Decision-procedures for invariant properties of short algorithms

For a class of algorithms R satisfying sufficiently general conditions and an enumeration of the algorithms of this class, it is proved that if the algorithm from R with code m in this enumeration algorithmically decides a property, nontrivial to N₁ and invariant (with respect to extensional equality) up to N, then for N max (t²(c)t⁵(N₁, t²(a)) one has m t^–3(N), where the constantsa, c and the function t are indicated in the text of the paper, t^–3(N) is used instead of t^–1(t^–1(t^–1(N))), and finally, t^–1(x)=yx[t(y)x]. For natural enumerations the constantsa, c are not large, and the function t does not grow too rapidly. From the result obtained also follows a generalization of a theorem of Rice in a form close to that proved in [2].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 73–76, 1979.In conclusion, the author thanks the participants in the Leningrad seminar on mathematical logic for valuable comments.     

Some Conditions for a C0-Semigroup to Be Asymptotically Finite-Dimensional

We study the class of bounded C ₀-semigroups T=(T _t ) _t0 on a Banach space X satisfying the asymptotic finite dimensionality condition: codim X ₀(T)<, where X ₀(T):={x X:lim_t T _t x=0}. We prove a theorem which provides some necessary and sufficient conditions for asymptotic finite dimensionality.     

Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeu _t–R¦u¦^m+u^q=0 in ⁿ× ₊, u(·,0⁺)=u_o(·) in ^N, whereR > 0, 0 <q < 1,m 1, andu _o is a positive uniformly continuous function verifying –R¦u _o¦^m+u ₀ ^q 0 in ^N . We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t(·) defined byu(x, t) > 0 if 0<t<t (x) andu(x, t)=0 ift t (x). Regularity, extinction rate, and asymptotic behavior of t(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(u _o(x – t))^1–q –(1–q)t]₊)^1/(1–q): ¦¦R}, (x, t) ₊ ^N+1 .Partially supported by the DGICYT No. 86/0405 project.     

On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity

We prove the existence of continuously differentiable solutions with required asymptotic properties as t +0 and determine the number of solutions of the following Cauchy problem for a functional differential equation:

Ont ∞ quasi-similarity of linear systems

Summary Conti defined t similarity of systems (1)x=tA(t)xand (2)y=B(t)y, and showed that it is an equivalence relation which preserves uniform and strict stability. Here the definition is weakened by imposing less stringent integrability conditions, some in terms of perhaps conditionally convergent improper integrals, on the matrix function relating A and B. The extended relation, t quasi-similarity, is not symmetric or transitive; however, it is shown that if (2) is t quasi-similar to (1)and (1)is uniformly, uniformly asymptotically, or strictly stable, then so is (2).Results are also given concerning linear asymptotic equilibrium of (2)in the case where (1)is stricly stable or has linear asymptotic equilibrium.     

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.     

On a zero-crossing probability

Let {X(t), 0E{exp (–sX(t))}=exp (–t(s)), where (s)=(1–(s)), is the intensity of the Poisson process, and (s) is the Laplace transform of the distribution of nonnegative jumps. Consider the zero-crossing probability =P{X(t)–t=0 for some t,0<t<}. We show that =() where is the largest nonnegative root of the equation (s)=s. It is conjectured that this result holds more generally for any stochastic process with stationary independent increments and with sample paths that are nondecreasing step functions vanishing at 0.     

Boundary-layer averaging for standard systems with lag

Thekth-order asymptotic solution of a standard system with lag is constructed along trajectories calculated according to the averaging scheme of A. N. Filatov. If the perturbation parameter 1, then the use of the step method for finding the solution is connected with cumbersome calculations because the number of required steps is inversely proportional to . We suggest another approach in which the step method is used onlyk times fort [0,k] and justify the asymptotic method.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10 pp. 1362–1368, October, 1994.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

L ∞ — Decay estimations of the spherical mean value on symmetric spaces

LetM _t[](x) be the spherical mean value operator applied to a function on a symmetric Riemannian space of the non-compact type.L —decay estimations forM _t [](x) as well as for its derivatives with respect to (t, x) are given, provided that belongs to a Banach space with suitable weighted supremum norm. This leads to estimates of the solutions to the wave equation in certain cases in which Huygens' principle is valid.     

An integral representation for decomposable measures of measurable functions

Summary We start with a measurem on a measurable space (,A), decomposable with respect to an Archimedeant-conorm on a real interval [0,M], which generalizes an additive measure. Using the integral introduced by the second author, a Radon-Nikodym type theorem, needed in what follows, is given.The integral naturally leads to a -decomposable measurem on the space of all measurable functions from to [0, 1]. The main result of the present paper is the converse of this, namely that, under natural conditions, any -decomposable measurem on can be represented as an integral of a certain Markov-kernelK. We extend this representation to measures on which have values in a set of distribution functions.These results generalize the work done by the first author in the case of additive measures.     

Remarks on multiple Fourier series

: (1) ( , , ), (2) ( —, , ). , .     

Interpolation Sequences for the Class of Functions of Finite η-Type Analytic in the Unit Disk

We establish conditions for the existence of a solution of the interpolation problem f( _n ) = b _n in the class of functions f analytic in the unit disk and such that

On the one-dimensional two-phase inverse Stefan problems

New formulations of the inverse nonstationary Stefan problems are considered: (a) forx [0,1] (the inverse problem IP₁; (b) forx [0, (t)] with a degenerate initial condition (the inverse problem IP). Necessary conditions for the existence and uniqueness of a solution to these problems are formulated. On the first phase {x [0, y(t)]{, the solution of the inverse problem is found in the form of a series; on the second phase {x [y(t), 1] orx [y(t), (t)]{, it is found as a sum of heat double-layer potentials. By representing the inverse problem in the form of two connected boundary-value problems for the heat conduction equation in the domains with moving boundaries, it can be reduced to the integral Volterra equations of the second kind. An exact solution of the problem IP is found for the self similar motion of the boundariesx=y(t) andx=(t).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1058–1065, August, 1993.     

L 2(R n )-extremal problems on the basis of incomplete information

. ( ), R ⁿ L ₂(R ²).

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The author is supported by the National Natural Science Found of China.     

Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature

Summary In this article we develop a method of deriving asymptotic formulae for the orbital counting function for the action of certain discrete groups of isometries of simply connected negatively curved manifolds. We consider the particular case of normal subgroups ₀ of a co-compact group ₀ for which the quotient ₀/ ^k . Even in the special case of manifolds ofconstant negative curvature, this leads to new results. In particular, we have asymptotic estimates for some groups which arenot geometrically finite.Oblatum III-1993 & 18-VIII-1993The first author was supported by The Royal Society through a University Research Fellowship. The second author was supported by the UK SERC under grant number GR/G51930     

The Ramsey property for collections of sequences not containing all arithmetic progressions

A family of sequences has the Ramsey property if for every positive integerk, there exists a least positive integerf (k) such that for every 2-coloring of {1,2, ...,f (k)} there is a monochromatick-term member of . For fixed integersm > 1 and 0 q < m, let _q(m) be the collection of those increasing sequences of positive integers {x ₁,..., x_k} such thatx _i+1 – x_i q(modm) for 1 i k – 1. Fort a fixed positive integer, denote byA _t the collection of those arithmetic progressions having constant differencet. Landman and Long showed that for allm 2 and 1 q < m, _q(m) does not have the Ramsey property, while _q(m) A _m does. We extend these results to various finite unions of _q(m) 's andA _t 's. We show that for allm 2, _q=1 ^m–1 _q(m) does not have the Ramsey property. We give necessary and sufficient conditions for collections of the form _q(m) ( _{t T} A _t) to have the Ramsey property. We determine when collections of the form _a(m₁) _b(m₂) have the Ramsey property. We extend this to the study of arbitrary finite unions of _q(m)'s. In all cases considered for which has the Ramsey property, upper bounds are given forf .     

Best approximations of continuous functions by spline functions

An investigation of the approximation on [0, 1] of functionsf (x) by spline functions s(f,; x) of degree 2r-1 and of deficiency r (r>1) depending on the vector function = ₁ (x),..., _r-1(x) and interpolatingf (x) at fixed points. For the optimal choice of the vector ₀, exact estimates are obtained of the norms f(x)-s (f, ₀; x)_C[0,1] and f (x)-s (f, ₀; x)_{L[0, 1]} on the function classes H Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 41–46, July, 1970.In conclusion we would like to thank N. P. Korneichuk for suggesting this problem and for his valuable advice.     

Fundamental equation of information revisited

Summary This paper presents a new, shorter and more direct proof of the following result of J. Aczél and C. T. Ng: IfM: J R (J =]0, 1[ ^k ) is both multiplicative and additive, then the general solution: J R of(x) + M(1 – x)(y/1 – x) = (y) + M(1 – y)(x/1 – y) (x, y, x + y J) is given by(x) = ifM = 0,(x) = M(x)[L(x) + ] + M(1 – x)L(1 – x) ifM 0,where is an arbitrary constant andL: J R is an arbitrary solution of the logarithmic functional equationL(xy) = L(x) + L(y) (x, y J). Also, some extensions of this result to fields more general than the reals are given.     

A Gaussian fluid model

A fluid model with infinite buffer is considered. The total net rate is a stationary Gaussian process with mean –c and covariance functionR(t). Let (x) be the probability that in steady state conditions the buffer content exceedsx. Under the condition ₀ t ² ¦R(t)¦dt< we show that admits a logarithmic linear upper bound, i.e. (x)Cexp[–x]+o(exp[–x]) and find and C. Special cases are worked out whenR is as in a Gauss-Markov or AR-Gaussian process.