Sums of the squares of terms of sequence {<Emphasis Type="Italic">u</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>}

In this paper, we consider generalized Fibonacci type second order linear recurrence {u _n }. We derive a generating matrix for both the sums of squares, ∑ _i=0 ⁿ u _i ² and the products of the form u _n u _n+2. We also derive explicit formulas for the sums and products by using matrix methods. Then we give a matrix method to generate the sums of product of two consecutive terms u _n u _n+1 as well as the product, u _n u _n+2. Further we give generating functions and combinatorial representations of the sums of squares of terms of {u _n } and the product, u _n u _n+2.     

Unitarily covariant maps in approximately finite-dimensional <Emphasis Type="Italic">C</Emphasis>*-algebras

We consider maps defined on a real space Asa of all self-adjoint elements of a C*-algebra A commuting with the conjugation by unitaries: F(u* au) = u* F(a)u for any a ∈ A _sa, u ∈ (A). In the case where A is a full matrix algebra, there is a functional realization of these maps (in terms of multivariable functions) and analytical properties of these maps can be expressed in terms of corresponding functions. In the present work, these results are generalized to the class of uniformly hyperfinite C*-algebras and to the algebra of all compact operators in a Hilbert space. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 213–227, 2007.     

Comparison of solutions of nonlinear evolution problems with different nonlinear terms

We obtain the comparison of solutions (formulated in terms, of some functions of them) of two nonlinear evolution problems with different nonlinear terms. First this is obtained for the equationsu ₁ – Δφ _i(u)=0, and then for the abstract equationsdu/dt+A _i u=0 (i=1,2) on a normal Banach lattice. Different applications of our abstract result are given in the last section.     

The Camassa–Holm Equation on the Half-Line: a Riemann–Hilbert Approach

We consider the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation u _t −u _txx +2u _x +3uu _x =2u _x u _xx +uu _xxx on the half-line x≥0. In this article, we aim to provide a characterization of the solution of the IBV problem in terms of the solution of a matrix Riemann–Hilbert (RH) factorization problem in the complex plane of the spectral parameter. The data of this RH problem are determined in terms of spectral functions associated to initial and boundary values of the solution. The construction requires more boundary data than those needed for a well-posed IBV problem. Their dependence is expressed in terms of an algebraic relation to be satisfied by the spectral functions. This RH formulation gives us the long-time asymptotics of a solution of the CH-equation. Dedicated to Gennadi Henkin in great admiration.     

<Emphasis Type="Italic">BV</Emphasis> functions and parabolic initial boundary value problems on domains

Given a uniformly elliptic second order operator on a possibly unbounded domain , let (T(t)) _t≥0 be the semigroup generated by in L ¹(Ω), under homogeneous co-normal boundary conditions on ∂Ω. We show that the limit as t → 0 of the L ¹-norm of the spatial gradient D _x T(t)u ₀ tends to the total variation of the initial datum u ₀, and in particular is finite if and only if u ₀ belongs to BV(Ω). This result is true also for weighted BV spaces. A further characterization of BV functions in terms of the short-time behaviour of (T(t)) _t≥0 is also given.      

Su un problema esterno per la equazione delle onde

Sunto Si considera il problema diCauchy per la equazione u_tt = u_xx + u_yy con dati iniziali sui due piani caratteristici t = ±y. Si dà un teorema di esistenza ed unicità della soluzione nella ?regione esterna? |t| ≤ |y|, e si prova che la soluzione non dipende in modo continuo dai dati.

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Summary We consider theCauchy problem for the wave equation u_tt = u_xx + u_yy with initial data on the characteristic planes t = ±y. We discuss the problem related to the ?exterior region? |t| ≤ |y|; we prove uniqueness of the solution, its instability with respect to the data, and the existence of the solution if the data are analytic functions of x (this hypothesis is essential, as we prove with an example).

     

On the representation theory of an algebra of braids and ties

We consider the algebra ℰ _n (u) introduced by Aicardi and Juyumaya as an abstraction of the Yokonuma–Hecke algebra. We construct a tensor space representation for ℰ _n (u) and show that this is faithful. We use it to give a basis of ℰ _n (u) and to classify its irreducible representations.     

On one representation of analytic functions by harmonic functions

Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝ ⁿ . We study the representation of this function in the form of a series u(x) = u ₀(x) + |x|² u ₁(x) + |x|⁴ u ₂(x) + …, where u _k (x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula. Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162.     

Asymptotic Behaviors of the Solutions to Scalar Viscous Conservation Laws on Bounded Interval

Abstract This paper concerns the asymptotic behaviors of the solutions to the initial-boundary value prob-lem for scalar viscous conservations laws u_t+f(u)_x=u_(xx) on[0,1],with the boundary condition u(0,t) =u_,u(1,t)=u_+ and the initial data u(x,0)=u_0(x,0)=u_0(x),where u_≠u_+ and f is a given function satisfyingf'(u)>0 for u under consideration.By means of energy estimates method and under some more regular condi-tions on the initial data,both the global existence and the asymptotic behavior are obtained.When u_u_+, which corresponds to shock waves in inviscid conservation laws, it is established for weak shockwaves,which means that │u_-u_+│is small.Moreover,exponential decay rates are both given.     

The Navier–Stokes flow around the linearly growing steady state with bounded disturbance

The existence and uniqueness of locally-in-time solutions of the Cauchy problem to the incompressible Navier–Stokes equations is established. The initial velocity U ₀ is of the form U ₀(x) := u ₀(x)−M x, where M is a real-valued matrix and u ₀ is a bounded smooth function. Our solutions satisfy the equations in the classical sense, even though the semigroup is not analytic. If M is skew-symmetric, and u ₀ is small and a sum of trigonometric functions, then obtained solutions can be extended globally-in-time with the exponential decay in time.     

Operator–valued Fourier multiplier theorems and maximal $L_p$-regularity

We prove a Mihlin–type multiplier theorem for operator–valued multiplier functions on UMD–spaces. The essential assumption is R–boundedness of the multiplier function. As an application we give a characterization of maximal –regularity for the generator of an analytic semigroup in terms of the R–boundedness of the resolvent of A or the semigroup . Received July 19, 1999 / Revised July 13, 2000 / Published online February 5, 2001     

Initial boundary value problem for generalized boussinesque type equations with nonlinear source

A comparison principle for solutions of the first initial boundary value problem for the generalized Boussinesque equation with a nonlinear sourceu _t-Δψ(u)-Δu _t+q(u)=0 is established. By using this comparison principle, we prove new existence and nonexistence theorems for solutions of the first initial boundary value problem in the case of power-law functions ψ (ξ) andq (ξ). Translated fromMathematicheskie Zametki, Vol. 65, No. 1, pp. 70–75, January, 1999.     

Controlled stability

Summary A rest solution, p, of a control system is said to possess controlled stability if for each t _1>0 a full nbd. of points of p can be steered to p in time t ₁ by solutions of the system. Equivalently, if the system acts in a linear space and p is the origin, this states the domain of ? null controllability ? is open. We consider systems of the form with either each u_i measurable with |u_i(t)| ⩽1, or |u_i| ≤1 inL ₁ or ‖u_i‖ ≤1 in the space of regular countably additive measures, and X, Y ² , …, Y^m analytic vector fields on an analytic n-manifold. If X(p)=0 a necessary condition that the rest solution p possess controlled stability is that the dimension, at p, of the Lie algebra generated by X, Y ² , …, Y^m be n. First order sufficient conditions are well known and can be stated in terms of subsets of the elements of this Lie algebra. This paper provides two higher order sufficiency tests, together with examples of their applications. Entrata in Redazione il 14 giugno 1976. This research was supported by the National Science Foundation under grant MCS76-04419.     

Solution of two-dimensional boundary value problems corresponding to initial-boundary value problems of diffusion on a right cylinder

We consider boundary value problems for the differential equations Δ₂ u + B u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation Δ₃ u − pu = ∂ _t u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for such solutions in the class of functions with continuous L ₂-norm.     

Classification of solutions to the higher order Liouville’s equation on {\mathbb{R}^{2m}}

We classify the solutions to the equation (−Δ) ^m u = (2m − 1)!e ^2mu on giving rise to a metric with finite total Q-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Δu at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric at infinity, and we observe that the pull-back of this metric to S ^2m via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.     

Darboux-integrable discrete systems

We extend Laplace’s cascade method to systems of discrete “hyperbolic” equations of the form u_i+1,j+1 = f(u_i+1,j, u_i,j+1 , u_i,j), where u_ij is a member of a sequence of unknown vectors, i, j ∊ ℤ. We introduce the notion of a generalized Laplace invariant and the associated property of the system being “Liouville.” We prove several statements on the well-definedness of the generalized invariant and on its use in the search for solutions and integrals of the system. We give examples of discrete Liouville-type systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 207–219, August, 2008.     

Global-in-time Uniform Convergence for Linear Hyperbolic-Parabolic Singular Perturbations

We consider the Cauchy problem εu^″ε ＋ δu′ε ＋ Auε = 0, uε（0） = uo, u′ε（0） = ul, where ε 〉 0, δ 〉 0, H is a Hilbert space, and A is a self-adjoint linear non-negative operator on H with dense domain D（A）. We study the convergence of （uε） to the solution of the limit problem ,δu＇ ＋ Au = 0, u（0） = u0. For initial data （u0, u1） ∈ D（A1/2）× H, we prove global-in-time convergence with respect to strong topologies. Moreover, we estimate the convergence rate in the case where （u0, u1）∈ D（A3/2） ∈ D（A1/2）, and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for ｜u′ε（t）｜ which does not depend on ε.     

Reconstruction of functions from their integrals over<Emphasis Type="Italic">k</Emphasis>-planes

Thek-plane Radon transform assigns to a functionsf(x) on ℝ ⁿ the collection of integralsf(τ)=∫ _τ f over allk-dimensional planesτ. We give a systematic treatment of two inversion methods for this transform, namely, the method of Riesz potentials, and the method of spherical means. We develop new analytic tools which allow to invertf(τ) under minimal assumptions forf. It is assumed thatfεL ^p , 1≤p<n/k, orf is a continuous function with minimal rate of decay at infinity. In the framework of the first method, our approach employs intertwining fractional integrals associated to thek-plane transform. Following the second method, we extend the original formula of Radon for continuous functions on ℝ² tofεL ^p (ℝ ⁿ ) and all 1≤k<n. New integral formulae and estimates, generalizing those of Fuglede and Solmon, are obtained. The work was supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

<Emphasis Type="Italic">Q</Emphasis>-measures on <Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">κ</Emphasis></Subscript>λ

 We give a characterization of strongly compact cardinals in terms of Q _κ λ. We also prove that weakly normal Q-measures on Q _κ λ are ⊂κ-normal. Received: 29 September 2000 / Revised version: March 2002 Published online: 5 November 2002 This project is supported by the New Zealand Marsden Fund. The author wishes to thank the referee for numerous comments and suggestions which have been incorporated into this version of the paper.