On the classification of APN functions up to dimension five

We classify the almost perfect nonlinear (APN) functions in dimensions 4 and 5 up to affine and CCZ equivalence using backtrack programming and give a partial model for the complexity of such a search. In particular, we demonstrate that up to dimension 5 any APN function is CCZ equivalent to a power function, while it is well known that in dimensions 4 and 5 there exist APN functions which are not extended affine (EA) equivalent to any power function. We further calculate the total number of APN functions up to dimension 5 and present a new CCZ equivalence class of APN functions in dimension 6.     

几乎完全非线性多项式函数间的EA等价性

简述了几乎完全非线性(APN)多项式函数的研究现状,讨论了两类几乎完全非线性多项式函数间的扩张仿射(EA)等价性,给出了验证EA等价的一般方法.     

带幂非线性项变系数非线性波动方程的李对称分类与等价性变换

从微分方程群理论分析角度,研究了一类含有3个任意函数和2个幂非线性项的变系数非线性波动方程.由于方程具有很强的任意性和非线性项,可通过等价性变换寻找方程的不变对称分类.首先给出了等价性变换的一般结果,其中包括一些包含任意元的非局部变换.然后对所研究的方程,利用广义扩展等价群和条件等价群给出了方程的完全对称分类.最后获得并分析了方程的特殊类相似解.     

A new class of generalized almost perfect nonlinear power function

As a generalization of almost perfect nonlinear functions, generalized almost perfect nonlinear (GAPN) functions defined over finite fields of odd characteristic have also important applications in cryptography and finite geometry. In this paper, we mainly focus on GAPN power functions, and a new class of such function (up to EA equivalence) is obtained. Furthermore, using our method, almost all the known GAPN power functions can be confirmed in a simple way.     

Equivalence classes of functions between finite groups

Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser (“graph”) equivalence are inequivalent under the finer (“bundle”) equivalence. Here we relate graphs to transversals and splitting relative difference sets (RDSs) and introduce an intermediate relation, canonical equivalence, to aid in distinguishing the classes. We identify very precisely the conditions under which a graph equivalence determines a bundle equivalence, using transversals and extensions. We derive a new and easily computed algebraic measure of nonlinearity for a function f, calculated from the image of its coboundary ∂f. This measure is preserved by bundle equivalence but not by the coarser equivalences. It takes its minimum value if f is a homomorphism, and takes its maximum value if the graph of f contains a splitting RDS.     

Equivalence classes of Niho bent functions

Equivalence classes of Niho bent functions are in one-to-one correspondence with equivalence classes of ovals in a projective plane. Since a hyperoval can produce several ovals, each hyperoval is associated with several inequivalent Niho bent functions. For all known types of hyperovals we described the equivalence classes of the corresponding Niho bent functions. For some types of hyperovals the number of equivalence classes of the associated Niho bent functions are at most 4. In general, the number of equivalence classes of associated Niho bent functions increases exponentially as the dimension of the underlying vector space grows. In small dimensions the equivalence classes were considered in detail.

     

Weaving hadamard matrices with maximum excess and classes with small excess

Weaving is a matrix construction developed in 1990 for the purpose of obtaining new weighing matrices. Hadamard matrices obtained by weaving have the same orders as those obtained using the Kronecker product, but weaving affords greater control over the internal structure of matrices constructed, leading to many new Hadamard equivalence classes among these known orders. It is known that different classes of Hadamard matrices may have different maximum excess. We explain why those classes with smaller excess may be of interest, apply the method of weaving to explore this question, and obtain constructions for new Hadamard matrices with maximum excess in their respective classes. With this method, we are also able to construct Hadamard matrices of near‐maximal excess with ease, in orders too large for other by‐hand constructions to be of much value. We obtain new lower bounds for the maximum excess among Hadamard matrices in some orders by constructing candidates for the largest excess. For example, we construct a Hadamard matrix with excess 1408 in order 128, larger than all previously known values. We obtain classes of Hadamard matrices of order 96 with maximum excess 912 and 920, which demonstrates that the maximum excess for classes of that order may assume at least three different values. Since the excess of a woven Hadamard matrix is determined by the row sums of the matrices used to weave it, we also investigate the properties of row sums of Hadamard matrices and give lists of them in small orders. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 233–255, 2004.     

Dyck Paths with Peaks Avoiding or Restricted to a Given Set

In this paper we focus on Dyck paths with peaks avoiding or restricted to an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We also discuss a special case that requires all peak heights to either lie on or avoid a congruence class (or classes) modulo k . The case when   k = 2  is especially interesting. The two sequences for this case are proved, combinatorially as well as algebraically, to be the Motzkin numbers and the Riordan numbers. We introduce the concept of shift equivalence on sequences, which in turn induces an equivalence relation on avoiding and restricted sets. Several interesting equivalence classes whose representatives are well-known sequences are given as examples.     

A transfer algorithm for hierarchical clustering

Transfer algorithms are usually used to optimize an objective function that is defined on the set of partitions of a finite set X. In this paper we define an equivalence relation ? on the set of fuzzy equivalence relations on X and establish a bijection from the set of hierarchies on X to the set of equivalence classes with respect to ?. Thus, hierarchies can be identified with fuzzy equivalence relations and the transfer algorithm can be modified in order to optimize an objective function that is defined on the set of hierarchies on X.     

Note on Morita Equivalence in Ring Extensions

It seems that Morita invariance judges of the importance of classes of ring extensions concerned. Miyashita introduced the notion of Morita equivalence in ring extensions, and he showed that the classes of G-Galois extensions and Frobenius extensions are Morita invariant. After that, Ikehata showed that the classes of separable extensions, Hirata separable extensions, symmetric extensions, and QF-extensions are Morita invariant. In this article, we shall prove that the classes of several extensions are Morita invariant. Further, we will give an example of the class of ring extensions which is not Morita invariant.     

Equivalence classes of regularly varying functions

A useful method to derive limit results for partial maxima and record values of independent, identically distributed random variables is to start from one specific probability distribution and to extend the result for this distribution to a class of distributions.This method involves an extended theory of regularly varying functions. In this paper, equivalence classes of regularly varying functions (in the extended sense) are studied, which is relevant to the problems mentioned above.     

Universal Affine Classification of Boolean Functions

In this paper we advance a practical solution of the classification problem of Boolean functions by the affine group – the largest group of linear transformations of variables. We show that the affine types (equivalence classes) can be arranged in a unique infinite sequence which contains all previous lists of types. The types are specified by their minimal representatives, spectral invariants, and stabilizer orders. A brief survey of the fundamental transformation groups is included.     

Independence equivalence classes of cycles

The independence equivalence class of a graph G is the set of graphs that have the same independence polynomial as G. Beaton, Brown and Cameron (2019) found the independence equivalence classes of even cycles, and raised the problem of finding the independence equivalence class of odd cycles. The problem is completely solved in this paper.     

The eight variable homogeneous degree three bent functions

We determine the affine equivalence classes of the eight variable degree three homogeneous bent functions using a new algorithm. Our algorithm applies to general bent functions and can systematically determine the automorphism groups. We provide a partial verification of the enumeration of eight variable degree three homogeneous bent functions obtained by Meng et al. We determine the affine equivalence classes of these functions.     

Admissible equivalence transformations for linearization of nonlinear wave type equations

In the present paper, we consider a general family of two‐dimensional wave equations, which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated the existence problem of point transformations that lead mappings between linear and nonlinear members of particular families and determined the structure of the nonlinear terms of linearizable equations. We have also given examples about some equivalence transformations between linear and nonlinear equations and obtained exact solutions of nonlinear equations via the linear ones.     

Algebras of multiplace functions for signatures containing antidomain

We define antidomain operations for algebras of multiplace partial functions. For all signatures containing composition, the antidomain operations and any subset of intersection, preferential union and fixset, we give finite equational or quasiequational axiomatisations for the representation class. We do the same for the question of representability by injective multiplace partial functions. For all our representation theorems, it is an immediate corollary of our proof that the finite representation property holds for the representation class. We show that for a large set of signatures, the representation classes have equational theories that are coNP-complete.     

Multiresolution Equivalence and Path Connectedness

An equivalence relation between multiresolution analyses was first introduced in 1996; an analogous definition for generalized multiresolution analyses was given in 2010. This article describes the relationship between the two notions and shows that both types of equivalence classes are path connected in an operator-theoretic sense. The GMRA paths are restricted to canonical GMRAs, and it is shown that whenever two MRAs in L ²(?) are equivalent, the GMRA path construction between their corresponding canonical GMRAs yields the natural analog of the MRA path. Examples are provided of GMRA paths that are distinct from MRA paths.     

亏格为4黎曼曲面上循环群作用的拓扑分类

本文主要考虑循环群作用 Riemann曲面的分类问题 ,我们列出了所有的循环群作用亏格为 4Riem ann曲面的拓扑分类和弱拓扑分类     

A class of gap functions for variational inequalities

Recently Auchmuty (1989) has introduced a new class of merit functions, or optimization formulations, for variational inequalities in finite-dimensional space. We develop and generalize Auchmuty's results, and relate his class of merit functions to other works done in this field. Especially, we investigate differentiability and convexity properties, and present characterizations of the set of solutions to variational inequalities. We then present new descent algorithms for variational inequalities within this framework, including approximate solutions of the direction finding and line search problems. The new class of merit functions include the primal and dual gap functions, introduced by Zuhovickii et al. (1969a, 1969b), and the differentiable merit function recently presented by Fukushima (1992); also, the descent algorithm proposed by Fukushima is a special case from the class of descent methods developed in this paper. Through a generalization of Auchmuty's class of merit functions we extend those inherent in the works of Dafermos (1983), Cohen (1988) and Wu et al. (1991); new algorithmic equivalence results, relating these algorithm classes to each other and to Auchmuty's framework, are also given.Corresponding author.     

Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions

We define an equivalence relation on integer compositions and show that two ribbon Schur functions are identical if and only if their defining compositions are equivalent in this sense. This equivalence is completely determined by means of a factorization for compositions: equivalent compositions have factorizations that differ only by reversing some of the terms. As an application, we can derive identities on certain Littlewood-Richardson coefficients.Finally, we consider the cone of symmetric functions having a nonnnegative representation in terms of the fundamental quasisymmetric basis. We show the Schur functions are among the extremes of this cone and conjecture its facets are in bijection with the equivalence classes of compositions.