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 .     

Finite Representability of ℓp in Quotients of Banach Spaces

A typical result of the paper states that if X is a Banach space with a basis and for some 1pq, the spaces _p and _q are finitely block representable in every block subspace of X, then every block subspace of X admits a block quotient Z such that for every r[p,q], the space _r is finitely block representable in Z. Results of a similar nature are also established for ^N _p-block-sequences and asymptotic spaces.     

A construction of complete minimal,but not uniformly minimal,exponential systems with real separable spectrum inL p andC

We construct real separable sequences { _n } such that the corresponding systems of exponentials exp(i _n t) are complete and minimal, but not uniformly minimal, in the spacesL ^p (–, ), 1p<, orC[–, ].Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 582–595, October, 1995.The work was supported by the Russian Foundation for Basic Research under grant No. 93-011-205     

The regularity of Lagrangiansf(x, ξ)=254-01254-01254-01with Hölder exponents α(x)

In this paper the regularity of the Lagrangiansf(x, )=||^(x)(1< ₁(x)₂< +) is studied. Our main result: If(x) is Holder continuous, then the Lagrangianf(x, )=f(x, )=||^(x) is regular. This result gives a negative answer to a conjecture of V. Zhikov.Supported by the National Natural Science Foundation of China.     

On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

Quadratically constrained quadratic programming: Some applications and a method for solution

The problem (QPQR) considered here is: minimizeQ ₁ (x) subject toQ _i (x) 0,i M ₁ {2,...,m},x P R ⁿ, whereQ _i (x), i M {1} M ₁ are quadratic forms with positive semi-definite matrices, andP a compact nonempty polyhedron of Rⁿ. Applications of (QPQR) and a new method to solve it are presented.Letu S={u R ^m;u 0, u _i= l}be fixed;then the problem:iM minimize u _iQ_i (x (u)) overP, always has an optimal solutionx (u), which is either feasible, iM i.e. u C₁ {u S;Q _i (x (u)) 0,i M ₁} or unfeasible, i.e. there exists ani M ₁ withu C {u S; Q_i(x(u)) 0}.Let us defineC _i C_i S _i withS _i {u S; u _i=0}, i M. A constructive method is used to prove that C _i is not empty and thatx (û) withiM û C _i characterizes an optimal solution to (QPQR). Quite attractive numerical results have been reached with this method.

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Zusammenfassung Die vorliegende Arbeit befaßt sich mit Anwendungen und einer neuen Lösungsmethode der folgenden Aufgabe (QPQR): man minimiere eine konvexe quadratische ZielfunktionQ _i (x) unter Berücksichtigung konvexer quadratischer RestriktionenQ _i (x) 0, iM ₁ {2,...,m}, und/oder linearer Restriktionen.·Für ein festesu S {u R ^m;u 0, u _i=1},M {1} M₁ besitzt das Problem:iM minimiere die konvexe quadratische Zielfunktion u _i Q_i (x (u)) über dem durch die lineareniM Restriktionen von (QPQR) erzeugten, kompakten und nicht leeren PolyederP R ⁿ, immer eine Optimallösungx (u), die entweder zulässig ist: u C₁ {u S;Q ₁ (x (u)) 0,i M ₁} oder unzulässig ist, d.h. es existiert eini M ₁ mitu C_i {u S;Q _i (x(u))0}.Es seien folgende MengenC _i C_i S _i definiert, mitS _i {u S;u _i=0}, i M. Es wird konstruktiv bewiesen, daß C _i 0 undx (û) mitû C _i eine Optimallösung voniM iM (QPQR) ist; damit ergibt sich eine Methode zur Lösung von (QPQR), die sich als sehr effizient erwiesen hat. Ein einfaches Beispiel ist angegeben, mit dem alle Schritte des Algorithmus und dessen Arbeitsweise graphisch dargestellt werden können.

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An earlier version of this paper was written during the author's stay at the Institute for Operations Research, Swiss Federal Institute of Technology, Zürich.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

Sukzessive Minima mit und ohne Nebenbedingungen

Given semi-normsf andg on ⁿ and a real number >0. Then the successive minima off under the constraintg are defined by _j : = inf {: there existj linear independent vectors inZ ⁿ withf andg}. The main theorem of this paper (Lagrange multiplier theorem) states that the successive minima of a certainnorm h on ⁿ (without constraints) coincide with the _j 's up to bounded factors. Moreover, this norm is constructed explicitly. Using Minkowski's wellknown theorem on successive minima and our result certain inequalities on simultaneous Diophantine approximations are derived.     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

On measures that agree on balls

The fundamental result: if and v are two finite Borel measures, defined in the spaceL ^p[0, 1] (1p<) or in C(K) (K is a metric compactum without isolated points), then from the equalities (B)=v(B) for all balls B of radius 1 there follows that =v. In addition, in the spaces C(K) and _p (1p<) from the inequalities (B) v(B) there follows that v.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 122–128, 1989.     

Characterization theorem of generalized polynomial of best approximation having bounded coefficients

Let the set of generalized polynomials having bounded coefficients beK={p= _jg_j. _{j j j},j=1, 2, ...,n}, whereg ₁,g ₂, ...,g _n are linearly independent continuous functions defined on the interval [a, b], _j, _j are extended real numbers satisfying _j<+, _j>-, and _{j j}. Assume thatf is a continuous function defined on a compact setX [a, b]. This paper gives the characterization theorem forp being the best uniform approximation tof fromK, and points out that the characterization theorem can be applied in calculating the approximate solution of best approximation tof fromK.     

Factorizations of the Complete Graph into C 5-Factors and 1-Factors

In this paper, we show that K_10n can be factored into C₅-factors and 1-factors for all non-negative integers and satisfying 2+=10n–1.Research partially supported by an NSF-AWM Mentoring Travel Grant     

Ergodicity of nonlinear first order autoregressive models

Criteria are derived for ergodicity and geometric ergodicity of Markov processes satisfyingX _n+1 =f(X _n )+(X _n ) _n+1 , wheref, are measurable, { _n } are i.i.d. with a (common) positive density,E| _n |>. In the special casef(x)/x has limits, , asx– andx+, respectively, it is shown that <1, <1, <1 is sufficient for geometric ergodicity, and that <-1, 1, 1 is necessary for recurrence.     

Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score

Let (X _n ) _{n 0} be a real random walk starting at 0, with centered increments bounded by a constant K. The main result of this study is: |P(S _n n x)–P( sup_{0 u 1} B _u x)| C(n,K) n/n, where x 0, ² is the variance of the increments, S _n is the supremum at time n of the random walk, (B _u ,u 0) is a standard linear Brownian motion and C(n,K) is an explicit constant. We also prove that in the previous inequality S _n can be replaced by the local score and sup₀ u 1 B _u by sup₀ u 1|B _u |.     

Bang-bang optimal control for the dam problem

The dam problem with general geometry is considered. Fluid is drawn from the bottomS ₁ at a ratek where 0 k N, _{S 1} k M; the objective is to minimize the total pressure of the fluid in the dam. A bang-bang principle is established for any optimal controlk ₀, that is,k ₀ = 0 on a setA andk ₀ =N on the complement setS ₁ A. In the case of a rectangular dam the structure ofA is determined and the uniqueness of the minimizerk ₀ is established.This work is partially supported by National Science Foundation Grants DMS-8501397 and DMS-8420896.     

A symmetry problem in the calculus of variations

Let be a ball in ^N, centered at zero, and letu be a minimizer of the nonconvex functional over one of the classesC _M := {w W _loc ^1, () 0 w(x) M in,w concave} orE _M := {w W _loc ^1,2 () 0 w(x) M in,w 0 inL()}of admissible functions. Thenu is not radial and not unique. Therefore one can further reduce the resistance of Newton's rotational body of minimal resistance through symmetry breaking.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

Efficient Regular Polygon Dissections

We study the minimum number g(m,n) (respectively, p(m,n)) of pieces needed to dissect a regular m-gon into a regular n-gon of the same area using glass-cuts (respectively, polygonal cuts). First we study regular polygon-square dissections and show that n/2 -2 g(4,n) (n/2) + o(n) and n/4 g(n,4) (n/2) + o(n) hold for sufficiently large n. We also consider polygonal cuts, i.e., the minimum number p(4,n) of pieces needed to dissect a square into a regular n-gon of the same area using polygonal cuts and show that n/4 p(4,n) (n/2) + o(n) holds for sufficiently large n. We also consider regular polygon-polygon dissections and obtain similar bounds for g(m,n) and p(m,n).     

The existence of probability measures with specified projections

Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (_x, _y P (X)×P(Y), a probability measure P(F) such that _X = _X ^–1 , _Y = _Y ^–1 it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition _XA+ _YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973.