Asymptotic behaviour of the solution of a semilinear parabolic equation

We study the asymptotic behaviour ast tends to + of the solution of (u/t)–Lu+(u)–0,u _|=0 whereL is a second order self-adjoint elliptic operator and a maximal monotone graph of . If |(r)|/|r|²L ¹ (-1, 1) and ₁ is the first eigenvalue ofL we prove that u(.,t) converges uniformly on to some element of Ker (L + ₁ I) and that the limit is nonzero if |(r)|/|r| is nondecreasing. We give also some properties of the limit (monotonicity, continuity, range).     

Nonlinear semigroups and age-dependent population models

Summary This paper treats the nonlinear age-dependent population problem (1)(0,a)=(a), a I; (2)(t, 0)=F((t, ·)), t0; (3) ,t0,where I is the age range of the population, (t, ·) is the unknown age density at time t, is the known initial age distribution, and the functionals F and G are nonlinear. The problems of existence, uniqueness, continuous dependence upon initial values, and the positivity of solutions are investigated using the method of nonlinear semigroups.Supported in part by the National Science Foundation Grant NSF 75-06332A01.     

Integral averages and oscillation of second order sublinear differential equations

New oscillation criteria are given for the second order sublinear differential equation

On relations among moduli of continuity in Lorentz spaces

Estimates are obtained among moduli of continuity of functions in several variables that belong to various Lorentz spaces. The functions considered are periodic with period 1 in each variable. More exactly the following theorem is proved: If 0<,<(t) and (t) are so-called -functions such that _;>₁>1, and

On minimax filtration of vector processes

We study the problem of optimal linear estimation of the transformation of a stationary random process (t) with values in a Hilbert space by observations of the process (t) + (t) fort0. We obtain relations for computing the error and the spectral characteristic of the optimal linear estimate of the transformationA for given spectral densities of the processes (t) and (t). The minimax spectral characteristics and the least favorable spectral densities are obtained for various classes of densities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 389–397, March, 1993.     

The Discrete Spectrum Asymptotics with Respect to a Large Coupling Constant under Strong Nonnegative Perturbations

Let A be a self-adjoint operator, let (, ) be an inner gap in the spectrum of the operator A, and let B(t) = A + tW ^* W, where the operator W(A – iI)^-1 is not necessarily bounded. Conditions are obtained under which the spectrum of B(t) in (, ) is discrete. Let N(, A, W, ), (, ), > 0, be the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to . The asymptotics of N(, A, W, ) as + is obtained in terms of the spectral asymptotics of a certain self-adjoint compact operator. Bibliography: 5 titles.     

Positive solutions to certain classes of singular nonlinear second order boundary value problems

This paper is devoted to the problem of existence of solutions to the nonlinear singular two point boundary value problem , withy satisfying either mixed boundary datay(1)=Lim_y0+p(t)y(t)=0 or dirichlet boundary datay(0)=y(1)=0. Throughout our nonlinear termqf is allowed to be singular att=0,t=1,y=0 and/orpy=0.     

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 ().     

An asymptotic theorem for a class of nonlinear neutral differential equations

The neutral differential equation is considered under the following conditions: n 2, > 0, = ±1, F(t, u) is nonnegative on [t ₀, ) × (0, ) and is nondecreasing in u (0, infin;), and lim g(t) = as t . It is shown that equation (1.1) has a solution x(t) such that 0}}{\text{.}} \hfill \\ \end{gathered} $$ " align="middle" border="0"> Here, k is an integer with 0 k n–1. To prove the existence of a solution x(t) satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.     

On the Ranked Excursion Heights of a Kiefer Process

Let (K(s,t), 0s1, t1) be a Kiefer process, i.e., a continuous two-parameter centered Gaussian process indexed by [0,1]×₊ whose covariance function is given by (K(s₁,t₁) K(s₂,t₂))=(s₁s₂-s₁s₂)t₁t₂, 0s₁, s₂1, t₁, t₂ 0. For each t>0, the process K(·,t) is a Brownian bridge on the scale of . Let M ₁ ^* (t) M ₂ ^* (t) M _j ^* (t) 0 be the ranked excursion heights of K(,t). In this paper, we study the path properties of the process tM _j ^* (t). Two laws of the iterated logarithm are established to describe the asymptotic behaviors of M _j ^* (t) as t goes to infinity.     

Two limit theorems on ARIMA models

Let the time series {X(t), t=1, 2, ...} satisfy (B)(1–B) ^d X(t)=(B)e(t), whereB is a backward shift operator, defined byBX(t)=X(t–1), and (z)=1+₁ z+...+ _p z ^p , (z)=1+₁ z+...+ _q z ^q , and all the roots of (z) lie outside the unit circle; {e(t)} is a sequence of iid random variables with mean zero andE|e(t)|^4+r < (r>0). In this paper, the limit properties of , where the integerd1, have been considered.     

On the Existence of Positive Solutions of Nonlinear Elliptic Equations

We study the existence of positive solutions of the nonlinear elliptic problem in D with u=0 on D, where and are two Randon's measures belonging to a Kato subclass and D is an unbounded smouth domain in ^d(d3). When g is superlinear at 0 and 0f(t)t for t(0,b), then probabilistic methods and fixed point argument are used to prove the existence of infinitely many bounded continuous solutions of this problem.     

Uniqueness of kernel functions of the heat equation

We consider the heat equation on ={(x,t) R ²;t<0, ¦x¦<(–t)} and give the uniqueness of kernel functions at the infinity (see Theorem 5). For the proof, we examine the continuity of the density of the parabolic measure onD ={(x,t);t>x}, closely related to . By this theorem, we can decide the Martin boundary of (<1) with respect to the heat equation.     

Asymptotics of the generating function for the volume of the Wiener sausage

Summary We consider the generating function of the voltime of the Wiener sausageC (t), which is the -neighbourhood of the Wiener path in the time interval [0,t]. For <0, the limiting behavior fort, up to logarithmic equivalence, had been determined in a celebrated work of Donsker and Varadhan. For >0 it had been investigated by van den Berg and Tóth, but in contrast to the case <0, there is no simple expression for the exponential rate known. We determine the asymptotic behaviour of this rate for small and large .     

Asymptotics of the partial sums of a set of integral transforms

In this paper, we explore the asymptotic distribution of the zeros of the partial sums of the family of entire functions of order 1 and type 1, defined by G(,,z)=₀ ¹(t)t ^–1×(1–t)^–1e ^zt dt, where Re,Re>0, is Riemann-integrable on [0,1], continuous at t=0, 1 and satisfies (0)(1)0.     

Über einen Satz von Shl. Sternberg in der Theorie der analytischen Iterationen

Let X=X ₁,...,X _n be the ring of formal power series inn indeterminates over . LetF:XAX+B(X)=(F ⁽¹⁾(X),...,F ⁽ⁿ⁾(X))(X) ⁿ denote an automorphism of X and let ₁,..., _n be the eigenvalues of the linear partA ofF. We will say thatF has an analytic iteration (a. i.) if there exists a family (F _t (itX)) _t of automorphisms such thatF _t(X) has coefficients analytic int and such thatF ₀=X,F ₁=F,F _t+t=F_tF_t for allt,t. Let now a set=(ln₁,...,ln _n ) of determinations of the logarithms be given. We ask if there exists an a. i. ofF such that the eigenvalues of the linear partA(t) ofF _t(X) are . We will give necessary and sufficient conditions forF to have such an a. i., namely thatF is conjugate to a semicanonical formN=T ^–1FT such that inN ^(k)(X) there appear at most monomialsX ₁ ¹ ...X _n ⁿ . This generalizes a result of Shl.Sternberg.

---

---

Herrn Prof. Dr. E. Hlawka zum 60. Geburtstag gewidmet     

An Erdős-Révész type law of the iterated logarithm for stationary Gaussian processes

Summary Let {X(t),t 0} be a stationary Gaussian process withEX(t)=0,EX ²(t)=1 and covariance function satisfying (i)r(t) = 1 2212;C |t | + o (|t|)ast0 for someC>0, 0<2; (ii)r(t)=0(t ^–2) as t for some >0 and (iii) sup_ts|r(t)|<1 for eachs>0. Put (t)= sup {s:0 s t,X(s) (2logs)^1/2}. The law of the iterated logarithm implies a.s. This paper gives the lower bound of (t) and obtains an Erds-Rèvèsz type LIL, i.e., a.s. if 0<<2 and . Applications to infinite series of independent Ornstein-Uhlenbeck processes and to fractional Wiener processes are also given.Research supported by the Fok Yingtung Education Foundation of China and by Charles Phelps Taft Postdoctoral Fellowship of the University of Cincinnati     

Matroid Shellability, β-Systems,and Affine Hyperplane Arrangements

The broken-circuit complex is fundamental to the shellability and homology of matroids, geometric lattices, and linear hyperplane arrangements. This paper introduces and studies the -system of a matroid, nbc(M), whose cardinality is Crapo's -invariant. In studying the shellability and homology of base-pointed matroids, geometric semilattices, and afflne hyperplane arrangements, it is found that the -system acts as the afflne counterpart to the broken-circuit complex. In particular, it is shown that the -system indexes the homology facets for the lexicographic shelling of the reduced broken-circuit complex , and the basic cycles are explicitly constructed. Similarly, an EL-shelling for the geometric semilattice associated with M is produced,_and it is shown that the -system labels its decreasing chains.Basic cycles can be carried over from The intersection poset of any (real or complex) afflnehyperplane arrangement is a geometric semilattice. Thus the construction yields a set of basic cycles, indexed by nbc(M), for the union of such an arrangement.     

Some asymptotic results related to the law of iterated logarithm for Brownian motion

For 0<<1, let . The questions addressed in this paper are motivated by a result due to Strassen: almost surely, lim sup _t U ((t))=1–exp{–4(^–1)–1}. We show that Strassen's result is closely related to a large deviations principle for the family of random variablesU (t), t>0. Also, when =1,U (t)0 almost surely and we obtain some bounds on the rate of convergence. Finally, we prove an analogous limit theorem for discounted averages of the form as 0, whereD is a suitable discount function. These results also hold for symmetric random walks.     

Asymptotic behavior of solutions of a nonlinear difference equation with continuous argument

We consider the difference equation with continuous argument