Arithmetic properties of bipartitions with even parts distinct

In this work, we consider various arithmetic properties of the function ped _?2(n) that denotes the number of bipartitions of n with even parts distinct. We prove two infinite families of congruences for p _?2(n) modulo 3. We also give characterizations of ped _?2(n) modulo 2 and 4. Furthermore, for a fixed positive integer k, we show that ped _?2(n) is divisible by 2 ^k for almost all n.     

Cesàro asymptotics for the orders of SLk(Zn) and GLk(Zn) as n→∞

Given an integer k>0, our main result states that the sequence of orders of the groups $\text{SL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$ (respectively, of the groups $\text{GL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$) is Cesàro equivalent as n→∞ to the sequence C₁(k)n^k2?1 (respectively, C₂(k)n^k2), where the coefficients C₁(k) and C₂(k) depend only on k; we give explicit formulas for C₁(k) and C₂(k). This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function ?(n) is Cesàro equivalent to $\text{n}\text{6}\text{π}{}^2$. We present some experimental facts related to the main result. To cite this article: A.G. Gorinov, S.V. Shadchin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A note: Every homogeneous difference equation of degree one admits a reduction in order

Every difference equation x _n+1 = f _n (x _n ,x _n ? 1,…,x _n ? k ) of order k+1 with each mapping f _n being homogeneous of degree 1 on a group G is shown to be equivalent to a system consisting of an equation of order k and a linear equation of order 1.     

Complexities of normal bases constructed from Gauss periods

Let q be a power of a prime p, and let \(r=nk+1\) be a prime such that \(r\not \mid q\), where n and k are positive integers. Under a simple condition on q, r and k, a Gauss period of type (n, k) is a normal element of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\); the complexity of the resulting normal basis of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\) is denoted by C(n, k; p). Recent works determined C(n, k; p) for \(k\le 7\) and all qualified n and q. In this paper, we show that for any given \(k>0\), C(n, k; p) is given by an explicit formula except for finitely many primes \(r=nk+1\) and the exceptional primes are easily determined. Moreover, we describe an algorithm that allows one to compute C(n, k; p) for the exceptional primes \(r=nk+1\). Our numerical results cover C(n, k; p) for \(k\le 20\) and all qualified n and q.     

Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n ? 2. These conditions are best possible. A related conjecture also is proposed.     

Central and local limit theorems applied to asymptotic enumeration

Let a double sequence a_n(k) ? 0 be given. We prove a simple theorem on generating functions which can be used to establish the asymptotic normality of a_n(k) as a function of k. Next we turn our attention to local limit theorems in order to obtain asymptotic formulas for a_n(k). Applications include constant coefficient recursions, Stirling numbers, and Eulerian numbers.     

Non-cover generalized Mycielski, Kneser, and Schrijver graphs

A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. We prove that the generalized Mycielski graphs M_m(C_2t+1) of an odd cycle, Kneser graphs KG(n,k), and Schrijver graphs SG(n,k) are not cover graphs when m?0,t?1, k?1, and n?2k+2. These results have consequences in circular chromatic number.     

On the Cotorsion Pair (𝒫1, 𝒟)

Abstract

We classify the C ₂ × C ₂-gradings on M ₂(k), k an arbitrary field. If char(k) ≠ 2 our approach relies on the duality between gradings and actions for finite abelian groups, and if k is algebraically closed we find precisely one isomorphism type of grading which is not isomorphic to a grading with all the matrix units being homogeneous elements. If char(k) = 2 we use a computational approach and we find that any C ₂ × C ₂-grading is induced by a C ₂-grading.     

Some dimensional results for homogeneous Moran sets

Let ?(¦n _k ¦k?1,¦c _k ¦k?1) be the collection of homogeneous Moran sets determined by ¦n _k ¦k?1 and ¦c _k ¦k?1, where ¦n _k ¦k?1 is a sequence of positive integers and ¦c _k ¦k?1 a sequence of positive numbers. Then the maximal and minimal values of Hausdorff dimensions for elements in ? are determined. The result is proved that for any values between the maximal and minimal values, there exists an element in ?(¦n _k ¦k?1,¦c _k¦k?1) such that its Hausdorff dimension is equal tos. The same results hold for packing dimension. In the meantime, some other properties of homogeneous Moran sets are discussed.     

Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds

We suggest a method for describing some types of degenerate orbits of orthogonal and unitary groups in the corresponding Lie algebras as level surfaces of a special collection of polynomial functions. This method allows one to describe orbits of the types SO(2n)/SO(2k)×SO(2) ^n?k , SO(2n+1)/SO(2k+1)×SO(2) ^n?k , and (S)U(n)/(S)(U(2k)×U(2) ^n?k ) in so(2n), so(2n+1), and (s)u(n), respectively. In addition, we show that the orbits of minimal dimensions of the groups under consideration can be described in the corresponding algebras as intersections of quadries. In particular, this approach is used for describing the orbit CP ^n?1?u(n).     

On the vanishing of negative K-groups

We show that for a d-dimensional scheme X essentially of finite type over an infinite perfect field k of characteristic p > 0, the negative K-groups K _q (X) vanish for q < ?d provided that strong resolution of singularities holds over the field k.     

On even-cycle-free subgraphs of the hypercube

It is shown that the size of any C4k+2-free subgraph of the hypercube Qn, k?3, is o(e(Qn)).     

A recursion formula for k-Schur functions

The Bernstein operators allow one to build recursively the Schur functions. We present a recursion formula for k-Schur functions at t=1 based on combinatorial operators that generalize the Bernstein operators. The recursion leads immediately to a combinatorial interpretation for the expansion coefficients of k-Schur functions at t=1 in terms of homogeneous symmetric functions.     

Domino tilings of rectangles with fixed width

Let t(k,n) denote the number of ways to tile a 1 × n rectangle with 1 × 2 rectangles (called dominoes). We show that for each fixed k the sequence t_k=(t(k,0), t(k,1),…) satisfies a difference equation (linear, homogeneous, and with constant coefficients). Furthermore, a computational method is given for finding this difference equation together with the initial terms of the sequence. This gives rise to a new way to compute t(k,n) which differs completely with the known Pfaffian method. The generating function of t_k is a rational function F_k, and F_k is given explicitly for k=1,…,8. We end with some conjectures concerning the form of F_k based on our computations.     

Hopf Algebra Approaches to Nonlinear Algebraic Equations

Abstract

Let M be a k-vector space and R ∈ Hom(M ^?p , M ^?q ), we present a general version of the FRT-construction, we provide a method for examining whether an FRT-bialgebra A(R) has a pre-braided structure and whether M can be regarded as an A(R)-dimodule. We show that the FRT-relation plays a fundamental role in determining the algebra structure on the FRT-bialgebra and the compatibility condition of relevant dimodule. As an example, we give a Hopf algebra approach for solving both homogeneous and non-homogeneous nonlinear (algebraic) equations.     

Further results on the existence of nested orthogonal arrays

Nested orthogonal arrays provide an option for designing an experimental setup consisting of two experiments, the expensive one of higher accuracy being nested in a larger and relatively less expensive one of lower accuracy. We denote by OA_(λ, μ)(t, k, (v, w)) (or OA(t, k, (v, w)) if λ = μ = 1) a (symmetric) orthogonal array OA _λ (t, k, v) with a nested OA _μ (t, k, w) (as a subarray). It is proved in this article that an OA(t, t + 1,(v, w)) exists if and only if v ≥ 2w for any positive integers v, w and any strength t ≥ 2. Some constructions of OA_(λ, μ)(t, k, (v, w))′s with λ ≠ μ and k ? t > 1 are also presented.     

Tensor Product Representations of the Lie Superalgebra 𝔭(n) and Their Centralizers

Abstract

We construct an associative algebra A _k and show that there is a representation of A _k on V ^?k , where V is the natural 2n-dimensional representation of the Lie superalgebra 𝔭(n). We prove that A _k is the full centralizer of 𝔭(n) on V ^?k , thereby obtaining a “Schur-Weyl duality” for the Lie superalgebra 𝔭(n). This result is used to understand the representation theory of the Lie superalgebra 𝔭(n). In particular, using A _k we decompose the tensor space V ^?k , for k = 2 or 3, and show that V ^?k is not completely reducible for any k ≥ 2.     

APPLICATIONS OF CLIFFORD ALGEBRAS TO INVOLUTIONS AND QUADRATIC FORMS

ABSTRACT

Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form ? ∈ I ²(F) such that dim ? = 8, [C(?)] = [D], and ? does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)/F is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution σ, then σ is hyperbolic if and only if σ _K(A) is hyperbolic. Finally, let σ be a decomposable orthogonal involution on the algebra M _{2 ^m} (K). In the case m ≤ 5 we give another proof of the fact that σ is a Pfister involution. If m ≥ 2 ^n?2 ? 2 and n ≥ 5, we show that q _σ ∈ I ⁿ (K), where q _σ is a quadratic form corresponding to σ. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms.     

Simple continued fractions for the Fredholm numbers

Let {a_i} with a₁ ≥ 2 be an infinite bounded sequence of positive integers, and d₁ = 1, d_i = ±1 for i = 2, 3,…. Let {Q_i} be another sequence defined by the recursion Q₁ = 1, Q_i = a_i?1Q_i?1^k for i = 2, 3,…, where k ≥ 2 an integer. Put C_k(a) = Σ_{i = 1}^∞d_iQ_i^?1. In this paper we shall determine the simple continued fraction expansion for the real numbers C_k(a).     

On the Stability of a k-Cubic Functional Equation in Intuitionistic Fuzzy n-Normed Spaces

In this paper, we investigate some stability results concerning the k-cubic functional equation f(kx + y) + f(kx?y) = kf(x + y) + kf(x?y) + 2k(k²?1)f(x) in the intuitionistic fuzzy n-normed spaces.