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.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

Radii of convexity and close-to-convexity of certain integral representations

Strict upper bounds are determined for ¦s(z)¦, ¦Re s(z)¦, and ¦Im s(z) ¦ in the class of functions s(z)=a _nzⁿ+a_n+1zⁿ⁺¹+... (n¹) regular in ¦z¦<1 and satisfying the condition ¦u (₁) –u (₂) ¦K¦ ₁- ₂¦, where U()=Re s (eⁱ ), K>0, and ₁ and ₂ are arbitrary real numbers. These bounds are used in the determination of radii of convexity and close-to-convexity of certain integral representations.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 581–592, May, 1970.The author wishes to thank L. A. Aksent'ev for his guidance in this work.     

DLR measures for one-dimensional harmonic systems

Summary A one-dimensional chain of nearest neighbor linearly interacting oscillators {q _x } _x is studied. The set of all its extremal DLR measures is characterized in terms of a parameter ². For each there is a Gaussian DLR measure with support on the set of configurations determined by the rate of growth of¦q _x¦. It is then finally proved that there is only one translationally invariant DLR measure. This proves the following conjecture: invariant DLR measures give uniformly finite first moment to ¦q _x¦.     

Separation of hypersurfaces

We give a generalization of results obtained in [15]. LetK ⁿ denote the set of embedded hypersurfaces in ⁿ⁺¹; for all xSⁿ and MK ⁿ we denote by C _x ^M the apparent contour ofM in the directionx. Then we give a sufficient condition on WSⁿ such that the map _W ^{K n}:K ⁿ P(T Sⁿ) , defined by _W ^{K n} (M)={C _w ^M ¦ wW}, is injective.     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

Estimates of the initial coefficients in the classes of univalent functions

Let S be the class of functions, regular and univalent in the circle ¦Z¦ < 1. Assume that D_n (), n=2,3,..., are defined by the expansion, ¦z¦<1,-11. In the paper one obtains sharp estimates for D₄() in the class S_R of functions from S with real coefficients C₂,C₃,... for all –1</1. In particular, as a consequence, one obtains sharp estimates for the coefficients C_3k+1 in the class S^K/R of K-symmetric functions from S_R for all K=2,3,... For k=2, the last result strengthens a result of Leeman.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 166–183, 1983.     

The capacity of a channel with arbitrarily varying channel probability functions and binary output alphabet

Summary Let X={1,..., a} be the input alphabet and Y={1,2} be the output alphabet. Let X ^t =X and Y ^t =Y for t=1,2,..., X _n = X ^t and Y _n = Y ^t . Let S be any set, C=={w(·¦·¦)s)¦sS} be a set of (a×2) stochastic matrices w(··¦s), and S ^t=S, t=1,..., n. For every s _n =(s ¹,...,s ⁿ ) S ^t define P(·¦·¦s _n)= w(y ^t ¦x ^t ¦s ^t ) for every x _n=x ¹, , x ⁿX _n and every y _n=(y ¹, , y ⁿ)Y _n. Consider the channel C _n ={P(·¦·¦)s _n )¦s _n S _n } with matrices (·¦·¦s), varying arbitrarily from letter to letter. The authors determine the capacity of this channel when a) neither sender nor receiver knows s _n, b) the sender knows s _n, but the receiver does not, and c) the receiver knows s _n, but the sender does not.Research of both authors supported by the U.S. Air Force under Grant AF-AFOSR-68-1472 to Cornell University.     

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¦<+ .     

The Müntz-Szasz problem

A condition is obtained on the placement of point _n (in some sense, the final point) with which completeness of the system of functionsexp (– _n x), Re_n>0, in spaces L^p, 1p<2. is equivalent to divergence of the series re_n(1+¦_n¦²)^–1.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 91–103, January, 1978.Deceased.     

Some monotonicity properties of symmetric pólya densities and their exponential families

Summary In this note we observe that for independent symmetric random variables X and Y, when the pdf of X is PF, the conditional distributions of ¦Y¦ given S = X + Y form a MLR family. We then show that for a function : R ⁿR that is symmetric in each coordinate and increasing on (0, )ⁿ, E((S₁,...,S_n)¦S_n = s) is even and increasing in ¦s¦. Here S₁,...,S_n are partial sums with independent symmetric PF summands. Application is made to sequential tests that minimize the maximum expected sample size when the model is a one-parameter exponential family generated by a symmetric PF density.Work supported by NSF grants MPS 72-05082 AO2 and MCS 75-23344     

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.     

Implicit elliptic boundary-value problems with discontinuous nonlinearities

Let be a bounded domain in ⁿ (n3) having a smooth boundary, let be an essentially bounded real-valued function defined on × ^h, and let be a continuous real-valued function defined on a given subset Y of Y ^h. In this paper, the existence of strong solutions u W ^2,p (, ^h) W _o ^1,p (n/2<p<+) to the implicit elliptic equation (–u)=(x,u), with u=(u₁, u₂, ..., u_h) and u=(u ₁, u ₂, ..., u _h), is established. The abstract framework where the problem is placed is that of set-valued analysis.     

Аппроксимация субга рмонических функций

The following statement is proved. Letu be a subharmonic function in the region and _u the associated measure. Then there exists a functionf holomorphic in and such that if _f is the associated measure of the function in ¦f¦, then ¦u(z)–ln¦f(z)¦ A¦ln s¦+B¦ln diam¦+ s(¦lns¦+1)+C. hold at every point z for which the setsD(z, t)={w: ¦w–z¦},t(0,s) lie in and satisfy(D(z, t))t both for= _u and for= _f . In the case where is an unbounded region, In diam should be replaced by ln ¦z¦. The constants, , do not depend on andu.

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On extremal contractions from threefolds to surfaces: The case of one non-Gorenstein point and a nonsingular base surface

In this paper, we study the local structure of extremal contractionsfXS from threefoldsX with only terminal singularities onto a surfaceS. If the surfaceS is nonsingular andX has a unique non-Gorenstein point on a fiber, we prove that either the linear system ¦–K _X ¦, ¦–2K _X ¦, or ¦–3K _X ¦contains a good divisor.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 45, Algebraic Geometry-8, 1997.     

Simplices by point-sliding and the Yamnitsky-Levin algorithm

Yamnitsky and Levin proposed a variant of Khachiyan's ellopsoid method for testing feasibility of systems of linear inequalities that also runs in polynomial time but uses simplices instead of ellipsoids. Starting with then-simplexS and the half-space {x¦a ^Tx }, the algorithm finds a simplexS _YL of small volume that enclosesS {x¦a ^Tx }. We interpretS _YL as a simplex obtainable by point-sliding and show that the smallest such simplex can be determined by minimizing a simple strictly convex function. We furthermore discuss some numerical results. The results suggest that the number of iterations used by our method may be considerably less than that of the standard ellipsoid method.     

On the Exact Asymptotics of the Best Relative Approximations of Classes of Periodic Functions by Splines

We obtain the exact asymptotics (as n ) of the best L ₁-approximations of classes of periodic functions by splines s S _{2n, r – 1} and s S _{2n, r + k – 1} (S _{2n, r} is the set of 2-periodic polynomial splines of order r and defect 1 with nodes at the points k/n, k Z) under certain restrictions on their derivatives.     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

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 ¦ < .     

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.