Equivalence Classes Of Representations Research Articles

Geometric bijections between spanning subgraphs and orientations of a graph

AbstractLet be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy‐to‐describe bijections between spanning trees of and ‐compatible orientations, where the ‐compatible orientations are the representatives of equivalence classes of orientations up to cycle–cocycle reversal that are determined by a cycle signature and a cocycle signature . Their bijections are geometric because the construction comes from zonotopal subdivisions. In this paper, we extend the geometric bijections to subgraph‐orientation correspondences. Moreover, we extend the geometric constructions accordingly. Our proofs are combinatorial, which do not make use of the zonotopes. We also provide geometric proofs for partial results, which make use of zonotopal tiling, relate to Backman, Baker, and Yuen's method, and motivate our combinatorial constructions. Finally, we explain that the main results hold for regular matroids.

Algebraic Structures in Group-theoretical Fusion Categories

It was shown by Ostrik (Int. Math. Res. Not. 2003(27), 1507–1520 2003) and Natale (SIGMA Symmetry Integrability Geom. Methods Appl. 13, 042 2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We generalize this result by constructing explicit Morita equivalence class representatives of indecomposable, separable algebras in group-theoretical fusion categories. This is achieved by providing the free functor Φ from fusion category to a category of bimodules in the original category with a (Frobenius) monoidal structure. Our algebras of interest are then constructed as the image of twisted group algebras under Φ. We also show that twisted group algebras admit the structure of Frobenius algebras in a pointed fusion category, and as a consequence, our algebras are Frobenius algebras in a group-theoretical fusion category. They also enjoy several good algebraic properties.

Медиана для нечётного числа отношений линейного порядка и её использование в задачах группового выбора

We consider the problem of constructing a median for an odd set of linear order relations defined on a finite set A = {a1,a2,..., an}, which is also sought in the class of linear order relations. We arrive at a similar problem when considering some group choice problems. The distance between binary relations is the Hamming distance between their adjacency matrices. In the case under consideration, the binary relation ρ, which has the minimum total distance to the given set of binary relations, is the median for these relations and, moreover, is unique. However, this median is not always transitive (and in this case is neither linear order, nor even a quasi-order), and therefore cannot be taken as a solution to a given problem. However, the median ρ necessarily belongs to the set LA[n] (of linear asymmetric binary relations on A), to which, in particular, all linear orders on A also belong. Some properties of binary relations from LA[n] are investigated. The concepts of “almost optimal” and Δ-optimal relations are introduced, which are linear orders and, at the same time, exact solutions of the stated problem. Algorithms for finding them are given, based on the obtained statements about binary relations from LA[n] and having polynomial computational complexity. An equivalence relation on the set LA[n] is considered, which allows one to divide this set into equivalence classes, the number of which Kn is much less than the number of elements in LA[n]. For example, |LA[5] | = 1024, K5 = 12. Thus, each binary relation from LA[n] is equivalent to exactly one of the Kn representatives of the equivalence classes and, therefore, has its main properties. But then the study of a wide class of problems can be reduced to considering a relatively small set of them. The process of finding the specified set of equivalence class representatives is illustrated for n = 2,3,4, 5. A method for solving the problem posed is also given, using the representation of binary relations in the form of graphs (the method of selecting the minimum sets of contour representatives in the median ρ), which has exponential computational complexity.

Enumeration and Generation of $\rm PSL$ Equivalence Classes for Quad-Phase Codes of Odd Length

Quad-phase codes of a given length can be grouped into equivalence classes based on operations preserving autocorrelation peak sidelobe level, or “ ${\rm PSL}$ -preserving operators.” The task of enumerating these equivalence classes is facilitated by establishing a relationship with the problem of enumerating equivalence classes of $2\times N$ binary grids with respect to a pair of binary grid symmetries. We show this connection by a one-to-one mapping between even-length quad-phase codes and $2\times N$ binary grids. When $N$ is even, this mapping allows the known enumeration of $2\times N$ binary grids to be applied to the enumeration of even-length quad-phase code PSL equivalence classes. Furthermore, this equivalence instructs the development of a simpler algorithm for visiting single representatives of quad-phase-code equivalence classes relative to the operations that preserve PSL. Lastly, we conduct exhaustive searches to determine the optimal ${\rm PSL}$ for quad-phase codes of lengths 25 and 26 (for the smaller of these lengths, the search space contains $4^{25}=2^{50}$ , or roughly 1 quadrillion codes). One of the devices used to narrow the search space while achieving an exhaustive search is to seed the search with the equivalence class representatives of length $N=6$ . The optimal ${\rm PSL}$ for length-25 quad-phase codes is determined to be 2, achieved by the codes in a single size-32 equivalence class. For length 26, the optimal ${\rm PSL}$ is found to be $\sqrt{5}$ , achieved by $604\,480$ codes belonging to 9445 equivalence classes of size 64.

Symmetry reduction in CSP model checking

We present an extension of FDR, the model checker for the process algebra CSP, that exploits symmetry to reduce the size of the state space searched. We define what it means for a process to be symmetric with respect to a group of permutations on the transition labels. We factor the state space of the search by symmetry equivalence, mapping each state to a representative of its equivalence class, thereby considering all symmetric states together. We prove a powerful syntactic result, identifying conditions under which a process will be symmetric in a particular type. We show how to implement such a search using the powerful technique of supercombinators used in the implementation of FDR: we identify conditions on a supercombinator for it to be symmetric and explain how to apply a permutation to a state. Finally, we present a novel efficient technique for calculating representatives of equivalence classes, which normally finds unique representatives; our experiments suggest that this technique typically works faster than other techniques and in particular scales better.

Control Systems on the Engel Group

We consider control affine systems, as well as cost-extended control systems, on the (four-dimensional) Engel group. Specifically, we classify the full-rank left-invariant control affine systems (under both detached feedback equivalence and strongly detached feedback equivalence). The cost-extended control systems with quadratic cost are then classified (under cost equivalence), as are their associated Hamilton-Poisson systems (up to affine isomorphism). In all cases, we exhibit a complete list of equivalence class representatives.

K-Abelian Equivalence and Rationality

Two words u and v are said to be k-abelian equivalent if, for each word x of length at most k, the number of occurrences of x as a factor of u is the same as for v. We study some combinatorial properties of k-abelian equivalence classes. Our starting point is a characterization of k-abelian equivalence by rewriting, so-called k-switching. We show that the set of lexicographically least representatives of equivalence classes is a regular language. From this we infer that the sequence of the numbers of equivalence classes is $$\mathbb {N}$$-rational. We also show that the set of words defining k-abelian singleton classes is regular.

Enumeration and Generation of PSL Equivalence Classes for Quad-Phase Codes of Even Length

Quad-phase codes of a given length can be grouped into equivalence classes based on operations preserving autocorrelation peak sidelobe level, or “PSL-preserving operators.” The task of enumerating these equivalence classes is facilitated by establishing a relationship with the problem of enumerating equivalence classes of 2 x N binary grids with respect to a pair of binary grid symmetries. We show this connection by a one-to-one mapping between even-length quadphase codes and 2 x N binary grids. When N is even, this mapping allows the known enumeration of 2 x N binary grids to be applied to the enumeration of even-length quad-phase code PSL equivalence classes. Furthermore, this equivalence instructs the development of a simpler algorithm for visiting single representatives of quad-phasecode equivalence classes relative to the operations that preserve PSL. Lastly, we conduct exhaustive searches to determine the optimal PSL for quad-phase codes of lengths 25 and 26 (for the smaller of these lengths, the search space contains 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">25</sup> = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">50</sup> , or roughly 1 quadrillion codes). One of the devices used to narrow the search space while achieving an exhaustive search is to seed the search with the equivalence class representatives of length N = 6. The optimal PSL for length-25 quad-phase codes is determined to be 2, achieved by the codes in a single size-32 equivalence class. For length 26, the optimal PSL is found to be √5, achieved by 604480 codes belonging to 9445 equivalence classes of size 64.

Equivalence of Control Systems on the Pseudo-Orthogonal Group SO (2, 1)0

Abstract We consider left-invariant control affine systems on the matrix Lie group SO (2, 1)0. A classification, under state space equivalence, of all such full-rank control systems is obtained. First, we identify certain subsets on which the group of Lie algebra automorphisms act transitively. We then systematically identify equivalence class representatives (for single-input, two-input and three-input control systems). A brief comparison of these classification results with existing results concludes the paper.

The classification of real singularities using Singular. Part II: The structure of the equivalence classes of the unimodal singularities

In the classification of real singularities by Arnold et al. (1985), normal forms, as representatives of equivalence classes under right equivalence, are not always uniquely determined. We describe the complete structure of the equivalence classes of the unimodal real singularities of corank 2. In other words, we explicitly answer the question which normal forms of different type are equivalent, and how a normal form can be transformed within the same equivalence class by changing the value of the parameter. This provides new theoretical insights into these singularities and has important consequences for their algorithmic classification.

Product Vacua with Boundary States and the Classification of Gapped Phases

We address the question of the classification of gapped ground states in one dimension that cannot be distinguished by a local order parameter. We introduce a family of quantum spin systems on the one-dimensional chain that have a unique gapped ground state in the thermodynamic limit that is a simple product state but which on the left and right half-infinite chains, have additional zero energy edge states. The models, which we call Product Vacua with Boundary States (PVBS), form phases that depend only on two integers corresponding to the number of edge states at each boundary. They can serve as representatives of equivalence classes of such gapped ground states phases and we show how the AKLT model and its $SO(2J+1)$-invariant generalizations fit into this classification.

A Classification of Sub-Riemannian Structures on the Heisenberg Groups

We apply Williamson's theorem for the diagonalization of quadratic forms by symplectic matrices to sub-Riemannian (and Riemannian) structures on the Heisenberg groups. A classification of these manifolds, under isometric Lie group automorphisms, is obtained. A (parametrized) list of equivalence class representatives is identified; a geometric characterization of this equivalence relation is provided. A corresponding classification of (drift-free) invariant optimal control problems is exhibited.

Strict intersection types for the Lambda Calculus

This article will show the usefulness and elegance of strict intersection types for the Lambda Calculus, that are strict in the sense that they are the representatives of equivalence classes of types in the BCD-system [Barendregt et al. 1983]. We will focus on the essential intersection type assignment; this system is almost syntax directed, and we will show that all major properties hold that are known to hold for other intersection systems, like the approximation theorem, the characterization of (head/strong) normalization, completeness of type assignment using filter semantics, strong normalization for cut-elimination and the principal pair property. In part, the proofs for these properties are new; we will briefly compare the essential system with other existing systems.

Growing words in the free group on two generators

This paper is concerned with minimal-length representatives of equivalence classes of words in $F_2$ under Aut $F_2$. We give a simple inequality characterizing words of minimal length in their equivalence class. We consider an operation that “grows” words from other words, increasing the length, and we study root words—minimal words that cannot be grown from other minimal words. Root words are “as minimal as possible” in the sense that their characterization is the boundary case of the minimality inequality. The property of being a root word is respected by equivalence classes, and the length of each root word is divisible by 4.

Solving Some Quadratic Diophantine Equations with Clifford Algebra

In this work, the equivalence class representatives of integer solutions of the Diophantine equation of the type \({{a_1x_1^2+ .\,.\,. + a_px_p^2 = a_{p+1}x^2_{p+1} + .\,.\,. +a_{p+q}x^2_{p+q} +a_1x^2_{n+1} (a_i > 0,i=1, .\,.\,.\,,p+q,x_{n+1}\neq0)}}\) are found using simple reflections of orthogonal vectors, manipulated using the Clifford algebra over orthogonal spaces Rp,q. These solutions are obtained from the application of a useful Lemma: given two different non-zero vectors of the same norm, we can map one onto the other, or its negative, by means of a simple reflection. With this Lemma, we extend and improve a previous work [1] concerning generalized Pythagorean numbers, which now can be obtained as a Corollary. We also show that our technique is promising for solving others Diophantine equations.

Efficient Symmetry Reduction and the Use of State Symmetries for Symbolic Model Checking

One technique to reduce the state-space explosion problem in temporal logic model checking is symmetry reduction. The combination of symmetry reduction and symbolic model checking by using BDDs suffered a long time from the prohibitively large BDD for the orbit relation. Dynamic symmetry reduction calculates representatives of equivalence classes of states dynamically and thus avoids the construction of the orbit relation. In this paper, we present a new efficient model checking algorithm based on dynamic symmetry reduction. Our experiments show that the algorithm is very fast and allows the verification of larger systems. We additionally implemented the use of state symmetries for symbolic symmetry reduction. To our knowledge we are the first who investigated state symmetries in combination with BDD based symbolic model checking.

Vector Symmetry Reduction

Symmetry reduction is an effective state-space reduction technique for model checking, and works by restricting search to equivalence class representatives with respect to a group of symmetries for a model. A major problem with symmetry reduction techniques is the time taken to compute the representative of a state, which can be prohibitive. In efficient implementations of symmetry reduction, a symmetry is applied to a state as a sequence of operations which swap component identities. We show that vector processing technology, common to modern computers, can be used to implement a vectorised swap operation, which can be incorporated into the representative computation algorithm to accelerate symmetry reduction. Via a worked example, we present details of this vector symmetry reduction method. We have implemented our techniques in the TopSpin symmetry reduction package for the Spin model checker, and present experimental results showing the speedups obtained via vectorisation for two case-studies running on a PowerPC vector processor.

Hilbert’s theorem on positive ternary quartics: A refined analysis

Let X X be an integral plane quartic curve over a field k k , let f f be an equation for X X . We first consider representations ( ∗ ) (*) c f = p 1 p 2 − p 0 2 cf=p_1p_2-p_0^2 (where c ∈ k ∗ c\in k^* and the p i p_i are quadratic forms), up to a natural notion of equivalence. Using the general theory of determinantal varieties we show that equivalence classes of such representations correspond to nontrivial globally generated torsion-free rank one sheaves on X X with a self-duality which are not exceptional, and that the exceptional sheaves are in bijection with the k k -rational singular points of X X . For k = C k=\mathbb {C} , the number of representations ( ∗ ) (*) (up to equivalence) depends only on the singularities of X X , and is determined explicitly in each case. In the second part we focus on the case where k = R k=\mathbb {R} and f f is nonnegative. By a famous theorem of Hilbert, such f f is a sum of three squares of quadratic forms. We use the Brauer group and Galois cohomology to relate identities ( ∗ ∗ ) (**) f = p 0 2 + p 1 2 + p 2 2 f=p_0^2+p_1^2+p_2^2 to ( ∗ ) (*) , and we determine the number of equivalence classes of representations ( ∗ ∗ ) (**) for each f f . Both in the complex and in the real definite case, our results are considerably more precise since they give the number of representations with any prescribed base locus.

Intrinsic geometry on the class of probability densities and exponential families

We present a way of thinking of exponential families as geodesic surfaces in the class of positive functions considered as a (multiplicative) sub-group $G^+$ of the group $G$ of all invertible elements in the algebra $\mathcal{A}$ of all complex bounded functions defined on a measurable space. For that we have to study a natural geometry on that algebra. The class $\mathcal{D}$ of densities with respect to a given measure will happen to be representatives of equivalence classes defining a projective space in $\mathcal{A}$. The natural geometry is defined by an intrinsic group action which allows us to think of the class of positive, invertible functions $G^+$ as a homogeneous space. Also, the parallel transport in $G^+$ and $\mathcal{D}$ will be given by the original group action. Besides studying some relationships among these constructions, we examine some Riemannian geometries and provide a geometric interpretation of Pinsker's and other classical inequalities. Also we provide a geometric reinterpretation of some relationships between polynomial sequences of convolution type, probability distributions on $\mathbb{N}$ in terms of geodesics in the Banach space $\ell_1(\alpha)$.

The design of composite permutations with applications to DES-like S-boxes

This paper presents an iterative construction method for building composite permutations. Its efficiency is based on the concepts of pre-computation and equivalence classes. Equivalence class representatives of permutations on four bits are pre-computed. These class representatives can serve as input to the construction method, however, the results are also of independent interest for applications in cryptography. A well-known example of a cryptosystem using composite permutations for its Substitution boxes (S-boxes) is the Data Encryption Standard (DES). Throughout the paper, DES-like S-boxes are defined as mappings satisfying all design criteria as disclosed by one of the designers of DES. All permutations on four bits with DES-like properties are identified. Starting with pre-computed representatives of classes with such permutations, two iterations of a specialized version of the algorithm are applied to obtain bounds on the minimum differential uniformity and minimum non-linear uniformity of DES-like S-boxes. It is established that the two values cannot be less than eight, and that DES-like S-boxes for which the values are both equal to 12 do exist. In addition, if the non-linear uniformity of each of the four permutations in a DES-like S-box is at most six, as in all DES S-boxes, then its non-linear uniformity cannot be less than ten and its minimum differential uniformity equals 12.