pinecone/core-2020-05-10-deduplication huggingface dataset:
Optimal Design Problems for Two-Phase Conducting Composites with Weakly Discontinuous Objective Functionals

AbstractWe consider the relatively simple but non-trivial example of an optimal design problem with a weakly discontinuous objective functional. The objective functional is quadratic and was suggested by Tartar in his “Remarks on Optimal Design” paper. We analyze a problem of finding a layout of a conducting composite such that the fields in both phases provide least squares fit to a given field. The main result of the paper is an explicit formula for the relaxed optimal design problem, suitable for numerical solution. Further analysis of our explicit formula shows that the optimal layout is a rank one laminate locally, lending some support for Tartar's conjecture that the minimizing sequences always converge strongly
A Structured Family of Clustering and Tree Construction Methods

AbstractA cluster A is an Apresjan cluster if every pair of objects within A is more similar than either is to any object outside A. The criterion is intuitive, compelling, but often too restrictive for applications in classification. We therefore explore extensions of Apresjan clustering to a family of related hierarchical clustering methods. The extensions are shown to be closely connected with the well-known single and average linkage tree constructions. A dual family of methods for classification by splits is also presented. Splits are partitions of the set of objects into two disjoint blocks and are widely used in domains such as phylogenetics. Both the cluster and split methods give rise to progressively refined tree representations. We exploit dualities and connections between the various methods, giving polynomial time construction algorithms for most of the constructions and NP-hardness results for the rest
Geometry of the Space of Phylogenetic Trees

AbstractWe consider a continuous space which models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature, giving a way of measuring distance between phylogenetic trees and providing some procedures for averaging or combining several trees whose leaves are identical. This geometry also shows which trees appear within a fixed distance of a given tree and enables construction of convex hulls of a set of trees. This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made
A Time-Fractional Step Method for Conservation Law Related Obstacle Problems

AbstractWe are interested in approximating the solution of a first-order quasi-linear equation associated with a forced unilateral obstacle condition. With this view, we make use of the time-splitting method developed classically to compute discontinuous solutions of nonhomogeneous scalar conservation laws. Here, one proves that this fractional step method converges in L1 to the weak entropy solution of the considered obstacle problem. In the case of the Cauchy problem, an L1-error bound in O(Δt) is established
On Acyclic Orientations and Sequential Dynamical Systems

AbstractWe study a class of discrete dynamical systems that consists of the following data: (a) a finite (labeled) graph Y with vertex set {1,…,n}, where each vertex has a binary state, (b) a vertex labeled multi-set of functions (Fi,Y: F2n→F2n)i, and (c) a permutation π∈Sn. The function Fi,Y updates the binary state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves the states of all other vertices fixed. The permutation π represents a Y-vertex ordering according to which the functions Fi,Y are applied. By composing the functions Fi,Y in the order given by π we obtain the sequential dynamical system (SDS):[FY,π]=Fπ(n),Y∘⋯∘Fπ(1),Y: F2n→F2n.In this paper we first establish a sharp, combinatorial upper bound on the number of non-equivalent SDSs for fixed graph Y and multi-set of functions (Fi,Y). Second, we analyze the structure of a certain class of fixed-point-free SDSs
Rado Numbers for the Equation ∑i=1m−1xi+c=xm, for Negative Values of c

AbstractFor every integer m≥3 and every integer c, let r(m,c) be the least integer, if it exists, such that for every 2-coloring of the set {1,2,…,r(m,c)} there exists a monochromatic solution to the equation∑i=1m−1xi+c=xm.The values of r(m,c) were previously known for all values of m and all nonnegative values of c. In this paper, exact values of r(m,c) are found for all values of m and all values of c such that −m+2<c<0 or c<−(m−1)(m−2). Upper and lower bounds are given for the remaining values of c
A Computer Proof of a Polynomial Identity Implying a Partition Theorem of Göllnitz

AbstractIn this paper we give a computer proof of a new polynomial identity, which extends a recent result of Alladi and the first author. In addition, we provide computer proofs for new finite analogs of Jacobi and Euler formulas. All computer proofs are done with the aid of the new computer algebra package qMultiSum developed by the second author. qMultiSum implements an algorithmic refinement of Wilf and Zeilberger's multi-q-extension of Sister Celine's technique utilizing additional ideas of Verbaeten and Wegschaider
Shuffle Invariance of the Super-RSK Algorithm

AbstractAs in the (k,l)-RSK (Robinson–Schensted–Knuth) of A. Berele and A. Regev (1987, Adv. Math.64, 118–175), other super-RSK algorithms can be applied to sequences of variables from the set {t1,…,tk,u1,…,ul}, where t1<⋯<tk and u1<⋯<ul. While the (k,l)-RSK is the case where ti<uj for all i and j, these other super-RSK's correspond to all the [formula] shuffles of the t's and u's satisfying the above restrictions that t1<⋯<tk and u1<⋯<ul. We show that the shape of the tableaux produced by any such super-RSK is independent of the particular shuffle of the t's and u's
The Laurent Phenomenon

AbstractA composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes—quite unexpectedly—it does. We suggest a unified treatment of this phenomenon, which covers a large class of applications. In particular, we settle in the affirmative a conjecture of D. Gale and R. Robinson on integrality of generalized Somos sequences, and prove the Laurent property for several multidimensional recurrences, confirming conjectures by J. Propp, N. Elkies, and M. Kleber
Tree Reconstruction from Multi-State Characters

AbstractIn evolutionary biology, a character is a function χ from a set X of present-day species into a finite set of states. Suppose the species in X have evolved according to a bifurcating tree T. Biologists would like to use characters to infer this tree. Assume that χ is the result of an evolutionary process on T that has not involved reverse or parallel transitions; such characters are called homoplasy-free. In this case, χ provides direct combinatorial information about the underlying evolutionary tree T for X. We consider the question of how many homoplasy-free characters are required so that T can be correctly reconstructed. We first establish lower bounds showing that, when the number of states is bounded, the number of homoplasy-free characters required to reconstruct T grows (at least) linearly with the size of X. In contrast, our main result shows that, when the state space is sufficiently large, every bifurcating tree can be uniquely determined by just five homoplasy-free characters. We briefly describe the relevance of this result for some new types of genomic data, and for the amalgamation of evolutionary trees
Annihilators of Orlik–Solomon Relations

AbstractThe Orlik–Solomon algebra A of a matroid is isomorphic to the quotient of an exterior algebra E by a defining ideal I. We find an explicit presentation of the annihilator ideal of I or, equivalently, the E-module dual to A. As an application of that we provide a necessary, combinatorial condition for the algebra A to be quadratic. We show that this is stronger than the matroid being line-closed, thereby resolving (negatively) a conjecture by M. Falk. We also show that our condition is not sufficient for quadraticity
Line-Closed Matroids, Quadratic Algebras, and Formal Arrangments

AbstractLet G be a matroid on ground set A. The Orlik–Solomon algebra A(G) is the quotient of the exterior algebra E on A by the ideal I generated by circuit boundaries. The quadratic closure [formula](G) of A(G) is the quotient of E by the ideal generated by the degree-two component of I. We introduce the notion of the nbb set in G, determined by a linear order on A, and show that the corresponding monomials are linearly independent in the quadratic closure [formula](G). As a consequence, A(G) is a quadratic algebra only if G is line-closed. An example of S. Yuzvinsky proves the converse false. [G. Denham and S. Yuzvinsky, Adv. in Appl. Math.28, 2002, doi:10.1006/aama.2001.0779]. These results generalize to the degree r closure of A(G).The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for A to be free and for the complement M of A to be a K(π,1) space. Formality of A is also necessary for A(G) to be a quadratic algebra. We clarify the relationship between formality, line-closure, and other matroidal conditions related to formality. We give examples to show that line-closure of G is not necessary or sufficient for M to be a K(π,1) or for A to be free
Spaces of Lattice Diagram Polynomials in One Set of Variables

AbstractFor a partition μ of n, let Mμ be the space span of all partial derivatives of the alternate associated to μ in two n-sets of variables X and Y. The n!-theorem of [M. Haiman, J. Amer. Math. Soc.14 (2001), 941–1006] states that the dimension of Mμ is n!. In [F. Bergeron, N. Bergeron, A. Garsia, M. Haiman, and G. Tesler, Adv. Math.142 (1999), 244–334], we introduced slightly more general spaces Mμ/ij as a tool for an elementary proof of the n!-theorem. For this one needs a formula for the bigraded characters in terms of Macdonald polynomials that would follow from a representation theoretic proof of our four-term recurrence. Here we give an explicit basis for the Y-free component of the space Mμ/ij, and using this description we prove the Y-free analog of the four-term recurrence. Our basis of Mμ/ij has the further nice feature that it is a natural generalization of the Artin basis for the space of harmonic (covariant) polynomials for the symmetric group
Special Functions on the Sphere with Applications to Minimal Surfaces

AbstractA function which is homogeneous in x,y,z of degree n and satisfies Vxx+Vyy+Vzz=0 is called a spherical harmonic. In polar coordinates, the spherical harmonics take the form rnfn, where fn is a spherical surface harmonic of degree n. On a sphere, fn satisfies ▵fn+n(n+1)fn=0, where ▵ is the spherical Laplacian. Bounded spherical surface harmonics are well studied, but in certain instances, unbounded spherical surface harmonics may be of interest. For example, if X is a parameterization of a minimal surface and n is the corresponding unit normal, it is known that the support function, w=X·n, satisfies ▵w+2w=0 on a branched covering of a sphere with some points removed. While simple in form, the boundary value problem for the support function has a very rich solution set. We illustrate this by using spherical harmonics of degree one to construct a number of classical genus-zero minimal surfaces such as the catenoid, the helicoid, Enneper's surface, and Hennenberg's surface, and Riemann's family of singly periodic genus-one minimal surfaces
Pattern Frequency Sequences and Internal Zeros

AbstractLet q be a pattern and let Sn,q(c) be the number of n-permutations having exactly c copies of q. We investigate when the sequence (Sn,q(c))c≥0 has internal zeros. If q is a monotone pattern it turns out that, except for q=12 or 21, the nontrivial sequences (those where n is at least the length of q) always have internal zeros. For the pattern q=1(l+1)l…2 there are infinitely many sequences which contain internal zeros and when l=2 there are also infinitely many which do not. In the latter case, the only possible places for internal zeros are the next-to-last or the second-to-last positions. Note that by symmetry this completely determines the existence of internal zeros for all patterns of length at most 3
The Asymptotics of Almost Alternating Permutations

AbstractThe goal of this paper is to study the asymptotic behavior of almost alternating permutations, that is, permutations that are alternating except for a finite number of exceptions. Let β(l1,…,lk) denote the number of permutations which consist of l1 ascents, l2 descents, l3 ascents, and so on. By combining the Viennot triangle and the boustrophedon transform, we obtain the exponential generating function for the numbers β(L,1n−m−1), where L is a descent–ascent list of size m. As a corollary we have β(L,1n−m−1)∼c(L)·En, where En=β(1n−1) denotes the nth Euler number and c(L) is a constant depending on the list L. Using these results and inequalities due to Ehrenborg–Mahajan, we obtain β(1a,2,1b)∼2/π·En, when min(a,b) tends to infinity and where n=a+b+3. From this result we obtain that the asymptotic behavior of β(L1,1a,L2,1b,L3) is the product of three constants depending respectively on the lists L1, L2, and L3, times the Euler number Ea+b+m+1, where m is the sum of the sizes of the Li's
Almost-Graphic Matroids

AbstractA nongraphic matroid M is said to be almost-graphic if, for all elements e, either M\e or M/e is graphic. We determine completely the class of almost-graphic matroids, thereby answering a question posed by Oxley in his book “Matroid Theory.” A nonregular matroid is said to be almost-regular if, for all elements e, either M\e or M/e is regular. An element e for which both M\e and M/e are regular is called a regular element. We also determine the almost-regular matroids with at least one regular element
Curious Characterizations of Projective and Affine Geometries

AbstractWe prove the following characterization theorem: If any three of the following four matroid invariants—the number of points, the number of lines, the coefficient of λn−2 in the characteristic polynomial, and the number of three-element dependent sets—of a rank-n combinatorial geometry (or simple matroid) are the same as those of a rank-n projective geometry, then it is a projective geometry (of the same order). To do this, we use a lemma which is of independent interest: If H is a geometry in which all the lines have exactly ℓ−1 or ℓ points and G is a geometry with at least three of the four matroid invariants the same as H, then all the lines in G also have exactly ℓ−1 or ℓ points. An analogue of the characterization theorem holds for affine geometries. Our methods also yield inequalities amongst the four matroid invariants
New Lower Bounds for Ramsey Numbers of Graphs and Hypergraphs

AbstractLet G be an r-uniform hypergraph. The multicolor Ramsey number rk(G) is the minimum n such that every k-coloring of the edges of the complete r-uniform hypergraph K(r)n yields a monochromatic copy of G. Improving slightly upon results from M. Axenovich, Z. Füredi, and D. Mubayi (J. Combin. Theory Ser. B79 (2000), 66–86), we prove thattk2+1≤rk(K2,t+1)≤tk2+k+2,where the lower bound holds when t and k are both powers of a prime p. When t=1, we improve the lower bound by 1, proving that rk(C4)≥k2+2 for any prime power k. This extends the result of F. Lazebnik and A. J. Woldar (J. Combin. Theory Ser. B79 (2000), 172–176), which proves the same bound when k is an odd prime power. These results are generalized to hypergraphs in the following sense. Fix integers r,s,t≥2. Let H(r)(s,t) be the complete r-partite r-graph with r−2 parts of size 1, one part of size s, and one part of size t (note that H(2)(s,t)=Ks,t). We prove thattk2−k+1≤rk(H(r)(2,t+1))≤tk2+k+r,where the lower bound holds when t and k are both powers of a prime p and(1−o(1))ks≤rk(H(r)(s,t))≤O(ks),for fixedt,s≥2,t>(s−1)!,rk(H(r)(3,3))=(1+o(1))k3.Some of our lower bounds are special cases of a family of more general hypergraph constructions obtained by algebraic methods. We describe these, thereby extending results of F. Lazebnik and A. J. Woldar (J. Graph Theory, 2001) about graphs
A Bijective Approach to the Area of Generalized Motzkin Paths

AbstractFor fixed positive integer k, let En denote the set of lattice paths using the steps (1,1), (1,−1), and (k,0) and running from (0,0) to (n,0) while remaining strictly above the x-axis elsewhere. We first prove bijectively that the total area of the regions bounded by the paths of En and the x-axis satisfies a four-term recurrence depending only on k. We then give both a bijective and a generating function argument proving that the total area under the paths of En equals the total number of lattice points on the x-axis hit by the unrestricted paths running from (0,0) to (n−2,0) and using the same step set as above
Determinants of Period Matrices and an Application to Selberg's Multidimensional Beta Integral

AbstractIn work on critical values of linear functions and hyperplane arrangements, A. Varchenko (Izv. Akad. Nauk SSSR Ser. Mat.53 (1989), 1206–1235; 54 (1990), 146–158) defined certain period matrices whose entries are Euler-type integrals representing hypergeometric functions of several variables and derived remarkable closed-form expressions for the determinants of those matrices. In this article, we present elementary proofs of some of Varchenko's determinant formulas. By the same method, we obtain proofs of variations of Varchenko's determinants. As an application, we deduce new proofs of the multidimensional beta integrals of Selberg and of Aomoto. Further, we obtain a new proof of a determinant formula of A. Varchenko (Funct. Anal. Appl.25 (1999), 304–305) in which the entries are multidimensional Selberg-type integrals
Generalized Bernstein Polynomials and Symmetric Functions

AbstractWe begin by classifying all solutions of two natural recurrences that Bernstein polynomials satisfy. The first scheme gives a natural characterization of Stancu polynomials. In Section 2, we identify the Bernstein polynomials as coefficients in the generating function for the elementary symmetric functions, which gives a new proof of total positivity for Bernstein polynomials, by identifying the required determinants as Schur functions. In the final section, we introduce a new class of approximation polynomials based on the symplectic Schur functions. These polynomials are shown to agree with the polynomials introduced by Vaughan Jones in his work on subfactors and knots. We show that they have the same fundamental properties as the usual Bernstein polynomials: variation diminishing (whose proof uses symplectic characters), uniform convergence, and conditions for monotone convergence
Geometry of Affine Time–Frequency Distributions

AbstractBertrands' bilinear affine time-frequency distributions are considered from the point of view of their geometry in the time-frequency plane. General construction rules are established for interference terms, with further interpretations in terms of localization properties, generalized means and symmetries. In the case of frequency modulated signals, it is shown how the pointwise application of these rules can be refined by the study of a critical manifold and stationary phase-type approximations. Theoretical results are supported by both analytical and numerical examples
Wavelets Associated with Periodic Basis Functions

AbstractIn this paper, we investigate a class of nonstationary, orthogonal periodic scaling functions and wavelets generated by continuously differentiable periodic functions with positive Fourier coefficients; such functions are termedperiodic basis functions.For this class of wavelets, the decomposition and reconstruction coefficients can be computed in terms of the discrete Fourier transform, so that FFT methods apply for their evaluation. In addition, decomposition at thenth level only involves 2 terms from the higher level. Similar remarks apply for reconstruction. We apply a periodic “uncertainty principle” to obtain an angle/frequency uncertainty “window” for these wavelets, and we show that for many wavelets in this class the angle/frequency localization is good
Some Remarks about Compactly Supported Spline Wavelets

AbstractIn this paper we propose an extended family of almost orthogonal spline wavelets with compact support. These functions provide snug basesL2(R), preserving semiorthogonal properties. As it is well known, orthogonality is a desirable quality while finite support has attractive features for numerical applications. This work represents an effort to combine these conditions in the spline case and to enhance previous results of Chui and Unseret al.We start by reviewing the concept of semiorthogonal wavelets and we discuss their performance. Next, we give a brief description of the general technique for computing compactly supported spline wavelets. Finally we expose these functions. We also develop several formulas in accord with our purposes
Gibbs Phenomenon for Wavelets

AbstractWhen a Fourier series is used to approximate a function with a jump discontinuity, an overshoot at the discontinuity occurs. This phenomenon was noticed by Michelson [6] and explained by Gibbs [3] in 1899. This phenomenon is known as the Gibbs effect. In this paper, possible Gibbs effects will be looked at for wavelet expansions of functions at points with jump discontinuities. Certain conditions on the size of the wavelet kernel will be examined to determine if a Gibbs effect occurs and what magnitude it is. An if and only if condition for the existence of a Gibbs effect is presented, and this condition is used to prove existence of Gibbs effects for some compactly supported wavelets. Since wavelets are not translation invariant, effects of a discontinuity will depend on its location. Also, computer estimates on the sizes of the overshoots and undershoots were computed for some compactly supported wavelets with small support
A Fast Algorithm for Adapted Time–Frequency Tilings

AbstractWe first consider orthonormal bases of open face RNconsisting of discretized rescaled Walsh functions, whereNis a power of two. Given a vector, the best basis with respect to an additive cost function is found with an algorithm of orderO(NlogN). The algorithm operates in the time–frequency plane by constructing a tiling of minimal cost among all possible tilings with dyadic rectangles of area one. Then we discuss generalizations replacing the Walsh group, which controls the structure of the time–frequency plane, by other finite abelian groups. The main example here involves the Fast Fourier Transform
Wavelet Coefficients Measuring Smoothness inHp(open face Rd)

AbstractWe investigate the connection between Besov spaces and certain approximation subspaces of the Hardy spacesHp(open face Rd), 0 <p≤1. In particular, we establish the wavelet decompositions for theHp(open face Rd) spaces and we characterize the homogeneous Besov spacesBsp,q(open face Rd) in terms of the wavelet coefficients of such decompositions, under minimal decay and smoothness conditions on the wavelet set Ψ. In the process, we also obtain a new characterization of theBsp,q(open face Rd) spaces in terms of the modulus of smoothness measured in the norm ofHp(open face Rd)
Fast Algorithms for Calderón–Zygmund Singular Integral Operators

AbstractMany people would say that Calderón–Zygmund operators have almost diagonal matrices in orthonormal wavelet bases. We will show that this statement is not true as stated. In contrast, the "nonstandard matrix representation" of Calderón–Zygmund operators always yields almost diagonal matrices. The Beyklin–Coifman–Rokhlin fast algorithm amounts to replacing these almost diagonal matrices by banded ones. We compute the operator norm of the error term in this approximation and give sharp estimates