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.     

Orthogonal line packings of PG2m − 1(2)

Baker (Discrete Math., 15 (1976), 205–211) has shown that there exists a packing of the lines of each odd dimensional projective space over the field of two elements as a corollary to a theorem asserting the existence of a 2-resolution of the Steiner quadruple system of planes in an even dimensional affine space over the field of two elements. Two packings are orthogonal if any two of their spreads have at most one line in common. A variation of the previous construction gives alternate packings so that, for example, the existence of orthogonal packings of PG_{2m ? 1}(2) when three does not divide 2m ? 1 can be demonstrated.     

Partitioning the planes of AG2m(2) into 2-designs

A t-design S_λ(t, k, v) is an arrangement of v elements in blocks of k elements each such that every t element subset is contained in exactly λ blocks. A t-design S_λ(t, k, v) is called t′-resolvable if the blocks can be partitioned into families such that each family is the block system of a S_λ(t′, k, v). It is shown that the S₁(3, 4, 2^2m) design of planes on an even dimensional affine space over the field of two elements is 2-resolvable. Each S₁(2, 4, 2^2m) given by the resolution is itself 1-resolvable. As a corollary it is shown that every odd dimensional projective space over the field of two elements admits a 1-packing of 1-spreads, i.e. a partition of its lines into families of mutually disjoint lines whose union covers the space. This 1-packing may be generated from any one of its spreads by repeated application of a fixed collineation.     

Zur Bestimmung vonE n [x n+2m ]

Remark on the estimation ofE _n [x ^n+2m ]. Let be $$E_n [f]: = \mathop {\inf }\limits_{p \in P_n } \mathop {\sup }\limits_{x \in [ - 1, 1]} |f(x) - p(x)|$$ (P _n : set of all polynomials of degreen). Riess-Johnson [4] proved (3) $$E_n [x^{n + 2m} ] = \frac{{n^{m - 1} }}{{2^{n + 2m - 1} (m - 1)!}}[1 + O(n^{ - 1} )],n even.$$ This degree of approximation is realized by expansion in Chebyshev polynomials and by interpolation at Chebyshev nodes. The purpose of this paper is to give a more precise estimation by constructing the polynomial of best approximation on a finite set. This construction is easily done and one obtains the result, that the termO(n ^?1) in (3) may be replaced by 1/2(m ? 1) (3m + 2)n ^?1 + O(n ^?2).     

Quantization for an elliptic equation of order 2m with critical exponential non-linearity

On a smoothly bounded domain ${\Omega\subset\mathbb{R}^{2m}}$ we consider a sequence of positive solutions ${u_k\stackrel{w}{\rightharpoondown}0}$ in H ^m (Ω) to the equation ${(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}}$ subject to Dirichlet boundary conditions, where 0 < λ _k → 0. Assuming that $$0 < \Lambda:=\lim_{k\to\infty}\int\limits_\Omega u_k(-\Delta)^m u_k dx < \infty,$$ we prove that Λ is an integer multiple of Λ₁ := (2m ? 1)! vol(S ^2m ), the total Q-curvature of the standard 2m-dimensional sphere.     

The exponential Lebesgue-Nagell equation X 2 + P 2m = Y n

Let p be an odd prime. In this paper, a complete classification of all positive integer solutions (x, y, m, n) of the equation x ²+p ^2m = y ⁿ , gcd(x, y) = 1, n > 2, is given. As a consequence, we solve the equation for certain interesting cases.     

Asymptotics of the spectrum of nonsmooth perturbations of differential operators of order 2m

On a finite closed interval, we obtain the asymptotics of the eigenvalues of a differential operator of order 2m perturbed by a differential operator of order 2m ? 2 given by a quasidifferential expression. We also consider the case of multiple eigenvalues.     

Bounds for the mean queue length of the M/K2/m queue

This paper deals with the mean queue length E(Q) of the M/K₂/m queue. In particular we study the behaviour of E(Q) when the coefficient of variation (C_x) for the service time is constant. Two limiting cases are studied (when C_x2> 1) and it is conjectured that these cases give bounds for E(Q), when C_x² is fixed. A simple approximation for E(Q) is suggested for a particular subclass of M/G/m queues (including M/D/m, M/E_k/m and certain M/H₂/m cases).     

Optimal partially balanced fractional 2 m 1+m 2 factorial designs of resolution IV

Summary This paper investigates some partially balanced fractional 2 ^m ₁+m ₂ factorial designs of resolution IV derived from partially balanced arrays, which permit estimation of the general mean, all main effects, all two-factor interactions within each set of them _k factors (k=1, 2) and some linear combinations of the two-factor interactions between the sets of them _k ones. In addition, optimal designs with respect to the generalized trace criterion defined by Shirakura (1976,Ann. Statist.,4, 723–735) are presented for each pair (m ₁,m ₂) with 2≦m ₁≦m ₂ andm ₁+m ₂≦6, and for values ofN (the number of observations) in a reasonable range. Partially supported in part by Grants 56530009 (C) and 57530010 (C).     

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.     

Nonlinear elliptic equations of order 2m and subdifferentials

LetA be an operator of the calculus of variations of order 2m onW ^m,p (Ω) andj a normal convex integrand. Forf ∈L ^p (Ω), the equation $$\mathcal{A}u + \partial j(x,u) \ni f, in \Omega , u - \phi \in W_0^{m,p} (\Omega ),$$ may have no strong solutions whenm>1, even ifj is independent ofx and φ=0. However, we obtain existence results whenj is everywhere finite and $$\int_\Omega {j(x,\phi ) dx< + \infty ,} $$ by the study of the subdifferential of the function $$\upsilon \mapsto \int_\Omega {j(x,\upsilon + \phi ) dx on W_0^{m,p} (\Omega ).} $$      

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.     

On primitive elements of trace equal to 1 in GF(2m)

The object of this paper is to present a simple proof for the existence of primitive elements of trace equal 1, in GF(2^m).     

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.     

Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space $L_{2}^{(m)}(0,1)$

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the \(L_{2}^{(m)}(0,1)\) space for numerical calculation of Fourier coefficients. Using the S.L.Sobolev’s method, we obtain new optimal quadrature formulas of such type for N+1≥m, where N+1 is the number of nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formula for the case m=1. The obtained optimal quadrature formulas in the \(L_{2}^{(m)}(0,1)\) space are exact for P _m?1(x), where P _m?1(x) is a polynomial of degree m?1. Furthermore, we present some numerical results, which confirm the obtained theoretical results.     

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.     

Exact solutions of the generalized K(m, m) equations

Family of equations, which is the generalization of the K(m, m) equation, is considered. Periodic wave solutions for the family of nonlinear equations are constructed.     

(m)-covering of a triangulation

Let G be a graph triangularly imbedded into a surface S, G_(m) is the graph constructed from G by replacing each vertex x by m vertices (xx,0), (x, 1), ..., (x, m − 1) and joining two vertices (x, i) and (y, j) by an edge if and only if x and y are joined in G. The main result is that the construction of G_(m) is possible whenever n is an odd prime and a well separating cycle (mod m) can be determined.