On Maps Preserving Zero Jordan Products

Let R be a ring, A = M _n (R) and θ: A → A a surjective additive map preserving zero Jordan products, i.e. if x,y ∈ A are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains \frac12\frac{1}{2} and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: A → A is a Jordan homomorphism.     

On the number of real eigenvalues of a certain boundary-value problem for a second-order equation with fractional derivative

The asymptotics as α → 0+ of the number of real eigenvalues λ _n (α) of the problem y″(x)+λD ₀ ^α (x) = 0, 0 < x < 1, y(0) = y(1) = 0, is obtained. The minimization of real eigenvalues is carried out. It is proved that . __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 137–155, 2006.     

Realization of the Perron effect whereby the characteristic exponents of solutions of differential systems change their values

We realize the Perron effect of change of values of characteristic exponents: for arbitrary parameters λ ₁ <- λ ₂ < 0, β ₂ ≥ β ₁ ≥ λ ₂, and m > 1, we prove the existence of a linear differential system $ \dot x $ \dot x = A(t)x, x ∈ R ₂, t ≥ t ₀, with bounded infinitely differentiable coefficients and with characteristic exponents λ ₁(A) = λ ₁ <- λ ₂(A) = λ ₂ and of an m-perturbation f: [t ₀,+∞) × R ₂ → R ₂ infinitely differentiable in time, continuously differentiable with respect to the phase variables y ₁ and y ₂, (y ₁, y ₂) = y ∈ R ₂ (infinitely differentiable with respect to the variables y ₁ ≠ 0 and y ₂ ≠ 0 and with respect to all of these variables in the case of a positive integer m > 1), satisfying the condition ‖f(t, y)‖ ≤ const × ‖y‖ ^m , y ∈ R ², t ≥ t ₀, and such that all nontrivial solutions y(t, c) of the perturbed system

Algébres topologiquement uniformes et algébres fortement topologiquement uniformes

We characterize locally convex topological algebrasA satisfying: a sequence (x _n) inA converges to 0 if, and only if, (x _n ²) converges to 0. We also show that a real Banach algebra such thatx _n ²+y _n ²→0 if, and only if,x _n → 0 andy _n → 0, for every sequences (x _n) and (y _n) inA, is isomorphic to, whereX is a compact space.      

On Cesáro summability of fourier series with respect to double complete orthonormal systems

Let {ϕn(x), n = 1, 2,...} be an arbitrary complete orthonormal system on the interval I:= [0, 1]which consists of a.e. bounded functions. Suppose that E ₀ ⊂ I ² is any Lebesgue measurable set such that μ₂ E ₀ > 0, and φ, φ(0) = 0, is an increasing continuous function on [0, ∞) with φ(u) = o(u ln u) as u → ∞. Then there exist a function f ∈ L¹(I ²) and a set E ₀ ^′ , ⊂ E ₀, μ₂ E ₀ ^′ > 0, such that

Estimation of Parameters for Diffusion Processes with Jumps from Discrete Observations

In this paper, we consider a multidimensional diffusion process with jumps whose jump term is driven by a compound Poisson process. Let a(x,θ) be a drift coefficient, b(x,σ) be a diffusion coefficient respectively, and the jump term is driven by a Poisson random measure p. We assume that its intensity measure q^θ has a finite total mass. The aim of this paper is estimating the parameter α = (θ,σ) from some discrete data. We can observe n + 1 data at t_iⁿ = ih_n, . We suppose h_n → 0, nh_n → ∞, nh_n² → 0. Final version 20 December 2004     

One-Sided Ideal Growth of Free Associative Algebras

Let R be a finitely generated associative algebra with unity over a finite field . Denote by a _n (R) the number of left ideals J ⊂ R such that dim R/J = n for all n ≥ 1. We explicitly compute and find asymptotics of the left ideal growth for the free associative algebra A _d of rank d with unity over , where d ≥ 1. This function yields a bound a _n (R) ≤ a _n (A _d ), , where R is an arbitrary algebra generated by d elements. Denote by m _n (R) the number of maximal left ideals J ⊂ R such that dim R/J = n, for n ≥ 1. Let d ≥ 2, we prove that m _n (A _d ) ≈ a _n (A _d ) as n → ∞.     

An implicit function theorem: Comment

In Ref. 1, Jittorntrum proposed an implicit function theorem for a continuous mappingF:R ⁿ ×R ^m R ⁿ, withF(x ⁰,y ⁰)=0, that requires neither differentiability ofF nor nonsingularity of _x F(x ⁰,y ⁰). In the proof, the local one-to-one condition forF(·,y):A R ⁿ R ⁿ for ally B is consciously or unconsciously treated as implying thatF(·,y) mapsA one-to-one ontoF(A, y) for ally B, and the proof is not perfect. A proof can be given directly, and the theorem is shown to be the strongest, in the sense that the condition is truly if and only if.     

Homomorphisms between JC＊-algebras and Lie C＊-algebras

It is shown that every almost ＊-homomorphism h ： A→B of a unital JC＊-algebra A to a unital JC＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x∈A, and that every almost linear mapping h ： A→B is a ＊-homomorphism when h（2^nu o y） - h（2^nu） o h（y）, h（3^nu o y） - h（3^nu） o h（y） or h（q^nu o y） = h（q^nu） o h（y） for all unitaries u ∈A, all y ∈A, and n = 0, 1,.... Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings. We prove that every almost ＊-homomorphism h ： A→B of a unital Lie C＊-algebra A to a unital Lie C＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x ∈A.     

Linear functional differential equations of neutral type asymptotically autonomous

Summary It is studied the relationship between the solutions of the linear functional differential equations(1) (d/dx) D(x_t)=L(x_t) and its perturbed equation(2) [(d/dx) D(x_t)−G(t, x_t)]= =L(x_t)+F(t, x_t) and is proved, under certain hypotheses which will be precised bellow that, if μ is a simple characteristic root of(1), then there exist a σ > 0 and a non zero vector a such that system(2) has a solution satisfying where δ(t)=αd{F(t, ϕ_μ)+μG(t, ϕ_μ)+F(t, X₀G(t, ϕ_μ))}, ϕ_μ(θ)=c·exp (μθ), −r⩾θ⩾0 and α, d, X₀ are given constants. Entrata in Redazione il 5 gennaio 1972.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

Sopra una curva gobba luogo di certi punti parabolici di una rete di superficie,generale, dell’ordinen

Abstract   we prove that the operator maps into itself for where and k(x,y)=ϕ(x,y) e^ig(x,y), ϕ(x,y) satisfies (5), (e.g. ϕ(x,y)=|x–y|^iτ,τ real) and the phase g(x,y)=x^a⋅ y^b +Φ_**(x^a,y^b). We obtain L^p estimates for operators with more general phases than in [5] and for these operators we require that b₁ b₂>1, and and a_l≥ b_l≥ 1, which remained open from [4]. Keywords Oscillatory integrals, L^p mappings Mathematics Subject Classification (2000) Primary 42B20, Secondary 46B70, 47G10     

Representation of Contractive Solutions of a Class of Algebraic Riccati Equations as Characteristic Functions of Maximal Dissipative Operators

Let

Distribution of values of bounded generalized polynomials

A generalized polynomial is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a generalized polynomial mapping is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial mapping u: Z ^d  → R ^l has a representation u(n) = f(ϕ(n)x), n ∈ Z ^d , where f is a piecewise polynomial function on a compact nilmanifold X, x ∈ X, and ϕ is an ergodic Z ^d -action by translations on X. This fact is used to show that the sequence u(n), n ∈ Z ^d , is well distributed on a piecewise polynomial surface (with respect to the Borel measure on that is the image of the Lebesgue measure under the piecewise polynomial function defining ). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine approximations extending the work of van der Corput and of Furstenberg–Weiss.     

Growth and Hölder conditions for the sample paths of Feller processes

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C _c ^∞(ℝ ⁿ )⊂D(A) and A|C _c ^∞(ℝ ⁿ ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|_∞≤c(1+|ξ|²) and |Imp(x,ξ)|≤c ₀Rep(x,ξ). We show that the associated Feller process {X _t } _{t ≥0} on ℝ ⁿ is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β_∞ ^x :={λ>0:lim _{|ξ|→∞ | x − y |≤2/|ξ|}|p(y,ξ)|/|ξ|^λ=0} or δ_∞ ^x :={λ>0:liminf _{|ξ|→∞ | x − y |≤2/|ξ| |ε|≤1}|p(y,|ξ|ε)|/|ξ|^λ=0}, and obtain a.s. (ℙ ^x ) that lim _{t →0} t ^−1/λ _{s ≤ t} |X _s −x|=0 or ∞ according to λ>β_∞ ^x or λ<δ_∞ ^x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27]. Received: 21 July 1997 / Revised version: 26 January 1998     

Equations fonctionnelles et estimations de normes de matrices

We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(x−y)/θ(x−y) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(x−y)+τ(x)τ(y), τ(0)=0. In both cases we find θ²=aθ⁴+bθ²+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x _r −x _s )) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex _r are equidistant.

Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux     

A note on Sobolev orthogonality for Laguerre matrix polynomials

Let {Ln(A,λ)(x)}n≥0 be the sequence of monic Laguerre matrix polynomials defined on [0, ∞) by Ln(A,λ)(x)=n!/(-λ)n∑nk=0(-λ)κ/k!(n-1)! (A I)n[(A I)k]-1 xk,where A ∈ Cr×r. It is known that {Ln(A,λ)(x)}n≥0 is orthogonal with respect to a matrix moment functional when A satisfies the spectral condition that Re(z) ＞ - 1 for every z ∈σ(A).In this note we show that forA such that σ(A) does not contain negative integers, the Laguerre matrix polynomials Ln(A,λ) (x) are orthogonal with respect to a non-diagonal SobolevLaguerre matrix moment functional, which extends two cases: the above matrix case and the known scalar case.     

Reduction of CalderON Commutators

Let A₁,..., A_n be Lipschitz functions on R such that A'₁,...,A'_nVMO. We show that on any bounded interval, the Calderóncommutator associated with the kernel (A₁(x)–A₁(y)) ...(A_n(x) – A_n(y))/(x–y) ⁿ¹ is a compact perturbationof , where H is the Hilberttransform. 1991 Mathematics Subject Classification 47B38, 47B47,47G10, 45E99.