On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives

For 2-periodic functions and arbitrary q [1, ] and p (0, ], we obtain the new exact Kolmogorov-type inequality which takes into account the number of changes in the sign of the derivatives (x ^(k)) over the period. Here, = (r – k + 1/q)/(r + 1/p), _r is the Euler perfect spline of degree r, and . The inequality indicated turns into the equality for functions of the form x(t) = a _r (nt + b), a, b R, n N. We also obtain an analog of this inequality in the case where k = 0 and q = and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines.     

Admissibility of linear estimators of regression coefficients under quadratic loss

For the general fixed effects linear model:Y=X+, N(0,V),V0, we obtain the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS in the class of all estimators under the loss function (d -S)D(d -S), whereD0 is known. For the general random effects linear model: =XV ₁₁ X+XV ₁₂+V ₂₁ X+V ₂₂0, we also get the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS+Q in the class of all estimators under the loss function (d -S -Q)D(d -S -Q), whereD0 is known.     

Complexity of the realization of a linear function in the class of II-circuits

It is proved that the linear function gn(x₁,..., x_n) = x₁ + ... + x_nmod 2 is realized in the class of II-circuits with complexity L(g_n) n². Combination of this result with S. V. Yablonskii's upper bound yields L(g_n) n².Translated from Matematicheskie Zametki, Vol. 9, No. 1, pp. 35–40, January, 1971.     

On positive solutions of quasilinear elliptic systems

In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems where –_p is the p-Laplace operator, p > 1 and is a C ^1,-domain in . We prove an analogue of [7, 16] for the eigenvalue problem with and obtain a non-existence result of positive solutions for the general systems.     

On a Problem of P. Gallagher

Let p be an odd prime. For each integer a with x < a x + u and (a,p) = 1, we define by a 1 (mod p) and 1 p - 1. Let r(p,u,x) be the number of integers with x < a x + u and (a,p) = 1 for which a and are of opposite parity, and let E(n,u,x) = r(n,u,x) - 1/2 1, where denotes summation over all a such that (a,p) = 1. The main purpose of this paper is to prove that for any positive integer 1 u we have the asymptotic formula

Concentration phenomena in weakly coupled elliptic systems with critical growth*

In this paper we consider the weakly coupled elliptic system with critical growth

An Upper Bound for the Cardinality of an <Emphasis Type="Italic">s</Emphasis>-Distance Set in Euclidean Space

In this paper we show that if X is an s-distance set in ^m and X is on p concentric spheres then Moreover if X is antipodal, then .     

Zur Stetigkeit der mengenwertigen metrischen Projektion in endlichdimensionalen Räumen

Let X be a real normed linear space, X^* its dual, V a linear subspace of X and S(V) the unit sphere in the orthogonal space In this note we prove in the case of finite-dimensional X the following sufficient condition for the continuity of the set-valued metric projection in terms of the mapping and : If the restriction of T to S(V) is lower semi-continuous then P_V is lower semi-continuous.     

A 2-extension of the field ℚ of rational numbersof rational numbers

It is proved that the rational number field has one, and only one, normal 2-extension (_{2, t8)}/with group isomorphic to .If is the maximal subfield of a real-closed field, which does not contain ,then the algebraic closure of is isomorphic to the field .Bibliography: 7titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 192–196.     

Mean quadratic variations and Fourier asymptotics of self-similar measures

We show that the mean quadratic variation of a self-similar measure under certain open set condition exhibits asymptotic periodicity. Through a generalized Wiener's Tauberian Theorem, we obtain some new identities and equivalences of the mean quadratic variation of a bounded measurev and its Fourier average . They are used to sharpen some recent results of Strichartz concerning the asymptotic behavior ofH _a(T); ) asT, where is the self-similar measure as above. In the development some results concerning the open set condition are also obtained.     

Block elimination with one refinement solves bordered linear systems accurately

Linear systems with a fairly well-conditioned matrixM of the form , for which a black box solver forA is available, can be accurately solved by the standard process of Block Elimination, followed by just one step of Iterative Refinement, no matter how singularA may be — provided the black box has a property that is possessed by LU- and QR-based solvers with very high probability. The resulting Algorithm BE + 1 is simpler and slightly faster than T.F. Chan's Deflation Method, and just as accurate. We analyse the case where the black box is a solver not forA but for a matrix close toA. This is of interest for numerical continuation methods.Dedicated to the memory of J. H. Wilkinson     

Approximation by infinitely divisible distributions in the multidimensional case

Let be the collection of parallelepipeds in R with edges parallel with the coordinate axes and let be the collection of closed sets in R. Let (G, H)=inf {G{A}H{A}+, H{A}G{A}+ for any; L(G, H)= inf {G{A}H{A}+, H{A}G{A}+ for any, where G, H are distributions in . In the paper one gives the proofs of results announced earlier by the author (Dokl. Akad. Nauk SSSR,253, No. 2, 277–279 (1980)). One considers the problem of the approximation of the distributions of sums of independent random vectors with the aid of infinitely divisible distributions. One obtains estimates for the distances (·, ·), L(·, ·) and. It is proved that, where 0p_i1, ; E is the distribution concentrated at zero; V_i(i=1, ..., n) are arbitrary distributions; the products and the exponentials are understood in the sense of convolution.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 89–103, 1983.     

Asymptotic properties of local densities of measures generated by semimartingales

For families of probability measures (P , )) generated by semimartingales, we consider the local density)(y, )= _t (y, )) _t0 of a, measureP _y with respect to the measureP whose logarithm is the difference of a local martingale and a positive predictable increasing locally bounded process. Conditions are obtained under which the relations and hold, wherey _t depends in some way ont, while _t ast . Applications of these relations are exhibited and an example is given when the hypotheses of the theorems proved can be verified.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 48–55, 1986.     

Multiple Wick Product Chaos Processes

Let u(x) xR ^q be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p _x, (·) be the density of an R ^q valued canonical normal random variable with mean x and variance and let {G _x, ; (x, )R ^q ×[0,1 ]} be the mean zero Gaussian process with covariance

A proof of Shelah's partition theorem

A self contained proof of Shelah's theorem is presented: If is a strong limit singular cardinal of uncountable cofinality and 2 > ⁺ then .     

Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Let = (₁,...,_d) be a vector with positive components and let D be the corresponding mixed derivative (of order _j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that

Symmetric Squares of Hecke L-Functions and Fourier Coefficients of Cusp Forms

Let S _k (₀(N)) be the space of cusp forms of even weight k for ₀ (N), let be the set of all newforms in S _k ( ₀ (N)), and let be the symmetric square of the Hecke L-function of a form . It is proved that for N=p we have

über Wurzeln und Logarithmen beschrÄnkter Ma\e

Ohne ZusammenfassungBezeichnungen und Symbole G lokalkompakte topologische Gruppe - M(G)/R(G)/P(G)/ regulÄre komplexe/reelle/positive Ma\e/ - Q(G)/W(G) Ma\e mit · l/Wahrscheinlichkeitsma\e - _x Punktma\: _x(f)=f(x) - v Faltung, — Bekanntlich bildet M(G) bezüglich der Faltung eine Banachalgebra; - Involution in M(G), , wobei — die Komplexkonjugierte bezeichnet - × diskreter Anteil eines Ma\es, - T _gm Faltungsoperator auf L ² (G) (bezüglich des linken Haarschen Ma\es), f.ü. - p(·)/q(·)/u(·) - exp(·.) Exponentialfunktion, exp - normal/unitÄr/symmetrisch/positiv definit bezeichnet man ein Ma\ , wenn der Faltungsoperator T diese Eigenschaft besitzt - invertierbar hei\t M(G), wenn ein vM(G) existiert, so da\ v = v= _e - _1/n n-te Wurzel von ₁ hei\t wenn( _1/n)ⁿ= ₁ - ₁ hei\t unendlich teilbar wenn zu jedem natürlichen n eine n-te Wurzel _1/n von existiert - N Menge der natürlichen Zahlen     

The product of finitely based varieties of lattice-ordered groups

The question as to whether a product of two finitely based varieties of lattice-ordered groups is finitely based is considered. It is proved that varieties and are finitely based; here is a variety of lattice-ordered groups defined by identities [x ⁿ,y ⁿ] =e and [[x,y] z, [x ₁,y ₁] z ₁] =e; is a variety of lattice-ordered nilpotent groups of class s, defined by an identity [x ₁,x ₂,...,x _(s+1)] =e; V is an arbitrary finitely based variety of lattice-ordered groups. Translated fromAlgebra i Logika, Vol. 33, No. 3, pp. 255–263, May–June, 1994.Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-1524.     

Variants of the Maurey–Rosenthal Theorem for Quasi Köthe Function Spaces

The Maurey–Rosenthal theorem states that each bounded and linear operator T from a quasi normed space E into some L_p()(0) which satisfies a vector-valued norm inequality