Semigroup identities in the monoid of two-by-two tropical matrices

We show that the monoid $M_{2}(\mathbb {T})$ of 2×2 tropical matrices is a regular semigroup satisfying the semigroup identity $$A^2B^4A^2A^2B^2A^2B^4A^2=A^2B^4A^2B^2A^2A^2B^4A^2.$$ Studying reduced identities for subsemigroups of $M_{2}(\mathbb {T})$ , and introducing a faithful semigroup representation for the bicyclic monoid by 2×2 tropical matrices, we reprove Adjan’s identity for the bicyclic monoid in a much simpler way.     

Classification of self dual quadratic bent functions

We classify all self dual and anti self dual quadratic bent functions in 2n variables under the action of the orthogonal group O(2n,\mathbb F₂){{O}(2n,\mathbb F_2)} . This is done through a classification of all 2n × 2n involutory alternating matrices over \mathbb F₂{\mathbb F_2} under the action of the orthogonal group. The sizes of the O(2n,\mathbb F₂){{O}(2n,\mathbb F_2)} -orbits of self dual and anti self dual quadratic bent functions are determined explicitly.     

A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2^2d+2, 2^2d+1±2 ^d , 2^2d ±2 ^d ). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2 ^d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2^2d+2 with exponent 2 ^d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.     

Triangle in a triangle: On a problem of Steinhaus

A necessary and sufficient condition on the sidesp, q, r of a trianglePQR and the sidesa, b, c of a triangleABC in order thatABC contains a congruent copy ofPQR is the following: At least one of the 18 inequalities obtained by cyclic permutation of {a, b, c} and arbitrary permutation of {itp, q, r} in the formula $$\begin{array}{l} Max\{ F(q^2 + r^2 - p^2 ), F'(b^2 + c^2 - a^2 )\} \\ + Max\{ F(p^2 + r^2 - q^2 ), F'(a^2 + c^2 - b^2 )\} \le 2Fcr \\ \end{array}$$ is satisfied. In this formulaF andF′ denote the surface areas of the triangles, i.e. $$\begin{array}{l} F = {\textstyle{1 \over 4}}(2a^2 b^2 + 2b^2 c^2 + 2c^2 a^2 - a^4 - b^4 - c^4 )^{1/2} \\ F' = {\textstyle{1 \over 4}}(2p^2 q^2 + 2q^2 r^2 + 2r^2 p^2 - p^4 - q^4 - r^4 )^{1/2} . \\ \end{array}$$      

Dinuclear copper (II) and nickel (II) systems with planar M2N2O2 bridge

The tridentate ligand systemb (abbreviated as inkR₂) readily yield copper (II) and nickel (II) species of the formula M₂ (inkR₂)₂(CLO₄)₂. 2xH₂O (x=0–1). Dinuclear formulation is based on variable temperature magnetic susceptibility and conductivity data and on the known structure of some related systems. The Cu₂ (inkR₂) ₂ ²⁺ species are strongly antiferromagnetic (?2J=600–800 cm^?1) while the Ni₂(inkR₂) ₂ ²⁺ species are diamagnetic. The major coordination sphere is planar around each metal (II). The metal ions in a dimer are linked by planar M₂N₂O₂ bridge. The copper (II) and nickel (II) species freely form solid solutions. In these statistical scrambling of copper and nickel occur among the metal ion sites of the dimeric structure. Powder epr spectra of such mixed crystals are indicative of axial geometry around copper (II) ion.     

On the distribution of integral points on cones

New results on the distribution of integral points on the cones

On the classification of nonsingular 2×2×2×2 hypercubes

As a first step in the classification of nonsingular 2×2×2×2 hypercubes up to equivalence, we resolve the case where the base field is finite and the hypercubes can be written as a product of two 2×2×2 hypercubes. (Nonsingular hypercubes were introduced by D. Knuth in the context of semifields. Where semifields are related to hypercubes of dimension 3, this paper considers the next case, i.e., hypercubes of dimension 4.) We define the notion of ij-rank (with 1 ≤ i < j ≤ 4) and prove that a hypercube is the product of two 2×2×2 hypercubes if and only if its 12-rank is at most 2. We derive a ‘standard form’ for nonsingular 2×2×2×2 hypercubes of 12-rank less than 4 as a first step in the classification of such hypercubes up to equivalence. Our main result states that the equivalence class of a nonsingular 2×2×2×2 hypercube M of 12-rank 2 depends only on the value of an invariant δ ₀(M) which derives in a natural way from the Cayley hyperdeterminant det₀ M and another polynomial invariant det M of degree 4. As a corollary we prove that the number of equivalence classes is (q + 1)/2 or q/2 depending on whether the order q of the field is odd or even.     

Sharp bounds for the Bernoulli numbers

We determine the best possible real constants a\alpha and b\beta such that the inequalities [(2(2n)!)/((2p)²ⁿ)] [1/(1-2^a-2n)] \leqq |B_2n| \leqq [(2(2n)!)/((2p)²ⁿ)] [1/(1-2^b-2n)]{2(2n)! \over(2\pi)^{2n}} {1 \over 1-2^{\alpha -2n}} \leqq |B_{2n}| \leqq {2(2n)! \over (2\pi )^{2n}}\, {1 \over 1-2^{\beta -2n}}hold for all integers n\geqq 1n\geqq 1. Here, B₂, B₄, B₆,... are Bernoulli numbers.     

On the free boundary problem of magnetohydrodynamics

In the paper, the solvability of the free boundary problem of magnetohydrodynamics for a viscous incompressible fluid in a simply connected domain is proved. The solution is obtained in the Sobolev–Slobodetskii spaces W₂^{2 + l,1 + l/2},1/2 < l < 1 W_2^{2 + l,1 + l/2},1/2 < l < 1 . Bibliography: 15 titles.     

Computing the number of ways of representing primes by a norm form

Formulae for the number of different integral solutions ofa ²+b²+c²+d²+ac+bd=p are given wherep is a prime and the solution satisfies certain natural congruence conditions. Similar formulae are given for the case of the quadratic forma ²+b²+2c²+2d²+ac+bd.     

Coding and Counting Arrangements of Pseudolines

Arrangements of lines and pseudolines are important and appealing objects for research in discrete and computational geometry. We show that there are at most 2^{0.657&gt; n2}2^{0.657\> n^{2}} simple arrangements of n pseudolines in the plane. This improves on previous work by Knuth who proved an upper bound of 3^\binomn2 @ 2^{0.792&gt; n2}3^{\binom{n}{2}} \cong 2^{0.792\> n^{2}} in 1992 and the first author, who obtained 2^{0.697&gt; n2}2^{0.697\> n^{2}} in 1997. The argument uses surprisingly little geometry. The main ingredient is a lemma that was already central to the argument given by Knuth.     

Explicit Formulas of the Bergman Kernel for Some Reinhardt Domains

In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for Reinhardt domains \(\{|z_3|^{\lambda } < |z_1|^{2p} + |z_2|^2, \ |z_1|^{2p} + |z_2|^2 < |z_1|^{p} \}\) and \(\{|z_4|^{\lambda } < (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2, \ (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2 < (|z_1|^2 + |z_2|^2 )^{p/2} \}\).     

A new class of integrable systems

This paper concerns the integrability of Hamiltonian systems with two degrees of freedom whose Hamiltonian has the form¶ H=¹/₂(x₁²+x₂²) +V(y₁,y₂) H={1\over2}(x_{1}^{2}+x_{2}^{2}) +V(y_{1},y_{2}) where¶¶ V(y₁,y₂)=¹/₂(a₁y₁²+a₂y₂²) + ¹/₄b₁y₁⁴ + ¹/₄b₂y₂⁴ + ¹/₂b₃y₁²y₂² + ?_k=1³g_k(y₁²+y₂²) ^k+2 V(y_{1},y_{2})={1\over2}\big(\alpha _{1}y_{1}^{2}+\alpha_{2}y_{2}^{2}\big) + {1\over4}\beta _{1}y_{1}^{4} + {1\over4}\beta_{2}y_{2}^{4} + {1\over2}\beta _{3}y_{1}^{2}y_{2}^{2} + \sum_{k=1}^{3}\gamma_{k}\big(y_{1}^{2}+y_{2}^{2}\big) ^{k+2} ¶¶ which, constitues a generalization of some well-known integrable systems. We give new values of the vector (a₁,a₂,b₁,b₂,b₃,g₁,g₂,g₃) (\alpha _{1},\alpha_{2},\beta _{1},\beta _{2},\beta _{3},\gamma _{1},\gamma _{2},\gamma _{3}) for which this system is completely integrable and we show that the system is linearized in the Jacobian variety Jac(G \Gamma ) of a smooth genus 2 hyperelliptic Riemann surface G \Gamma .     

On Tame Kernel and Class Group in Terms of Quadratic Forms

The paper is to investigate the structure of the tame kernel K₂O_F for certain quadratic number fields F, which extends the scope of Conner and Hurrelbrink (J. Number Theory88 (2001), 263-282). We determine the 4-rank and the 8-rank of the tame kernel, the Tate kernel, and the 2-part of the class group. Our characterizations are in terms of binary quadratic forms X²+32Y,X²+64Y²,X²+2Py²,2X²+Py²,X²−2Py²,2X²−Py². The results are very useful for numerical computations.     

Double stage estimation of population variance

Summary Consider a normal population with mean μ and variance σ². We are interested in the estimation of population variance with the help of guess value σ ₀ ² and a sample of observations. In this paper, a double stage shrinkage estimator based on the shrinkage estimatorks ₁ ² +(1-k)σ ₀ ² ifs ₁ ² ∈R and the usual estimator ifs ₁ ² ∋R, whereR is some specified region, have been proposed. The expressions for bias and mean squared error have been obtained. Comparison with the usual estimators ² have been made. It was found that though the largest gain is obtained fork=0, we can use with 0≦k≦1/2 even when σ² is very close to σ ₀ ²     

On the BIB design having the minimum p-rank

It is shown that (i) among BIB designs with parameters (2^t+1 ? 1, 2^t+1 ? 1, 2^t ? 1, 2^t ? 1, 2^t?1 ? 1), the incidence matrix of the BIB design PG(t, 2):t ? 1 derived from a finite projective geometry PG(t, 2) has the minimum 2-rank and (ii) among BIB designs with parameters (2^t, 2^t+1 ? 2, 2^t ? 1, 2^?1, 2^t?1 ? 1), the incidence matrix of the BIB design EG(t, 2):t ? 1 derived from an affine geometry EG(t, 2) has the minimum 2-rank.     

On a Żołądek theorem

We prove the existence of cubic systems of the form $$ \begin{gathered} \dot x = y[1 - 2r(5 + 3r^2 )x + \gamma \lambda ^2 x^2 ] + a_0 x + a_1 x^2 + a_2 xy + a_3 y^2 + a_4 x^3 + a_5 x^2 y + a_6 xy^2 , \hfill \\ \dot y = - x(1 - 8rx)(1 - 3r\gamma x) - 2x[2(1 - 3r^2 ) - r\gamma (7 - 15r^2 )x]y \hfill \\ - [r(11 + r^2 ) + \gamma (1 - 22r^2 - 3r^4 )x]y^2 \hfill \\ - 2r\gamma \delta y^3 + a_0 y + a_7 x^2 + a_8 xy + a_9 y^2 + a_{10} x^3 + a_{11} x^2 y, \hfill \\ \end{gathered} $$ where α = 3r ² + 17, γ = r ² + 3, δ = 1 ? r ², and λ = 3r ² + 1, that have at least eleven limit cycles in a neighborhood of the point O(0, 0).     

Uniqueness of Positive Solutions for a Nonlinear Elliptic System

In this paper we study the uniqueness of nontrivial positive solutions for the following second order nonlinear elliptic system:

Banach Spaces of Solutions of the Helmholtz Equation in the Plane

The purpose of this article is to study the Hilbert space W²\mathcal{ W}^2 consisting of all solutions of the Helmholtz equation Du+u=0\Delta u+u=0 in \BbbR²\Bbb{R}^2 that are the image under the Fourier transform of L²L^2 densities in the unit circle. We characterize this space as a close subspace of the Hilbert space H²\mathcal{ H}^2 of all functions belonging to L²( | x | ^-3dx) L^2( | x | ^{-3}dx) jointly with their angular and radial derivatives, in the complement of the unit disk in \BbbR²\Bbb{R}^2. We calculate the reproducing kernel of W²\mathcal{ W}^2 and study its reproducing properties in the corresponding spaces H^p\mathcal{H}^p, for $p>1$p>1.     

On the restriction of cross characteristic representations of 2 F 4(q) to proper subgroups

We prove that the restriction of any nontrivial representation of the Ree groups ² F ₄(q), q = 2²ⁿ⁺¹ ≥ 8 in odd characteristic to any proper subgroup is reducible. We also determine all triples (K, V, H) such that ${K \in \{^2F_4(2), ^2F_4(2)'\} }We prove that the restriction of any nontrivial representation of the Ree groups ² F ₄(q), q = 2²ⁿ⁺¹ ≥ 8 in odd characteristic to any proper subgroup is reducible. We also determine all triples (K, V, H) such that K ? {²F₄(2), ²F₄(2)￠}{K \in \{^2F_4(2), ^2F_4(2)'\} } , H is a proper subgroup of K, and V is a representation of K in odd characteristic restricting absolutely irreducibly to H.