Characterizations of Lie derivations of triangular algebras

Let T be a triangular algebra over a commutative ring R. In this paper, under some mild conditions on T, we prove that if δ:T→T is an R-linear map satisfying

δ([x,y])=[δ(x),y]+[x,δ(y)]     

Holomorphic Solutions of a Functional Equation Related to Nonlinear Difference Systems

Here we consider the following functional equation, $$\Psi(X(x,\Psi(x)))=Y(x, \Psi(x)),$$ where X(x, y) and Y(x, y) are holomorphic functions in |x| < δ ₁, |y| < δ ₁. When we consider a nonlinear simultaneous system of two variables difference equations, we can reduce it to a single difference equation of first order by a solution Ψ of the above functional equation. We obtain a matrix by the linear terms of functions X and Y. When the all eigenvalues of the matrix are equal to 1, it is difficult to have a solution of the above functional equation. In the present paper, we derive a formal solution of the above functional equation under the condition. Further we prove the existence of a solution which is holomorphic and have an asymptotically expansion of the formal solution. Moreover, we will show an example of nonlinear difference system such that our results are applicable.     

Two special cases of the assignment problem

The assignment problem may be stated as follows: Given finite sets of points S and T, with|S| ? |T|, and given a “metric” which assigns a distance d(x, y) to each pair (x, y) such that x ∈ T and y ∈ S find a 1?1 function Q: T→ S which minimizes Σ_x∈Td(x, Q(x)) We consider the two special cases in which the points lie (1) on a line segment and (2) on a circle, and the metric is the distance along the line segment or circle, respectively. In each case, we show that the optimal assignment Q can be computed in a number of steps (additions and comparisons) proportional to the number of points. The problem arose in connection with the efficient rearrangement of desks located in offices along a corridor which encircles one floor of a building.     

Compact causal data interpolation

Consider a Hilbert space equipped with a time-structure, i.e., a resolution E of the identity on defined on subsets of some linearly ordered set Λ. For which x and y in is it possible to find a causal (time respecting) compact operator T, so that Tx = y? When T is required to be a Hilbert-Schmidt operator and (Λ, E) is sufficiently regular, this question is answered in terms of the “time-densities” of x and y. The condition is that the integral ∝_gLμ_x({s t})⁻¹ dμ_y(t) should be finite, where μ_x and μ_y are the measures on Λ given by μ_x(Ω) = ¦|E(Ω)x¦|² and μ_y(Ω) = ¦|E(Ω)y¦|². Further a solution is given for the related problem of minimizing the sum of ¦|Tx − y¦|² and the squared Hilbert-Schmidt norm ¦|R¦|₂² of T.     

The Sprague-Grundy function of the game Euclid

We show that the Sprague-Grundy function of the game Euclid is given by g(x,y)=⌊|y/x-x/y|⌋ for x,y≥1.     

Periodic solutions to one-dimensional wave equation with x-dependent coefficients

In this paper, we study the nonlinear one-dimensional periodic wave equation with x-dependent coefficients u(x)ytt−(ux(x)yx)+g(x,t,y)=f(x,t) on (0,π)×R under the boundary conditions a₁y(0,t)+b₁yx(0,t)=0, a₂y(π,t)+b₂yx(π,t)=0 ( for i=1,2) and the periodic conditions y(x,t+T)=y(x,t), yt(x,t+T)=yt(x,t). Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. A main concept is the notion “weak solution” to be given in Section 2. For T is the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.     

On Skew Derivations in Semiprime Rings

We investigate the commutativity in a (semi-)prime ring R which admits skew derivations δ ₁, δ ₂ satisfying [δ ₁(x), δ ₂(y)]?=?[x, y] for all x, y in a nonzero right ideal of R. This result is a natural generalization of Bell and Daif’s theorem on strong commutativity preserving derivations and a recent result by Ali and Huang.     

Max-min of polynomials and exponential diophantine equations

In this paper we study the maximum-minimum value of polynomials over the integer ring Z. In particular, we prove the following: Let F(x,y) be a polynomial over Z. Then, maxx∈Z(T)miny∈Z|F(x,y)|=o(T1/2) as T→∞ if and only if there is a positive integer B such that maxx∈Zminy∈Z|F(x,y)|?B. We then apply these results to exponential diophantine equations and obtain that: Let f(x,y), g(x,y) and G(x,y) be polynomials over Q, G(x,y)∈(Q[x,y]−Q[x])∪Q, and b a positive integer. For every α in Z, there is a y in Z such that f(α,y)+g(α,y)bG(α,y)=0 if and only if for every integer α there exists an h(x)∈Q[x] such that f(x,h(x))+g(x,h(x))bG(x,h(x))≡0, and h(α)∈Z.     

Asymptotic estimates of sums involving the Moebius function

Let p(n) denote the smallest prime factor of an integer n>1 and let p(1)=∞. We study the asymptotic behavior of the sum M(x,y)=Σ_{1≤n≤x,p(n)>y}μ(n) and use this to estimate the size of A(x)=max_|f|≤1|Σ_2≤n<xμ(n)f(p(n))|, where μ(n) is the Moebius function. Applications of bounds for A(x), M(x,y) and similar quantities are discussed.     

On a multivariate normal probability minimization problem

Given a 2k component (column) vector z=(x′,y′)′, x′=(x₁,…, x_k), y′= (y₁,…,y_k), having a 2k variate multivariate normal distribution with mean vector zero, and an unknown 2k×2k positive definite correlation matrix Σ, Shafer and Olkin (1983) conjecture that P(x,y)=P{|x₁|<c,…,|x_k|<c,|y₁|<c,…,|y_k|<c} is minimized when Σ has a certain pattern. This conjecture is proved to be correct. A known inequality is generalized.     

Control problems for parabolic equations on controlled domains

Given the one-dimensional heat equation v_t = v_xx on the controlled domain Q(y) = {(t, x); 0 < x < y(t), 0 < t < T} subject to some initial-boundary conditions, we study the problem of optimally selecting y(·) from some admissible class so as to maximize a given payoff of fixed duration. Q(y) is thus a controlled domain. We also study the problem in which the heat equation holds in Q(y, z) = {z(t) < x < y(t), 0 < t < T}; z minimizing, y maximizing, i.e., the differential game. The principle techniques involved are (i) transforming the controlled domain to an uncontrolled domain and then (ii) using the method of lines for parabolic equations to enable us to use known results for control systems governed by ordinary differential equations. Sufficient conditions for existence in an admissible class is given and the method of lines allows numerical techniques to be applied to determine the optimal control in our class.     

Near automorphisms of cycles

Let f be a permutation of V(G). Define δ_f(x,y)=|d_G(x,y)-d_G(f(x),f(y))| and δ_f(G)=∑δ_f(x,y) over all the unordered pairs {x,y} of distinct vertices of G. Let π(G) denote the smallest positive value of δ_f(G) among all the permutations f of V(G). The permutation f with δ_f(G)=π(G) is called a near automorphism of G. In this paper, we study the near automorphisms of cycles C_n and we prove that π(C_n)=4⌊n/2⌋-4, moreover, we obtain the set of near automorphisms of C_n.     

Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation

In this work we study the period function T of solutions to the conservative equation x^″(t)+f(x(t))=0. We present conditions on f that imply the monotonicity and convexity of T. As a consequence we obtain the criterium established by C. Chicone and find conditions easier to apply. We also get a condition obtained by Cima, Gasull and Mañosas about monotonicity and, following some of their calculations, present results on the period function of Hamiltonian systems where H(x,y)=F(x)+n^-1|y|ⁿ. Using the monotonicity of T, we count the homogeneous solutions to the semilinear elliptic equation Δu=γu^γ-1 in two dimensions.     

On the construction of cyclic quasigroups

Let F(x,y) be the free groupoid on two generators x and y. Define an infinite class of words in F(x,y) by w₀(x,y) = x,w₁(x,y) = y and w_i+2(x,y) = w_i(x,y)w_i+1(x,y). An identity of the form w_3n(x,y) = x is called a cyclic identity and a quasigroup satisfying a cyclic identity is called a cyclic quasigroup. The most extensively studied cyclic quasigroups have been models of the identity y(xy) = x. The more general notion of cyclic quasigroups was introduced by N.S. Mendelsohn. In this paper a new construction for cyclic quasigroups is given. This construction is useful in constructing large numbers of nonisomorphic quasigroups satisfying a given cyclic identity or a consequence of a cyclic identity. The construction is based on a generalization of A. Sade's singular direct product of quasigroups.     

Boundary regularity for viscosity solutions to degenerate elliptic PDEs

Sufficient conditions are given for the solutions to the (fully nonlinear, degenerate) elliptic equation F(x,u,Du,D²u)=0 in Ω to satisfy |u(x)−u(y)|?Cα|x−y| for some α∈(0,1) when x∈Ω and y∈∂Ω.     

On the Cauchy Problem for Integro-differential Operators in Hölder Classes and the Uniqueness of the Martingale Problem

The existence and uniqueness in Hölder spaces of classical solutions of the Cauchy problem to parabolic integro-differential equation of the order α ∈ (0, 2) is investigated. The principal part of the operator has kernel m(t, x, y)/|y| ^d?+?α with a bounded nondegenerate m, Hölder in x and measurable in y. The result is applied to prove the uniqueness of the corresponding martingale problem.     

Constructive existence and uniqueness for two-point boundary-value problems with a linear gradient term

Constructive existence and uniqueness theorems are established for nonlinear two-point boundary-value problems governed by the equation y″=f(x, y)+p(x)y′, 0?x?1. The existence and uniqueness is established for functions y(x) that satisfy a certain constraint, i.e., that sfncy(x)sfnc is bounded by a known function or that 0?y(x)?M for some M. Applications to scientific problems in the literature are given.     

Multistability and cyclic attractors in duopoly games

A dynamic Cournot duopoly game, whose time evolution is modeled by the iteration of a map T:(x,y)→(r₁(y),r₂(x)), is considered. Results on the existence of cycles and more complex attractors are given, based on the study of the one-dimensional map F(x)=(r₁∘r₂)(x). The property of multistability, i.e. the existence of many coexisting attractors (that may be cycles or cyclic chaotic sets), is proved to be a characteristic property of such games. The problem of the delimitation of the attractors and of their basins is studied. These general results are applied to the study of a particular duopoly game, proposed in M. Kopel [Chaos, Solitons & Fractals, 7 (12) (1996) 2031–2048] as a model of an economic system, in which the reaction functions r₁ and r₂ are logistic maps.     

A new generalization of Hardy–Berndt sums

In this paper, we construct a new generalization of Hardy–Berndt sums which are explicit extensions of Hardy–Berndt sums. We express these sums in terms of Dedekind sums s _r (h, k : x, y|λ) with x?=?y?=?0 and obtain corresponding reciprocity formulas.