On the best approximations and Kolmogorov widths of besov classes of periodic functions of many variables

Order estimates are obtained for the best approximations of the classesB _1, ^r in the spaceL _q with 1<q< and classesB _, ^r in a uniform metric. The behavior of Kolmogorov widths of the classesB _p, ^r ,1<p, in the metric of L is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 79–92, January, 1995.     

Some Problems for Cubic Nonlinear Equations on a Half-Line

The paper deals with a problem of developing an inverse-scattering based formalism for solving problems for the cubic nonlinear (or the modified Korteweg–de Vries (KdV)) equations: q _t +q _xxx +6q ² q _x =0, 0x<, –<t<,q _t +q _xxx –6q ² q _x =0, with the given initial and boundary conditions: q(x,0)=q(x),q(0,t)=p(t), p(t)L ₁(–,). The relation between the solution of the initial-boundary value problem (1), (3), (4) and that of the KdV equation on the half-line is shown. The Cauchy problem for the cubic nonlinear equation: q _t +q _xxx –6|q|² q _x =0, 0x<, –<t<, with the given initial condition (3) is considered also. Here we solve the above problems on the half-line 0x< but with –<t<.     

Characterization of a Two-Weighted Vector-Valued Inequality for Fractional Maximal Operators

We give a characterization of the weights u(·) and v(·) for which the fractional maximal operator M _s is bounded from the weighted Lebesgue spaces L ^p(l ^r, vdx) into L ^q(l ^r, udx) whenever 0 s < n, 1 < p, r < , and 1 q < .     

Equidistribution of Gauss sums and Kloosterman sums

Let m be a positive integer. Fix a nontrivial additive character for each finite field F_q. To state the first result of this paper, we also fix r distinct multiplicative characters ₁,...,_r for each finite field F_q with more than r elements. We shall prove that when varies over multiplicative characters of F_q other than the m-th roots of the r-tuples of angles of Gauss sums are asymptotically equidistributed on the r-dimensional torus (S¹)^r as q goes to infinity.The n-dimensional Kloosterman sum over F_q at a F_q^× is One can define the angle (q,a) of Kl_n(q,a) in a suitable way. We shall prove that when a varies over nonzero elements of F_q, the q–1 angles (q,a^m) of Kloosterman sums are asymptotically equidistributed as q goes to infinity.Mathematics Subject Classification (2000) 11L05, 14F20     

Comparison of optimal quadrature formulas

Summary We prove the monotonicity of the error of the optimal quadrature formula of a given quasi-Hermitian type inW _q ^r [0.1] (1<q) with respect to the order of the derivatives appearing in the end point terms.     

Evolution in a Gaussian Random Field

We consider an evolution process in a Gaussian random field V(q) with the mean ‹V(q)› = 0 and the correlation function W(|q – q|) ‹V(q)V(q)›, where q ^d and d is the dimension of the Euclidean space ^d . For the value ‹G(q,t;q ₀)›, t > 0, of the Green's function of the evolution equation averaged over all realizations of the random field, we use the Feynman–Kac formula to establish an integral equation that is invariant with respect to a continuous renormalization group. This invariance property allows using the renormalization group method to find an asymptotic expression for ‹G(q,t;q ₀)› as |q – q ₀| and t .     

A q-Analogue of Weber-Schafheitlin Integral of Bessel Functions

In an attempt to find a q-analogue of Weber and Schafheitlin's integral ₀ x ^– J (ax) J (bx) dx which is discontinuous on the diagonal a = b the integral ₀ x ^– J ⁽²⁾ (a(1 – q)x; q)J ⁽¹⁾ (b(1 – q)x; q) dx is evaluated where J ⁽¹⁾ (x; q) and J ⁽²⁾ (x; q) are two of Jackson's three q-Bessel functions. It is found that the question of discontinuity becomes irrelevant in this case. Evaluations of this integral are also made in some interesting special cases. A biorthogonality formula is found as well as a Neumann series expansion for x in terms of J ⁽²⁾ _+1+2n ((1 – q)x; q). Finally, a q-Lommel function is introduced.     

On stationary queue length distributions for G/M/s/r queues

Consider aG/M/s/r queue, where the sequence{A _n } _n=– of nonnegative interarrival times is stationary and ergodic, and the service timesS _n are i.i.d. exponentially distributed. (SinceA _n =0 is possible for somen, batch arrivals are included.) In caser < , a uniquely determined stationary process of the number of customers in the system is constructed. This extends corresponding results by Loynes [12] and Brandt [4] forr= (with=ES₀/EA₀<s) and Franken et al. [9], Borovkov [2] forr=0 ors=. Furthermore, we give a proof of the relation min(i, s)¯p(i)=p(i–1), 1ir + s, between the time- and arrival-stationary probabilities¯p(i) andp(i), respectively. This extends earlier results of Franken [7], Franken et al. [9].     

Partial applicability of Moser's method to nonlinear elliptic systems

We consider nonlinear elliptic systems of divergent-type second-order partial differential equations with solutionsu W _p ¹ . It is proved thatDu L _q with someq (p; +) and it is explicitly shown howq depends on the ellipticity modulus of the system. Some conditions on the ellipticity modulus are obtained under which the solutions satisfy the Hölder conditions and the Liouville theorem holds.Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 547–557, October, 1995.     

Tauberian Theorems for Weighted Means of Double Sequences

Let p := {p _j } _j=0 and q := {q _k } _k–0 be complex (or real) sequences with the property that P _m := _j–0 ^m p _j 0 for all m 0, Q _n := _k–0 ⁿ q _k 0 for all n 0, and both of {P _m } _m=0 and {Q _n } _n=0 are varying away from 1. Assume that {s _mn } is a double sequence in C(or one of R, a Banach space, and an ordered linear space), which is (N¯,p,q; ,) summable to a finite limit, where (,) =(1,1), (1,0), or (0,1). We give necessary and sufficient conditions under which {s _mn } converges in Pringsheim's sense. These conditions are weaker than the two-dimensional analogues of Landau's condition and Schmidt's slow decrease condition. Our results generalize and extend [1 4, 12 15]. We also solve the problems posed in [3, 13, 14].     

A Criterion for Convergence of Weak Greedy Algorithms

We find a class V of sequences such that the condition V is necessary and sufficient for convergence of weak greedy algorithm with weakness sequence for each f and all Hilbert spaces H and dictionaries D. We denote by V the class of sequences x={x _{k k=1} , x _k 0, k=1,2,..., with the following property: there exists a sequence 0=q ₀<q ₁< such that _s=1 2 ^s /q _s )< and _s=1 2^–s _k=1 ^q s ^x _k ²<, where q _s :=q _s –q _s–1.     

Criteria for the Boundedness and Compactness of Operators with Power-Logarithmic Kernels

In the present work, necessary and sufficient conditions are given in terms of a nonnegative Borel measure which ensure the boundedness and compactness of operators with power-logarithmic kernels from L ^p (0, a) to L ^p (0, a) (or to L ^q (0, a)), where 0 < a < , 1 < p, q < , > 1/p and 0.     

A relation between spline approximation and the problem of the approximation of one class by another

An investigation of the approximation in L_q(–, ) of differentiable functions whose k-th derivatives belong to L_p(–, ), by splines S_m (x) with nonfixed nodes, under the extra assumption that the norms in L_s(–, ) of theirl-th derivatives have a common bound. A relation is established with the problem of approximating functions of one class by functions of another class.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 501–510, May, 1971.     

Best Orthogonal Trigonometric Approximations of Classes of Functions of Many Variables Lβ, pψ

We obtain order estimates for the best orthogonal trigonometric approximations of classes of functions of many variables L _{, p} in the space L _q, 1 < p < q < , q > 2.     

Laurent expansions for certain functions defined by Dirichlet series

Summary The Poisson summation formula is employed to find the Laurent expansions of the Dirichlet seriesF(s, c) = _{n = 0} exp[–(n + c)^1/2 s] andG(s, c) = _{n = 0} (–1) ⁿ exp[–(n + c)^1/2 s] (0c<1) abouts = 0. The Laurent expansions ofF(s, c) andG(s, c) are convergent respectively for 0 < |s| < and |s| < , and define the analytic continuation of the Dirichlet series to the half-plane Res < 0.     

Positivity and Negativity of Solutions to a Schrödinger Equation in ℝ N

Weak L ² -solutions u of the Schrödinger equation, –u + q(x) u – u = f(x) in L ² , are represented by a Fourier series using spherical harmonics in order to prove the following strong maximum and anti-maximum principles in (N 2): Let ₁ denote the positive eigenfunction associated with the principal eigenvalue ₁ of the Schrödinger operator . Assume that the potential q(x) is radially symmetric and grows fast enough near infinity, and f is a `sufficiently smooth' perturbation of a radially symmetric function, f 0 and 0 f / C const a.e. in . Then u is ₁-positive for - < < ₁ (i.e., u c ₁ with c const > 0) and ₁-negative for ₁ < < ₁ + (i.e., u –c₁ with c const > 0), where > 0 is a number depending on f. The constant c > 0 depends on both and f.     

The interpolation ofℓ r sequences by hp functions

The sequence spaceH ^P (z)={{f (z_h)}:f H ^p} is defined for a fixed sequence Z={z_k} of different points of the open unit disk and the Hardy class HP of analytic functions in the disk. For an arbitrary p[1, ) is constructed a point sequence Z= {z_k} such that ¹h ^p(z), but ^r h^p (Z) for r > 1. It follows from a well-known result of L. Carleson that the inclusions ^r h (Z) for all r[1,] are equivalent.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 503–508, April, 1977.     

Vergleich zweierS ∞-Methoden zur Lösung der Neutronentransportgleichung

Zusammenfassung Es wird eine semi-analytische Methode (S _/0) zur Lösung der monoenergetischen Transportgleichung entwickelt. Diese Methode wird mit der bekanntenS ¹ -Methode verglichen. Bei Anwendung letzterer auf Reaktorabschirmungen können negative Flüsse vorkommen. Dieser unphysikalische Effekt wird durch die neue Methode vollständig vermieden. Numerische Ergebnisse zeigen die Begrenzungen der Anwendbarkeit vonS ¹ und bestätigen die numerische Stabilität vonS ⁰ .

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A semianalytic method (S ⁰ ) is presented for solving the monoenergetic multi-region transport equation. This method is compared with results fromS ¹ -theory given in the literature. Application ofS ¹ -theory to reactor shields may lead to negative neutron fluxes and to flux oscillations. These unphysical effects are completely avoided by the new method. Numerical results demonstrate the limitation ofS ¹ and confirm the numerical stability ofS ⁰ .

     

Steady vortex rings with swirl in an ideal fluid: asymptotics for some solutions in exterior domains

In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder (rd) where (r, , z) denotes the cylindrical co-ordinates in ³ is considered. The motion is with swirl (i.e. the -component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. [9] that for the problem without swirl (f _q = 0 in (f)) in the whole space, as the flux constant k tends to 1) dist(0z, A) = O(k ^1/2); diam A = O(exp(–c ₀ k ^3/2));2) k^1/2)_k converges to a vortex cylinder U _m (see (1.2)).We show that for the problem with swirl, as k , 1) holds; if m q + 2 then 2) holds and if m > q + 2 it holds with U _q+2 instead of U _m. Moreover, these results are independent of f ₀, f _q and d > 0.     

Moser-type inequalities in Lorentz spaces

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH _inf0 ^sup1 of Au=(f _i ) _{x i} , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).