the plane is π/2 times the mean crack diameter in these dimensions. This is consistent with the well-known observation that small cracks have a lower probability of being intercepted by a plane than larger cracks.The indirect problem is to infer the three-dimensional distribution of cracks from the distribution on a plane section. This problem is solved by deriving an integral equation relating the three-dimensional distribution of cracks to the distribution of line segments in a plane and showing that it can be solved for an arbitrary distribution of segments. The special case of the Hankel distribution in the plane leads to an exponential distribution in three dimensions, verifying the general solution of the indirect problem
Gröbner Bases for Spaces of Quadrics of Low Codimension

AbstractLet R=⊕i≥0Ri be a quadratic standard graded K-algebra. Backelin has shown that R is Koszul provided dimR2≤2. One may wonder whether, under the same assumption, R is defined by a Gröbner basis of quadrics. In other words, one may ask whether an ideal I in a polynomial ring S generated by a space of quadrics of codimension ≤2 always has a Gröbner basis of quadrics. We will prove that this is indeed the case with, essentially, one exception given by the ideal I=(x2, xy, y2−xz, yz)⊂K[x, y, z]. We show also that if R is a generic quadratic algebra with dim R2<dimR1 then R is defined by a Gröbner basis of quadrics
Graphs of Some CAT(0) Complexes

AbstractIn this note, we characterize the graphs (1-skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov's CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addition, all maximal cells are either regular Euclidean cubes or right Euclidean triangles glued in a special way, then the underlying graph G(K) is either a median graph or a hereditary modular graph without two forbidden induced subgraphs. We also characterize the simplicial complexes arising from bridged graphs, a class of graphs whose metric enjoys one of the basic properties of CAT(0) spaces. Additionally, we show that the graphs of all these complexes and some more general classes of graphs have geodesic combings and bicombings verifying the 1- or 2-fellow traveler property