Combinatorial Relations Research Articles

B - 115 How Training Conditions Impact Performance on a Theoretically-Derived Learning Instrument

Abstract Objective Few neuropsychological measures were developed using an explicit theory that accounts for individual performance. We developed a novel associative learning measure to address this limitation and investigated whether training conditions could influence accuracy and response time. Methods Fifty young, community-dwelling adult participants completed the novel relational learning task, which explicitly trained two stimulus pairs (A- > B, A- > C) and involved a single common stimulus. Participants identify six unique relationships between stimuli: trained relations (A- > B, A- > C), mutually entailed relations (B- > A, C- > A), and combinatorial relations (B- > C, C- > B). Participants were randomly assigned to a congruent (A is opposite B, A is opposite C) or incongruent (A is opposite B, B is the same as C) training condition and completed the procedure when stimuli were observably (e.g., color-coded shapes) or arbitrarily related (e.g., letters). Results Comparisons of accuracy between training conditions using Bayesian independent samples t-tests found moderate evidence of no reliable difference between training groups when stimuli were observably related (BF01 = 3.4, δ = 0.06) and arbitrarily related (BF01 = 3.5, δ = 0.03). Comparisons of response time found strong evidence that participants took longer to identify relations when receiving incongruent training when the stimuli were visually related (BF10 = 38.8, δ = 0.9) but inconclusive evidence when the stimuli were arbitrarily related (BF10 = 1.5, δ = 0.49) (See Table 1). Conclusion These findings suggest that manipulating training conditions does not appreciably impact the ability to identify relationships between stimuli accurately. However, training appears to increase the response time needed to identify visibly related stimuli accurately. While this is a promising theoretically-derived learning paradigm, further research is required for validation in clinical practice.

Combinatorial relations among relations for level 2 standard Cn(1)-modules

For an affine Lie algebra ĝ, the coefficients of certain vertex operators that annihilate level k standard ĝ-modules are the defining relations for level k standard modules. In this paper, we study a combinatorial structure of the leading terms of these relations for level k = 2 standard ĝ-modules for affine Lie algebras of type Cn(1) and the main result is the construction of combinatorially parameterized relations among the coefficients of annihilating fields. It is believed that the constructed relations among relations will play a key role in the construction of Gröbner-like basis of the maximal ideal of the universal vertex operator algebra Vgk for k = 2.

Immune sets in monotone infection rules. Characterization and complexity

Many dissemination processes in graphs can be described as follows at a basic level. At each step of the process, some vertices of the graph are coloured blue, and the remaining are coloured white, and a well-defined infection rule acts locally on a chosen element of the graph. As an outcome of this action, perhaps one or more white vertices are forced to become blue. Zero forcing, power domination and bootstrap percolation are some examples of widely studied infection rules.This paper presents a general view of infection rules on graphs, paying particular attention to monotone rules. We state several results referring to the final stable set of blue vertices at the end of the dissemination process driven by the infection rule R, and to the combinatorial transversal relation between the families of inclusion-minimal R-forcing and R-immune sets of the graph. Our results apply to many infection rules considered in the literature, as well as to new ones introduced in this paper. Besides, for each one of these infection rules, we provide a characterization of their R-immune sets formulated in terms of neighbourhood, so without referring to the iterative dissemination process acting on the graph.In the second part of the paper, and for the particular rules treated in the first part (k-PUSH, (kb,kw)-PUSH, α-PUSH, k-PULL, α-PULL, and k-wPULL), we prove the NP-Completeness of the decision problem associated to the corresponding R-immune number of the graph.

A heuristic algorithm solving the mutual-exclusivity-sorting problem.

Binary (or Boolean) matrices provide a common effective data representation adopted in several domains of computational biology, especially for investigating cancer and other human diseases. For instance, they are used to summarize genetic aberrations-copy number alterations or mutations-observed in cancer patient cohorts, effectively highlighting combinatorial relations among them. One of these is the tendency for two or more genes not to be co-mutated in the same sample or patient, i.e. a mutual-exclusivity trend. Exploiting this principle has allowed identifying new cancer driver protein-interaction networks and has been proposed to design effective combinatorial anti-cancer therapies rationally. Several tools exist to identify and statistically assess mutual-exclusive cancer-driver genomic events. However, these tools need to be equipped with robust/efficient methods to sort rows and columns of a binary matrix to visually highlight possible mutual-exclusivity trends. Here, we formalize the mutual-exclusivity-sorting problem and present MutExMatSorting: an R package implementing a computationally efficient algorithm able to sort rows and columns of a binary matrix to highlight mutual-exclusivity patterns. Particularly, our algorithm minimizes the extent of collective vertical overlap between consecutive non-zero entries across rows while maximizing the number of adjacent non-zero entries in the same row. Here, we demonstrate that existing tools for mutual-exclusivity analysis are suboptimal according to these criteria and are outperformed by MutExMatSorting. https://github.com/AleVin1995/MutExMatSorting. Supplementary data are available at Bioinformatics online.

A q-analogue of the bipartite distance matrix of a nonsingular tree

All graphs considered here are simple and finite. Let G be a labeled, connected, bipartite graph and let (L,R) be a vertex bipartition of G. The bipartite adjacency matrix A of G is the submatrix of its adjacency matrix determined by the rows corresponding to L and the columns corresponding to R, see [6]. In a similar way, one can consider the bipartite distance matrix of G. It is the submatrix of the usual distance matrix D of G determined by the rows corresponding to L and the columns corresponding to R. Let G=T be a tree with a unique perfect matching. Very recently, in [2], it was shown that this matrix possesses many interesting properties and has a very deep combinatorial relation with the tree structure. One of those results, is the elegant description of the determinant of the bipartite distance matrix of T by a combinatorial sum over the collection of all alternating paths. Another result is reminiscent of the well known result of Graham and Pollack, which tells that the determinant of the usual distance matrix a tree of order 2p is a multiple of 22p−2. It was shown that the determinant of the bipartite distance matrix of a tree of order 2p with a unique perfect matching is a multiple of 2p−1.There are many studies of distance-like matrices available in the literature. In this article, we consider two of them. The first one is the q-distance matrix which is a generalization of the usual distance matrix. We show that the determinant of the q-bipartite distance matrix of a tree with a unique perfect matching can still be expressed as a combinatorial sum over the set of alternating paths with respect to a suitable sequence. The result of [2] follows as a corollary when q=1.The second distance-like matrix we study is the exponential distance matrix, which has a very simple expression for the determinant. We show that an extremely similar result hold here too. This is unexpected, as our matrix is of half the order.Recall that, the determinant of the usual distance matrix of a tree is just dependent on the number of vertices and is independent of the tree structure. In [2], it was shown that the determinant of the bipartite distance matrix of a tree with a unique perfect matching was dependent on the tree structure and on some smaller classes it was independent of the tree structure. We prove similar results for the q-bipartite distance matrix.However, as another surprise, we show that the determinant of the exponential bipartite distance matrix of a tree with a unique perfect matching is independent of the tree structure.

A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions

We give a new characterization of the vertical-strip LLT polynomials GP(x;q) as the unique family of symmetric functions that satisfy certain combinatorial relations. This characterization is then used to prove an explicit combinatorial expansion of vertical-strip LLT polynomials in terms of elementary symmetric functions. Such formulas were conjectured independently by A. Garsia et al. and the first named author, and are governed by the combinatorics of orientations of unit-interval graphs. The obtained expansion is manifestly positive if q is replaced by q+1, thus recovering a recent result of M. D'Adderio. Our results are based on linear relations among LLT polynomials that arise in the work of D'Adderio, and of E. Carlsson and A. Mellit. To some extent these relations are given new bijective proofs using colorings of unit-interval graphs. As a bonus we obtain a new characterization of chromatic quasisymmetric functions of unit-interval graphs.

Combinatorial Relation of Optimized Combination with Permutation

Combinatorial Relation of Optimized Combination with Permutation

The genetic informational network: how DNA conveys semantic information.

The question of whether "genetic information" is a merely causal factor in development or can be made sense of semantically, in a way analogous to a language or other type of representation, has generated a long debate in the philosophy of biology. It is intimately connected with another intense debate, concerning the limits of genetic determinism. In this paper I argue that widespread attempts to draw analogies between genetic information and information contained in books, blueprints or computer programs, are fundamentally inadequate. In development, gene exons are the central part of an intricate and densely ramified semantic Genetic Informational Network. DNA in the entire genome is in a state of continuous positive and negative feedback with itself and with its 'environment', and is 'read' and acted upon by the cell in various alternative and complementary ways. The linear combinatorial coding relation between codons and amino acids is but one aspect of semantic genetic information, which is, when considered in its entirety, a far wider and richer concept.

On restriction of unitarizable representations of general linear groups and the non-generic local Gan–Gross–Prasad conjecture

We prove the first direction of a recently posed conjecture by Gan, Gross and Prasad, which predicts branching laws that govern restriction from p -adic GL_n to GL_{n-1} of irreducible smooth representations within the Arthur-type class. We extend this prediction to the full class of unitarizable representations, by exhibiting a combinatorial relation that must be satisfied for any pair of irreducible representations, in which one appears as a quotient of the restriction of the other. We settle the full conjecture for the cases in which either one of the representations in the pair is generic. The method of proof involves a transfer of the problem, using the Bernstein decomposition and the quantum affine Schur–Weyl duality, into the realm of quantum affine algebras. This restatement of the problem allows for an application of the combined power of a result of Hernandez on cyclic modules together with the Lapid–Mínguez criterion from the p -adic setting.

Combinatorial relations on skew Schur and skew stable Grothendieck polynomials

We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in earlier joint work with López Martín and Teixidor i Bigas concerning Brill–Noether curves, and it generalizes a 2000 formula of Lenart and a recent result of Reiner–Tenner–Yong to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms of skew Grothendieck polynomials.

Level curve portraits of rational inner functions

We analyze the behavior of rational inner functions on the unit bidisk near singularities on the distinguished boundary $\mathbb{T}^2$ using level sets. We show that the unimodular level sets of a rational inner function $\phi$ can be parametrized with analytic curves and connect the behavior of these analytic curves to that of the zero set of $\phi$. We apply these results to obtain a detailed description of the fine numerical stability of $\phi$: for instance, we show that $\frac{\partial \phi}{\partial z_1}$ and $\frac{\partial \phi}{\partial z_2}$ always possess the same $L^{\mathfrak{p}}$-integrability on $\mathbb{T}^2$, and we obtain combinatorial relations between intersection multiplicities at singularities and vanishing orders for branches of level sets. We also present several new methods of constructing rational inner functions that allow us to prescribe properties of their zero sets, unimodular level sets, and singularities.

A Linear-Time Algorithm for Seeds Computation

A seed in a word is a relaxed version of a period in which the occurrences of the repeating subword may overlap. Our first contribution is a linear-time algorithm computing a linear-size representation of all seeds of a word (the number of seeds might be quadratic). In particular, one can easily derive the shortest seed and the number of seeds from our representation. Thus, we solve an open problem stated in a survey by Smyth from 2000 and improve upon a previous O ( n log n )-time algorithm by Iliopoulos et al. from 1996. Our approach is based on combinatorial relations between seeds and subword complexity (used here for the first time in the context of seeds). In previous papers, compact representations of seeds consisted of two independent parts operating on the suffix tree of the input word and the suffix tree of its reverse, respectively. Our second contribution is a novel and significantly simpler representation of all seeds that avoids dealing with the suffix tree of the reversed word. This result is also of independent interest from a combinatorial point of view. A preliminary version of this work, with a much more complex algorithm constructing a representation of seeds on two suffix trees, was presented at the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’12).

Cross-multidimensional spaces and environments as factors of innovative renewal of educational activities

The article proposes and reveals the scientific idea and typology of interactions and intersections of various spaces and environments, generating new cross-multidimensional realities, causing the emergence of new didactic opportunities and properties of educational systems. In relation to educational organizations of general education, the most effective combinations and combinatorial relations of the educational environment in the framework of its cross-multidimensional interaction with other spaces and environments have been identified.

On random polynomials generated by a symmetric three-term recurrence relation

ABSTRACTWe investigate the sequence of random polynomials generated by the three-term recurrence relation , , with initial conditions , , assuming that is a sequence of positive i.i.d. random variables. is a sequence of orthogonal polynomials on the real line, and is the characteristic polynomial of a Jacobi matrix . We investigate the relation between the common distribution of the recurrence coefficients and two other distributions obtained as weak limits of the averaged empirical and spectral measures of . Our main result is a description of combinatorial relations between the moments of the aforementioned distributions in terms of certain classes of coloured planar trees. Our approach is combinatorial, and the starting point of the analysis is a formula of P. Flajolet for weight polynomials associated with labelled Dyck paths.

New relation for Witten diagrams

In this paper, we present a simple and iterative algorithm that computes Witten diagrams. We focus on the gauge correlators in AdS in four dimensions in momentum space. These new combinatorial relations will allow one to generate tree level amplitudes algebraically, without having to do any explicit bulk integrations; hence, leading to a simple method of calculating higher point gauge amplitudes.

New symmetries of -invariant Bethe vectors

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing -invariant R-matrix. We study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of Bethe vectors are identical up to normalization and reshuffling of the Bethe parameters. To prove this correspondence we use the current approach. This identity gives new combinatorial relations for the scalar products of the Bethe vectors. The q-deformed case, as well as the superalgebra case, are also evoked in the conclusion.

MGSEA \u2013 a multivariate Gene set enrichment analysis

BackgroundGene Set Enrichment Analysis (GSEA) is a powerful tool to identify enriched functional categories of informative biomarkers. Canonical GSEA takes one-dimensional feature scores derived from the data of one platform as inputs. Numerous extensions of GSEA handling multimodal OMIC data are proposed, yet none of them explicitly captures combinatorial relations of feature scores from multiple platforms.ResultsWe propose multivariate GSEA (MGSEA) to capture combinatorial relations of gene set enrichment among multiple platform features. MGSEA successfully captures designed feature relations from simulated data. By applying it to the scores of delineating breast cancer and glioblastoma multiforme (GBM) subtypes from The Cancer Genome Atlas (TCGA) datasets of CNV, DNA methylation and mRNA expressions, we find that breast cancer and GBM data yield both similar and distinct outcomes. Among the enriched functional categories, subtype-specific biomarkers are dominated by mRNA expression in many functional categories in both cancer types and also by CNV in many functional categories in breast cancer. The enriched functional categories belonging to distinct combinatorial patterns are involved different oncogenic processes: cell proliferation (such as cell cycle control, estrogen responses, MYC and E2F targets) for mRNA expression in breast cancer, invasion and metastasis (such as cell adhesion and epithelial-mesenchymal transition (EMT)) for CNV in breast cancer, and diverse processes (such as immune and inflammatory responses, cell adhesion, angiogenesis, and EMT) for mRNA expression in GBM. These observations persist in two external datasets (Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) for breast cancer and Repository for Molecular Brain Neoplasia Data (REMBRANDT) for GBM) and are consistent with knowledge of cancer subtypes. We further compare the characteristics of MGSEA with several extensions of GSEA and point out the pros and cons of each method.ConclusionsWe demonstrated the utility of MGSEA by inferring the combinatorial relations of multiple platforms for cancer subtype delineation in three multi-OMIC datasets: TCGA, METABRIC and REMBRANDT. The inferred combinatorial patterns are consistent with the current knowledge and also reveal novel insights about cancer subtypes. MGSEA can be further applied to any genotype-phenotype association problems with multimodal OMIC data.

Condition for zero and nonzero discord in graph Laplacian quantum states

This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph theoretic conditions for zero and nonzero quantum discord for these states. We apply these results on some important pure two qubit states, as well as a number of mixed quantum states, such as the Werner, Isotropic, and [Formula: see text]-states. We also consider graph Laplacian states corresponding to simple graphs as a special case.

Structural relations of symmetry among players in strategic games

ABSTRACTThe notions of symmetry and anonymity in strategic games have been formalized in different ways in the literature. We propose a combinatorial framework to analyze these notions, using group actions. Then, the same framework is used to define partial symmetries in payoff matrices. With this purpose, we introduce the notion of the role a player plays with respect to another one, and combinatorial relations between roles are studied. Building on them, we define relations directly between players, which provide yet another characterization of structural symmetries in the payoff matrices of strategic games.

Many-body spin echo

We predict a universal echo phenomenon present in the time evolution of many-body states of interacting quantum systems described by Fermi-Hubbard models. It consists of the coherent revival of transition probabilities echoing a sudden flip of the spins that, contrary to its single-particle (Hahn) version, is not dephased by interactions or spin-orbit coupling. The many-body spin echo signal has a universal shape independent of the interaction strength, and an amplitude and sign depending only on combinatorial relations between the number of particles and the number of applied spin flips. Our analytical predictions, based on semiclassical interfering amplitudes in Fock space associated with chaotic mean-field solutions, are tested against extensive numerical simulations confirming that the coherent origin of the echo lies in the existence of anti-unitary symmetries.