Combinatorial Representation Research Articles

Combinatorial rank of [formula omitted

In general, an intersection of two Hopf ideals of a Hopf algebra is not a Hopf ideal. By this reason, one may not define a Hopf ideal generated by a set of elements, and the Hopf algebras do not admit a usual combinatorial representation by generators and relations. Nevertheless, Heyneman–Radford theorem implies that each nonzero Hopf ideal of a pointed Hopf algebra has nonzero skew primitive element. Each ideal generated by skew primitive elements is a Hopf ideal. Therefore, the Heyneman–Radford theorem allows one to define a combinatorial representation step-by step by means of skew primitive relations. By definition the combinatorial rank is the minimal number of steps in that representation. We prove that the combinatorial rank of the multiparameter version of the Frobenius–Lusztig kernel of type Dn, n>3 equals ⌊log2⁡(2n−3)⌋+1, provided that q has a finite multiplicative order greater than 2. It is already known that the combinatorial rank of the multiparameter version of the Frobenius–Lusztig kernel uq(sln+1) of type An equals ⌊log2⁡n⌋+1, whereas in case uq(so2n+1) of type Bn, it is equal to ⌊log2⁡(n−1)⌋+2.

Classical conformal blocks via AdS/CFT correspondence

We continue to develop the holographic interpretation of classical conformal blocks in terms of particles propagating in an asymptotically $AdS_3$ geometry. We study $n$-point block with two heavy and $n-2$ light fields. Using the worldline approach we propose and explicitly describe the corresponding bulk configuration, which consists of $n-3$ particles propagating in the conical defect background produced by the heavy fields. We test this general picture in the case of five points. Using the special combinatorial representation of the Virasoro conformal block we compute $5$-point classical block and find the exact correspondence with the bulk worldline action. In particular, the bulk analysis relies upon the special perturbative procedure which treats the $5$-point case as a deformation of the $4$-pt case.

Data-Driven Scene Understanding with Adaptively Retrieved Exemplars

This article investigates a data-driven approach for semantic scene understanding, without pixelwise annotation or classifier training. The proposed framework parses a target image in two steps: first, retrieving its exemplars (that is, references) from an image database, where all images are unsegmented but annotated with tags; second, recovering its pixel labels by propagating semantics from the references. The authors present a novel framework making the two steps mutually conditional and bootstrapped under the probabilistic Expectation-Maximization (EM) formulation. In the first step, the system selects the references by jointly matching the appearances as well as the semantics (that is, the assigned labels) with the target. They process the second step via a combinatorial graphical representation, in which the vertices are superpixels extracted from the target and its selected references. Then they derive the potentials of assigning labels to one vertex of the target, which depend upon the graph edges that connect the vertex to its spatial neighbors of the target and to similar vertices of the references. The proposed framework can be applied naturally to perform image annotation on new test images. In the experiments, the authors validated their approach on two public databases, and demonstrated superior performance over the state-of-the-art methods in both semantic segmentation and image annotation tasks.

The Yamada polynomial of lens spatial graphs

Let [Formula: see text] be a spatial graph in the 3-sphere which covers a graph in the lens space. We introduce a combinatorial representation of [Formula: see text] which reflects the symmetry of the spatial graph. This representation involves a ribbon n-graph and the generator of the center of the n-braid group. We apply this result to study the Yamada polynomial of a class of lens spatial graphs.

A duality formula for Feynman–Kac path particle models

This Note and its extended version [10] present a new duality formula between genetic type genealogical tree based particle models and Feynman–Kac measures on path spaces. Among others, this formula allows us to design reversible Gibbs–Glauber Markov chains for Feynman–Kac integration on path spaces. Our approach yields new Taylor series expansions of the particle Gibbs–Glauber semigroup around its equilibrium measure w.r.t. the size of the particle system, generalizing the recent work of Andrieu, Doucet, and Holenstein [1]. We analyze the rate of convergence to equilibrium in terms of the ratio of the length of the trajectories to the number of particles. The analysis relies on a tree-based functional and combinatorial representation of a class of Feynman–Kac particle models with a frozen ancestral line. We illustrate the impact of these results in the context of Quantum and Diffusion Monte Carlo methods.

Searching combinatorial optimality using graph-based homology information

This paper analyses the topological information of a digital object \(O\) under a combined combinatorial-algebraic point of view. Working with a topology-preserving cellularization \(K(O)\) of the object, algebraic and combinatorial tools are jointly used. The combinatorial entities used here are vector fields, \(V\)-paths and directed graphs. In the algebraic side, chain complexes with extra \(2\)-nilpotent operators are considered. By mixing these two perspectives we are able to explore the problems of combinatorial and homological optimality. Combinatorial optimality is understood here as the problem for constructing a discrete gradient vector field (DGVF) in the sense of Discrete Morse Theory, such that it has the least possible number of critical cells. Fixing \({\mathbb {Z}}/2{\mathbb {Z}}\) as field of coefficients, by homological ‘optimality’ we mean the problem of constructing a \(2\)-nilpotent codifferential map \(\phi :C_*(K(O)) \rightarrow C_{*+1}(K(O))\) for finite linear combinations of cells in \(K(O)\), called homology integral operator. The homology groups associated to the chain complex \((C(K(O)), \phi )\) are isomorphic to those of \((C(K(O)), \partial )\), being \(\partial \) the canonical boundary or differential operator of the cell complex \(K(O)\). Relations between these two problems are tackled here by using a type of discrete graphs associated to a homology integral operator, called Homological Spanning Forests (HSF for short). Informally, an HSF for a cell complex can be seen as a kind of combinatorial compressed representation of a homology integral operator. As main result, we refine the heuristic for computing DGVFs based on the iterative Morse complex reduction technique of [1], reducing the search space for an optimal DGVF to an HSF associated to a homology integral operator.

Combinatorial representation of tetrahedral chains

Combinatorial representation of tetrahedral chains

On Mardia skewness and kurtosis of Soules basis matrices in ROM simulation

Relying on an interesting connection between generalized Helmert–Ledermann matrices and Soules basis matrices, as well as a complete binary tree classification of the latter, a recursive determination of the Mardia skewness and kurtosis coefficients of Soules basis matrices is proposed. It is based on a convenient combinatorial representation of Soules basis matrices by restricted binary partitions of a natural number into two ordered summands with recursive repetition of this binary partitioning on the summands. The obtained simple composition and invariance formulas enable the recursive calculation of these skewness and kurtosis coefficients.

An olfactory cocktail party: figure-ground segregation of odorants in rodents.

In odorant-rich environments, animals must be able to detect specific odorants of interest against variable backgrounds. However, several studies have suggested that both humans and rodents are very poor at analyzing the components of odorant mixtures, leading to the idea that olfaction is a synthetic sense in which mixtures are perceived holistically. We have developed a behavioral task to directly measure the ability of mice to perceive mixture components and found that mice can be easily trained to detect target odorants embedded in unpredictable and variable mixtures. We imaged the responses of olfactory bulb glomeruli to the individual odors used in the task in mice expressing the Ca++ indicator GCaMP3 in olfactory receptor neurons. By relating behavioral performance to the glomerular response patterns, we found that the difficulty of segregating the target from the background was strongly dependent on the extent of overlap between the representations of the target and the background odors by olfactory receptors. Our study indicates that the olfactory system has powerful analytic abilities that are constrained by the limits of combinatorial neural representation of odorants at the level of the olfactory receptors.

Conformal blocks of W N $$ {\mathcal{W}}_N $$ minimal models and AGT correspondence

We study the AGT correspondence between four-dimensional supersymmetric gauge field theory and two-dimensional conformal field theories in the context of W_N minimal models. The origin of the AGT correspondence is in a special integrable structure which appears in the properly extended conformal theory. One of the basic manifestations of this integrability is the special orthogonal basis which arises in the extended theory. We propose modification of the AGT representation for the W_N conformal blocks in the minimal models. The necessary modification is related to the reduction of the orthogonal basis. This leads to the explicit combinatorial representation for the conformal blocks of minimal models and employs the sum over N-tupels of Young diagrams with additional restrictions.

Perceptual Color Map in Macaque Visual Area V4

Colors distinguishable with trichromatic vision can be defined by a 3D color space, such as red-green-blue or hue-saturation-lightness (HSL) space, but it remains unclear how the cortex represents colors along these dimensions. Using intrinsic optical imaging and electrophysiology, and systematically choosing color stimuli from HSL coordinates, we examined how perceptual colors are mapped in visual area V4 in behaving macaques. We show that any color activates 1-4 separate cortical patches within "globs," millimeter-sized color-preferring modules. Most patches belong to different hue or lightness clusters, in which sequential representations follow the color order in HSL space. Some patches overlap greatly with those of related colors, forming stacks, possibly representing invariable features, whereas few seem positioned irregularly. However, for any color, saturation increases the activity of all its patches. These results reveal how the color map in V4 is organized along the framework of the perceptual HSL space, whereupon different multipatch activity patterns represent different colors. We propose that such distributed and combinatorial representations may expand the encodable color space of small cortical maps and facilitate binding color information to other image features.

Painlevé Functions and Conformal Blocks

We outline recent developments relating Painleve equations and 2D conformal field theory. Generic tau functions of Painleve VI and Painleve III3 are written as linear combinations of c=1 conformal blocks and their irregular limits. This provides explicit combinatorial series representations of the tau functions, and helps to establish a connection formula for the tau function in the Painleve VI case.

Equalization of odor representations by a network of electrically coupled inhibitory interneurons

Robustness of neuronal activity patterns against variations in input intensity is critical for neuronal computations. We found that odor representations in the olfactory bulb were stabilized by interneurons that were densely coupled to the output neurons by electrical and GABAergic synapses. This interneuron network modulated responses of output neurons as a function of stimulus intensity in two ways: it globally boosted responses to weak odors, but attenuated responses to strong odors, and it increased the sensitivity of some output neurons, but decreased the sensitivity of others. These effects are closely related to strategies used in engineering to increase dynamic range. Together, they maintained not only the mean, but also the distribution, of activity across the population of output neurons within narrow limits, which is important for pattern classification. Neuronal circuits in the olfactory bulb therefore stabilize combinatorial sensory representations against variations in stimulus intensity by generic mechanisms.

Cluster and Feature Modeling from Combinatorial Stochastic Processes

One of the focal points of the modern literature on Bayesian nonparametrics has been the problem of clustering, or partitioning, where each data point is modeled as being associated with one and only one of some collection of groups called clusters or partition blocks. Underlying these Bayesian nonparametric models are a set of interrelated stochastic processes, most notably the Dirichlet process and the Chinese restaurant process. In this paper we provide a formal development of an analogous problem, called feature modeling, for associating data points with arbitrary nonnegative integer numbers of groups, now called features or topics. We review the existing combinatorial stochastic process representations for the clustering problem and develop analogous representations for the feature modeling problem. These representations include the beta process and the Indian buffet process as well as new representations that provide insight into the connections between these processes. We thereby bring the same level of completeness to the treatment of Bayesian nonparametric feature modeling that has previously been achieved for Bayesian nonparametric clustering.

How instanton combinatorics solves Painlevé VI, V and IIIs

We elaborate on a recently conjectured relation of Painlevé transcendents and 2D conformal field theory. General solutions of Painlevé VI, V and III are expressed in terms of c = 1 conformal blocks and their irregular limits, AGT related to instanton partition functions in supersymmetric gauge theories with Nf = 0, 1, 2, 3, 4. The resulting combinatorial series representations of Painlevé functions provide an efficient tool for their numerical computation at finite values of the argument. The series involves sums over bipartitions which, in the simplest cases, coincide with Gessel expansions of certain Toeplitz determinants. Considered applications include Fredholm determinants of classical integrable kernels, scaled gap probability in the bulk of the Gaussian Unitary Ensemble, and all-order conformal perturbation theory expansions of correlation functions in the sine–Gordon field theory at the free-fermion point.

Realistic roofs over a rectilinear polygon

Given a simple rectilinear polygon P in the xy-plane, a roof over P is a terrain over P whose faces are supported by planes through edges of P that make a dihedral angle π/4 with the xy-plane. According to this definition, some roofs may have faces isolated from the boundary of P or even local minima, which are undesirable for several practical reasons. In this paper, we introduce realistic roofs by imposing a few additional constraints. We investigate the geometric and combinatorial properties of realistic roofs and show that the straight skeleton induces a realistic roof with maximum height and volume. We also show that the maximum possible number of distinct realistic roofs over P is ((n−4)/2⌊(n−4)/4⌋) when P has n vertices. We present an algorithm that enumerates a combinatorial representation of each such roof in O(1) time per roof without repetition, after O(n4) preprocessing time. We also present an O(n5)-time algorithm for computing a realistic roof with minimum height or volume.

A LOWER BOUND FOR THE NUMBER OF FORBIDDEN MOVES TO UNKNOT A LONG VIRTUAL KNOT

Nelson and Kanenobu showed that forbidden moves unknot any virtual knot. Similarly a long virtual knot can be unknotted by a finite sequence of forbidden moves. Goussarov, Polyak and Viro introduced finite type invariants of virtual knots and long virtual knots and gave combinatorial representations of finite type invariants. We introduce Fn-moves which generalize the forbidden moves. Assume that two long virtual knots K and K′ are related by a finite sequence of Fn-moves. We show that the values of the finite type invariants of degree 2 of K and K′ are congruent modulo n and give a lower bound for the number of Fn-moves needed to transform K to K′.

Rigorous Computation of the Global Dynamics of Integrodifference Equations with Smooth Nonlinearities

Topological tools, such as Conley index theory, have inspired rigorous computational methods for studying dynamics. These methods rely on the construction of an outer approximation, a combinatorial representation of the system that incorporates small, bounded error. In this work, we present an automated approach to constructing outer approximations for systems in a class of integrodifference operators with smooth nonlinearities. Chebyshev interpolants and Galerkin projections form the basis for the construction, while analysis and interval arithmetic are used to incorporate explicit error bounds. This represents a significant advance to the approach given by Day, Junge, and Mischaikow [SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 117--160], extending the nonlinearities that may be studied from low degree polynomials to smooth functions and the studied portion of phase space from a simulated attracting region to the global maximal invariant set. As a demonstration of the techniques, a Morse decomposition of the global dynamics, a list of validated periodic orbits, semiconjugate symbolic dynamics, and a lower bound on topological entropy are computed for the Kot--Schaffer integrodifference operator from ecology with the exponential Ricker nonlinearity.

2-Modem Pursuit-Evasion Problem

In this paper we introduce a new version of the Pursuit-Evasion problem in which the pursuer is a 2-modem which pursues an unpredictable evader in a polygonal environment. A 2-modem searcher is a wireless device whose radio signal can penetrate two walls. We will present a new cell decomposition of a given polygon P for the 2-modem searcher such that the combinatorial representation of the invisible regions of the searcher remains unchanged. In other words, when the searcher moves inside a cell, no evader can move from an invisible region to another one without detecting by the pursuer.

Combinatorial representations

This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then prove that any graph is representable over all alphabets of size larger than some number depending on the graph. We also provide a characterisation of families representable over a given alphabet. Then, we associate a rank function and a closure operator to any representation which help us determine some criteria for the functions used in a representation. While linearly representable matroids can be viewed as having representations via matrices with only one row, we conclude this paper by an investigation of representations via matrices with only two rows.