pinecone/core-2020-05-10-deduplication huggingface dataset:
On the Construction of Complete Sets of Geometric Invariants for Algebraic Curves

AbstractWe provide a solution to the important problem of constructing complete independent sets of Euclidean and affine invariants for algebraic curves. We first simplify algebraic curves through polynomial decompositions and then use some classical geometric results to construct functionally independent sets of invariants. The results presented here represent some new contributions to algebraic curve theory that can be used in many application areas, such model-based vision, object recognition, graphics, geometric modeling, and CAD
The Semi-Baire-like Three-Space Problems

AbstractLocally convex Baire spaces have poor permanence properties, as demonstrated by Valdivia's non-Baire product of two Baire spaces, which also proves that Baire is not a three-space property. The opposite is true of Baire-like (BL) and others in a long list of weaker space properties whose three-space status is known. We add semi-Baire-like (sBL) to the list. It has long been known that quasi-Baire (QB) is a three-space property, since a space is QB if and only if it is barrelled and non-Sσ, and the latter two are three-space properties. Since BL spaces are just those barrelled spaces that are sBL, the result would be similarly immediate for BL if sBL were a three-space property. Such a proof is impossible, for we show that sBL is not a three-space property, and much more: in the extended context, our results answer with optimal efficiency all remaining 631 questions of the form: “if F has property a and E/F has property b, then must the third space E have property c?
A Combinatorial Theory of Higher-Dimensional Permutation Arrays

AbstractWe define a class of hypercubic (shape [n]d) arrays that in a certain sense are d-dimensional analogs of permutation matrices with our motivation from algebraic geometry. Various characterizations of permutation arrays are proved, an efficient generation algorithm is given, and enumerative questions are discussed although not settled. There is a partial order on the permutation arrays, specializing to the Bruhat order on Sn when d equals 2, and specializing to the lattice of partitions of a d-set when n equals 2
Forbidden Words in Symbolic Dynamics

AbstractWe introduce an equivalence relation≃between functions from N to N. By describing a symbolic dynamical system in terms of forbidden words, we prove that the≃-equivalence class of the function that counts the minimal forbidden words of a system is a topological invariant of the system. We show that the new invariant is independent from previous ones, but it is not characteristic. In the case of sofic systems, we prove that the≃-equivalence of the corresponding functions is a decidable question. As a more special application, we show, by using the new invariant, that two systems associated to Sturmian words having “different slope” are not conjugate
An Infinite Family of Engel Expansions of Rogers–Ramanujan Type

AbstractThe extended Engel expansion is an algorithm that leads to unique series expansions of q-series. Various examples related to classical partition theorems, including the Rogers–Ramanujan identities, have been given recently. The object of this paper is to show that the new and elegant Rogers–Ramanujan generalization found by Garrett, Ismail, and Stanton also fits into this framework. This not only reveals the existence of an infinite, parameterized family of extended Engel expansions, but also provides an alternative proof of the Garrett, Ismail, and Stanton result. A finite version of it, which finds an elementary proof, is derived as a by-product of the Engel approach
Z8-Cyclic Codes and Quadratic Residue Codes

AbstractA set of n-tuples over Z8 is called a code over Z8 or a Z8 code if it is a Z8 module. A particularly interesting family of Z8-cyclic codes are quadratic residue codes. We define such codes in terms of their idempotent generators and show that these codes also have many good properties which are analogous in many respects to properties of quadratic residue codes over a field. In particular we show that the quadratic residuce codes over Z8 have large automorphism groups which will be useful in decoding these codes by using the powerful permutation decoding methods described by F. J. MacWilliams and N. J. A. Sloane (1978, “Theory of Error-Correcting Codes,” North-Holland, Amsterdam). We also define a distance preserving map from ZN8 (Lee distance) to Z4N2 (Hamming distance)
Perfect Two-Fault Tolerant Search with Minimum Adaptiveness

AbstractOur aim is to minimize the number of answer/question alternations in a perfect two-fault-tolerant search. Ulam and Rényi posed the problem of searching for an unknown m-bit number by asking the minimum number of yes–no questions, when up to ℓ of the answers may be erroneous/mendacious. Berlekamp considered the same problem in the context of error-correcting communication with feedback. Among others, he proved that at least qℓ(m) questions are necessary, where qℓ(m) is the smallest integer q satisfying 2q−m≥∑ℓj=0(qj). When all questions are asked in advance, and adaptiveness has no role, finding a perfect strategy (i.e., a strategy with qℓ(m) questions) amounts to finding an ℓ-error-correcting code with 2m codewords of length qℓ(m). From coding theory it is known that such perfect non-adaptive searching strategies are rather the exception, for ℓ≥2. At the other extreme, in a fully adaptive search, where the (t+1)th question is asked knowing the answer to the tth question, perfect strategies are known to exist for all sufficiently large m.What happens if we impose restrictions on the amount of adaptiveness avaliable to the questioner? Focusing attention on the case ℓ=2, we shall prove that, for each m≠2, perfect searching strategies still exist even if the questioner is allowed to adapt his strategy only once. All our results are constructive and explicitly yield perfect two-error-correcting codes with the least possible feedback. We finally generalize our results to k-ary search
The Cycle-Path Indicator Polynomial of a Digraph

AbstractA cycle-path cover of a digraph D is a spanning subgraph made of disjoint cycles and paths. In order to count such covers by types we introduce the cycle-path indicator polynomial of D. We show that this polynomial can be obtained by a deletion–contraction recurrence relation. Then we study some specializations of the cycle-path indicator polynomial, such as the geometric cover polynomial (the geometric version of the cover polynomial introduced by Chung and Graham), the cycle cover polynomial, and the path cover polynomial.As an application, we consider chromatic arrangements of non-attacking rooks and the associated polynomial which is symmetric and generalizes the usual rook polynomial
A Decomposition of Fl(n)d Indexed by Permutation Arrays

AbstractWe study a decomposition of Fl(n)d−1, where Fl(n) denotes the flag manifold over Cn. The strata are defined by the dimensions of intersections of one space from each flag, so for d equal to 2 this is the usual Bruhat cell decomposition. The strata are indexed by “permutation arrays,” which are d-dimensional analogs of permutation matrices. We present a partial order on these permutation arrays, specializing to the Bruhat order on Sn when d equals 2 and to the lattice of partitions of a d-set when n equals 2
On Quantum Algorithms for Noncommutative Hidden Subgroups

AbstractQuantum algorithms for factoring and finding discrete logarithms have previously been generalized to finding hidden subgroups of finite Abelian groups. This paper explores the possibility of extending this general viewpoint to finding hidden subgroups of noncommutative groups. We present a quantum algorithm for the special case of dihedral groups which determines the hidden subgroup in a linear number of calls to the input function. We also explore the difficulties of developing an algorithm to process the data to explicitly calculate a generating set for the subgroup. A general framework for the noncommutative hidden subgroup problem is discussed and we indicate future research directions
Foundations of a Connectivity Theory for Simplicial Complexes

AbstractThis paper lays the foundations of a combinatorial homotopy theory, called A-theory, for simplicial complexes, which reflects their connectivity properties. A collection of bigraded groups is constructed, and methods for computation are given. A Seifert–Van Kampen type theorem and a long exact sequence of relative A-groups are derived. A related theory for graphs is constructed as well. This theory provides a general framework encompassing homotopy methods used to prove connectivity results about buildings, graphs, and matroids
A Technology for Reverse-Engineering a Combinatorial Problem from a Rational Generating Function

AbstractIn this paper, we tackle the problem of giving, by means of a regular language, a combinatorial interpretation of a positive sequence (fn) defined by a linear recurrence with integer coefficients. We propose two algorithms able to determine if the rational generating function of (fn), f(x), is the generating function of some regular language, and, in the affirmative case, to find it. We illustrate some applications of this method to combinatorial object enumeration problems and bijective combinatorics and discuss an open problem regarding languages having a rational generating function
The Yuri Manin Ring and Its Bn-Analogue

AbstractThe Manin ring is a family of quadratic algebras describing pointed stable curves of genus zero whose homology gives the solution of the Commutativity Equations. This solution was first observed by the physicist Losev. We show the Manin ring is the Stanley–Reisner ring of the standard triangulation of the n-cube modulo a system of parameters. Thus, the Hilbert series of the Manin ring is given by the Eulerian polynomial. One can also view the Manin ring as the Stanley–Reisner ring of the dual of the permutahedron modulo a system of parameters. Furthermore, we develop a Bn-analogue of the Manin ring. In this case the signed Manin ring is the Stanley–Reisner ring of the barycentric subdivision of the n-cube (equivalently, the dual of the signed permutahedron) modulo a system of parameters and its Hilbert series is the descent polynomial of augmented signed permutations
On Circuit Valuation of Matroids

AbstractThe concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of (R∪{−∞})-valued vectors defined on the circuits of the underlying matroid, where R is a totally ordered additive group. The dual of a valuated matroid is characterized by an orthogonality of (R∪{−∞})-valued vectors on circuits. Minty's characterization for matroids by the painting property is generalized for valuated matroids
Off-Diagonal Generalized Schur Numbers

AbstractWe determine all values of the 2-colored off-diagonal generalized Schur numbers (also called Issai numbers), an extension of the generalized Schur numbers. These numbers, denoted S(k,l), are the minimal integers such that any red and blue coloring of the integers from 1 to S(k,l) must admit either a solution to ∑k−1i=1xi=xk consisting of only red integers, or a solution to ∑l−1i=1xi=xl consisting of only blue integers. We show that S(3,l)=3l−4 for odd l≥3,S(3,l)=3l−5 for even l≥4, and S(k,l)=kl−l−1 for 4≤k≤l
The Diameter of Sparse Random Graphs

AbstractWe consider the diameter of a random graph G(n,p) for various ranges of p close to the phase transition point for connectivity. For a disconnected graph G, we use the convention that the diameter of G is the maximum diameter of its connected components. We show that almost surely the diameter of random graph G(n,p) is close to lognlog(np) if np→∞. Moreover if nplogn=c>8, then the diameter of G(n,p) is concentrated on two values. In general, if nplogn=c>c0, the diameter is concentrated on at most 2⌊1/c0⌋+4 values. We also proved that the diameter of G(n,p) is almost surely equal to the diameter of its giant component if np>3.6
Global Existence for a Parabolic Chemotaxis Model with Prevention of Overcrowding

AbstractIn this paper we study a version of the Keller–Segel model where the chemotactic cross-diffusion depends on both the external signal and the local population density. A parabolic quasi-linear strongly coupled system follows. By incorporation of a population-sensing (or “quorum-sensing”) mechanism, we assume that the chemotactic response is switched off at high cell densities. The response to high population densities prevents overcrowding, and we prove local and global existence in time of classical solutions. Numerical simulations show interesting phenomena of pattern formation and formation of stable aggregates. We discuss the results with respect to previous analytical results on the Keller–Segel model
Mathematical Theory of Phenotypical Selection

AbstractA general concept of phenotypical structure over a genotypical structure is developed. The direct decompositions of multilocus phenotypical structures are considered. Some aspects of phenotypical heredity are described in terms of graph theory. The acyclic phenotypical structures are introduced and studied on this base. The evolutionary equations are adjusted to the phenotypical selection. It is proved that if a phenotypical structure is acyclic then the set of fixed points of the corresponding evolutionary operator is finite except for a proper algebraic subset of the operator space. Some applications of this theorem are given
MacMahon's Partition Analysis VIII. Plane Partition Diamonds

AbstractIn his famous book “Combinatory Analysis” MacMahon introduced Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. However, MacMahon failed in his attempt to use his method for a satisfactory treatment of plane partitions. It is the object of this article to show that nevertheless Partition Analysis is of significant value when treating non-standard types of plane partitions. To this end “plane partition diamonds” are introduced. Applying Partition Analysis a simple closed form for the full generating function is derived. In the discovering process the Omega package developed by the authors has played a fundamental role
Walks on the Slit Plane: Other Approaches

AbstractLet S be a finite subset of Z2. A walk on the slit plane with steps in S is a sequence (0,0)=w0,w1,…,wn of points of Z2 such that wi+1−wi belongs to S for all i, and none of the points wi, i≥1, lie on the half-line H={(k,0):k≤0}. In a recent paper [Walks on the Slit Plane, preprint 2000, arXiv: match.CO/0012230] G. Schaeffer and the author computed the length generating function S(t) of walks on the slit plane for several sets S. All the generating functions thus obtained turned out to be algebraic: for instance, on the ordinary square lattice, St=1+1+4t1/21+1−4t1/2/21−4t3/4. The combinatorial reasons for this algebraicity remain obscure. In this paper, we present two new approaches for solving slit plane models. One of them simplifies and extends the functional equation approach of the original paper. The other one is inspired by an argument of Lawler; it is more combinatorial, and explains the algebraicity of the product of three series related to the model. It can also be seen as an extension of the classical cycle lemma. Both methods work for any set of steps S. We exhibit a large family of sets S for which the generating function of walks on the slit plane is algebraic, and another family for which it is neither algebraic, nor even D-finite. These examples give a hint at where the border between algebraicity and transcendence lies, and call for a complete classification of the sets S
Random Partitions with Non-negative rth Differences

AbstractLet Pr(n) be the set of partitions of n with non-negative rth differences. Let λ be a partition of an integer n chosen uniformly at random among the set Pr(n). Let d(λ) be a positive rth difference chosen uniformly at random in λ. The aim of this work is to show that for every m≥1, the probability that d(λ)≥m approaches the constant m−1/r as n→∞. This work is a generalization of a result on integer partitions and was motivated by a recent identity from the Omega package of G. E. Andrews et al. (European J. Combin., MacMahon's partition analysis. III. The Omega package). To prove this result we use bijective, asymptotic/analytic, and probabilistic combinatorics
Bijections behind the Ramanujan Polynomials

AbstractThe Ramanujan polynomials were introduced by Ramanujan in his study of power series inversions. In an approach to the Cayley formula on the number of trees, Shor discovers a refined recurrence relation in terms of the number of improper edges, without realizing the connection to the Ramanujan polynomials. On the other hand, Dumont and Ramamonjisoa independently take the grammatical approach to a sequence associated with the Ramanujan polynomials and have reached the same conclusion as Shor's. It was a coincidence for Zeng to realize that the Shor polynomials turn out to be the Ramanujan polynomials through an explicit substitution of parameters. Shor also discovers a recursion of Ramanujan polynomials which is equivalent to the Berndt–Evans–Wilson recursion under the substitution of Zeng and asks for a combinatorial interpretation. The objective of this paper is to present a bijection for the Shor recursion, or the Berndt–Evans–Wilson recursion, answering the question of Shor. Such a bijection also leads to a combinatorial interpretation of the recurrence relation originally given by Ramanujan
Note on the Lamp Lighting Problem

AbstractWe answer some questions concerning the so-called σ-game of Sutner [Linear cellular automata and the Garden of Eden, Math. Intelligencer11 (1989), 49–53]. It is played on a graph where each vertex has a lamp, the light of which is toggled by pressing any vertex with an edge directed to the lamp. For example, we show that every configuration of lamps can be lit if and only if the number of complete matchings in the graph is odd. In the special case of an orthogonal grid one gets a criterion for whether the number of monomer-dimer tilings of an m×n grid is odd or even
Babson–Steingrı&#x0301;msson Statistics are Indeed Mahonian (and Sometimes Even Euler–Mahonian)

AbstractBabson and Steingrı&#x0301;msson have recently introduced seven new permutation statistics, that they conjectured were all Mahonian (i.e., equi-distributed with the number of inversions). We prove their conjecture for the first four and also prove that the first and the fourth are even Euler–Mahonian. We use two different, in fact, opposite, techniques. For three of them we give a computer-generated proof, using the Maple package ROTA, that implements the second author's “Umbral Transfer Matrix Method.” For the fourth one a geometric permutation transformation is used that leads to a further refinement of this Euler–Mahonian distribution study
Rook Theory for Perfect Matchings

AbstractIn classical rook theory there is a fundamental relationship between the rook numbers and the hit numbers relative to any board. In that theory the k-th hit number of a board B can be interpreted as the number of permutations whose intersection with B is of size k. In the case of Ferrers boards there are q-analogues of the hit numbers and the rook numbers developed by A. M. Garsia and J. B. Remmel (1986, J. Combin. Theory, Ser. A41, 246–275) M. Dworkin (1996, “Generalizations of Rook Polynomials,” Ph. D. Thesis, Brandeis University”; 1998, J. Combin. Theory, Ser. A81, 149–175) and J. Haglund (1998, Adv. Appl. Math.20, 450–487). In this paper we develop a rook theory appropriate for shifted partitions, where hit numbers can be interpreted as the number of perfect matchings in the complete graph whose intersection with the board is of size k. We show there is also analogous q-theory for the rook and hit numbers for these shifted Ferrers boards
Permutations with Restricted Patterns and Dyck Paths

AbstractWe exhibit a bijection between 132-avoiding permutations and Dyck paths. Using this bijection, it is shown that all the recently discovered results on generating functions for 132-avoiding permutations with a given number of occurrences of the pattern 12…k follow directly from old results on the enumeration of Motzkin paths, among which is a continued fraction result due to Flajolet. As a bonus, we use these observations to derive further results and a precise asymptotic estimate for the number of 132-avoiding permutations of {1,2,…,n} with exactly r occurrences of the pattern 12…k. Second, we exhibit a bijection between 123-avoiding permutations and Dyck paths. When combined with a result of Roblet and Viennot, this bijection allows us to express the generating function for 123-avoiding permutations with a given number of occurrences of the pattern (k−1)(k−2)…1k in the form of a continued fraction and to derive further results for these permutations
MacMahon's Master Theorem, Representation Theory, and Moments of Wishart Distributions

AbstractD. Foata and D. Zeilberger (1988, SIAM J. Discrete Math.4, 425–433) and D. Vere-Jones (1988, Linear Algebra Appl.111, 119–124) independently derived a generalization of MacMahon's master theorem. In this article we apply their result to obtain an explicit formula for the moments of arbitrary polynomials in the entries of X, a real random matrix having a Wishart distribution. In the case of the complex Wishart distributions, the same method is applicable. Furthermore, we apply the representation theory of GL(d,C), the complex general linear group, to derive explicit formulas for the expectation of Kronecker products of any complex Wishart random matrix
Bijective Recurrences for Motzkin Paths

AbstractConsider the lattice paths on Z2 with steps (1,1), (1,−1), and (1,0). For n≥2, let En denote the set of such paths running from (0,0) to (n,0) and remaining strictly above the x-axis except initially and terminally. The cardinalities, fn=|En| for n≥2, are the Motzkin numbers, 1,1,2,4,9,21,51,127,…. We define a bijection yielding the recurrence (n+1)fn+1=(2n−1)fn+3(n−2)fn−1, for n≥3. A modification of the bijection proves that the sum of the areas under the paths of En, denoted by An, satisfies An+1=2An+3An−1, for n≥3. A second modification yields a recurrence for a second moment for the paths of En which agrees with Euler's recurrence for the central trinomial numbers
Generalized Parking Functions, Tree Inversions, and Multicolored Graphs

AbstractA generalized x-parking function associated to a positive integer vector of the form (a,b,b,…,b) is a sequence (a1,a2,…,an) of positive integers whose nondecreasing rearrangement b1≤b2≤⋯≤bn satisfies bi≤a+(i−1)b. The set of x-parking functions has the same cardinality as the set of sequences of rooted b-forests on [n]. We construct a bijection between these two sets. We show that the sum enumerator of complements of x-parking functions is identical to the inversion enumerator of sequences of rooted b-forests by generating function analysis. Combinatorial correspondences between the sequences of rooted forests and x-parking functions are also given in terms of depth-first search and breadth-first search on multicolored graphs
Skew Diagrams and Ordered Trees

AbstractWe use a known combinatorial argument to prove that among all ordered trees the ratio of the total number of vertices to leaves is two. We introduce a new combinatorial bijection on the set of these trees that shows why this must be so. Ordered trees are then enumerated by number of leaves, total path length, and number of vertices to obtain q-analogs of Catalan numbers. The results on ordered trees are then readily transferred by the skew diagrams to help enumerate parallelogram polyominoes by their area, perimeter, and other statistics