Localization of the spectrum of certain non-self-adjoint operators

Let the self-adjoint operator A and the bounded operator B be specified in Hilbert space We let denote the spectral family of the operator A. If (E – E _N ) B ₂+E–_NB ² 0 npnN , then in the complex plane z=+ there will exist the curve ¦ ¦ =f (), limf () = 0 for ± such that the entire spectrum of the operator A+B lies within the region ¦ ¦ f(). In particular, the condition of the theorem will be satisfied when B is a completely continuous operator.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 415–420, April, 1968.The author expresses his appreciation to R. S. Ismagilov for his discussion of the results.     

Growth along a ray of a subharmonic function,having a mass distributed on the negative axis

Let U be a subharmonic function in C with a Riesz mass , distributed on the negative semiaxis without some neighborhood of zero, let and be its order and lower order, and let B(r, U) be the maximum of U(z) for ¦z¦=r. Estimates are obtained for the measure of sets of those values of r 0 for which certain inequalities hold. The following result is typical. LetE = {r:u(re ^l)–cosB<(r,U) > 0}. If < < 1, ¦¦=., then the lower logarithmic density of the set E is at least 1 – /. If < > 1,¦¦ ., then the upper logarithmic density of the set E is at least 1 – /.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 31–38, 1988.     

Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeu _t–R¦u¦^m+u^q=0 in ⁿ× ₊, u(·,0⁺)=u_o(·) in ^N, whereR > 0, 0 <q < 1,m 1, andu _o is a positive uniformly continuous function verifying –R¦u _o¦^m+u ₀ ^q 0 in ^N . We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t(·) defined byu(x, t) > 0 if 0<t<t (x) andu(x, t)=0 ift t (x). Regularity, extinction rate, and asymptotic behavior of t(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(u _o(x – t))^1–q –(1–q)t]₊)^1/(1–q): ¦¦R}, (x, t) ₊ ^N+1 .Partially supported by the DGICYT No. 86/0405 project.     

On the width of ordered sets and Boolean algebras

Thewidth (chain number) of a partial order P, < is the smallest cardinal such that ¦A¦< 1 + whenever A is an antichain (chain) in P. We prove that, if a partial order (P, <) has width and cf()=, then P contains antichains A_n (n<) such that ¦A ₀¦<¦A₁¦ <...<={¦A_n¦: n < < } and either A₀1 A₂< ... or A₀>A₁ >A₂> ... A similar structure result is obtained for partial orders with chain number if cf()=. As an application we solve a problem of van Douwen, Monk and Rubin [1] by showing that if a Boolean algebra has width , thencf() .This work has been partially supported by NATO grant No. 339/84.Presented by Bjarni Jonsson.     

Local partial regularity theorems for suitable weak solutions of a class of degenerate systems

A partial regularity theorem is established for a particular class of weak solutions to the systemu/t– div(K(u)u)=(u)¦¦², div((u))=0 on a bounded domain inR ^N . Under our assumptions, (u) may exhibit exponential decay, and thus the system may be degenerate. Our proof is based upon a blow-up argument.This work was supported in part by NSF Grant DMS9424448.     

The measure of a translated ball in uniformly convex spaces

Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r ^q , whereq depends on the properties of the norm. We specify it in the case ofL spaces, >1.     

Asymptotic behaviour of Gaussian random fields

Summary Let X={X(t), t ^N} be a centred Gaussian random field with covariance X(t)X(s)=r(t–s) continuous on ^N×^N and r(0)=1. Let (t,s)=((X(t)–X(s)) ²)^1/2; (t,s) is a pseudometric on ^N. Assume X is -separable. Let D ₁ be the unit cube in ^N and for 0<k, D _k= {x^N: k ^–1 xD₁}, Z(k)=sup{X(t),tD _k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦) as ¦t¦8 then Z(k)-(2Nlogk) ^1/20 as k a.s.     

A short proof of the nonuniform Ray-Chaudhuri-Wilson inequality

LetL be a set ofs nonnegative integers and a family of subsets of ann-element setX. Suppose that for any two distinct membersA,B we have¦A B¦ L. Assuming in addition that, is uniform, i.e. each member of has the same cardinality, a celebrated theorem of D. K. Ray-Chaudhuri and R. M. Wilson asserts that ¦¦ P. Frankl and R. M. Wilson proved that without the uniformity assumption, we have.We give a short proof of this latter result.     

Growth of analytic functions

For a function f analytic in a disk {z ¦ z ¦ <R }, 0 <R + , a criterion is obtained, in terms of the Taylor coefficients of the function, for the validity of the relation (r) ln⁺ M (r)dr < + , whereM (r)= max {¦ (f (z) ¦ ¦z ¦=r < R}, and is a function positive on [0, R) and satisfying certain conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 668–673, May, 1990.     

Relative stability and the strong law of large numbers

Summary LetX ₁,X ₂,..., be i.i.d. random variables andS _n=X ₁+X ₂+. +X _n. In this paper we simplify Rogozin's condition forS _n/B _n ±1for someB _n+, which generalises Hinin's condition for relative stability ofS _n. We also consider convergence of subsequences ofS _n/B _n. As an application of our methods, we extend a result of Chow and Robbins to show thatS _n/B _n±1 a.s. for someB _n + if and only if 0<¦EX¦E¦X¦<+ .     

On the set of positive deflections of functions meromorphic in the unit disk

For an arbitrary (0, 1/2) we construct a functionf(z) of lower order [f]=. that is meromorphic in the disk D = {z ¦z¦< 1} and such that the set (f) of positive deflections of the functionf (in the sense of V. P. Petrenko) has positive logarithmic capacity.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 48–53.     

Solution of two sequencing problems

The solution of the following problems is offered. Suppose a multiset J (¦J¦=p) is given. For each pair of elements and J, a number 1 P is given. Moreover, if 1 < x<p then x is undefined. If x=1, then x=p. Problem 1. Find the permutation ₁..._F of elements of the multiset J satisfying the following conditions. Let _i, _i=. If i,j < x, thenj <i. If i,j > x, then i<j. Such a permutation is called a PC-schedule. Problem 2. Find a PC-schedule in which the following property holds: if i < x < j, _i=, _j=, then. Such a PC-schedule is called an SC-schedule. The conditions under which these problems have solutions are studied. For their solution an algorithm of shifts is used with the complexity O(¦B(J)¦²¦J¦).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 44–72, 1983.     

Correctness of multi-dimensional Darboux problems for the wave equation

It is proved that multi-dimensional Darboux problems for the wave equation are correct in the domain bounded by the surfaces ¦x ¦=t + and ¦x ¦=1- t and the planet = 0, 0 < 1. The behavior of the solutions as 0 is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1299–1306, September, 1993.     

Strong solution for an elastic-plastic plate

Given a regular bounded open set R ²,, >0 andg L ^q () withq>2, we prove, under compatibility and safe load conditions ong, the existence of a minimizing pair for the functional, over closed setsK ² and functionsu C⁰( ) C²(/K); here ¦[Du]¦ denotes the jump ofDu acrossK and ¹ is the 1-dimensional Hausdorff measure.Dedicated to Enrico Magenes for his 70th birthday     

The problem of conformal transformations of a circle into nonoverlapping regions

Let a, a0, a, be a fixed point in the z-plane, (a, 0, ), the class of all systemsf _k()_l ³ of functions z=f ^k(), k=1, 2, 3, of which the first two map conformally and in a s ingle-sheeted manner the circle ¦¦<1, and the third maps in a similar manner the region ¦¦>1, into pair-wise nonintersecting regions B_k, k=1, 2, 3, containing the points a, 0, and , respectively, so thatf ₁(0)=a,f ₂(0)=0 andf ₃()=. The region of values (a, 0, ) of the system M(¦f ₁'(0)¦, ¦f ₂'(0)¦, 1/¦f ₃'()¦) in the class (a, 0, ) is determined.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 417–424, October, 1969.     

The caratheodory number for thek-core

Thek-core of the setS ⁿ is the intersection of the convex hull of all setsA S with ¦SA¦<-k. The Caratheodory number of thek-core is the smallest integerf (d,k) with the property thatx core _{kS, S} ⁿ implies the existence of a subsetT S such thatx core_kT and ¦T¦f (d, k). In this paper various properties off(d, k) are established.Research of this author was partially supported by Hungarian National Science Foundation grant no. 1812.     

Оценки норм тригоном етрических полиномо в на интервалах и множе ствах

Relations between various norms of the polynomial are investigated. Estimates for the norm of the derivative ofP and for its integral are obtained. A typical result:For arbitrary polynomial P and segment J, ¦J¦>0, inequalities hold with a factor (), positive and depending on only. The following statement is also proved. Suppose that a measurable and bounded set E lies in a segment J with ¦J¦>0, (x) is nonnegative and nondecreasing on [0, +), andP x satisfies the condition: for alln ₁n ₂.Then the inequality holds.

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Estimates of the norms of trigonometric polynomials on intervals and sets

     

Sets with fixed congruence in a steiner system

In this paper we study particular sets of a Steiner systemS. More precisely, we study the setsA such that ¦A ¦ d modh for all lines ofS, withd andh integers satisfyingd 0,h 2.Dedicated to Professor M. Scafati Tallini on the occasion of her sixtyfifth birthday     

Some inequalities of the V. A. Markov type

Let B be a domain in the complex plane, let p_n(z) and P_n(z) be polynomials of degree n where the zeros of P_n(z) lie in , let(z) be a finite function,(z) 0, z . We consider the problem of estimating from above the functions L[p_n(z)]=(z)p_n(z) – wp_n(z), z , if ¦p_n(z)¦ ¦P_n(z)¦ for zB. Under some very general conditions on B, z, (z), and w we prove the inequality ¦L[p_n(z)]¦ ¦L[P_n(z)]¦.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 431–440, April, 1968.     

The absolute convergence of weighted sums of dependent sequences of random variables

Summary The sum a _n X _n of a weighted series of a sequence {X _n } of identically distributed (not necessarily independent) random variables (r.v.s.) is a.s. absolutely convergent if for some in 0<1, ¦a _n ¦ < and E¦X _n ¦ < ; if a _n =z ⁿ for some ¦z¦<1 then it suffices that E(log¦X _n ¦)₊<. Examples show that these sufficient conditions are not necessary. For mutually independent {X _n } necessary conditions can be given: the a.s. absolute convergence of X _n z ⁿ (all ¦z¦<1) then implies E(log¦X _n ¦)₊ < , while if the X _n are non-negative stable r.v.s. of index , ¦a _n X _n ¦< if and only if ¦a _n ¦ < .