Method for symmetrization and estimation of solutions of the Neumann problem for the equation of a porous medium in domains with noncompact boundary for infinitely increasing time

We consider the initial boundary-value Neumann problem for the equation of a porous medium in a domain with noncompact boundary. By using a symmetrization method, we obtain exactL _p-estimates, 1p, for solutions as t.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 147–157, February, 1995.     

Two-sided polynomial approximation of functions

Conditions are established when the collocation polynomials P_m(x) and P_M(x), m M, constructed respectively using the system of nodes x_j of multiplicities a_j 1, j = O,, n, and the system of nodes x_-r,,x_o,,x_n,,x_n+r1, r O, r₁ O, of multiplicities a_-r,,(a_o + y_o),,(a_n + y_n),,a_n+r1, a_j + y_j 1, are two sided-approximations of the function f on the intervals , x_j[, j = O,...,n + 1, and on unions of any number of these intervals. In this case, the polynomials P_m (x), P_M ^(l) (x) with l a_j are two-sided approximations of the function f⁽¹⁾ in the neighborhood of the node x_j and the integrals of the polynomials P_m(x), P_M(x) over D_j are two-sided approximations of the integral of the function f (over D_j). If the multiplicities a_j a_j + y_j of the nodes x_j are even, then this is also true for integrals over the set _j= _µ ^k D_j µ 1, k n. It is shown that noncollocation polynomials (Fourier polynomials, etc.) do not have these properties.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 31–37, 1989.     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

Fast Summation of Power Series with Coefficients Analytic at Infinity

We propose a fast summation algorithm for slowly convergent power series of the form _{j=j 0} z ^j _j j _i=1 ^s (j+ _i )^– _i , where R, _i 0 and _i C, 1is, are known parameters, and _j =(j), being a given real or complex function, analytic at infinity. Such series embody many cases treated by specific methods in the recent literature on acceleration. Our approach rests on explicit asymptotic summation, started from the efficient numerical computation of the Laurent coefficients of . The effectiveness of the resulting method, termed ASM (Asymptotic Summation Method), is shown by several numerical tests.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

Tangential Limits of Potentials on Homogeneous Trees

Let T be a homogeneous tree of homogeneity q+1. Let denote the boundary of T, consisting of all infinite geodesics b=[b ₀,b ₁,b ₂,] beginning at the root, 0. For each b, 1, and a0 we define the approach region _,a (b) to be the set of all vertices t such that, for some j, t is a descendant of b _j and the geodesic distance of t to b _j is at most (–1)j+a. If >1, we view these as tangential approach regions to b with degree of tangency . We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition _T f ^p (t)q ^–|t|<, where p>1 and 0<<1, or p=1 and 0<1. For 11/, we show that Gf(s) has limit zero as s approaches a boundary point b within _,a (b) except for a subset E of of -dimensional Hausdorff measure 0, where H (E)=sup_>0inf _i q ^{–|t i|}:E a subset of the boundary points passing through t _i for some i,|t _i |>log _q (1/).     

First order properties of random posets

Let =(n,p) be a binary relation on the set [n]={1, 2, ..., n} such that (i,i) for every i and (i,j) with probability p, independently for each pair i,j [n], where i<j. Define as the transitive closure of and denote poset ([n], ) by R(n, p). We show that for any constant p probability of each first order property of R(n, p) converges as n .     

Isomorphismen von rechtseiträumen

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation and whose set of point pairs P² is equipped with a congruence relation , such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism of two rectangular spaces (P,G, , ) and (P,G, , ) we mean a bijection of the point setP onto P which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, , ) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P is in fact an isomorphism from (P,G, , ) onto (P,G, , ) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, , ) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, , ). By a motion of(P. G,, ) we mean a bijection ofP which maps lines onto lines, preserves parallelism and satisfies the condition((x), (y)) (x,y) for allx, y P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, , ) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).

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Herrn Professor Burau zum 85. Geburtstag gewidmet     

Stabilization of ill-posed Cauchy problems for parabolic equations

Summary In this paper we study the noncharacteristic Cauchy problem, u_t–(a(x)u_x)_x=0, x(0, l), t.(0, T], u(0, t)=(t), u_x(0,t)=0, 0tT, assuming only L for a. In the case of weak a priori bounds on u, we derive stability estimates on u of Hölder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.

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Sunto Nel presente lavoro consideriamo il problema di Cauchy non ben posto u_t= (a(x)u_x)_x, x(0, l), t(0, T), u(0, t)=(t), u_x(0, t)=0, 0tT. Supponiamo che a sia misurabile e limitato inferiormente e superiormente da constanti positive. Introduciamo delle limitazioni a priori su u e dimostriamo la dipendenza continua di u rispetto al dato sia in (0, l)×(0, T) (di tipo hölderiano) sia per x=l (di tipo logaritmico). Consideriamo, inoltre, la dipendenza continua di u da a.

     

Boundary control of heat conduction

The paper considers control of the heat conduction process u_t — u = g from the initial state u(x, 0) to the final state u(x, t₁) in a fixed (finite) time t₁ via the coefficient (z) in the boundary condition Bu = (u/n) + (x)u. A uniqueness theorem is proved for the problem to find the process—control pair (u, ). The control problem is posed in terms of the coefficient in a boundary condition of the form Bu = (u/n) + (t)u.Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 3, pp. 93–97, 1993.     

Central Limit Theorem for a Class of Nonhomogeneous Random Walks

A spatially nonhomogeneous random walk _t on the grid =^m X ⁿ is considered. Let _t ⁰ be a random walk homogeneous in time and space, and let _t be obtained from it by changing transition probabilities on the set A= X ⁿ, || < , so that the walk remains homogeneous only with respect to the subgroup ⁿ of the group . It is shown that if >m 2 or the drift is distinct from zero, then the central limit theorem holds for _t.     

Limits of certain iterates of Post-Widder operators

The paper is a study of the limiting behaviour of the [n t]-th iterates of the well-known Post-Widder operatorsL _{n, x} used in the real inversion of the Laplace transform. It is shown that the limiting operators constitute a semigroup T _t;t0 of class (C ₀) on a family C _,; , >0 of Banach spaces. Applications of the semigroup structure lead to a pointwise saturation theorem forL _{n, x} and a characterization of convex functions inC _, through an inequality involving the action ofL _{n, x}.     

Über nullstellenfreie Gebiete von ζ(itk) (s)

Let ^(itk) (s) denote thek-th derivative of the Riemann Zeta-function,s=+it, ,t real numbers,k1 rational integers. Using ideas fromT. C. Titchmarsh and from a paper ofR. Spira, lower bounds are derived for |^(itk)(s)|, |^(itk)(1-s) for >1 and some infinitely many, sufficiently large values oft. Further let be an algebraic number of degreen and heightH; then a lower bound for |^(itk)(its)|, dependent onn, H, k is established for alln,H1,k3, 2+7k/4 and all realt.     

Integral representation of linear operators

In this paper we obtain necessary and sufficient conditions in order that a linear operator, acting in spaces of measurable functions, should admit an integral representation. We give here the fundamental results. Let (T_i, _i) (i=1,2) be spaces of finite measure, and let (T,) be the product of these spaces. Let E be an ideal in the space S(T₁, ₁) of measurable functions (i.e., from |e₁||e₂|, e₁ S (T₁, ₁), e₂E it follows that e₁E). THEOREM 2. Let U be a linear operator from E into S(T₂, ₂). The following statements are equivalent: 1) there exists a-measurable kernel K(t,S) such that (Ue)(S)=K(t,S) e(t)d(t) (eE); 2) if 0e_nE (n=1,2,...) and e_n0 in measure, then (Ue_n)(S) 0 ₂ a.e. THEOREM 3. Assume that the function (t,S) is such that for any eE and for s a.e., the ₂-measurable function Y(S)=(t,S)e(t)d ₁(t) is defined. Then there exists a-measurable function K(t,S) such that for any eE we have (t,S)e(t)d ₁(t)=K(t,S)e(t)d ₁(t) ₁a.e.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 5–14, 1974.     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

On the maximum of a Wiener process and its location

Summary Consider a Wiener process {W(t),t0}, letM(t)=max |W(s)| andv(t) be the location of the maximum of the absolute value of in [0,t] i.e.|W(v(t))|=M(t). We study the limit points of ( _t M(t), _t v(t)) ast where _t and _t are positive, decreasing normalizing constants. Moreover, a lim inf result is proved for the length of the longest flat interval ofM(t).Research supported by Hungarian National Foundation for Scientific Research Grant n. 1808     

A Note on a Boundary Value Problem

In this note, we prove that, for Robins boundary value problem, a unique solution exists if f_x(t, x, x), f_x(t, x, x), (t), and (t) are continuous, and f_x -(t), f_x -(t), 4(t) ² + ²(t) ++ 2(t), and 4(t) ² + ²(t) + 2(t).AMS Subject Classification (2000) 34B15     

ALTERNATION THEORY IN APPROXIMATION BY POLYNOMIALS HAVING BOUNDED COEFFICIENTS

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An automatic integration procedure for infinite range integrals involving oscillatory kernels

Let the real functionsK(x) andL(x) be such thatM(x)=K(x)+iL(x)=e^ix g(x), whereg(x) is infinitely differentiable for all largex and is non-oscillatory at infinity. We develop an efficient automatic quadrature procedure for numerically computing the integrals _a K(t)f(t) and _a L(t)f(t)dt, where the functionf(t) is smooth and nonoscillatory at infinity. One such example for which we also provide numerical results is that for whichK(x)=J (x) andL(x)=Y (x), whereJ (x) andY (x) are the Bessel functions of order . The procedure involves the use of an automatic scheme for Fourier integrals and the modified W-transformation which is used for computing oscillatory infinite integrals.     

Multipliers with singularities along a curve in ℝ n

We consider in ⁿ ,n2, the curve = (t,t ² ,...,t ⁿ ), ₀t₀,₀>0 a small number. We study the boundedness of operatorsT ,>0, defined by multipliers which present singularities along . Our results are derived from a sharp estimate on a suitable maximal function. In the casen=2 theT 's are Bochner-Riesz operators and our results coincide with the known ones.