Solution of an open problem for Schur convexity or concavity of the Gini mean values

The Schur convexity or concavity problem of the Gini mean values S(a, b; x, y) with respect to (x, y) ∈ (0, ∞) × (0, ∞) for fixed (a, b) ∈ R × R is still open. In this paper, we prove that S(a, b; x, y) is Schur convex with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b) : a 0, b 0, a + b 1}, and Schur concave with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b) : b 0, b a, a + b 1} ∪ {(a, b) : a 0, a b, a + b 1}.     

Note on the solution of a certain boundary-value problem

A simple algorithm is described for inverting the operatorD _x D _y (D _x andD _y here and subsequently denote partial differentiation with respect tox andy respectively) which occurs in the iterative solution of the equationD _x D _y f (x, y)=g (x, y, f, D _x f, D _x ² f,D _x D _y f, D _y ² f) when boundary values off(x,y) are given along the sides of the rectangle in thexy-plane whose corners are at the points (a,b); (a+(n+1)k,b); (a+(n+1)k,b+(n+1)h); (a,b+(n+1)h).Communication M. R. 43 of the Computation Department of the Mathematical Centre, Amsterdam.     

Hilbert functions and Betti numbers in a flat family

Summary This paper is dedicated to the study of Hilbert functions and Betti numbers of the projective varieties in a flat family. We prove that the Hilbert function H(X _y ,n),y Y-a parameter scheme-is lower semicontinuous for any fixed n. In case Y is integral and noetherian we obtain the well-known fact that the set V Y where H(X _y ,n)is maximal for all n's is open and nonempty. We show also that b_i(X _y )-the i- th Betti number of X_y—is upper semicontinuous for y V. The paper contains also a number of results concerning the relations among the various Betti numbers.Member of G.N.S.A.G.A.-C.N.R. Supported in part by M.P.I. (Italian Minstry of Education).     

Some new perfect Steiner triple systems

In a Steiner triple system STS(v) = (V, B), for each pair {a, b} ⊂ V, the cycle graph G_a,b can be defined as follows. The vertices of G_a,b are V \ {a, b, c} where {a, b, c} ∈ B. {x, y} is an edge if either {a, x, y} or {b, x, y} ∈ B. The Steiner triple system is said to be perfect if the cycle graph of every pair is a single (v − 3)-cycle. Perfect STS(v) are known only for v = 7, 9, 25, and 33. We construct perfect STS (v) for v = 79, 139, 367, 811, 1531, 25771, 50923, 61339, and 69991. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 327–330, 1999     

Polar factorization of maps on Riemannian manifolds

Let (M,g) be a connected compact manifold, C³ smooth and without boundary, equipped with a Riemannian distance d(x,y). If s : M ? M s : M \to M is merely Borel and never maps positive volume into zero volume, we show s = t °u s = t \circ u factors uniquely a.e. into the composition of a map t(x) = exp_x[-?y(x)] t(x) = {\rm exp}_x[-\nabla\psi(x)] and a volume-preserving map u : M ? M u : M \to M , where y: M ? \bold R \psi : M \to {\bold R} satisfies the additional property that (y^c)^c = y (\psi^c)^c = \psi with y^c(y) :=inf{c(x,y) - y(x) | x ? M} \psi^c(y) :={\rm inf}\{c(x,y) - \psi(x)\,\vert\,x \in M\} and c(x,y) = d²(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields.¶The results are obtained by solving a Riemannian version of the Monge--Kantorovich problem, which means minimizing the expected value of the cost c(x,y) for transporting one distribution f 3 0 f \ge 0 of mass in L¹(M) onto another. Parallel results for other strictly convex cost functions c(x,y) 3 0 c(x,y) \ge 0 of the Riemannian distance on non-compact manifolds are briefly discussed.     

A Lower Bound for {a+b:aA, bB, and P(a,b)≠0}

Let A and B be two finite subsets of a field . In this paper, we provide a non-trivial lower bound for {a+b:aA, bB, and P(a,b)≠0} where P(x,y) [x,y].     

On the ratio of values of a polynomial

For given positive integersa andb, the equationa(x + 1)… (x + k) =b(y+1)… (y + k) in positive integers is considered. More general equations are also considered.     

A Numerical solution to the linear and nonlinear Fredholm integral equations using Legendre wavelet functions

Two methods for solving the Fredholm integral equation of the second kind in linear case, i.e. f (x) – λ ∫_a^b K (x,y)f (y)dy = g (x), and nonlinear case, i.e., f (x) = g (x) + λ ∫_a^b K (x,y)F (f (y))dy, are proposed. In order to solve the linear equation, the kernel K (x,y) as well as the functions f and g are initially approximated through Legendre wavelet functions. This leads to a system of linear equations its solution culminates in a solution to the Fredholm integral equation. In nonlinear case only K (x,y) is approximated by Legendre wavelet base functions. This leads to a separable kernel and makes it possible to employ a number of earlier methods in solving nonlinear Fredholm integral equation with separable kernels. Another feature of the proposed method is that it finds the solution as a function instead of specific solution points, what is done by the majority of the existing methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Einige unstetige stochastische Prozesse

Summary Stochastic processes of the following type are considered. At random time points, the variablex(t) jumps fromy tox, say. The heightsx–y of the jumps have a given distributionG ^*(x–y) that may depend ony ort. Between the jumps,x(t) is a solution to a given differential equationdx/dt=x(x, t). We look for the distributionF(x, t) ofx at timet>0,F(x, 0) being given. In the stationary case, stable distributions are investigated.If there is a lower boundaryx ₀ and ifF(x ₀)>0, the problem is similar to the queueing problem. We solve it in the stationary case with integral equations of the Volterra type. Other problems can be transformed to differential equations for the moment generating functions. These equations are partial in the non stationary and ordinary in the stationary case.     

Pseudo-riemannian manifolds with commuting jacobi operators

We study the geometry of pseudo-Riemannian manifolds which are Jacobi-Tsankov, i.e. ℊ(x)ℊ(y)=ℊ(y)ℊ(x) for allx, y. We also study manifolds which are 2-step Jacobi nilpotent, i.e. ℊ(x)ℊ(y)=0 for allx, y.     

Differential transformations of parabolic second-order operators in the plane

Darboux’s classical results about transformations of second-order hyperbolic equations by means of differential substitutions are extended to the case of parabolic equations of the form Lu = (D _x ² + a(x, y)D _x + b(x, y)D _y + c(x, y))u = 0. We prove a general theorem that provides a way to determine admissible differential substitutions for such parabolic equations. It turns out that higher order transforming operators can always be represented as a composition of first-order operators that define a series of consecutive transformations. The existence of inverse transformations imposes some differential constrains on the coefficients of the initial operator. We show that these constraints may imply famous integrable equations, in particular, the Boussinesq equation.     

Distortion-suppression optimization for a class of nonlinear optical systems with feedback

We consider the initial boundary-value problem for the quasilinear diffusion equation ∂ _t u+u − DΔu = K(x, y)(1 + γ(u + ϕ(x, y))) describing the dynamics of optical systems with controlled feedback wave intensity modulation K(x, y) in the presence of incoming-wave phase perturbations ϕ(x, y). The control problem for the parameter K(x, y) is formulated with the objective of smoothing out the spatial nonhomogeneities of the total output phase u(x, y, T) + ϕ(x, y). We prove existence and uniqueness theorems for the generalized solutions of the direct and conjugate problems, solvability theorems for the optimization problems, and Frechet-differentiability of the objective functional. A formula for the functional gradient is derived and the efficiency of the gradient projection method is demonstrated numerically. __________ Translated from Prikladnaya Matematika i Informatika, No. 20, pp. 80–99, 2005.     

A note on k-th commutators in an associative ring

Il commutatorek-esimo dix, y in un anelloR è definito induttivamente da: [x, y]₁=[x, y]=xy−yx and [x, y] _k =[[x, y] _k−1 ,y]. Sia oraR un anello libero da 2-torsione e senza ideali destri nil non nulli. Si prova che se [[a, b], [c, d]] _k è nilpotente, per ogni scelta dia, b, c, d inR, alloraR è commutativo, se [[a, b], [c, d]] _k ha potenze nel centro diR alloraR soddisfa l'identità standard di grado 4. Inoltre si caratterizzano gli anelli in cui [[a, b], [c, d]] _k è nilpotente o regolare per ogni scelta dia, b, c, d inR.      

Implicit function theorem and its application to a Ulam’s problem for exact differential equations

We prove a global version of the implicit function theorem under a special condition and apply this result to the proof of a modified Hyers-Ulam-Rassias stability of exact differential equations of the form, g（x, y） ＋ h（x, y）y＇ =0.     

Coalescing and annihilating random walk with ‘action at a distance’

We consider a system of particles which perform continuous time random walks onZ ^d . These random walks are independent as long as no two particles are at the same site or adjacent to each other. When a particle jumps from a site x to a sitey and there is already another particle aty or at some neighbory′ ofy, then there is an interaction. In the coalescing model, either the particle which just jumped toy is removed (or, equivalently, coalesces with a particle aty ory′) or all the particles at the sites adjacent toy (other thanx) are removed. In the annihilating random walk, the particle which just jumped toy and one particle aty ory′ annihilate each other. We prove that when the dimensiond is at least 9, then the density of this system is asymptotically equivalent toC/t for some constant C, whose value is explicitly given.     

Functions with Nonzero Wronskian form a fundamental solution set

This self-contained note could find classroom use in a course on differential equations. It is proved that if y₁(x) and y₂(x) are C ² -functions whose Wronskian is never zero for α < x < β, then y₁ and y₂ form a fundamental solution set for a uniquely determined second-order linear homogeneous ordinary differential equation, y″ + p(x)y′ + q(x)y = 0, whose coefficients, p(x) and q(x), are continuous on (α, β).     

Multiple solutions of nonlinear functional boundary value problems

A class of nonlinear functional boundary conditions for the system of functional differential equations x"(t)=(F(x,y))(t)x'(t)=(F(x,y))(t), y"(t)=(H(x,y))(t)y'(t)=(H(x,y))(t) is introduced. Here F, H:C¹([a,b]) ×C¹([a,b]) ? L₁([a,b])F,\,H:C^1([a,b]) \times C^1([a,b]) \rightarrow L_1([a,b]) are nonlinear continuous operators. Sufficient conditions for the existence of at least four solutions are given. Results are proved by the Bihari lemma, the Leray-Schauder degree theory and the Borsuk theorem.     

Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation

We consider the equation y ^m u _xx − u _yy − b ² y ^m u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u _y (x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u _x (0, y) = 0 or u _x (0, y) = u _x (1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems     

Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences

We consider the family f _a,b (x,y)=(y,(y+a)/(x+b)) of birational maps of the plane and the parameter values (a,b) for which f _a,b gives an automorphism of a rational surface. In particular, we find values for which f _a,b is an automorphism of positive entropy but no invariant curve. The Main Theorem: If f _a,b is an automorphism with an invariant curve and positive entropy, then either (1) (a,b) is real, and the restriction of f to the real points has maximal entropy, or (2) f _a,b has a rotation (Siegel) domain. Research supported in part by the NSF.     

Finite difference methods for two-point boundary value problems involving high order differential equations

We discuss the construction of finite difference schemes for the two-point nonlinear boundary value problem:y ⁽²ⁿ⁾+f(x,y)=0,y ^(2j)(a)=A _2j ,y ^(2j)(b)=B _2j ,j=0(1)n–1,n2. In the case of linear differential equations, these finite difference schemes lead to (2n+1)-diagonal linear systems. We consider in detail methods of orders two, four and six for two-point boundary value problems involving a fourth order differential equation; convergence of these methods is established and illustrated by numerical examples.