Search

We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $\omega_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The combinatorial properties that we make use of are natural principles, and in particular for inaccessible cardinals, these principles are equivalent to their corresponding large cardinal properties. Our characterizations make use of the concepts of internal large cardinals introduced in this paper, and of the classical concept of generic elementary embeddings., Comment: 28 pages

Monkey alarm calls have shown that in the primate clade, combinatorial rules in acoustic communication are not exclusive to humans. A recent hypothesis suggests that the number of different call combinations in monkeys increases with increased number of predator species. However, the existence of combinatorial rules in great ape alarm calls remains largely unstudied, despite its obvious relevance to ideas about the evolution of human speech. In this paper, we examine the potential use of combinatorial rules in the alarm calls of the only Asian great ape: the orangutan. Alarm calls in orangutans are composed of syllables (with either one or two distinct elements), which in turn are organized into sequences. Tigers and clouded leopards are predators for Sumatran orangutans, but in Borneo, tigers are extinct. Thus, orangutans make a suitable great ape model to assess alarm call composition in relation to the size of the predator guild. We exposed orangutans on both islands to a tiger and control model. Response compositionality was analyzed at two levels (i.e., syllable and syllable sequences) between models and populations. Results were corroborated using information theory algorithms. We made specific, directed predictions for the variation expected if orangutans used combinatorial rules. None of these predictions were met, indicating that monkey alarm call combinatorial rules do not have direct homologues in orangutans. If these results are replicated in other great apes, this indicates that predation did not drive selection towards ever more combinatorial rules in the human lineage.

International audience; — A combinatorial approach has been recently proposed for multi-parametric quadratic programming and has shown to be more effective in finding the complete solution than existing geometric methods for higher-order systems. In this paper, we propose a method for exploring the combinatorial tree which exploits some of the underlying geometric properties of adjacent critical regions as the supplementary information in combinatorial approach to exclude a noticeable number of feasible candidate active sets from combinatorial tree. This method is particularly well-suited for cases where many combinations of active constraints are feasible but not optimal. Results indicate that this method can find all critical regions corresponding to non-degenerate multi-parametric programming. A post-processing algorithm can be applied to complete the proposed method in the cases in which some critical regions might not be enumerated due to degeneracies in the problem.

Human language can express limitless meanings from a finite set of words based on combinatorial rules (i.e., compositional syntax). Although animal vocalizations may be comprised of different basic elements (notes), it remains unknown whether compositional syntax has also evolved in animals. Here we report the first experimental evidence for compositional syntax in a wild animal species, the Japanese great tit (Parus minor). Tits have over ten different notes in their vocal repertoire and use them either solely or in combination with other notes. Experiments reveal that receivers extract different meanings from ‘ABC' (scan for danger) and ‘D' notes (approach the caller), and a compound meaning from ‘ABC–D' combinations. However, receivers rarely scan and approach when note ordering is artificially reversed (‘D–ABC'). Thus, compositional syntax is not unique to human language but may have evolved independently in animals as one of the basic mechanisms of information transmission., Animal vocalizations contain distinct elements, but it is not clear whether they convey combined meanings in the same way as human speech. Here, Suzuki et al. show that Japanese great tits can combine different elements of vocal signals so that they have compositional syntax.

We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a definable set. We prove that some consequences of Ramsey's Theorem for colorings of pairs could help in simplifying the definitions of some $\Delta^0_2$ sets, while some others could not. We also investigate some consequences of Ramsey's Theorem for colorings of longer tuples. These results of definability strength have some interesting consequences in reverse mathematics, including strengthening of known theorems in a more uniform way and also new theorems., Comment: 23 pages; a few changes of references; a corrected description of a result of Patey

In this paper, we will prove the inevitable non-uniformity of two constructions from combinatorial group theory related to the word problem for finitely generated groups and the Higman–Neumann–Neumann Embedding Theorem.

Given an orientable surface with boundary and a free homotopy class, we present a purely combinatorial algorithm which produces a representative of that homotopy class with minimal self intersection., 14 pages, 11 figures. The algorithm described in the paper is implemented at http://www.math.sunysb.edu/~moira/applets/chrisApplet.html

We consider the family of 3D minimal polyominoes inscribed in a rectanglar prism. These objects are polyominos and so they are connected sets of unitary cubic cells inscribed in a given rectangular prism of size $b\times k \times h$ and of minimal volume equal to $b+k+h-2$. They extend the concept of minimal 2D polyominoes inscribed in a rectangle studied in a previous work. Using their geometric structure and elementary combinatorial principles, we construct rational generating functions of minimal 3D polyominoes. We also obtain a number of exact formulas and recurrences for sub-families of these polyominoes., Nous considérons la famille des polyominos 3D de volume minimal inscrits dans un prisme rectangulaire. Ces objets sont des polyominos et sont donc des ensembles connexes de cubes unitaires. De plus ils sont inscrits dans un prisme rectangulaire de format $b\times k \times h$ donné et ont un volume minimal égal à $b+k+h-2$. Ces polyominos généralisent le concept de polyomino 2D étudié dans un travail précédent. Nous construisons des séries génératrices rationnelles de polyominos 3D minimaux et nous obtenons des formules exactes et des récurrences pour des sous-familles de ces polyominos.

We introduce combinatorial principles that characterize strong compactness and supercompactness for inaccessible cardinals but also make sense for successor cardinals. Their consistency is established from what is supposedly optimal. Utilizing the failure of a weak version of a square, we show that the best currently known lower bounds for the consistency strength of these principles can be applied.

We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks. The characterization is a true analogue of the Maxwell-Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorithms. To prove our rigidity theorem we introduce and develop periodic direction networks and Z2-graded-sparse colored graphs., Some typographical errors fixed. 61 pages, to appear in Advances in Math

Cover relations for orbits and bundles of controllability and observability pairs associated with linear time-invariant systems are derived. The cover relations are combinatorial rules acting on integer sequences, each representing a subset of the Jordan and singular Kronecker structures of the corresponding system pencil. By representing these integer sequences as coin piles, the derived stratification rules are expressed as minimal coin moves between and within these piles, which satisfy and preserve certain monotonicity properties. The stratification theory is illustrated with two examples from systems and control applications, a mechanical system consisting of a thin uniform platform supported at both ends by springs, and a linearized Boeing 747 model. For both examples, nearby uncontrollable systems are identified as subsets of the complete closure hierarchy for the associated system pencils.

Systems of polynomial equations over an algebraically-closed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert's Nullstellensatz certificates for polynomial systems arising in combinatorics and on large-scale linear-algebra computations over K. We report on experiments based on the problem of proving the non-3-colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges.

Although much of the information regarding genes' expressions is encoded in the genome, deciphering such information has been very challenging. We reexamined Beer and Tavazoie's (BT) approach to predict mRNA expression patterns of 2,587 genes in Saccharomyces cerevisiae from the information in their respective promoter sequences. Instead of fitting complex Bayesian network models, we trained naïve Bayes classifiers using only the sequence-motif matching scores provided by BT. Our simple models correctly predict expression patterns for 79% of the genes, based on the same criterion and the same cross-validation (CV) procedure as BT, which compares favorably to the 73% accuracy of BT. The fact that our approach did not use position and orientation information of the predicted binding sites but achieved a higher prediction accuracy, motivated us to investigate a few biological predictions made by BT. We found that some of their predictions, especially those related to motif orientations and positions, are at best circumstantial. For example, the combinatorial rules suggested by BT for the PAC and RRPE motifs are not unique to the cluster of genes from which the predictive model was inferred, and there are simpler rules that are statistically more significant than BT's ones. We also show that CV procedure used by BT to estimate their method's prediction accuracy is inappropriate and may have overestimated the prediction accuracy by about 10%., Author Summary Through binding to certain sequence-specific sites upstream of the target genes, a special class of proteins called transcription factors (TFs) control transcription activities, i.e., expression amounts, of the downstream genes. The DNA sequence patterns bound by TFs are called motifs. It has been shown in an article by Beer and Tavazoie (BT) published in Cell in 2004 that a gene's expression pattern can be well-predicted based only on its upstream sequence information in the form of matching scores of a set of sequence motifs and the location and orientation of corresponding predicted binding sites. Here we report a new naïve Bayes method for such a prediction task. Compared to BT's work, our model is simpler, more robust, and achieves a higher prediction accuracy using only the motif matching score. In our method, the location and orientation information do not further help the prediction in a global way. Our result also casts doubt on several biological hypotheses generated by BT based on their model. Finally, we show that the cross-validation procedure used by BT to estimate their method's prediction accuracy is inappropriate and may have overestimated the accuracy by about 10%.

We study combinatorial principles we call Homogeneity Principle HP(κ) and Injectivity Principle IP(κ, λ) for regular κ > @1 and λ • κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of C s (κ) and F s (κ) of I. Juhasz, L. Soukup and Z. Szentmiklossy (13). Generalizing, their results, we show e.g. that IP(@2,@1) (hence also IP(@2,@2) as well as HP(@2)) holds in a generic extension of a model of CH by Cohen forcing and IP(@2,@2) (hence also HP(@2)) holds in a generic extension by countable support side-by-side product of Sacks or Prikry-Silver forc- ing (Corollary 4.8). We also show that the latter result is optimal (Theorem 5.2). Relations between these principles and their influence on the val- ues of the variations b " , b h , b ⁄ , do of the bounding number b are studied. One of the consequences of HP(κ) besides C s (κ) is that there is no projective well-ordering of length κ on any subset of ω ω. We construct a model in which there is no projective well-ordering of length ω2 on any subset of ω ω (do = @1 in our terminology) while b ⁄ = @2 (Theorem 6.4).

For many families of combinatorial objects, a construction of Wilf (1977) allows the members of the family to be viewed as paths in a directed multigraph. Introducing a partition of these paths generates a number of known, but hitherto disparate, combinatorial identities. We include several examples.

We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory, that full classical chaos is paralleled by quantum energy spectra with universal spectral statistics, in agreement with random-matrix theory. For dynamics from all three Wigner-Dyson symmetry classes, we calculate the small-time spectral form factor $K(\tau)$ as power series in the time $\tau$. Each term $\tau^n$ of that series is provided by specific families of pairs of periodic orbits. The contributing pairs are classified in terms of close self-encounters in phase space. The frequency of occurrence of self-encounters is calculated by invoking ergodicity. Combinatorial rules for building pairs involve non-trivial properties of permutations. We show our series to be equivalent to perturbative implementations of the non-linear sigma models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs are one-to-one with Feynman diagrams known from the sigma model., Comment: 31 pages, 17 figures

In reverse mathematics, one formalizes theorems of ordinary mathematics in second order arithmetic and attempts to discover which set theoretic axioms are required to prove these theorems. Often, this project involves making choices between classically equivalent definitions for the relevant mathematical concepts. In this paper, we consider a number of equivalent definitions for the notions of well quasi-order and better quasi-order and examine how difficult it is to prove the equivalences of these definitions.As usual in reverse mathematics, we work in the context of subsystems of second order arithmetic and take RCA0 as our base system. RCA0 is the subsystem formed by restricting the comprehension scheme in second order arithmetic to formulas and adding a formula induction scheme for formulas. For the purposes of this paper, we will be concerned with fairly weak extensions of RCA0 (indeed strictly weaker than the subsystem ACA0 which is formed by extending the comprehension scheme in RCA0 to cover all arithmetic formulas) obtained by adjoining certain combinatorial principles to RCA0. Among these, the most widely used in reverse mathematics is Weak König's Lemma; the resulting theory WKL0 is extensively documented in [11] and elsewhere.We give three other combinatorial principles which we use in this paper. In these principles, we use k to denote not only a natural number but also the finite set {0, …, k − 1}.

We provide combinatorial proofs of some of the q-series identities considered by Andrews, Jimenez-Urroz and Ono [q-series identities and values of certain $L$-functions. Duke Math. J. 108 (2001), no. 3, 395--419]., 14 pages. Submitted to Journal of Combinatorial Theory A

In this paper two new combinatorial principles in nonstandard analysis are isolated and applications are given. The second principle provides an equivalent formulation of Henson's isomorphism property.

We present a ``method'' for bijective proofs for determinant identities, which is based on translating determinants to Schur functions by the Jacobi--Trudi identity. We illustrate this ``method'' by generalizing a bijective construction (which was first used by Goulden) to a class of Schur function identities, from which we shall obtain bijective proofs for Dodgson's condensation formula, Pl\"ucker relations and a recent identity of the second author., Comment: Co-author Michael Kleber added a new proof of his theorem by inclusion-exclusion

We prove that the theory IΔ 0 , extended by a weak version of the Δ 0 -Pigeonhole Principle, proves that every integer is the sum of four squares (Lagrange's theorem). Since the required weak version is derivable from the theory IΔ 0 + ∀ x ( x log( x ) exists), our results give a positive answer to a question of Macintyre (1986). In the rest of the paper we consider the number-theoretical consequences of a new combinatorial principle, the ‘Δ 0 -Equipartition Principle’ (Δ 0 EQ). In particular we give a new proof, which can be formalized in IΔ 0 + Δ 0 EQ, of the fact that every prime of the form 4 n + 1 is the sum of two squares.

In this paper we study the matrix A(n) which was defined by W. B. R. Lickorish [3]. We prove a result required by Lickorish which completes his topological and combinatorial approach to the 3-manifold invariants of Witten-Reshetikhin-Turaev [4], [5]. This matrix arises from a pairing on a set of geometric configurations. These are the configurations of n nonintersecting arcs in the disk with 2n specified boundary points. There are Cn such configurations where Cn is the nth Catalan number so the matrix increases in size very rapidly. The Catalan numbers were discovered by Euler who considered the ways to partition a polygon into triangles [1]. These two counting problems correspond naturally by considering "restricted sequences". The matrix has entries in Z[δ]. Lickorish needed that det A(n) = 0 if δ = ±2 cos j^γ. We find a recursive formula for det A(n) and show that all the roots are of the form 2 cos -£fγ for 1 < m < n and 1 < k < m and verify the result. Using this formula, we derive a simple rule that allows one to recursively compute detA(n) by generating all of its factors. There have been three approaches to study polynomial invariants of classical links: the topological and combinatorial approach considered by Kauffman, Lickorish and many other topologists; the study of quantized Yang-Baxter equations and related Lie algebras by Reshetikhin and Turaev; and the study of subfactors and traces of von Neumann and Hecke algebras by Jones. We took a topological and combinatorial viewpoint. The authors have been informed that the essential result needed by Lickorish could have been obtained by pursuing the two other approaches.

We survey the definition and fundamental properties of the family of short core models, which extend the core model K of Dodd and Jensen to include α-sequences of measurable cardinals (α ϵ On). The theory is applied to various combinatorial principles to get lower bounds for their consistency strengths in terms of the existence of sequences of measurable cardinals. We consider instances of Chang's conjecture, ‘accessible’ Jonsson cardinals, the free subset property for small cardinals, a canonization property of ω ω , and a non-closure property of elementary embeddings of the universe. In some cases, equiconsistencies are obtained.

This work properly belongs to combinatorial group theory. But in its motivation and applications, it is concerned with the homotopy theory of two-dimensional cellular spaces. We describe both the combinatorial and homotopical aspects of this work in the following introduction.

In this paper we derive a general combinatorial identity in terms of polynomials with dual sequences of coefficients. Moreover, combinatorial identities involving Bernoulli and Euler polynomials are deduced. Also, various other known identities are obtained as particular cases.

We give combinatorial proofs of the primary results developed by Stanley for deriving enumerative properties of differential posets. In order to do this we extend the theory of combinatorial differential equations developed by Leroux and Viennot., Facultad de Ciencias Exactas, Laboratorio de Investigación y Formación en Informática Avanzada

In this paper, we present a general method for the random generation of some classes of combinatorial objects. Our basic idea is to translate ECO method (Enumerating Combinatorial Objects) from a method for the enumeration of combinatorial objects into a random generation method. The algorithms we illustrate are based on the concepts of succession rule and generating tree: the former is a law that predicts the combinatorial object class growth according to a given parameter. The generating tree related to a given succession rule is a particular labelled plane tree that represents the rule in an extensive way. Each node of a generating tree can also be seen as a particular combinatorial object and so a random path in the generating tree coincides with the random generation of that combinatorial object. The generation is uniform if we take the probability of each branch to be selected into account when the path is generated. We also give the formulae evaluating complexity. Finally, we take the class of m -ary trees into consideration in order to illustrate our general method. In this case, the average time complexity of the generating algorithm can be estimated as O( mn ).

Scattering amplitudes in Yang-Mills theory can be represented in the formalism of Cachazo, He and Yuan (CHY) as integrals over an auxiliary projective space---fully localized on the support of the scattering equations. Because solving the scattering equations is difficult and summing over the solutions algebraically complex, a method of directly integrating the terms that appear in this representation has long been sought. We solve this important open problem by first rewriting the terms in a manifestly Mobius-invariant form and then using monodromy relations (inspired by analogy to string theory) to decompose terms into those for which combinatorial rules of integration are known. The result is a systematic procedure to obtain analytic, covariant forms of Yang-Mills tree-amplitudes for any number of external legs and in any number of dimensions. As examples, we provide compact analytic expressions for amplitudes involving up to six gluons of arbitrary helicities., Comment: 29 pages, 43 figures; also included is a Mathematica notebook with explicit formulae. v2: citations added, and several (important) typos fixed

We analyse the solution set of first-order initial value differential problems of the form dy dx = ƒ(x, y), y(0) = 0 in the context of combinatorial species in the sense of A. Joyal ( Adv. in Math. 42 (1981), 1–82). It turns out that the situation is much richer than in the case of formal power series: many non-isomorphic combinatorial solutions are possible for a given problem, although they all have the same underlying generating series. We give many examples of this phenomenon and also elaborate a combinatorial Newton-Raphson iterative scheme for the construction of the solutions. The multidimensional case is treated explicitly.

Sarmanov, Sevast'yanov, and Tarakanov have proved certain combinatorial identities by a combinatorial argument, namely, by counting the number of weighted chains by two different methods. The identities are applied to the summation of infinite series and to the computation of the quartiles of discrete probability distributions. Egorychev and Yuzhakov have proved these and some further identities by applying the multidimensional generalization of the Lagrange expansion. In the present paper the identities are proved in a direct, elementary way.

Bijections are obtained between nonseparable planar maps and two different kinds of trees: description trees and skew ternary trees. A combinatorial relation between the latter and ternary trees allows bijective enumeration and random generation of nonseparable planar maps. The involved bijections take account of the usual combinatorial parameters and give a bijective proof of formulae established by Brown and Tutte. These results, combined with a bijection due to Goulden and West, give a purely combinatorial enumeration of two-stack-sortable permutations.

We study combinatorial principles weaker than Ramsey’s theorem for pairs over the RCA0 (recursive comprehension axiom) system with Σ20-bounding. It is shown that the cohesiveness (COH), ascending and descending sequence (ADS), and chain/antichain (CAC) principles are all Π11-conservative over Σ20-bounding. In particular, none of these principles proves Σ20-induction.

It has been shown by M. Marcus and others that, in regard to combinatorial matrix functions and combinatorial inequalities, it is frequently fruitful to pass immediately from the consideration of permutations to the consideration of their tensor representations. Such an approach embeds the combinatorial arguments into the framework of linear algebra and frequently results in deeper theorems. It is interesting to note that certain basic combinatorial identities concerned with pattern enumeration and combinatorial generating functions can also be put into this framework. In this paper we consider one possible way of doing this.

We study the positions in the Weihrauch lattice of parallel products of various combinatorial principles related to Ramsey's theorem. Among other results, we obtain an answer to a question of Brattka, by showing that Ramsey's theorem for pairs ($\mathsf{RT}^2_2$) is strictly Weihrauch below the parallel product of the stable Ramsey's theorem for pairs and the cohesive principle ($\mathsf{SRT}^2_2 \times \mathsf{COH}$).

This paper presents a combinatorial theory of formal power series. The combinatorial interpretation of formal power series is based on the concept of species of structures. A categorical approach is used to formulate it. A new proof of Cayley's formula for the number of labelled trees is given as well as a new combinatorial proof (due to G. Labelle) of Lagrange's inversion formula. Polya's enumeration theory of isomorphism classes of structures is entirely renewed. Recursive methods for computing cycle index polynomials are described. A combinatorial version of the implicit function theorem is stated and proved. The paper ends with general considerations on the use of coalgebras in combinatorics.

for 1

1. Preliminaries. A functor from a category of combinatorial geometries, or equivalently a category of geometric lattices, to a category of commutative algebras will be described, and some properties of this functor will be investigated. In particular, a cohomology will be associated to each point of a geometry and will be derived from the associated algebra. If (G, S) is a geometry on a set S [l , p. 2.4], then L(G), or simply L when no opportunity for confusion exists, denotes the associated geometric lattice of closed subsets of S. The rank function of L or G is denoted r. A morphism