The monotone follower problem in stochastic decision theory

We consider the problem of optimally tracking the random demandx+w _t, w. Brownian motion, by a nondecreasing process. adapted to the Brownian past, so as to minimize the expected lossE ₀ ^T (x+w_t–_t)dt. The decision problem is reduced to a free boundary one, and the latter is studied and solved for a large class of cost functions().This research was supported in part by the Air Force Office of Scientific Research, under AF-AFOSR 77-3063.     

Factorization of One-Generated Formations

A large class of one-generated factorable foliated formations including composition, bicanonical, and other formations is described. V. A. Vedernikov suggested to study foliated formations with directions such that , where and are also some directions, from a unified point of view. We study one-generated factorable foliated formations with directions from the interval [,], where and are the directions of the bicanonical and composition formations, respectively. In particular, all irreducible factorizations of one-generated composition formations are studied.     

About positive invariance and asymptotic stability

In this note we consider the differential equation , t0 with initial conditionx(0)G, whereG is a closed pointed convex cone. Our results pertain to positive invariance and asymptotic stability. A result by Stern, [3], will be improved. Subtangentiality will also be discussed as this notion is important in deriving results about positive invariance.     

Explicit expression for the transition density of the solution of a stochastic diffusion equation with piecewise-constant drift coefficient

The explicit form of the transition density is determined for the solution (t) of the stochastic diffusion equation d(t)=a((t))dt+dw(t), where a(z)= for x [a, b] and a(x)=0 for x [a, b], w(t) is a Wiener process.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 99–105, 1987.     

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.     

Normalized Solutions of Schrödinger Equations with Potentially Bounded Measures

Given a potentially bounded signed measure on a Brelot space (X,) with Green function G, it is well known that -harmonic functions (i.e., in the classical case, finely continuous versions of solutions to u–u=0) may be very discontinuous. In this paper it is shown that under very general assumptions on G (satisfied for large classes of elliptic second-order linear differential operators) normalized perturbation, however, leads to a Brelot space (X, ) admitting a Green function T (G) which is locally (or even globally) comparable with G and has all properties required of G before. In particular, iterated perturbation is possible. Moreover, intrinsic Hölder continuity of quotients of harmonic functions with respect to the local quasimetric :=(G ^–1+^* G ^–1)/2 yields -Hölder continuity for quotients of -harmonic functions as well.     

A generalization of Pal-type interpolation

. .     

Замечания о Сходкмости лагранжева интерполрования

( qP. )     

On unconditionally convergent basis expansions in the space of continuous functions

[0,1], - H .

---

---

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

The tree number of a graph with a given girth

A family of subtrees of a graphG whose edge sets form a partition of the edge set ofG is called atree decomposition ofG. The minimum number of trees in a tree decomposition ofG is called thetree number ofG and is denoted by(G). It is known that ifG is connected then(G) |G|/2. In this paper we show that ifG is connected and has girthg 5 then(G) |G|/g + 1. Surprisingly, the case wheng = 4 seems to be more difficult. We conjecture that in this case(G) |G|/4 + 1 and show a wide class of graphs that satisfy it. Also, some special graphs like complete bipartite graphs andn-dimensional cubes, for which we determine their tree numbers, satisfy it. In the general case we prove the weaker inequality(G) (|G| – 1)/3 + 1.     

Estimate of the modulus of continuity of the generalized solutions of certain singular parabolic equations

One considers singular parabolic equations of the form (u)/t–u0,where sign u is a multivalued function, equal to -I for u<0, to 1 for u>0, and to the segment [-I,I] for u=0. Such a class of equations contains, in particular, the model for the two-phase Stefan problem, the porous medium equation, and the plasma equation. For the bounded generalized solutions u(x,t) of the indicated equations (without the assumption u/L²one has established a qualified local estimate of the modulus of continuity.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Ins'tituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 49–71, 1985.     

Quasilinear elliptic equations with small boundary data

It is shown that if the order of non-uniformity of a quasilinear elliptic equation is h, 1<h2, then the critical norm separating existence and non-existence of a solution to the Dirichlet problem with small boundary data is the norm. For 0h1, existence of a solution is guaranteed without any smallness assumption on the given boundary data, provided that the usual a priori interior gradient bound for solution is available.     

Almost regular integer hexagons

We establish the sequence of all Equiangular Integer Hexagons that have alternating sides whose lengths differ by one.The author is indebted to T.G. Ostrom, Richard Guy, and a referee for their assistance and suggestions in the development of this paper     

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 .     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.     

Note on irregularities of distribution

Letf be a periodic function on with period 1, piecewise continuously differentiable, satisfying . For an arbitrary sequence = ( _i ) in [0,1) put and . If then _n (f,) >c· logn holds for some positive constantc (depending onf only) and almost alln. In a certain sense the converse is also true: there is a class of functionsf with such that _n (f,) =o (logn).Support has been received from Netherlands Organization for the Advancement of Pure Research (Z. W. O.).     

On the rate of convergence of an unstable solution of a stochastic differential equation

We study the rate of convergence of the process(tT)/T to the processw(t)/ asT , where(t) is a solution of the stochastic differential equationd(t)=a((t))dt+((t))dw(t) Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1424–1427, October, 1994.     

Notes on Interpolation. X (Two Boundary Value Problems)

In this paper we use (0, 2) interpolational polynomials to give an approximate solution of the differential equation y(x) + A(x)y(x) = F(x), x I := [-1, 1] j in case when the boundary values are y(-1) = and y(1) = , , R.     

On conditions for oscillation and nonoscillation of the solutions of a semilinear second-order differential equation

We establish sufficient conditions for oscillation and nonoscillation of regular solutions of the secondorder differential equation Sign y=0, where <1 and p: [a,[ , –< a < + is a locally summable function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 833–841, July, 1994.     

Asymptotics at t→∞ of the solution of the Cauchy problem for the Landau-Lifshitz equation

Conclusions The formulas obtained in the present paper for the leading term in the asymptotic behavior of the solution of the Cauchy problem for the LL equation subject to the boundary conditions L₃1, x± describe the solitonless sector. The transition to the general case, which takes into account the presence in the solution of soliton formations, can be made on the basis solely of algebraic considerations that use the procedure of soliton dressing developed in [17, 18] for the LL equation. In particular, applying to the obtained asymptotic formulas the procedure for a dressing of domain wall type (see [17]), we arrive at formulas that describe the asymptotic solution of the Cauchy problem for the LL equation with boundary conditions of the form L₃±1, x±.Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 76, No. 1, pp. 3–17, July, 1988.