Hall Polynomials for Affine Quivers of Type Am（m≥1）

It is well known that Hall polynomials as structural coefficients play an important role in the structure of Lie algebras and quantum groups. By using the properties of representation categories of affine quivers, the task of computing Hall polynomials for affine quivers can be reduced to counting the numbers of solutions of some matrix equations. This method has been applied to obtain Hall polynomials for indecomposable representations of quivers of type Am（m≥1）     

Partitions and sums of (m,p, c)-sets

The (m, p, c)-sets characterize those sets which contain solutions to every partition-regular system of homogeneous linear equations. We show that if N is partitioned into finitely many classes there is one class and for each (m, p, c) ϵ N × N × N, an (m, p, c) set B(m, p, c) such that all finite sums choosing at most one term from each B(m, p, c) lie in the given class.     

New exact solitary-wave special solutions for the nonlinear dispersive K(m, n) equations

In this paper, we study the nonlinear dispersive K(m, n) equations: u_t + (u^m)_x − (uⁿ)_xxx = 0 which exhibit solutions with solitary patterns. New exact solitary solutions are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the decomposition method in K(m, n) equations. The nonlinear equations K(m, n) are studied for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of K(m, n) equations are established.     

Exact special solitary solutions with compact support for the nonlinear dispersive K(m, n) equations

The nonlinear dispersive K(m, n) equations, u_t−(u^m)_x−(uⁿ)_xxx = 0 which exhibit compactons: solitons with compact support, are studied. New exact solitary solutions with compact support are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the decomposition method in K(m, n) equations. General formulas for the solutions of K(m, n) equations are established.     

Hadamard matrices of order 28m, 36m and 44m

We show that if four suitable matrices of order m exist then there are Hadamard matrices of order 28m, 36m, and 44m. In particular we show that Hadamard matrices of orders 14(q + 1), 18(q + 1), and 22(q + 1) exist when q is a prime power and q ≡ 1 (mod 4).Also we show that if n is the order of a conference matrix there is an Hadamard matrix of order 4mn.As a consequence there are Hadamard matrices of the following orders less than 4000: 476, 532, 836, 1036, 1012, 1100, 1148, 1276, 1364, 1372, 1476, 1672, 1836, 2024, 2052, 2156, 2212, 2380, 2484, 2508, 2548, 2716, 3036, 3476, 3892.All these orders seem to be new.     

New Kloosterman sums identities over F2m for all m

Recently, Shin and Sung found new identities for Kloosterman sums over F_2^m with odd m. They posed the question whether similar results could be obtained for even m. In this paper, we will give a positive answer to this question. We will present new results that hold for any m and include as special cases the results of Shin and Sung in the case where m is odd.     

2m-Cycle Systems of K 2m+1\C m

For all m ≥ 3 the edges of complete graph on 2m + 1 vertices can he partitioned into m 2m-cycles and an m-cycle.     

Equidistribution of numerical semigroup gaps modulo m

For a positive integer m, a finite set of integers is said to be equidistributed modulo m if the set contains an equal number of elements in each congruence class modulo m. In this paper, we consider the problem of determining when the set of gaps of a numerical semigroup S is equidistributed modulo m. Of particular interest is the case when the nonzero elements of an Apéry set of S form an arithmetic sequence. We explicitly describe such numerical semigroups S and determine conditions for which the sets of gaps of these numerical semigroups are equidistributed modulo m.     

The Mother Minkowski Algebra of Order m

It is found that all polynomials of up to degree m have an encoding as m-vectors in a geometric algebra referred to as the Mother Minkowski algebra of order m. It is then shown that all conformal transformations may be applied to these m-vectors, the results of which, when converted back into polynomial form, give us the transformed surfaces in terms of the zero sets of the original and final polynomials.     

Naturally graded Leibniz algebras with characteristic sequence (n − m, m)

We consider the classification problem for special classes of nilpotent Leibniz algebras. Namely, we consider “naturally” graded nilpotent n-dimensional Leibniz algebras for which the right multiplication operator (by the generic element) has two Jordan blocks of dimensionsm and n ? m. Earlier, the problem of classifying such algebras was studied form < 4. The present paper continues these studies for the case m ≥ 4.     

Permutation functions and (n,m)-commutativity of semigroups

By [4], a semigroupS is called an (n, m)-commutative semigroup (n, m ∈ ?⁺, the set of all positive integers) if $$x_1 x_2 \cdot \cdot \cdot x_n y_1 y_2 \cdot \cdot \cdot y_m = y_1 y_2 \cdot \cdot \cdot y_m x_1 x_2 \cdot \cdot \cdot x_n $$ holds for allx ₁,...,x _n ,y ₁,...,y _m ∈S It is evident that ifS is an (n, m)-commutative semigroup then it is (n′,m′)-commutative for alln′≥n andm′≥m. In this paper, for an arbitrary semigroupS, we determine all pairs (n, m) of positive integersn andm for which the semigroupS is (n, m)-commutative. In our investigation a special type of function mapping ?⁺ into itself plays an important role. These functions which are defined and discussed here will be called permutation functions.     

Remark on a theorem of Lindström

Let Ω denote the set of all n by n doubly stochastic matrices and let m be a positive integer. Then the set $\text{Ω}{}_{\mathrm{m}}\text{= {A ? Ω : 0 ? a}{}_{\mathrm{ij}}\text{?}\text{1}\text{m}\text{, 1 ? i, j ≤ n}}$ is the convex hull of the matrices in $\text{Ω}{}_{\mathrm{m}}$ having exactly m entries equal to $\text{1}\text{m}$ in each row and column and the other entries equal to zero.     

Commutative fundamental (m,n)-modules

In this paper, we introduce the concept of fundamental relation θ¹ on an (m, n)-hypermodule M as the smallest equivalence relation such that M/θ¹ is a commutative (m, n)-module, and then some related properties are investigated.     

Convergence and error estimates for (m,l,s)-splines

In some recent papers, Bouhamidi and Le Méhauté introduced L^m,l,s-splines as a natural extension of J. Duchon's (m,s)-splines. In the present work, some results on convergence and error estimates for smoothing and interpolating L^m,l,s-splines (called here (m,l,s)-splines) are given. These results extend those presented in several papers by Duchon, Arcangéli and López de Silances for (m,s)-splines and also for D^m-splines (i.e. (m,0)-splines).     

The theory of closed ordered differential fields with m commuting derivations

We generalize the work of M. Singer (1978) on the theory of closed ordered differential fields to the case of m-ODF, the theory of ordered fields equipped with m commuting derivations. We give an algebraic axiomatization of the model completion (denoted by m-CODF) of this theory and we can immediately deduce that m-CODF has quantifier elimination in the natural language of ordered Δ-rings. To cite this article: C. Rivière, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A strongly polynomial algorithm for solving two-sided linear systems in max-algebra

An algorithm for solving m×n systems of (max,+)-linear equations is presented. The systems have variables on both sides of the equations. After O(m⁴n⁴) iterations the algorithm either finds a solution of the system or finds out that no solution exists. Each iteration needs O(mn) operations so that the complexity of the presented algorithm is O(m⁵n⁵).     

On spline function approximations to the solution of volterra integral equations of the first kind

A procedure, using spline functions of degreem, for the solution of linear Volterra integral equations of the first kind is presented. The method produces an approximate solution of classC ^m-1, is order (m+1) and is shown to be numerically stable form≦4.     

EXISTENCE AND NONEXISTENCE OF GLOBAL POSITIVE SOLUTIONS FOR DEGENERATE PARABOLIC EQUATIONS IN EXTERIOR DOMAINS

This article deals with the degenerate parabolic equations in exterior domains and with inhomogeneous Dirichlet boundary conditions. We obtain that pc = (σ+m)n/(n-σ-2) is its critical exponent provided max{-1, [(1-m)n-2]/(n+1)} σ n-2. This critical exponent is not the same as that for the corresponding equations with the boundary value 0, but is more closely tied to the critical exponent of the elliptic type degenerate equations. Furthermore, we demonstrate that if max{1, σ + m} p ≤ pc, then every positive solution of the equations blows up in finite time; whereas for ppc, the equations admit global positive solutions for some boundary values and initial data. Meantime, we also demonstrate that its positive solutions blow up in finite time provided n ≤σ+2.