pinecone/core-2020-05-10-deduplication huggingface dataset:
Inversion Formula for Continuous Multifractals

AbstractIn a previous paper the authors introduced the inverse measure μ†of a probability measure μ on [0,1]. It was argued that the respective multifractal spectra are linked by the “inversion formula”f†(α)=αf(1/α). Here, the statements of the previous paper are put into more mathematical terms and proofs are given for the inversion formula in the case of continuous measures. Thereby,fmay stand for the Hausdorff spectrum, the packing spectrum, or the coarse grained spectrum. With a closer look at the special case of self-similar measures we offer a motivation of the inversion formula as well as a discussion of possible generalizations. Doing so we find a natural extension of the scope of the notation “self-similar” and a failure of the usual multifractal formalism
Random Induced Subgraphs of Generalizedn-Cubes

AbstractThe subject of this paper is to analyse connectivity and paths in random induced subgraphs of generalizedn-cubes over an alphabet A with |A|=α,Qαn. The vertices of Qαnaren-tuples (x1,…,xn), wherexi∈A,i=1,…,n, and two vertices are adjacent if they differ in exactly one coordinate. Random induced subgraphs, Γn<Qαn, are obtained by selecting each vertex of Qαnindependently with probability λn. The first theorem shows that for λn=(cln(n))/nthere exists a unique largest component in Γn<Qαnwhich contains almost all vertices and that the size of the second largest component is ≤Cn/ln(n),C>0. The second theorem describes connectivity and paths of Γnfor constant probability λ>0. It is proved that two verticesP,Qcontained in the largest component of Γn, that have distancekin Qα, have a distance ≤k+kin Γnand are typically connected by a large number of independent Γn-paths. Further, for any two verticesP,Qcontained in the largest Γncomponent there exists a constantcP,Q>0 such that their distance is ≤cP,Qn
Nonintersecting Paths, Pfaffians, and ▵-Matroids

AbstractIn this paper, we study ▵-matroids induced by nonintersecting paths in a directed graph. The association between nonintersecting paths and ▵-matroids is derived from Pfaffians. On the one hand, certain numbers ofk-tuples of nonintersecting paths may (often) be expressed as a Pfaffian, while, on the other hand, representability problems for ▵-matroids may be studied in terms of Pfaffians. It is shown that ▵-matroids induced by nonintersecting paths are representable over fields of any characteristic and that weightings defined on the edge set and with values in some linearly ordered abelian group give rise to valuated ▵-matroids
Aq-Analog of Approximate Inclusion–Exclusion

AbstractWe consider the lattice of subspaces of ann-dimensional vector spaceVnqover a finite fieldGF(q) and represent a family of such subspaces by elements of a setX. Theq-analog of the principle of inclusion–exclusion expresses the size of the union of elements ofXrepresenting subspaces ofVnqin terms of the sizes of subsets ofXwhose intersection contains a given subspace ofVnq. We study the problem of approximating the size of this union when intersection sizes are known only for some subspaces ofVnq. In particular, we consider the case where intersection sizes are given for subsets ofXwhose intersection contains a subspace ofVnqof dimension at mostk. We extend methods of Linial and Nisan (1990,Combinatorica10, No. 4, pp. 349–365), drawn from approximation theory, to show that the quality of approximation changes in a significant way around [equation]: if [equation], then any approximation may err by a factor of [equation]; while if [equation] , the size of the union may be approximated to within a multiplicative factor of [equation]. Our result, the firstq-analog of a computational property of the lattice of subsets of a finite set, answers in the affirmative a question posed by Linial and Nisan
On the Need of Convexity in Patchworking

AbstractViro's construction of real smooth hypersurfaces uses regular (also called convex or coherent) subdivisions of Newton polytopes. Nevertheless, Viro's construction, sometimes calledpatchworking, can be applied as well to arbitrary subdivisions as a purely combinatorial procedure. Are the schemes coming from nonregular subdivisions, still topological types of some real smooth hypersurfaces? In the first part of this paper we prove a combinatorial version of Hilbert's Lemma (a consequence of Bezout's Theorem) that bounds the depth of nests in aT-curve, and we use this result and a previous work by I. Itenberg to answer the question affirmatively forT-curves of degree less than 6.According to V. A. Rokhlin, a real algebraic scheme has complex orientation of type I (alternatively of type II) if any curve with this real scheme divides (does not divide) its complexification. A real algebraic scheme hasindefinite typeif there are type I and type II curves with that particular scheme. In the second part of this paper, we describe a combinatorial algorithm, due to I. Itenberg and O. Viro, that allows one to determine the type of aT-curve. We then present a partial list of indefinite schemes forT-curves of degrees 7 and 8
Tau-Function Constructions of the Recurrence Coefficients of Orthogonal Polynomials

AbstractIn this paper we compute the recurrence coefficients of orthogonal polynomials using τ-function techniques. It is shown that for polynomials orthogonal with respect to positive weight functions on a noncompact interval, the recurrence coefficient can be expressed as the change in the chemical potential which, for sufficiently largeNis the second derivative of the free energy with respect toN, the particle number. We give three examples using this technique: Freud weights, Erdős weights, and weak exponential weights
Sharp and Unsharp Quantum Effects

AbstractA survey of the algebraic and the statistical properties of sharp and unsharp quantum effects is presented. We begin with a discussion and a comparison of four types of probability theories, the sharp and unsharp classical and quantum theories. A structure called an effect algebra that generalizes and unifies all four of these probability theories, is then considered. Finally, we present some recent investigations on tensor products and quotients of effect algebras. Examples and representative results for the various theories are discussed
Algebraic Properties of Beta and Gamma Distributions, and Applications

AbstractThree new properties are derived. The first one relates to the distribution ofUG+G′, where the three variables are independent,G,G′ have gamma distributions andUis arbitrary; the second one concerns the case whereUhas a beta distribution, and the third one the case where 1/Uhas a beta distribution. These properties generalize the additivity of independent gamma distributions with the same scale parameter. Applications to Markov chains and to the stochastic equationX=(in law)U(X+C) are given. The last section shows how non-negative relationships may be transformed into new ones on the whole line. In particular, the limit distribution of a time series with random coefficients and Gaussian error terms is found
Eulerian Stratification of Polyhedra

AbstractIn this paper we introduce Eulerian stratifications, weight functions, and boundary weight functions to study linear conditions onf-vectors of stratified triangulations on arbitrary polyhedra. The Eulerian stratified spaces are characterized by the Euler characteristics of the links between strata. With the new concepts, the classical Dehn–Sommerville equations are generalized to weightedf-vectors of arbitrary polyhedra; and the linear conditions on weightedf-vectors of all stratified triangulations are classified. Moreover, the necessary and sufficient conditions are obtained for given numbers to be the Euler characteristics of the links between strata, and a procedure of constructing Eulerian stratifications from the given numbers on posets is provided.Our study of Eulerian stratifications and weight functions suggests that the underlying combinatorial information should become a rather general setup that includes many classical linear combinatorial theories. It also points out a possible approach toward the study onf-vectors of triangulations of more general spaces
q-Rook Polynomials and Matrices over Finite Fields

AbstractConnections betweenq-rook polynomials and matrices over finite fields are exploited to derive a new statistic for Garsia and Remmel'sq-hit polynomial. Both this new statisticmatand another statistic for theq-hit polynomial ξ recently introduced by Dworkin are shown to induce different multiset Mahonian permutation statistics for any Ferrers board. In addition, for the triangular boards they are shown to generate different families of Euler–Mahonian statistics. For these boards the ξ family includes Denert's statisticden, and gives a new proof of Foata and Zeilberger's Theorem that (exc,den) is equidistributed with (des,maj). Thematfamily appears to be new. A proof is also given that theq-hit polynomials are symmetric and unimodal
Topology and Invertible Maps

AbstractI study connected manifolds and prove that a proper mapf:M→Mis globally invertible when it has a nonvanishing Jacobian and the fundamental group π1(M) is finite. This includes finite and infinite dimensional manifolds. Reciprocally, if π1(M) is infinite, there exist locally invertible maps that are not globally invertible. The results provide simple conditions for unique solutions to systems of simultaneous equations and for unique market equilibrium. Under standard desirability conditions, it is shown that a competitive market has a unique equilibrium if its reduced excess demand has a nonvanishing Jacobian. The applications are sharpest in markets with limited arbitrage and strictly convex preferences: a nonvanishing Jacobian ensures the existence of a unique equilibrium in finite or infinite dimensions, even when the excess demand is not defined for some prices, and with or without short sales
Braids,q-Binomials, and Quantum Groups

AbstractThe classical identities between theq-binomial coefficients and factorials can be generalized to a context in which numbers are replaced by braids. More precisely, for every pairi,nof natural numbers, there is defined an elementb(n)iof the braid group algebrakBn, and these satisfy analogs of the classical identities for the binomial coefficients. By choosing representations of the braid groups, one obtains numerical or matrix realizations of these identities; in particular, one recovers theq-identities in this way. These binomial braidsb(n)iplay a crucial role in a simple definition of a family of quantum groups, including the quantum groupsU+q(C) of Drinfeld and Jimbo
The Diametric Theorem in Hamming Spaces—Optimal Anticodes

AbstractFor a Hamming space (Xnα,dH), the set ofn-length words over the alphabet Xα={0,1,…,α−1} endowed with the distancedH, which for two wordsxn=(x1,…,xn),yn=(y1,…,yn)∈Xnαcounts the number of different components, we determine the maximal cardinality of subsets with a prescribed diameterdor, in another language, anticodes with distanced. We refer to the result as the diametric theorem.In a sense anticodes are dual to codes, which have a prescribedlowerbound on the pairwise distance. It is a hopeless task to determine their maximal sizes exactly.We find it remarkable that the diametric theorem (for arbitrary α) can be derived from our recent complete intersection theorem, which can be viewed as a diametric theorem (for α=2) in the restricted case, where alln-length words considered have exactlykones
Explicit Computation of Generalized Hamming Weights for Some Algebraic Geometric Codes

AbstractThe generalized Hamming weights, introduced a few years ago by V. K. Wei, provide substantial information of codes and thus play a central role in coding theory. For algebraic geometric codes, there have been many works on their generalized Hamming weights (or weight hierarchy). However, for lots of codes from Hermitian curves and the Klein quartic, some generalized Hamming weights still have not yet been found explicitly. In this paper, we first prove a general result (Theorem 1.4) on the computation of generalized Hamming weights of geometric Goppa codes on plane curves, using the configuration ofFq-rational points on the curves. Then we give the exact values (Theorem 2.2) of the first and second generalized Hamming weights of some codes arising from the Klein quartic. Our main result (Theorem 2.3) gives the exact values of the second and third generalized Hamming weights of certain codes from Hermitian curves. In the Appendix, a previous known result of Yang, Kumar, and Stichtenoth for Hermitian codes is shown to follow from Theorem 1.4. We also give the exact values of the first three generalized Hamming weights for Fermat codes
Symmetry Properties of Average Densities and Tangent Measure Distributions of Measures on the Line

AbstractAnswering a question by Bedford and Fisher, we show that for the circular and one-sided average densities of a Radon measure μ on the line with positive lower and finite upper α-densities, the following relations hold μ-almost everywhere: [equation] and [equation] We infer the result from a more general formula, which is proved by means of a detailed study of the structure of the measure and which involves the notion of tangent measure distributions introduced by Bandt and Graf. We show that for μ-almost every pointx, the formula [equation] holds for every tangent measure distributionPof μ atxand all Borel functionsG: M(R)×R→[0,∞). HereTuν is the measure defined byTuν(E)=ν(u+E), and M(R) is the space of Radon measures with the vague topology. By this formula, the tangent measure distributions are Palm distributions and thus define α-self-similar random measures in the sense of Zähle
Acyclic Orientations of Random Graphs

AbstractAn acyclic orientation of an undirected graph is an orientation of its edges such that the resulting directed graph contains no cycles. The random graphGn,pis a probability space consisting of subgraphs ofKnthat are obtained by selecting eachKn-edge with independent probabilityp. The random graphQn2,pis defined analogously and consists of subgraphs of then-cube,Qn2. In this paper we first derive a bijection between certain equivalence classes of permutations and acyclic orientations. Second, we present a lower and an upper bound on the r.v.a(Gn,p) that counts the number of acyclic orientations ofGn,p. Finally we study the distribution ofa(Gn,p) anda(Qn2,p) and show that log2a(Gn,p) and log2a(Qn2,p) are sharply concentrated at their respective expectation values
Balancing Game with a Buffer

AbstractWe consider a two person perfect information game with a buffer. On each round, Player I selects a vectorv∈Rnwith |v|≤1, where |·| is thel2-norm, and Player II can either put the vector in the buffer or choose a sign ϵi=±1 for a given vectorvi. There are no more thandvectors that can be put in the buffer. Player II's object is to keep the cumulative sum |∑ϵivi| as small as possible. We prove that the value of the game goes to infinity ifd≤n−2. We give an upper bound of the value ifd≥n−1. The same results hold for a generalized problem
The Cycling of Partitions and Compositions under Repeated Shifts

AbstractIn “Bulgarian Solitaire,” a player divides a deck ofncards into piles. Each move consists of taking a card from each pile to form a single new pile. One is concerned only with how many piles there are of each size. Starting from any division into piles, one always reaches some cycle of partitions ofn. Brandt proved that forn=1+2+···+k, the cycle is just the single partition into piles of distinct sizes 1,2,…,k. LetDB(n) denote the maximum number of moves required to reach a cycle. Igusa and Etienne proved thatDB(n)≤k2−kwhenevern≤1+2+···+k, and equality holds whenn=1+2+···+k. We present a simple new derivation of these facts. We improve the bound toDB(n)≤k2−2k−1, whenevern<1+2+···+kwithk≥4. We present a lower bound forDB(n) that is likely to be the actual value. We introduce a new version of the game, Carolina Solitaire, in which the piles are kept in order, so we work with compositions rather than partitions. Many analogous results can be obtained
Generalized Binomial Coefficients and the Subset–Subspace Problem

AbstractGeneralized binomial coefficients of the first and second kind are defined in terms of object selection with and without repetition from weighted boxes. The combinatorial definition unifies the binomial coefficients, the Gaussian coefficients, and the Stirling numbers and their recurrence relations under a common interpretation. Combinatorial proofs for some Gaussian coefficient identities are derived and shown to reduce to the ordinary binomial coefficients whenq=1. This approach provides a different perspective on the subset–subspace analogy problem. Generating function relations for the generalized binomial coefficients are derived by formal methods
Weak Maps and Stabilizers of Classes of Matroids

AbstractLetFbe a field and letNbe a matroid in a class N ofF-representable matroids that is closed under minors and the taking of duals. ThenNis anF-stabilizer for N if every representation of a 3-connected member of N is determined up to elementary row operations and column scaling by a representation of any one of itsN-minors. The study of stabilizers was initiated by Whittle. This paper extends that study by examining certain types of stabilizers and considering the connection with weak maps.The notion of a universal stabilizer is introduced to identify the underlying matroid structure that guarantees thatNwill be anF′-stabilizer for N for every fieldF′ over which members of N are representable. It is shown that, just as withF-stabilizers, one can establish whether or notNis a universal stabilizer for N by an elementary finite check. IfNis a universal stabilizer for N, we determine additional conditions onNand N that ensure that ifNis not a strict rank-preserving weak-map image of any matroid in N, then no connected matroid in N with anN-minor is a strict rank-preserving weak-map image of any 3-connected matroid in N.Applications of the theory are given for quaternary matroids. For example, it is shown thatU2,5is a universal stabilizer for the class of quaternary matroids with noU3,6-minor. Moreover, ifM1andM2are distinct quaternary matroids withU2,5-minors but noU3,6-minors andM1is connected whileM2is 3-connected, thenM1is not a rank-preserving weak-map image ofM2
The Number of Rhombus Tilings of a “Punctured” Hexagon and the Minor Summation Formula

AbstractWe compute the number of all rhombus tilings of a hexagon with sidesa,b+1,c,a+1,b,c+1, of which the central triangle is removed, provideda,b,chave the same parity. The result isB(⌈a/2⌉,⌈b/2⌉,⌈c/2⌉)B(⌈(a+1)/2⌉,⌊b/2⌋, ⌈c/2⌉)B(⌈a/2⌉,⌈(b+1)/2⌉,⌊c/2⌋)B(⌊a/2⌋,⌈b/2⌉, ⌈(c+1)/2⌉), whereB(α,β,γ) is the number of plane partitions inside the α×β×γ box. The proof uses nonintersecting lattice paths and a new identity for Schur functions, which is proved by means of the minor summation formula of Ishikawa and Wakayama. [Proc. Japan Acad. Ser. A71(1995), 54–57]. A symmetric generalization of this identity is stated as a conjecture
Enumerative Questions on Rooted Weighted Necklaces

AbstractWe develop the language and the tools to create a new family of integer sequences. The concrete motivation for this article came from a combinatorial counting problem first posed by Cipra [Math. Mag.65(1) (1992), 56] and still open in full generality. LetCkbe the cyclic group {q0,q1,…,qk−1} and letZ[Ck] be the algebra of polynomials in the variablesq0,q1,q2,…,qk−1with coefficients fromZ. We introduce weighted rooted necklaces, which give a combinatorial interpretation ofZ[Ck]. Actions onZ[Ck] via a family of operators of the formR(f[q])=qt(f[q]+h[q]), wheret∈Zandh[q],f[q]∈Z[Ck], are introduced. In particular we study the nonlinear, invertible operatorR(f[q])=q−f[0](f[q]−f[0]+∑f[0]i=1σ(f[0])qi), wheref[q]=∑k−1i=0niqi∈Z[Ck] andf[0]=n0. The operatorRcreates equivalence classes onZ[Ck]. We determine a recursive function to count the equivalence class containingG[q]=∑ki=0qi∈Z[Ck] which solves a special case of the aforementioned counting problem posed by Cipra. We give a closed form for such recursive formula in some instances. We also develop a series of related questions
Combinatorics of Topological Posets Homotopy Complementation Formulas

AbstractWe show that the well-knownhomotopy complementation formulaof Björner and Walker [European J. Combin.4(1983), 11–19] admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian and configuration posets, Gn(R) and expn(X) which were introduced and studied by V. Vassiliev [St. Petersburg Math. J.3(4) (1991), 108–115]. Among other applications we present a reasonably complete description, in terms of more standard spaces, of homology types of configuration posets expn(Sm) which leads to a negative answer to a question of Vassiliev raised at the workshop “Geometric Combinatorics” MSRI, February 1997.11The main ideas of the article were born during the workshop “Geometric Combinatorics” MSRI, February 1997. We thank the organizers for the support and the organization of this inspiring conference. This work was partially supported by Grant 04M03 of the Ministry for Science and Technology of Serbia
Scaling Universalities ofkth-Nearest Neighbor Distances on Closed Manifolds

AbstractTakeNsites distributed randomly and uniformly on a smooth closed surface. We express the expected distance 〈Dk(N)〉 from an arbitrary point on the surface to itskth-nearest neighboring site, in terms of the functionA(l) giving the area of a disc of radiuslabout that point. We then find two universalities. First, for a flat surface, whereA(l)=πl2, 〈Dk(N)〉 is separable inkandN. Allkth-nearest neighbor distances thus scale the same way inN. Second, for a curved surface, 〈Dk(N)〉 averaged over the surface is a topological invariant at leading and subleading order in a largeNexpansion. The 1/Nscaling series then depends, up throughO(1/N), only on the surface's topology and not on its precise shape. We discuss the case of higher dimensions (d>2), and also interpret our results using Regge calculus
Markov Processes Related with Dunkl Operators

AbstractDunkl operators are parameterized differential-difference operators on RNthat are related to finite reflection groups. They can be regarded as a generalization of partial derivatives and play a major role in the study of Calogero–Moser–Sutherland-type quantum many-body systems. Dunkl operators lead to generalizations of various analytic structures, like the Laplace operator, the Fourier transform, Hermite polynomials, and the heat semigroup. In this paper we investigate some probabilistic aspects of this theory in a systematic way. For this, we introduce a concept of homogeneity of Markov processes on RNthat generalizes the classical notion of processes with independent, stationary increments to the Dunkl setting. This includes analogues of Brownian motion and Cauchy processes. The generalizations of Brownian motion have the càdlàg property and form, after symmetrization with respect to the underlying reflection groups, diffusions on the Weyl chambers. A major part of the paper is devoted to the concept of modified moments of probability measures on RNin the Dunkl setting. This leads to several results for homogeneous Markov processes (in our extended setting), including martingale characterizations and limit theorems. Furthermore, relations to generalized Hermite polynomials, Appell systems, and Ornstein–Uhlenbeck processes are discussed