On unconditionally convergent basis expansions in the space of continuous functions

[0,1], - H .

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This paper was written during the author's scholarship at the State University of Odessa in the USSR.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

Shift-coupling in continuous time

Summary The result linking shift-coupling to time-average total variation convergence and to the invariant -field is extended to continuous time and an analogous result established linking -couplings to smooth total variation convergence and to a smooth tail -field. Shift- and -coupling inequalities are presented.     

Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

On a continuous time stochastic approximation problem

This paper is concerned with a continuous time stochastic approximation/optimization problem. The algorithm is given by a pair of differential-integral equations. Our main effort is to derive the asymptotic properties of the algorithm. It is shown that ast , a suitably normalized sequence of the estimation error,t(¯x _tr–) is equivalent to a scaled sequence of the random noise process, namely, (1/t) ₀ ^tr _sds. Consequently, the asymptotic normality is obtained via a functional invariance theorem, and the asymptotic covariance matrix is shown to be the optimal one. As a result, the algorithm is asymptotically efficient.Supported in part by the National Science Foundation, and in part by Wayne State University.Supported in part by Wayne State University through a research assistantship.     

Sample path large deviations for super-Brownian motion

Summary We derive two large deviation principles of Freidlin-Wentzell type for rescaled super-Brownian motion. For one of the appearing rate functions an integral representation is given and interpreted as Kakutani-Hellinger energy. As a tool we develop estimates for the Laplace functionals of (historical) super-Brownian motion and certain maximal inequalities. Also it is shown that the Hölder norm of index <1/2 of the processtf, X _t possesses some finite exponential moments provided the functionf is smooth.This work was supported in part by the Graduiertenkolleg Algebraische, analytische und geometrische Methoden und ihre Wechselwirkung in der modernen Mathematik, Bonn     

Reflected Brownian motion in a cone: Semimartingale property

Summary In this paper, the object of study is reflected Brownian motion in a cone ind-dimensions (d3) with nonconstant oblique reflection on each radial line emanating from the vertex of the cone. The basic question considered here is When is this process a semimartingale?. Conditions for the existence and uniqueness of the process for which the vertex is an instantaneous state were given by Kwon, which is resolved in terms of a real parameter depending on the cone and the direction of reflection. It is shown that starting from any point of the cone, the process is a semimartingale if < 1, + 0 and not a semimartingale if < < 2.This research is supported by KOSEF grant 941-0100-011-1     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

D λ -cycles in 3-cyclable graphs

Letk and be positive integers, andG a 2-connected graph of ordern with minimum degree and independence number. A cycleC ofG is called aD -cycle if every component ofG – V(C) has order smaller than. The graphG isk-cyclable if anyk vertices ofG lie on a common cycle. A previous result of the author is that if k 2, G isk-connected and every connected subgraphH ofG of order has at leastn +k ² + 1/k + 1 – vertices outsideH adjacent to at least one vertex ofH, thenG contains aD -cycle. Here it is conjectured that k-connected can be replaced by k-cyclable, and this is proved fork = 3. As a consequence it is shown that ifn 4 – 6, or ifG is triangle-free andn 8 – 10, thenG contains aD ₃-cycle orG , where denotes a well-known class of nonhamiltonian graphs of connectivity 2. As an analogue of a result of Nash-Williams it follows that ifn 4 – 6 and – 1, thenG is hamiltonian orG . The results are all best possible and compare favorably with recent results on hamiltonicity of graphs which are close to claw-free.     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

Similarity solutions of radially symmetric two-phase flow

When both the diffusivityD and fractional flow functionf have a power law dependence on the water content , i.e.D=D _o andf=⁺¹, the nonlinear transport equation for radially symmetric two phase flow can, in certain circumstances, be reduced to a weakly coupled system of two first order nonlinear ordinary differential equations. Numerical solutions of these equations for a constant flux boundary conditionV _wo and comparison with experimental data are given. In particular, when the fluxV _wo and a are related byV _wo( + 1)/D _o=2, a new fully explicit analytical solution is found as (r, t)=(1 – r ²/4D _ot)^1/ forr ² < 4D _ot/ and (r, t)=0 forr ² 4D _ot/ We show that the existence of this exact soution is due to the presence of a Lagrangian symmetry.     

A characterization of semimartingales on nuclear spaces

Summary This work is devoted to prove the following fact: Suppose that is a nuclear space whose dual is nuclear under the strong topology. IfX is a weakly adapted mapping with values in such that for any,X'() has a modification which is a semimartingale then there exists a unique projective system of Hubert space-valued semimartingales indexed by the Hilbert-Schmidt neighbourhood base of the dual space whose projective limit isX.In the last part we study in detail a semimartingale defined as the convolution of a distribution by a random Dirac measure whose support is determined by the trajectories of a real-valued semimartingale.     

Exact error bound of approximation by interpolating splines inL-metric on the classesW p r (1≦p

Корнейчук  Н. П. 《Analysis Mathematica》1977,3(2):109-117

Spectral synthesis on a system of unbounded domains starlike in a common direction

, . . - ₁, ..., ₄, — ; =(₁,)×...×H(₄), — H(₁, ..., H(₄), r H(₁) — , ₁ ; D: HH- . , D. , ₁..., ₄ , (.. z₁ z+te^ia ₁ t>0), W H .     

Quelques propriétés des solutions de l'équation de Ginzburg-Landau sur un ouvert de ∝2

For a solution u of –u=u(1–|u|²) on the whole plane, |u|<1 holds everywhere unless u=eⁱ for some ; the derivatives of order k have moduli a constant M _kdepending only on k. For a solution u on an open set ², the moduli of u and its derivatives have upper bounds depending only on the distance to ²\ therefore the set of solutions on a given is compact in C() for the topology of uniform convergence on compact subsets of . For a solution u such that |u|<1, 1–|u| satisfies an estimation similar to the classical Harnack inequality for positive harmonic functions.Finally, if is bounded and |u| has a lim supm at each boundary point, the |u|m in if m1, but if m<1 then |u| admits only a majorant S _m with values in ]m, 1[ and sufficient conditions are given for lim S _m =0 or S _m =O(m) as m0.

     

On cubature formulas on the basis of lattices

Q (.. , L). Q . P(S_r(2)) — 2 (S _r(2) (r — ). , M(P(S _r(m=sup{t(·)t(·)₁:t P(S _r(2)),t 0}. , /4+(1)M(P(S _r(2)))/r ²15/17+(1)(r+). (Q), Q L.     

Measures induced on Wiener space by monotone shifts

Summary In this work we study the absolute continuity of the image of the Wiener measure under the transformations of the formT()=+u(), the shiftu is a random variable with values in the Cameron-Martin spaceH and is monotone in the sense that (T(+h-T(),h) _H 0 a.s. for allh inH.     

Black holes on the plane drawn by a Wiener process

Summary We say that the discD()R ², of radius , located around the origin isp-covered in timeT by a Wiener processW(·) if for anyzD() there exists a 0tT such thatW(t) is a point of the disc of radiusp, located aroundz. The supremum of those 's (0) is studied for which,D() isp-covered inT.     

On Fourier multipliers between Besov spaces with 0 <p o ≤ min{1,p 1}

, 0<0 min{1,p1}.     

On power series with positive coefficients

M. . , . , ^p () (). , , .