Abstract: Recently introduced cost-effective depth sensors coupled with the real-time skeleton estimation algorithm of Shotton et al. [16] have generated a renewed interest in skeleton-based human action recognition. Most of the existing skeleton-based approaches use either the joint locations or the joint angles to represent a human skeleton. In this paper, we propose a new skeletal representation that explicitly models the 3D geometric relationships between various body parts using rotations and translations in 3D space. Since 3D rigid body motions are members of the special Euclidean group SE(3), the proposed skeletal representation lies in the Lie group SE(3)×…×SE(3), which is a curved manifold. Using the proposed representation, human actions can be modeled as curves in this Lie group. Since classification of curves in this Lie group is not an easy task, we map the action curves from the Lie group to its Lie algebra, which is a vector space. We then perform classification using a combination of dynamic time warping, Fourier temporal pyramid representation and linear SVM. Experimental results on three action datasets show that the proposed representation performs better than many existing skeletal representations. The proposed approach also outperforms various state-of-the-art skeleton-based human action recognition approaches.

Abstract: Product quantization (PQ) is an effective vector quantization method. A product quantizer can generate an exponentially large codebook at very low memory/time cost. The essence of PQ is to decompose the high-dimensional vector space into the Cartesian product of subspaces and then quantize these subspaces separately. The optimal space decomposition is important for the PQ performance, but still remains an unaddressed issue. In this paper, we optimize PQ by minimizing quantization distortions w.r.t the space decomposition and the quantization codebooks. We present two novel solutions to this challenging optimization problem. The first solution iteratively solves two simpler sub-problems. The second solution is based on a Gaussian assumption and provides theoretical analysis of the optimality. We evaluate our optimized product quantizers in three applications: (i) compact encoding for exhaustive ranking [1], (ii) building inverted multi-indexing for non-exhaustive search [2], and (iii) compacting image representations for image retrieval [3]. In all applications our optimized product quantizers outperform existing solutions.

Abstract: 1 Aperitif.- 1 Aperitif.- 1.1 Hensel's Analogy.- 1.2 Solving Congruences Modulopn.- 1.3 Other Examples.- 2 Foundations.- 2.1 Absolute Values on a Field.- 2.2 Basic Properties.- 2.3 Topology.- 2.4 Algebra.- 3 p-adic Numbers.- 3.1 Absolute Values on ?.- 3.2 Completions.- 3.3 Exploring ?p.- 3.4 Hensel's Lemma.- 3.5 Local and Global.- 4 Elementary Analysis in ?p.- 4.1 Sequences and Series.- 4.2 Functions, Continuity, Derivatives.- 4.3 Power Series.- 4.4 Functions Defined by Power Series.- 4.5 Some Elementary Functions.- 4.6 Interpolation.- 5 Vector Spaces and Field Extensions.- 5.1 Normed Vector Spaces over Complete Valued Fields.- 5.2 Finite-dimensional Normed Vector Spaces.- 5.3 Finite Field Extensions.- 5.4 Properties of Finite Extensions.- 5.5 Analysis.- 5.6 Example: Adjoining a p-th Root of Unity.- 5.7 On to ?.- 6 Analysis in ?p.- 6.1 Almost Everything Extends.- 6.2 Deeper Results on Polynomials and Power Series.- 6.3 Entire Functions.- 6.4 Newton Polygons.- 6.5 Problems.- A Hints and Comments on the Problems.- B A Brief Glance at the Literature.- B.1 Texts.- B.2 Software.- B.3 Other Books.

Abstract: Finite Dimensional Division Algebras.- Structure of Finite Dimensional Algebras.- Modules and Vector Spaces.- Tensor Products.- Structure of Rings.- Noncommutative Polynomials.- Rings of Quotients and Structure of PI-Rings.

Abstract: Control theory uses "signal-flow diagrams" to describe processes where real-valued functions of time are added, multiplied by scalars, differentiated and integrated, duplicated and deleted. These diagrams can be seen as string diagrams for the symmetric monoidal category FinVect_k of finite-dimensional vector spaces over the field of rational functions k = R(s), where the variable s acts as differentiation and the monoidal structure is direct sum rather than the usual tensor product of vector spaces. For any field k we give a presentation of FinVect_k in terms of the generators used in signal flow diagrams. A broader class of signal-flow diagrams also includes "caps" and "cups" to model feedback. We show these diagrams can be seen as string diagrams for the symmetric monoidal category FinRel_k, where objects are still finite-dimensional vector spaces but the morphisms are linear relations. We also give a presentation for FinRel_k. The relations say, among other things, that the 1-dimensional vector space k has two special commutative dagger-Frobenius structures, such that the multiplication and unit of either one and the comultiplication and counit of the other fit together to form a bimonoid. This sort of structure, but with tensor product replacing direct sum, is familiar from the "ZX-calculus" obeyed by a finite-dimensional Hilbert space with two mutually unbiased bases.

Abstract: Metrics on shape spaces are used to describe deformations that take one shape to another, and to define a distance between shapes. We study a family of metrics on the space of curves, which includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics can be easily computed. This family consists of Sobolev-type Riemannian metrics of order one on the space Imm(S-1, R-2) of parameterized plane curves and the quotient space Imm(S-1,R-2)/Diff (S-1) of unparameterized curves. For the space of open parameterized curves we find an explicit formula for the geodesic distance and show that the sectional curvatures vanish on the space of parameterized open curves and are non-negative on the space of unparameterized open curves. For one particular metric we provide a numerical algorithm that computes geodesics between unparameterized, closed curves, making use of a constrained formulation that is implemented numerically using the RATTLE algorithm. We illustrate the algorithm with some numerical tests between shapes. (C) 2014 Elsevier B.V. All rights reserved.

Abstract: I n 1970, Gian-Carlo Rota posed a conjecture predicting a beautiful combinatorial characterization of linear dependence in vector spaces over any given finite field. We have recently completed a fifteen-year research program that culminated in a solution to Rota’s Conjecture. In this article we discuss the conjecture and give an overview of the proof. Matroids are a combinatorial abstraction of linear independence among vectors; given a finite collection of vectors in a vector space, each subset is either dependent or independent. A matroid consists of a finite ground set together with a collection of subsets that we call independent; the independent sets satisfy natural combinatorial axioms coming from linear algebra. Not all matroids can be represented by a collection of vectors and, ever since their introduction by Hassler Whitney [26] in 1935, mathematicians have sought ways to characterize those matroids that are. Rota’s Conjecture asserts that representability over any given finite field is characterized by a finite list of obstructions. We will formalize these notions, and the conjecture, in the next section. In the remainder of this introduction, we will describe the journey that led us to a solution. In the late 1990s, Rota’s Conjecture was already known to hold for fields of size two, three, and

Abstract: A Triple Helix (TH) network of bi- and trilateral relations among universities, industries, and governments can be considered as an ecosystem in which uncertainty can be reduced when functions become synergetic. The functions are based on correlations among distributions of relations, and therefore latent. The correlations span a vector space in which two vectors (P and Q) can be used to represent forward "sending" and reflexive "receiving," respectively. These two vectors can also be understood in terms of the generation versus reduction of uncertainty in the communication field that results from interactions among the three bi-lateral channels of communication. We specify a system of Lotka---Volterra equations between the vectors that can be solved. Redundancy generation can then be simulated and the results can be decomposed in terms of the TH components. Furthermore, we show that the strength and frequency of the relations are independent parameters in the model. Redundancy generation in TH arrangements can be decomposed using Fourier analysis of the time-series of empirical studies. As an example, the case of co-authorship relations in Japan is re-analyzed. The model allows us to interpret the sinusoidal functions of the Fourier analysis as representing redundancies.

Abstract: We elaborate on the simple alternative [1] to the matrix-factorization construction of Khovanov-Rozansky (KR) polynomials for arbitrary knots and links in the fundamental representation of arbitrary SL(N). Construction consists of two steps: with every link diagram with m vertices one associates an m-dimensional hypercube with certain q-graded vector spaces, associated to its 2 m vertices. A generating function for q-dimensions of these spaces is what we suggest to call the primary T -deformation of HOMFLY polynovmial — because, as we demonstrate, it can be explicitly reduced to calculations of ordinary HOMFLY polynomials, i.e. to manipulations with quantum R-matrices, what brings the story completely inside the ordinary Chern-Simons theory. The second step is a certain minimization of residues of this new polynomial with respect to T + 1. Minimization is ambiguous and is actually specified by the choice of commuting cut-and-join morphisms, acting along the edges of the hypercube — this promotes it to Abelian quiver, and KR polynomial is a Poincare polynomial of associated complex, just in the original Khovanov’s construction at N = 2. This second step is still somewhat sophisticated — though incomparably simpler than its conventional matrix-factorization counterpart. In this paper we concentrate on the first step, and provide just a mnemonic treatment of the second step. Still, this is enough to demonstrate that all the currently known examples of KR polynomials in the fundamental representation can be easily reproduced in this new approach. As additional bonus we get a simple description of the DGR relation between KR polynomials and superpolynomials and demonstrate that the difference between reduced and unreduced cases, which looks essential at KR level, practically disappears after transition to superpolynomials. However, a careful derivation of all these results from cohomologies of cut-and-join morphisms remains for further studies.

Abstract: Optimization without Calculus Chapter Summary The Arithmetic Mean-Geometric Mean Inequality An Application of the AGM Inequality: the Number e Extending the AGM Inequality Optimization Using the AGM Inequality The Holder and Minkowski Inequalities Cauchy's Inequality Optimizing using Cauchy's Inequality An Inner Product for Square Matrices Discrete Allocation Problems Geometric Programming Chapter Summary An Example of a GP Problem Posynomials and the GP Problem The Dual GP Problem Solving the GP Problem Solving the DGP Problem Constrained Geometric Programming Basic Analysis Chapter Summary Minima and Infima Limits Completeness Continuity Limsup and Liminf Another View Semi-Continuity Convex Sets Chapter Summary The Geometry of Real Euclidean Space A Bit of Topology Convex Sets in RJ More on Projections Linear and Affine Operators on RJ The Fundamental Theorems Block-Matrix Notation Theorems of the Alternative Another Proof of Farkas' Lemma Gordan's Theorem Revisited Vector Spaces and Matrices Chapter Summary Vector Spaces Basic Linear Algebra LU and QR Factorization The LU Factorization Linear Programming Chapter Summary Primal and Dual Converting a Problem to PS Form Duality Theorems The Basic Strong Duality Theorem Another Proof Proof of Gale's Strong Duality Theorem Some Examples The Simplex Method Yet Another Proof The Sherman-Morrison-Woodbury Identity An Example of the Simplex Method Another Example Some Possible Difficulties Topics for Projects Matrix Games and Optimization Chapter Summary Two-Person Zero-Sum Games Deterministic Solutions Randomized Solutions Symmetric Games Positive Games Example: The "Bluffing" Game Learning the Game Non-Constant-Sum Games Differentiation Chapter Summary Directional Derivative Partial Derivatives Some Examples Gateaux Derivative Frechet Derivative The Chain Rule Convex Functions Chapter Summary Functions of a Single Real Variable Functions of Several Real Variables Sub-Differentials and Sub-Gradients Sub-Gradients and Directional Derivatives Functions and Operators Convex Sets and Convex Functions Convex Programming Chapter Summary The Primal Problem From Constrained to Unconstrained Saddle Points The Karush-Kuhn-Tucker Theorem On Existence of Lagrange Multipliers The Problem of Equality Constraints Two Examples The Dual Problem Nonnegative Least-Squares Solutions An Example in Image Reconstruction Solving the Dual Problem Minimum One-Norm Solutions Iterative Optimization Chapter Summary The Need for Iterative Methods Optimizing Functions of a Single Real Variable The Newton-Raphson Approach Approximate Newton-Raphson Methods Derivative-Free Methods Rates of Convergence Descent Methods Optimizing Functions of Several Real Variables Auxiliary-Function Methods Projected Gradient-Descent Methods Feasible-Point Methods Quadratic Programming Simulated Annealing Solving Systems of Linear Equations Chapter Summary Arbitrary Systems of Linear Equations Regularization Nonnegative Systems of Linear Equations Regularized SMART and EMML Block-Iterative Methods Conjugate-Direction Methods Chapter Summary Iterative Minimization Quadratic Optimization Conjugate Bases for RJ The Conjugate Gradient Method Krylov Subspaces Extensions of the CGM Operators Chapter Summary Operators Contraction Operators Orthogonal-Projection Operators Two Useful Identities Averaged Operators Gradient Operators Affine-Linear Operators Paracontractive Operators Matrix Norms Looking Ahead Chapter Summary Sequential Unconstrained Minimization Examples of SUM Auxiliary-Function Methods The SUMMA Class of AF Methods Bibliography Index Exercises appear at the end of each chapter.

Abstract: We elaborate on the simple alternative from arXiv:1308.5759 to the matrix-factorization construction of Khovanov-Rozansky (KR) polynomials for arbitrary knots and links in the fundamental representation of arbitrary SL(N). Construction consists of 2 steps: first, with every link diagram with m vertices one associates an m-dimensional hypercube with certain q-graded vector spaces, associated to its 2^m vertices. A generating function for q-dimensions of these spaces is what we suggest to call the primary T-deformation of HOMFLY polynomial -- because, as we demonstrate, it can be explicitly reduced to calculations of ordinary HOMFLY polynomials, i.e. to manipulations with quantum R-matrices. The second step is a certain minimization of residues of this new polynomial with respect to T+1. Minimization is ambiguous and is actually specified by the choice of commuting cut-and-join morphisms, acting along the edges of the hypercube -- this promotes it to Abelian quiver, and KR polynomial is a Poincare polynomial of associated complex, just in the original Khovanov's construction at N=2. This second step is still somewhat sophisticated -- though incomparably simpler than its conventional matrix-factorization counterpart. In this paper we concentrate on the first step, and provide just a mnemonic treatment of the second step. Still, this is enough to demonstrate that all the currently known examples of KR polynomials in the fundamental representation can be easily reproduced in this new approach. As additional bonus we get a simple description of the DGR relation between KR polynomials and superpolynomials and demonstrate that the difference between reduced and unreduced cases, which looks essential at KR level, practically disappears after transition to superpolynomials. However, a careful derivation of all these results from cohomologies of cut-and-join morphisms remains for further studies.

Abstract: We describe a framework in which it is possible to develop and implement algorithms for the approximation of invariant measures of dynamical systems with a given bound on the error of the approximation. Our approach is based on a general statement on the approximation of fixed points for operators between normed vector spaces, allowing an explicit estimation of the error. We show the flexibility of our approach by applying it to piecewise expanding maps and to maps with indifferent fixed points. We show how the required estimations can be implemented to compute invariant densities up to a given error in the $L^{1}$ or $L^\infty $ distance. We also show how to use this to compute an estimation with certified error for the entropy of those systems. We show how several related computational and numerical issues can be solved to obtain working implementations and experimental results on some one dimensional maps.

Abstract: . Compact closed categories have found applications in modeling quantum information protocols by Abramsky-Coecke. They also provide semantics for Lambek's pregroup algebras, applied to formalizing the grammatical structure of natural language, and are implicit in a distributional model of word meaning based on vector spaces. Specifically, in previous work Coecke-Clark- Sadrzadeh used the product category of pregroups with vector spaces and provided a distributional model of meaning for sentences. We recast this theory in terms of strongly monoidal functors and advance it via Frobenius algebras over vector spaces. The former are used to formalize topological quantum field theories by Atiyah and Baez-Dolan, and the latter are used to model classical data in quantum protocols by Coecke-Pavlovic-Vicary. The Frobenius algebras enable us to work in a single space in which meanings of words, phrases, and sentences of any structure live. Hence we can compare meanings of different language constructs and enhance the applicability of the theory. We report on experimental results on a number of language tasks and verify the theoretical predictions. Introduction. Compact closed categories were first introduced by Kelly [19] in early 1970's. Some thirty years later they found applications in quantum mechanics [1], whereby the vector space foundations of quantum mechanics were recasted in a higher order language and quantum protocols such as teleportation found succinct conceptual proofs. Compact closed categories are complete with regard to a pictorial calculus [19, 35]; this calculus is used to depict and reason about information flows in entangled quantum states modeled in tensor spaces, the phenomena that were considered to be mysteries of quantum mechanics and the Achilles heel of quantum logic [4]. The pictorial calculus revealed the multi-linear algebraic level needed for proving quantum information protocols and simplified the reasoning thereof to a great extent, by hiding the underlying additive vector space structure. Most quantum protocols rely on classical, as well as quantum, data flow. In the work of [1], this classical data flow was modeled using bi-products defined over a compact closed category. However, the pictorial calculus could not extend well to bi-products, and their categorical axiomatization was not as clear as the built-in monoidal tensor of the category.

Abstract: Representation theory was created at the end of the 19th century through the work of F G Frobenius (1849–1917) and I Schur (1875–1941). Recall that groups often occur in nature not as an abstract entity but as the set of symmetries of some natural object X, i.e. G Aut(X). From this point of view, a representation ρ of a group G on a vector space V (over the field C of complex numbers, say) is an action of G on V as linear transformations, i.e. a group homomorphism

Abstract: We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2) at a root of unity indexed by generic complex numbers. This extends to cobordisms the non-semi-simple invariants defined in (arXiv:1202.3553) including the Kashaev invariant of links. Here the modular category framework does not apply and we use the ``universal construction''. Our TQFT provides a monoidal functor from a category of surfaces and their cobordisms into the category of graded finite dimensional vector spaces and their degree 0-morphisms and depends on the choice of a root of unity of order 2r. The functor is always symmetric monoidal but for even values of r the braiding on GrVect has to be the super-symmetric one, thus our TQFT may be considered as a super-TQFT. In the special case r=2 our construction yields a TQFT for a canonical normalization of Reidemeister torsion and we re-prove the classification of Lens spaces via the non-semi-simple quantum invariants defined in (arXiv:1202.3553). We prove that the representations of mapping class groups and Torelli groups resulting from our constructions are potentially more sensitive than those obtained from the standard Reshetikhin-Turaev functors; in particular we prove that the action of the bounding pairs generators of the Torelli group has always infinite order.

Abstract: When a sequence of numbers is slowly converging, it can be transformed into a new sequence which, under some assumptions, converges faster to the same limit. One of the best-known sequence transformations is the Shanks transformation, which can be recursively implemented by the $\varepsilon$-algorithm of Wynn. This transformation and this algorithm have been extended (in two different ways) to a sequence of elements of a topological vector space $E$. In this paper, we present new algorithms for implementing these topological Shanks transformations. They no longer require the manipulation of elements of the algebraic dual space $E^*$ of $E$, nor do they use the duality product inside the rules of the algorithms; they need the storage of fewer elements of $E$, and the stability is improved. They also allow us to prove convergence and acceleration results for some types of sequences. Various applications involving sequences of vectors or matrices show the interest of the new algorithms.

Abstract: In this article we study the Cauchy problem for a new class of parabolic-type pseudodifferential equations with variable coefficients for which the fundamental solutions are transition density functions of Markov processes in the four dimensional vector space over the field of p-adic numbers.

Abstract: We investigate the duality between algebraic and coalgebraic recognition of languages to derive a generalization of the local version of Eilenberg’s theorem. This theorem states that the lattice of all boolean algebras of regular languages over an alphabet Σ closed under derivatives is isomorphic to the lattice of all pseudovarieties of Σ-generated monoids. By applying our method to different categories, we obtain three related results: one, due to Gehrke, Grigorieff and Pin, weakens boolean algebras to distributive lattices, one due to Polak weakens them to join-semilattices, and the last one considers vector spaces over ℤ2.

Abstract: Canonical correlation analysis (CCA) is a widely used statistical technique to capture correlations between two sets of multi-variate random variables and has found a multitude of applications in computer vision, medical imaging and machine learning. The classical formulation assumes that the data live in a pair of vector spaces which makes its use in certain important scientific domains problematic. For instance, the set of symmetric positive definite matrices (SPD), rotations and probability distributions, all belong to certain curved Riemannian manifolds where vector-space operations are in general not applicable. Analyzing the space of such data via the classical versions of inference models is rather sub-optimal. But perhaps more importantly, since the algorithms do not respect the underlying geometry of the data space, it is hard to provide statistical guarantees (if any) on the results. Using the space of SPD matrices as a concrete example, this paper gives a principled generalization of the well known CCA to the Riemannian setting. Our CCA algorithm operates on the product Riemannian manifold representing SPD matrix-valued fields to identify meaningful statistical relationships on the product Riemannian manifold. As a proof of principle, we present results on an Alzheimer’s disease (AD) study where the analysis task involves identifying correlations across diffusion tensor images (DTI) and Cauchy deformation tensor fields derived from T1-weighted magnetic resonance (MR) images.

Abstract: Let $X$ be a first countable Hausdorff topological group. The limit of a sequence in $X$ defines a function denoted by $\lim$ from the set of all convergence sequences to $X$. This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing $\lim$ with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. Recently \c{C}akall{\i} has extended the concept to topological group setting and introduced the concepts of $G$-sequential compactness, $G$-sequential continuity and sequential connectedness. In this paper we give a further investigation of $G$-sequential continuity in topological groups.

Abstract: Every Grassmannian, in its Plucker embedding, is defined by quadratic polynomials. We prove a vast, qualitative, generalisation of this fact to what we call Plucker varieties. A Plucker variety is in fact a family of varieties in exterior powers of vector spaces that, like the Grassmannian, is functorial in the vector space and behaves well under duals. A special case of our result says that for each fixed natural number k, the k-th secant variety of any Plucker-embedded Grassmannian is defined in bounded degree independent of the Grassmannian. Our approach is to take the limit of a Plucker variety in the dual of a highly symmetric space known as the infinite wedge, and to prove that up to symmetry the limit is defined by finitely many polynomial equations. For this we prove the auxilliary result that for every natural number p the space of p-tuples of infinite-by-infinite matrices is Noetherian modulo row and column operations. Our results have algorithmic counterparts: every bounded Plucker variety has a polynomial-time membership test, and the same holds for Zariski-closed, basis-independent properties of p-tuples of matrices.

Abstract: This paper is devoted to investigate some characteristic features of complex numbers and functions in terms of non-Newtonian calculus. Following Grossman and Katz, (Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts, 1972), we construct the field of -complex numbers and the concept of -metric. Also, we give the definitions and the basic important properties of -boundedness and -continuity. Later, we define the space of -continuous functions and state that it forms a vector space with respect to the non-Newtonian addition and scalar multiplication and we prove that is a Banach space. Finally, Multiplicative calculus (MC), which is one of the most popular non-Newtonian calculus and created by the famous exp function, is applied to complex numbers and functions to investigate some advance inner product properties and give inclusion relationship between and the set of -differentiable functions.

Abstract: Let $q$ be an odd prime power and let $X(m,q)$ be the set of symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$. The partition of $X(m,q)$ induced by the action of the general linear group gives rise to a commutative translation association scheme. We give explicit expressions for the eigenvalues of this scheme in terms of linear combinations of generalised Krawtchouk polynomials. We then study $d$-codes in this scheme, namely subsets $Y$ of $X(m,q)$ with the property that, for all distinct $A,B\in Y$, the rank of $A-B$ is at least $d$. We prove bounds on the size of a $d$-code and show that, under certain conditions, the inner distribution of a $d$-code is determined by its parameters. Constructions of $d$-codes are given, which are optimal among the $d$-codes that are subgroups of $X(m,q)$. Finally, with every subset $Y$ of $X(m,q)$, we associate two classical codes over $\mathbb{F}_q$ and show that their Hamming distance enumerators can be expressed in terms of the inner distribution of $Y$. As an example, we obtain the distance enumerators of certain cyclic codes, for which many special cases have been previously obtained using long ad hoc calculations.

Abstract: Spatial transformations are mappings between locations of a $d$-dimensional space and are commonly used in computer vision and image analysis. Many of the spatial transformation sets have a group structure and can be represented by matrix groups. In the computer vision and image analysis fields there is a recent and growing interest in performing analyses on spatial transformations data. Differential and Riemannian geometry have been used as a framework to endow the set of spatial transformations with a metric space structure, allowing the extension of the standard analysis techniques defined on vector spaces. This paper presents a review of the concepts and an overview of approaches to computing Riemannian geodesics on spatial transformation groups. The paper is aimed at providing a bridge for researchers from computer vision and image analysis fields to fill in the gap between differential geometry and computer vision and imaging disciplines. Some application examples are shown to illustrate the use of ...

Abstract: We establish an equivalence of two systems of equations of one-dimensional shallow water models describing the propagation of surface waves over even and sloping bottoms. For each of these systems, we obtain formulas for the general form of their nondegenerate solutions, which are expressible in terms of solutions of the Darboux equation. The invariant solutions of the Darboux equation that we find are simplest representatives of its essentially different exact solutions (those not related by invertible point transformations). They depend on 21 arbitrary real constants; after “proliferation” formulas derived by methods of group theory analysis are applied, they generate a 27-parameter family of essentially different exact solutions. Subsequently using the derived infinitesimal “proliferation” formulas for the solutions in this family generates a denumerable set of exact solutions, whose linear span constitutes an infinite-dimensional vector space of solutions of the Darboux equation. This vector space of solutions of the Darboux equation and the general formulas for nondegenerate solutions of systems of shallow water equations with even and sloping bottoms give an infinite set of their solutions. The “proliferation” formulas for these systems determine their additional nondegenerate solutions. We also find all degenerate solutions of these systems and thus construct a database of an infinite set of exact solutions of systems of equations of the one-dimensional nonlinear shallow water model with even and sloping bottoms.

Abstract: We investigate the size of the distance set determined by two subsets of finite dimensional vector spaces over finite fields. A lower bound of the size is given explicitly in terms of cardinalities of the two subsets. As a result, we improve upon the results by Rainer Dietmann. In the case that one of the subsets is a product set, we obtain further improvement on the estimate.

Abstract: In this paper, first we show that $(\g,[\cdot,\cdot],\alpha)$ is a hom-Lie algebra if and only if $(\Lambda \g^*,\alpha^*,d)$ is an $(\alpha^*,\alpha^*)$-differential graded commutative algebra. Then, we revisit representations of hom-Lie algebras, and show that there are a series of coboundary operators. We also introduce the notion of an omni-hom-Lie algebra associated to a vector space and an invertible linear map. We show that regular hom-Lie algebra structures on a vector space can be characterized by Dirac structures in the corresponding omni-hom-Lie algebra. The underlying algebraic structure of the omni-hom-Lie algebra is a hom-Leibniz algebra, or a hom-Lie 2-algebra.

Abstract: It is well known that the edge vector space of an oriented graph can be decomposed in terms of cycles and cocycles (alias cuts, or bonds), and that a basis for the cycle and the cocycle spaces can be generated by adding and removing edges to an arbitrarily chosen spanning tree. In this paper we show that the edge vector space can also be decomposed in terms of cycles and the generating edges of cocycles (called cochords), or of cocycles and the generating edges of cycles (called chords). From this observation follows a construction in terms of oblique complementary projection operators. We employ this algebraic construction to prove several properties of unweighted Kirchhoff-Symanzik matrices, encoding the mutual superposition between cycles and cocycles. In particular, we prove that dual matrices of planar graphs have the same spectrum (up to multiplicities). We briefly comment on how this construction provides a refined formalization of Kirchhoff's mesh analysis of electrical circuits, which has lately been applied to generic thermodynamic networks.