pinecone/core-2020-05-10-deduplication huggingface dataset:
Necklaces, MSS Sequences, and DNA Sequences

AbstractBased on a unified approach via the classical Burnside Lemma in combinatorics, we deal with some problems arising from symbolic dynamics and biology. The MSS sequences play an important role in symbolic dynamics, and they are in one-to-one correspondence with necklaces of beads with two colors. This necklace problem was studied by Fine, Gilbert, and Riordan in the 1950s and 1960s. Recently, a problem on the classification of the DNA sequence arose in biology and was studied by Bell, Torney, and others. In this paper, we present a further treatment of the necklace problem which shows a closer connection with MSS sequences and brings more combinatorial insight into the properties of symbolic dynamics. We also give a clear derivation of the formula for the classification of DNA sequences. The techniques used in this paper originate principally from properties of cycle structures, which are related to the fixed points of then-cube recently studied by Chen and Stanley
Interval Partitions and Activities for the Greedoid Tutte Polynomial

AbstractThe two variable greedoid Tutte polynomialf(G;t,z), which was introduced in previous work of the authors, is studied via external activities. Two different partitions of the Boolean lattice of subsets are derived and a feasible set expansion off(G) is developed. All three of these results generalize theorems for matroids. One interval partition yields a characterization of antimatroids among the class of all greedoids. As an application, we prove that whenGis the directed branching greedoid associated with a rooted digraphD, then the highest power of (z+1) which dividesf(G) equals the minimum number of edges which can be removed fromDto produce an acyclic digraph in which all vertices ofDare still accessible from the root. The unifying theme behind these results is the idea of a computation tree
On a Generalization of the 3x+1 Problem

AbstractLetZbe the set of the integers andR(x)={0,1,…,x−1},x∈N=Z+. In this paper the sequences {mn}n≥0defined by the iterative formulam0=m∈Z,mn+1=armn+br∈Z,ifmn≡r(modp),r∈R(p), (S)wherep>1,ar=tr/p,tr∈N, andbris such thatmn∈Zfor everyn, are investigated.It is shown that if(tr,p)=gcd(tr,p)=1,r∈R(p), ((S1))anda=(a0···ap−1)1/p<1, ((S2))then, even ifar>1 for somer∈R(p), there existsc∈Nsuch that, for every ρ∈R+=(0,+∞), almost everym∈Zhas some iterate |mn|<ρ|m| or |mn|≤c, in the sense that the density of suchmeither inZ+or inZ−is one.Furthermore, it is conjectured that there existC∈Nands0∈Nsuch that, ifs≥s0and |m|≤ps, then |mn|≤Cfor everyn≥s(s+σ), where σ=1+[lnp].At the end, considering the case where instead of (S2) one hasa=(a0···ap−1)1/p>1 andar<1 for somer∈R(p), ((S2)′)it is shown when the sequence (S) belongs to an orbit of a permutation ofZfor everym∈Z
Inverse Measures, the Inversion Formula, and Discontinuous Multifractals

AbstractThe present paper is the first in a series of three closely related papers in which theinverse measureμ*(dt) of a given measure μ(dt) on [0,1] is introduced. In the first case discussed in detail, μ and μ* are multifractal in the usual sense, that is, both are linearly self-similar and continuous but not differentiable, and both are non-zero for every interval of [0,1]. Under these assumptions the Hölder spectra of μ(dt) and μ*(dt) are shown to be linked by the “inversion formula”f*(α)=αf(1/α). The inversion formula is then subjected to several diverse variations, which reveal telling details of interest to the full understanding of multifractals. The inverse of the uniform measure on a Cantor dust leads us to argue that this inversion formula applies to the Hölder spectrafHeven if the measures μ and μ* are not continuous while it may fail for the spectrumfLobtained by the Legendre path. This phenomenon goes along with a loss of concavity in the spectrumfH. Moreover, with the examples discussed it becomes natural to include the degenerate Hölder exponents 0 and ∞ in the Hölder spectra. This present paper is the first of three closely related papers on inverse measures, introducing the new notion in a language adopted for the physicist. The second and third papers in this series make rigorous what is argued with intuitive arguments here. The second paper extends the common scope of the notion of self-similar measures. With this broader class of invariant measures the third paper shows that the multifractal formalism may fail
From Association to Causation via Regression

AbstractFor nearly a century, investigators in the social sciences have used regression models to deduce cause-and-effect relationships from patterns of association. Path models and automated search procedures are more recent developments. In my view, this enterprise has not been successful. The models tend to neglect the difficulties in establishing causal relations, and the mathematical complexities tend to obscure rather than clarify the assumptions on which the analysis is based. Formal statistical inference is, by its nature, conditional. If maintained hypotheses A,B,C,… hold, then H can be tested against the data. However, if A,B,C,… remain in doubt, so must inferences about H. Careful scrutiny of maintained hypotheses should therefore be a critical part of empirical work—a principle honored more often in the breach than the observance. This paper focuses on modeling techniques that seem to convert association into causation. The object is to clarify the differences among the various uses of regression, as well as the source of the difficulty in making causal inferences by modeling. The discussion will proceed mainly by examples, ranging from71to63
A Phase Transition for the Minimum Free Energy of Secondary Structures of a Random RNA

AbstractThe free energy of a single-stranded RNA can be calculated by adding the free energies of the components: basepairs, bulges, and loops. Basepairs receive negative free energy while the unpaired bases receive positive free energy. The minimum free energy of a random RNA secondary structure with one domain has valueFnwhere the sequence length isn. Under simplifying assumptions, we show that for “small” values of bulge and loop penaltiesFnhas linear growth inn, while for “large” values of these parametersFnhas logarithmic growth inn. This phase transition generalizes results obtained for the local-alignment score of two random sequences. The random variableFnis conjectured to have a Poisson approximation. The multi-domain secondary structure minimum free energyEnhas linear growth innfor all values of the penalty functions. Nothing more is known about the distributional properties ofEn
Trees, Taxonomy, and Strongly Compatible Multi-state Characters

AbstractGiven a family ofbinarycharacters defined on a setX, a problem arising in biological and linguistic classification is to decide whether there is a tree structure onXwhich is “compatible” with this family. A fundamental result from hierarchical clustering theory states that there exists a tree structure onXfor such a family if and only if any two of the characters arecompatible. In this paper, we prove a generalization of this result. Namely, we show that given a family ofmulti-statecharacters onXwhich we denote by χ, there exists a tree structure onX, called an (X,χ)-tree, which is “compatible” with χ if and only if any two of the characters arestrongly compatible. To prove this result, we introduce the concept ofblock systems, set theoretical structures which arise naturally from, amongst other things,block graphs, and the related concepts ofblock interval systemsand Δ-systems
Absorption Processes: Models forq-Identities

AbstractSeveral extensions of Blomqvist's absorption process are presented. Inherent in some of the associated distributions is a method for establishingq-identities ranging from properties of Gaussian polynomials to product expansions of basic hypergeometric series to extensions of results on Mahonian statistics. One process links the comajor index to Russian roulette. Also given are examples involving the Rogers–Ramanujan identities that demonstrate howq-expressions may be modeled with absorption processes
Convex Polytopes and Enumeration

AbstractThis is an expository paper on connections between enumerative combinatorics and convex polytopes. It aims to give an essentially self-contained overview of five specific instances when enumerative combinatorics and convex polytopes arise jointly in problems whose initial formulation lies in only one of these two subjects. These examples constitute only a sample of such instances occurring in the work of several authors. On the enumerative side, they involved ordered graphical sequences, combinatorial statistics on the symmetric and hyperoctahedral groups, lattice paths, Baxter, André, and simsun permutations,q-Catalan andq-Schröder numbers. From the subject of polytopes, the examples involve the Ehrhart polynomial, the permutohedron, the associahedron, polytopes arising as intersections of cubes and simplices with half-spaces, and thecd-index of a polytope
New Euler–Mahonian Statistics on Permutations and Words

AbstractWe define new Mahonian statistics, called MAD, MAK, and ENV, on words. Of these, ENV is shown to equal the classical INV, that is, the number of inversions, while for permutations MAK has been already defined by Foata and Zeilberger. It is shown that the triple statistics (des, MAK, MAD) and (exc, DEN, ENV) are equidistributed over the rearrangement class of an arbitrary word. Here, exc is the number of excedances and DEN is Denert's statistic. In particular, this implies the equidistribution of (exc, INV) and (des, MAD). These bistatistics are not equidistributed with the classical Euler–Mahonian statistic (des, MAJ). The proof of the main result is by means of a bijection which, in the case of permutations, is essentially equivalent to several bijections in the literature (or inverses of these). These include bijections defined by Foata and Zeilberger, by Françon and Viennot and by Biane, between the symmetric group and sets of weighted Motzkin paths. These bijections are used to give a continued fraction expression for the generating function of (exc, INV) or (des, MAD) on the symmetric group
Conway Numbers and Iteration Theory

AbstractConway (“On Numbers and Games,” Academic Press, New York, 1976) has given an inductive procedure for generating the real numbers that extends in a natural way to a new class of numbers called the surreals. The number 0 is defined at the first step in terms of a pair of empty sets. At step 1, the number 1 and its negative are generated, giving the set {1,0,−1}; at step 2, the numbers 2,1/2 and their negatives are generated, giving the set {2,1,1/2,0,−1/2,−1,−2}; at step 3, the numbers 3,3/2,3/4,1/4 and their negatives are generated, giving the set {3,2,3/2,1,3/4,1/2,1/4,0,−1/4,−1/2,−3/4,−1,−3/2,−2,−3}, etc. It is shown that these numbers are generated in one-to-one correspondence with certain sequences of positive and negative sequences of integers: At step 0, the sequence (0) is introduced; at step 1, the sequence (1) and its negative (−1) are generated, giving the set {(1),(0),(−1)}; at step 2, the sequences (2),(1,1) and their negatives (−2),(−1,−1) are generated, giving the set {(2),(1),(1,1),(0),(−1,−1),(−1),(−2)}; at step 3, the sequences (3),(2,1),(1,1,1),(1,2) and their negatives (−3),(−2,−1),(−1,−1,−1),(−1,−2) are generated, giving the set {(3),(2),(2,1),(1),(1,1,1),(1,1),(1,2),(0),(−1,−2),(−1,−1),(−1,−1,−1),(−1),(−2,−1),(−2),(−3)}, etc. This generation of sequences in not ad hoc. The positive and negative sequences given here, and their generalizations, arise in iteration theory and in the theory of words associated with that theory. There is a natural order relation on these sequences that is one-to-one with the Conway numbers and which is rooted in the ordering of the inverse functions that arise in the description of the graph inverse to thenth iterate of certain classes of maps of an interval. A simple transformation of the points occurring in the cycles of thenth iterate of the trapezodal map of the interval [0,2] forn=1,2,… give all the dyadic Conway numbers, in one-to-one correspondence with sequences, and the inclusion of infinite iterates yields all the reals. The implication of this result is that all such Conway numbers arise as fixed points of the trapezodal map, one-to-one with the sequences that label uniquely these fixed points. However, because of certain “universality” properties, the sequences themselves have application to and significance for other maps of the interval. The various aspects of iteration theory and its relation to the subset of real Conway numbers are discussed in some detail. The question of whether the surreal Conway numbers have application to iteration theory, hence, possibly to chaos, is left open
Sparse Interpolation of Symmetric Polynomials

AbstractWe develop efficient algorithms for computing the expansion of a given symmetric polynomial into Schur functions. This problem frequently arises in applications as the problem of decomposing a given representation of the symmetric (or general linear) group into irreducible constituents. Our algorithms are probabilistic, and run in time which is polynomial in the sizes of the input and output. They can be used to compute Littlewood–Richardson coefficients, Kostka numbers, and irreducible characters of the symmetric group
The Cardinality of the Collection of Maximum Independent Sets of a Functional Graph

AbstractAn independent set (or stable set) of a graphG(V,E) is a subsetSof the vertices setVin which no two are adjacent. Let ψ(G) be the number of vertices in a stable set of maximum cardinality; ψ(G) is called the stability number ofG. Stability numbers of a graph have been well studied, but little has been done on the number of independent subsets whose cardinality is the stability number. In this paper we will provide an algorithm to find the number of independent subsets whose cardinality is the stability number
Transfinite Function Iteration and Surreal Numbers

AbstractLouck has developed a relation between surreal numbers up to the first transfinite ordinal ω and aspects of iterated trapezoid maps. In this paper, we present a simple connection between transfinite iterates of the inverse of the tent map and the class of all the surreal numbers. This connection extends Louck's work to all surreal numbers. In particular, one can define the arithmetic operations of addition, multiplication, division, square roots, etc., of transfinite iterates by conversion of them to surreal numbers. The extension is done by transfinite induction. Inverses of other unimodal onto maps of a real interval could be considered and then the possibility exists of obtaining different structures for surreal numbers
Bi-complement Reducible Graphs

AbstractWe introduce a new family of bipartite graphs which is the bipartite analogue of the class ofcomplement reduciblegraphs orcographs. Abi-complement reduciblegraph orbi-cographis a bipartite graphG=(W∪B,E) that can be reduced to single vertices by recursively bi-complementing the edge set of all connected bipartite subgraphs. Thebi-complementedgraphḠbipofGis the graph having the same vertex setW∪BasG, while its edge set is equal toW×B−E. The aim of this paper is to show that there exists an equivalent definition of bi-cographs by three forbidden configurations. We also propose a tree representation for this class of graphs
The Cayley Determinant of the Determinant Tensor and the Alon–Tarsi Conjecture

AbstractGiven a base on a vector space of dimensionn, we can represent a tensor of orderrwith a hypermatrix of dimensionnand orderr. Then, the standard determinant tensor is represented by a hypermatrixHof order and dimensionn. Gherardelli showed that the Cayley determinant ofH, timesn!, is equal to the number of even Latin squares of ordernminus the number of odd Latin squares of ordern. The Alon–Tarsi conjecture says that this difference is not zero, whenevernis even. Ifnis odd the difference is zero, but the conjecture can be extended to the odd case by computing the difference only for Latin squares which have the entries of the diagonal equal to 1. In this paper we use the Laplace rule in order to compute the Cayley determinant, and we prove that the difference between the number of even Latin squares and the number of odd Latin squares is nonnegative. We also prove the Alon–Tarsi conjecture for Latin squares of orderc2r, whereris a positive integer and eithercis an even integer for which the Alon-Tarsi conjecture is true, orcis an odd integer such that the extended Alon-Tarsi conjecture is true forcand forc+1
On a Likely Shape of the Random Ferrers Diagram

AbstractWe study the random partitions of a large integern, under the assumption that all such partitions are equally likely. We use Fristedt's conditioning device which connects the parts (summands) distribution to the one of ag-sequence, that is, a sequence of independent random variables, each distributed geometrically with a size-dependent parameter. Confirming a conjecture made by Arratia and Tavaré, we prove that the joint distribution of counts of parts with size at mostsn⪡n1/2(at leastsn⪢n1/2, resp.) is close—in terms of the total variation distance—to the distribution of the firstsncomponents of theg-sequence (of theg-sequence minus the firstsn−1 components, resp.). We supplement these results with the estimates for the middle-sized parts distribution, using the analytical tools revolving around the Hardy–Ramanujan formula for the partition function. Taken together, the estimates lead to an asymptotic description of the random Ferrers diagram, close to the one obtained earlier by Szalay and Turán. As an application, we simplify considerably and strengthen the Szalay–Turán formula for the likely degree of an irreducible representation of the symmetric groupSn. We show further that both the size of a random conjugacy class and the size of the centraliser for every element from the class are doubly exponentially distributed in the limit. We prove that a continuous time process that describes the random fluctuations of the diagram boundary from the deterministic approximation converges to a Gaussian (non-Markov) process with continuous sample path. Convergence is such that it implies weak convergence of every integral functional from a broad class. To demonstrate applicability of this general result, we prove that the eigenvalue distribution for the Diaconis–Shahshahani card-shuffling Markov chain is asymptotically Gaussian with zero mean, and variance of ordern−3/2
The Number of Permutations with Exactlyr132-Subsequences IsP-Recursive in the Size!

AbstractProving a first nontrivial instance of a conjecture of Noonan and Zeilberger we show that the numberSr(n) of permutations of lengthncontaining exactlyrsubsequences of type 132 is aP-recursive function ofn. We show that this remains true even if we impose some restrictions on the permutations. We also show the stronger statement that the ordinary generating functionGr(x) ofSr(n) is algebraic, in fact, it is rational in the variablesxand1−4x. We use this information to show that the degree of the polynomial recursion satisfied bySr(n) isr
An Algebraic Approach to Convolutions and Transform Methods

AbstractWe show that the Hopf algebra dual of the polynomials in one variable appears often in analysis, but under different disguises that include proper rational functions, exponential polynomials, shift invariant operators, Taylor functionals, and linearly recurrent sequences. The isomorphisms from the proper rational functions to the other algebras yield an explicit and general method for the solution of linear functional equations, which can be considered as an algebraic version of the usual integral transform methods. We also explain how some of the usual convolution products in spaces of functions arise in an algebraic way
Randomized Nonlinear Projections Uncover High-Dimensional Structure

AbstractWe consider the problem of investigating the “structure” of a set of points in high-dimensional space (npoints ind-dimensional Euclidean space) whenn⪡d. The analysis of such data sets is a notoriously difficult problem in both combinatorial optimization and statistics due to an exponential explosion ind. A randomized nonlinear projection method is presented that maps these observations to a low-dimensional space, while approximately preserving salient features of the original data. Classical statistical analyses can then be applied, and results from the multiple lower-dimensional projected spaces are combined to yield information about the high-dimensional structure. We apply our dimension reduction techniques to a pattern recognition problem involving PET scan brain volumes
A Family of Ideals of Minimal Regularity and the Hilbert Series ofCr(Δ)

AbstractFor a simplicial subdivision Δ of a region inR2, we analyze the dimension of the vector spaceCkr(Δ) ofCrpiecewise polynomial functions (splines) on Δ of degree at mostk. We find an exact sequence which allows us to prove that the dimension series for splines given by5does indeed agree with the bounds on the dimension of the spline space given by Alfeld and Schumaker [1;2]. We give sufficient conditions for the Alfeld–Schumaker bounds to be attained in all degrees, where Δ is a two-dimensional simplicial complex. The conditions are satisfied by the class of complexes considered by6, but also by a much broader class of complexes. Furthermore, for conditions which involve only local geometric data, this result is the strongest possible
A Spectral Sequence for Splines

AbstractWe define a complex R/J of graded modules on ad-dimensional simplicial complex Δ, so that the top homology module of this complex consists of piecewise polynomial functions (splines) of smoothnessron the cone of Δ. In a series of papers,4;5;6] used a similar approach to study the dimension of the spaces of splines on Δ, but with a complex substantially different from R/J. We obtain bounds on the dimension of the homology modulesHi(R/J) for alli<dand find a spectral sequence which relates these modules to the spline module. We use this to give simple conditions governing the projective dimension of the spline module. We also prove that if the spline module is free, then it is determined entirely by local data; that is, by the arrangements of hyperplanes incident to the various dimensional faces of Δ
Linear Conditions on the Number of Faces of Manifolds with Boundary

AbstractThe Euler equation and the Dehn–Sommerville equations are known to be the only (rational) linear conditions forf-vectors (number of simplices at various dimensions) of triangulations of spheres. We generalize this fact to arbitrary triangulations, linear triangulations of manifolds, and polytopal triangulations of Euclidean balls. We prove that for closed manifolds, the Euler equation and the Dehn–Sommerville equations remain the only linear conditions. We also prove that for manifolds with nonempty boundary, the Euler equation is the only linear condition. These results are proved not only over Q, but also over Z and Z/kZ
Clifford Analysis over Unbounded Domains

AbstractA modified Cauchy kernel is introduced over unbounded domains whose complement contains nonempty open sets. Basic results on Clifford analysis over bounded domains are now carried over to this more general context and to functions that are no longer assumed to be bounded. In particular Plemelj formulae are explicitly computed. Basic properties of the Cauchy transform over unbounded domains lying in a half space are investigated, and an orthogonal decomposition of theL2space for such a domain is set up. At the end a boundary value problem will be studied in the case of an unbounded domain without using weighted Sobolev spaces