Produktsätze für Verfahren zur Limitierung von Doppelfolgen

{p _mn } - ₀₀>0, (1, 1) (1.1) (1.2). {s _mn } J _p - ( bJ _p -lims _mn =), (1.3) 0<x,y<1 p _s (, )/p(x, y) x, y 1-. {r _mn } - , (1.5) 0<, <1. N _rp - , (1.6). , bJ _p -lims _mn = bJ _q -lim(N _rps) _mn =. J _p - . , .     

О простых идеалах первой степени

.      

Theorem on a convergence condition in the spaces

For a given -function (u), a condition on a -function (u) is found such that it is necessary and sufficient for the following to hold: if f_n(x) f(x) and f _n (x)M (n=1, 2, ...) where M>0 is an absolute constant, then f _n (x)–f(x)0(n). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.     

Net subgroups of Chevalley groups. II. Gauss decomposition

This paper is a continuation of RZhMat 1980, 5A439, where there was introduced the subgroup () of the Chevalley group G(,R) of type over a commutative ring R that corresponds to a net , i.e., to a set =(),, of ideals of R such that ₊ whenever ,,+ . It is proved that if the ring R is semilocal, then () coincides with the group ₀ considered earlier in RZhMat 1976, 10A151; 1977, 10A301; 1978, 6A476. For this purpose there is constructed a decomposition of () into a product of unipotent subgroups and a torus. Analogous results are obtained for sub-radical nets over an arbitrary commutative ring.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 62–76, 1982.In conclusion, the authors would like to thank Z. I. Borevich for his interest in this paper.     

Quasifractional Approximants in the Theory of Composite Materials

We study the effective heat conductivity of regular arrays of perfectly conducting spheres embedded in a matrix with unit conductivity. Quasifractional approximants allow us to derive an approximate analytical solution, valid for all values of the spheres volume fraction [0, _max] (_max is the maximum volume fraction of a spheres). As a starting point we use a perturbation approach for 0 and an asymptotic solution for _max. Three different spatial arrangements of the spheres, simple cubic, body centred and face centred cubic arrays, are considered. Results obtained give a good agreement with numerical data.     

Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis

Let ₁, ₂, ... be a sequence of independent identically distributed random variables with zero means. We consider the functional _n = _k=o ⁿ (S _k ) where S₁=0, S_k= _i=1 ^k _i (k1) and(x)=1 for x0,(x) = 0 for x<0. It is readily seen that _n is the time spent by the random walk S_n, n0, on the positive semi-axis after n steps. For the simplest walk the asymptotics of the distribution P (_n = k) for n and k, as well as for k = O(n) and k/n<1, was studied in [1]. In this paper we obtain the asymptotic expansions in powers of n^–1 of the probabilities P(h_n = nx) and P(nx_{1 n} nx₂) for 0<₁, x = k/n ₂<1, 0<₁x₁2₂<1.Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 613–620, April, 1974.The author wishes to thank B. A. Rogozin for valuable discussions in the course of his work.     

A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

Quasiclassical asymptotic behavior of the scattering cross section for asymptotically homogeneous potentials

One considers the total scattering cross section on the potential gV(x), x^m, m3, for large values of the coupling constant g and of the wave number k. One assumes that V(x)(x/|1x|)|x|^–, 2>m+1, as ¦x¦. It is shown that for gk^–1 , g^3–ak^2(a–2) the scattering cross section is equal asymptotically to _a(gk^–1), x=(m–1)(–1)^–1. Here the coefficient _a is determined only by the function and the number . Under the additional conditions >0, V>0, the indicated asymptotic behavior holds in the large domain gk^–1 , gk^a–z c(gk^–1), >0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 105–136, 1986.     

On the optimality of a characterization theorem for the gamma function using the multiplication formula

Summary The following Artin type characterization of : _{+ +} is proved: Assume thatf: _{+ +} satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim _{x 0} xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.     

On the bestL p multipoint local approximation

P (f) — , f L ^p - , k . f 02k–2 P (f) 0.     

Moduli of smoothness and best approximation in the spacesL p , 0<p<1

L ^p , 0<<1, .     

The Cesàro-Denjoy-Bochner scale of almost periodicity

, —— , . , f, ——, —. .     

Singular Eigenvalue Problems for the Equation y (n) + λp(x)y = 0

The paper studies singular eigenvalue problems for the equation y ⁽ⁿ⁾ +p(x)y=0 with boundary conditions imposed on the derivatives y ⁽ⁱ⁾ at the points x=a and x=. We look for singular problems which are analogous to regular problems on a finite interval. It is characterized when each eigenfunction has a finite number of zeros and when the spectrum is discrete or continuous, respectively.     

Improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set

We will establish the following improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set: For each k and d, 0 k d, define f(d,k) = d+1 if k = 0 and f(d,k) = max{d+1,2d–2k+2} if 1 k d.Theorem 1. Let S be a compact, connected, locally starshaped set in R^d, S not convex. Then for a k with 0 k d, dim ker S k if and only if every f(d, k) lnc points of S are clearly visible from a common k-dimensional subset of S.Theorem 2. Let S be a nonempty compact set in R^d. Then for a k with 0 k d, dim ker S k if and only if every f (d, k) boundary points of S are clearly visible from a common k-dimensional subset of S. In each case, the number f(d, k) is best possible for every d and k.     

Translation Invariant Positive Definite Hermitian Bilinear Ultradistributions

Every translation invariant positive definite Hermitian bilinear functional on the Gel'fand-Shilov space sMpMp(n×nK) of general type S is of the form B(,) = (x)(x)d(x), , sMpMp (n), where is a positive {M}-tempered measure, i.e., for every > 0 exp[-M(|x|)] d(x) < . To prove this we prove Schwartz kernel theorem for {M}-tempered ultradistributions and need Bochner-Schwartz theorem for {M}-tempered ultradistributions. Our result includes most of the quasianalytic cases. Also, we obtain parallel results for the case of Beurling type (Mp.     

On cubature formulas on the basis of lattices

Q (.. , L). Q . P(S_r(2)) — 2 (S _r(2) (r — ). , M(P(S _r(m=sup{t(·)t(·)₁:t P(S _r(2)),t 0}. , /4+(1)M(P(S _r(2)))/r ²15/17+(1)(r+). (Q), Q L.     

A generalized Laplace transform of generalized functions

- . . . , .     

Some inequalities of Hardy-Littlewood type

[3] , >0 n ^– a _n , . , . . , .

---

---

This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant #234.     

Global solutions describing the collapse of a spherical or cylindrical cavity

The collapse of a spherical (cylindrical) cavity in air is studied analytically. The global solution for the entire domain between the sound front, separating the undisturbed and the disturbed gas, and the vacuum front is constructed in the form of infinite series in time with coefficients depending on an appropriate similarity variable. At timet=0⁺, the exact planar solution for a uniformly moving cavity is assumed to hold. The global analytic solution of this initial boundary value problem is found until the collapse time (=(–1)/2) for 1+(2/(1+v)), wherev=1 for cylindrical geometry, andv=2 for spherical geometry. For higher values of , the solution series diverge at timet — 2(–1)/ (v(1+)+(1–)2) where =2/(–1). A close agreement is found in the prediction of qualitative features of analytic solution and numerical results of Thomaset al. [1].     

The Long Dimodules Category and Nonlinear Equations

Let H be a bialgebra and _H L^H be the category of Long H-dimodules defined, for a commutative and co-commutative H, by F. W. Long and studied in connection with the Brauer group of a so-called H-dimodule algebra. For a commutative and co-commutative H, _H L^H =_H YD^H (the category of Yetter–Drinfel'd modules), but for an arbitrary H, the categories _H L^H and _H YD^H are basically different. Keeping in mind that the category _H YD^H is deeply involved in solving the quantum Yang–Baxter equation, we study the category _H L^H of H-dimodules in connection with what we have called the D-equation: R¹² R²³ = R²³ R¹², where R End_k(M M) for a vector space M over a field k. The main result is a FRT-type theorem: if M is finite-dimensional, then any solution R of the D-equation has the form R = R_{(M, , )}, where (M, , ) is a Long D(R)-dimodule over a bialgebra D(R) and R_{(M, , )} is the special map R_{(M, , )}(m n) : = n₁ m n₀. In the last section, if C is a coalgebra and I is a coideal of C, we introduce the notion of D-map on C, that is a k-bilinear map : C C / I k satisfying a condition which ensures on the one hand that, for any right C-comodule, the special map R is a solution of the D-equation and, on the other, that, in the finite case, any solution of the D-equation has this form.