The trace to subsets ofR n of Besov spaces in the general case

R ⁿ. , , , F R ⁿ, F , R ⁿ R ⁿ . ^p,q (Rⁿ), >0, 1, q, — ( ) Rⁿ. , ^p,q (Rⁿ) ^F Rⁿ. , q B ^p,q (F), = – (n–)/, >0, — « », ad — F, . , . : , F=R ^d,F— « » F— R ⁿ, « », F. .

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This work has been supported in part by the Swedish Natural Science Research Council.     

Analysis of a class of fractional programming problems

We propose a solution strategy for fractional programming problems of the form max _xx g(x)/ (u(x)), where the function satisfies certain convexity conditions. It is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max _xx g(x) + u(x), where is an exogenous parameter. The proposed strategy combines fractional programming andc-programming techniques. A maximal mean-standard deviation ratio problem is solved to illustrate the strategy in action.     

Finite checkability for integer rounding properties in combinatorial programming problems

LetA be a nonnegative integral matrix with no zero columns. Theinteger round-up property holds forA if for each nonnegative integral vectorw, the solution value to the integer programming problem min{1 y: yA w, y 0, y integer} is obtained by rounding up to the nearest integer the solution value to the corresponding linear programming problem min{1 y: yA w, y 0}. Theinteger round-down property is similarly defined for a nonnegative integral matrixB with no zero rows by considering max{1 y: yB w, y 0, y integer} and its linear programming correspondent. It is shown that the integer round-up and round-down properties can be checked through a finite process. The method of proof motivates a new and elementary proof of Fulkerson's Pluperfect Graph Theorem.Research partially supported by NSF Grants ENG76-09936 and ENG78-09882.     

Maximal inequalities,convexity inequality,and their duality. II

, , . . . [1], , . , , ., , L logL. , , . . . . [5]. , .     

О Сходимости метода подобластей

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On convergence of function series

, a _n f _n (x) . .     

Convergence of simultaneous Padé approximants to two dilogarithms

. .     

Lower central series and derived series of net subgroups of the general linear group

Over a commutative ring R with invertible element 2 and with radical , nets (i.e., tables =(_ij) of ideals _ij such that _{irrj ij}) such that _ii are considered. Such nets are called pseudoradical. The groups of the lower central series and the derived series are explicitly constructed for the corresponding net subgroups G () (of the general linear group GL (n,R)) in terms of .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 180–186, 1982.     

On some problems of interpolation by ℒ-splines

_n (D) — ,s — _n (D), _v (v=1, 2, ...,s/2) — . _m={0x ₀<x ₁<...<x _2m–1<2,x _2m =x ₀+2} , x _j +1–x _j <(4s max _v )^–1,j=0, 1, ..., 2m –1, ( ) 2- - _n,m 2m , _m . , L _q - (1q) W ( _n )={f ₂ :f ^(n–1)AC ₂ , _n (D)f 1} 2- - (s _n f), _m . , - - _n,m .

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The author expresses his gratitude to Yu. N. Subbotin for a useful discussion on the results of this paper.     

On the Riemann derivatives ofC sP-integrable functions

[4] , .     

Замечания об ихтерполировании

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A chance-constrained programming algorithm

This paper describes a new algorithm solving the deterministic equivalents of chance-constrained problems where the random variables are normally distributed and independent of each other. In this method nonlinear chance-constraints are first replaced by uniformly tighter linear constraints. The resulting linear programming problem is solved by a standard simplex method. The linear programming problem is then revised using the solution data and solved again until the stopping rule of the algorithm terminates the process. It is proved that the algorithm converges and that the solution found is the -optimal solution of the chance-constrained programming problem.The computational experience of the algorithm is reported. The algorithm is efficient if the random variables are distributed independently of each other and if they number less than two hundred. The computing system is called CHAPS, i.e. Chance-ConstrainedProgrammingSystem.     

О лагранжевом интерполировании с узлами в корнях многочленов сонина-маркова

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Exact estimates for partially monotone approximation

f(x) — , - [–1,1], (f, ) — , as— f, . . (- ) (x _i,x _{i+ 1}) (i=0, 1, ...,s–1; =–1,x _s,=1), f(x) . , n=0,1,... _n() , [– 1,1] signf(x) sign _n(x) 0, ¦f(x)– _n(x)¦ C(s) (f, 1/n+1, C(s) s. , - , « » .     

Nonlocal problems in the theory of the motion equations of Kelvin-Voight fluids

For the motion equations of Kelvin-Voight fluids one proves: 1) a global theorem for the existence and uniqueness of a solution (v;{u_e}) of the initial-boundary value problem on the semiaxis t R⁺ from the class W ¹ (R⁺); W ₂ ² () H()) with initial condition v_o(x) W ₂ ² () H() when the right-hand side f(x, t) L(R ⁺; L₂()); 2) a global theorem for the existence and uniqueness of a solution (v; {u_l}) on the entire axisR from the classW ¹ (R; W ₂ ² () H()) when the right-hand side f(x, t) L(R; L₂()); 3) a global theorem for the existence of at least one solution (v; {u_l}), periodic with respect to t with period , from the class W ¹ (R ⁺; W ₂ ² () H()) when the right-hand side f(x, t) L(R ⁺; L₂()) is periodic with respect to t with period , and a local uniqueness theorem for such a solution; 4) a theorem for the existence and uniqueness in the small of a solution (v; {u_l}), almost periodic with respect to t R, from V. V. Stepanov's class S ¹ (R; W ₂ ² ()H()) when the right-hand side f(x, t) S(R; L₂()) is almost periodic with respect to t; 5) the linearization principle (Lyapunov's first method) is justified in the theory of the exponential stability of the solutions of an initial-boundary value problem in the space H() and conditions are given for the exponential stability of a stationary and periodic solution, with respect to t R, of the system (1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 146–185, 1990.     

A Generalization of the Hausdorff-Young Theorem

Considering mixed-norm sequence spaces lp,q, p, q 1, C. N. Kellogg proved the following theorem: if 1 < p 2 then lp,2 and lp,2 , where 1/p + 1/p = 1. This result extends the Hausdorff-Young Theorem.We introduce here multiple mixed-norm sequence spaces , examine their properties and characterize the multipliers of spaces of the form lp,[s;n],q, with the index s repeated n times. By an interpolation-type argument we prove that (l,[2;n],2, lp,[1;n],1) for 1 < p 2. Using these results we obtain a further generalization of the Hausdorff-Young Theorem: if 1 < p 2 then lp,[2;n] and lp,[2;n] for each n = 0, 1, 2, ¨. The spaces lp,[2;n] decrease and lp,[2;n] increase properly with n for 1 < p < 2 and 1/p + 1/p = 1. We also extend a theorem of J. H. Hedlund on multiplers of Hardy spaces and deduce other results.     

Sharp Nikolskii inequalities with exponential weights

w _a(x)=exp(–x^a), xR, a0. , N _n (a,p,q) — (2), _n P _nw_ap, CN_n(a,p, q)P_nw_aq. , — , {P _n}, .

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This material is based upon research supported by the National Science Foundation under Grant No. DMS-84-19525, by the United States Information Agency under Senior Research Fulbright Grant No. 85-41612, and by the Hungarian Ministry of Education (first author). The work was started while the second author visited The Ohio State University between 1983 and 1985, and it was completed during the first author's visit to Hungary in 1985.     

On the complex zeros of solutions of linear differential equations

Summary A classical result (see R.Nevanlinna, Acta Math.,58 (1932), p. 345) states that for a second-order linear differential equation, w + P(z) w + Q(z) w=0, where P(z) and Q(z) are polynomials, there exist finitely many rays, arg z=_j, for j=1,..., m, such that for any solution w=f(z) 0 and any > 0, all but finitely many zeros off lie in the union of the sectors ¦ arg z - _j¦ < for j=1,..., m. In this paper, we give a complete answer to the question of determining when the same result holds for equations of arbitrary order having polynomial coefficients. We prove that for any such equation, one of the following two properties must hold: (a) for any ray, arg z=, and any > 0, there is a solution f 0 of the equation having infinitely many zeros in the sector ¦arg z - ¦ <, or (b) there exist finitely many rays, arg z=_j, for j= 1,..., m, such that for any >0, all but finitely many zeros of any solution f 0 must lie in the union of the sectors ¦ arg z - _j¦ < for j=1, ..., m. In addition, our method of proof provides an effective procedure for determining which of the two possibilities holds for a given equation, and in the case when (b) holds, our method will produce the rays, arg z=_j. We emphasize that our result applies to all equations having polynomial coefficients, without exception. In addition, we mention that if the coefficients are only assumed to be rational functions, our results will still give precise information on the possible location of the bulk of the zeros of any solution.This research was supported in part by the National Science Foundation (DMS-84-20561 and DMS-87-21813).     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

A priori estimates of the solution of the dirichlet problem for the Monge-ampére equation in weight spaces

One proves that a priori boundedness of the norm of the solution of the problem det(U_xx)=f(x,u,u_x)>>0,u¦=0. The magnitudes of the exponents,() depends on whether the arguments u p occur or not in f (x,u,p).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 74–90, 1983.