Blow-up rates for parabolic systems

Let ⁿ be a bounded domain andB _R be a ball in ⁿ of radiusR. We consider two parabolic systems: ut=u +f(), i= +g(u) in × (0,T) withu=v=0 on × (0,T) andu _t =u, v _t =v inB _r × (0,T) withe/v=f (v), e/v=g(u) onB _R × (0,T). Whenf(v) andg(u) are power law or exponential functions, we establish estimates on the blow-up rates for nonnegative solutions of the systems.     

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 .     

On Decomposing an Abelian p-Group Under a p′-Operator Group

Let A be a p-group acting on a finite Abelian p-group P. We give a necessary and sufficient condition on an A-irreducible decomposition of ₁(P) = _i1U_i for the existence of an A-indecomposable decomposition of P = _i1R_i such that ₁(R_i) = U_i for all i I. This readily implies a main result of [1].2000 Mathematics Subject Classification: 20D15     

On the divisibility of homogeneous hypergraphs

We denote byK _k ,k, 2, the set of allk-uniform hypergraphsK which have the property that every element subset of the base ofK is a subset of one of the hyperedges ofK. So, the only element inK ₂ ² are the complete graphs. If is a subset ofK _k then there is exactly one homogeneous hypergraphH whose age is the set of all finite hypergraphs which do not embed any element of . We callH -free homogeneous graphsH _n have been shown to be indivisible, that is, for any partition ofH _n into two classes, oue of the classes embeds an isomorphic copy ofH _n . [5]. Here we will investigate this question of indivisibility in the more general context of-free homogeneous hypergraphs. We will derive a general necessary condition for a homogeneous structure to be indivisible and prove that all-free hypergraphs for K _k with 3 are indivisible. The-free hypergraphs with K _k ² satisfy a weaker form of indivisibility which was first shown by Henson [2] to hold forH _n . The general necessary condition for homogeneous structures to be indivisible will then be used to show that not all-free homogeneous hypergraphs are indivisible.This research has been supported by NSERC grant 69–1325.     

Some analytical properties of γ-convex functions on the real line

This paper deals with the analytical properties of -convex functions, which are defined as those functions satisfying the inequalityf(x ₁ )+f(x ₂ )f(x ₁)+f(x ₂), forx _i [x ₁,x ₂], |x _i –x _i |=, i=1,2, whenever |x ₁–x ₂|>, for some given positive . This class contains all convex functions and all periodic functions with period . In general, -convex functions do not have ideal properties as convex functions. For instance, there exist -convex functions which are totally discontinuous or not locally bounded. But -convex functions possess so-called conservation properties, meaning good properties which remain true on every bounded interval or even on the entire domain, if only they hold true on an arbitrary closed interval with length . It is shown that boundedness, bounded variation, integrability, continuity, and differentiability almost everywhere are conservation properties of -convex functions on the real line. However, -convex functions have also infection properties, meaning bad properties which propagate to other points, once they appear somewhere (for example, discontinuity). Some equivalent properties of -convexity are given. Ways for generating and representing -convex functions are described.This research was supported by the Deutsche Forschungsgemeinschaft. The first author thanks Prof. Dr. E. Zeidler and Prof. Dr. H. G. Bock for their hospitality and valuable support.     

Limit of Solutions of a SDE with a Large Drift Driven by a Poisson Random Measure

We consider a sequence of {X _n} of R ^d-valued processes satisfying a stochastic differential equation driven by a Brownian motion and a compensated Poisson random measure, with _n ~ _n with a large drift. Let be a m-dimensional submanifold (m<d), where F vanishes. Then under some suitable growth conditions for _n ~ _n, and some conditions for F, we show that dist(X _n, )0 before it exits any given compact set, that is, the large drift term forces X _n close to . And if the coefficients converge to some continuous functions, any limit process must actually stay on and satisfy a certain stochastic differential equation driven by Brownian motion and white noise.     

Decomposing symmetric exchanges in matroid bases

LetB,B be bases of a matroid, withX B, X B. SetsX,X are asymmetric exchange if(B – X) X and(B – X) X are bases. SetsX,X are astrong serial B-exchange if there is a bijectionf: X X, where for any ordering of the elements ofX, sayx _i ,i = 1, , m, bases are formed by the sets B₀ = B, B_i = (B_i–1 – x_i) f(x _i), fori = 1, , m. Any symmetric exchangeX,X can be decomposed by partitioning X = _i=1 ^m Y_i, X = _i=1 ^m Y_i, X, where (1) bases are formed by the setsB ₀ =B, B _i = (B _i–1 –Y _i ) Y _i ; (2) setsY _i ,Y _i are a strong serialB _i–1 -exchange; (3) properties analogous to (1) and (2) hold for baseB and setsY _i ,Y _i .     

G8 estimator of solutions of systems of linear algebraic equations

For linear forms of regularized solutions (x, c)=Re c' · Re[I + i)+A'A_n ^–1]^–1 A'_nb of systems of equations Ax=b, where A is an n×m matrix, x, c, b are vectors, and _n is a sequence of constants, we propose the estimator , where is any measurable solution of the equation ()Re[1+₁a(())]²+ (₁–₂)(1+₁(gq()))=, a(y)=n^–1 Sp[Iy+^–1Z_s'Z^s+ iI]^–1, , _i=n_n ²_n ^–1s_n ^–1, ⁿ=mI_n ²_n ^–1s_n ^–1, X_i are independent observations on the matrix A. Under certain conditions, it is proved that G₈ is a consistent estimator for n and 0.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 111–119, 1988.     

Compactness of probability measures and the uniform convergence of certain stochastic series

We give the conditions which ensure the compactness of the probability measures _n, n1, generated by Gaussian processes the realizations of which are continuous with unit probability in [0, 1]. We also give the conditions for the uniform convergence of stochastic series of the form _{k=1 2k}(t), where the _k(t) are independent Gaussian processes the realizations of which are continuous with unit probability in [0, 1].Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 443–451, October, 1972.In conclusion the author wishes to express his deep gratitude to Yu. V. Kozachenko for formulating the problem and for his attention to the paper.     

Perturbation of the poles of the scattering matrix under variation of the boundary condition

The behavior of the poles z_n(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |z_n()–z_n|=0(^(1/2)qn), where q_n is the order of the pole of the scattering matrix for the operator ₀u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.     

Remarks on some classes of Fourier coefficients

In this paper equivalent classes of the classes M' and S' _{p r}, p >1, 0,r {0,1,2,...,[]} defined by Sheng [5] are obtained. Then it is shown that the classes of Fourier coefficients S _p, S' _p(case r==0) and S _p(), p>1, defined by . V. Stanojevi, V. B. Stanojevi Sheng and the author of the present note are identical. As a corollary of this result, the L ¹-estimate for cosine series, obtained in [10], is refined.     

Stress relaxation: Experiment,theory, and computer simulation

Experimental evidence on stress relaxation is analyzed first for a wide variety of classes of materials: metals and their alloys, synthetic and natural polymers, glasses and frozen non-polymeric organic liquids. Common features of curves (t) of relaxation of stress a as a function of time t are discussed, and the importance of the internal stress _i() noted. Theoretical approaches are then reviewed, with particular attention to the cooperative model and its modifications; that model corresponds well to the experimental results. Some simulation results obtained with the method of molecular dynamics are reported for ideal metal lattices, metal lattices with defects, and for polymeric systems. In agreement with both experiments and the cooperative theory, the simulated (log t) curves exhibit three regions: initial, nearly horizontal, starting at ₀; central, descending approximately linearly; and final, corresponding to _i. In agreement with the theory, the slope of the simulated central part is proportional to the initial effective stress ₀*= ₀– _i. The time range taken by the central part is strongly dependent on the defect concentration: the lower the defect concentration, the shorter the range. Imposition in the beginning of a high strain destroys largely the resistance of a material to deformation, resulting in low values of the internal stress _i. On the joint basis of experimental, theoretical, and numerical results, we explain the mechanism of stress relaxation in terms of deformations occuring in the immediate environment of the defects. Simulations show several common features in the behavior of metals and polymers. Apart from the defect concentration, the amount of free volumev _f is also important.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 591–606, September–Ocotober, 1995.     

Isomorphisms of Some Complex Poisson Brackets

Given a complex manifold M_i, structures of Poisson algebras on (M_i)=C(M_i,C) which are associated with a nondegenerate -closed (2,0)-form _i on M_i are considered. It is shown that every isomorphism of Poisson structures (M₁) (M₂) is generated by a biholomorphic map :M₂ M₁ such that ₂ = *₁     

λ-Automorphisms of a Riemannian Foliation II

In Pak (Ann. Global Anal. Geom.13 (1995), 281–288), the notion of -automorphisms of a harmonic Riemannian foliation was extended to a general Riemannian foliation. By applying the characterization obtained in Pak of a -automorphism to be transversal Killing, we here consider the problem that a transversal conformal or projective field is transversal Killing. A special condition, namely _{B B} = 0, is discussed.     

On a Model Problem for the Orr–Sommerfeld Equation with Linear Profile

A model spectral problem of the form -i)y+xy= y on the finite interval [-1,1] with the Dirichlet boundary conditions is considered. Here is the spectral parameter and is positive. The behavior of the spectrum of this problem as 0 is completely investigated. The limit curves are found to which the eigenvalues concentrate and the counting eigenvalue functions along these curves are obtained.     

Folgenräume als universelle Generatoren für Klassen lokalkonvexer Räume

Motivated by the known characterizations of equicontinuity in the dual of a Schwartz space, a nuclear space, or a strongly nuclear space,we introduce the concepts of a -sequence and of a ()-sequence in the dual of an arbitrary lcs [E,], and we investigate the corresponding topologies and ₍₎ on E of uniform convergence on these sequences. Here is a normal sequence space such that . Under favorable enough conditions on , including the nuclearity of its normal topology , [,] acts as a universal generator for those lcs [E,] which satisfy =. Under somewhat weaker assumptions on , [,₍₎] is a universal generator for the lcs [E,] with =₍₎. These results cover e.g. the cases of -nuclear spaces and of nuclear spaces known from the recent literature. As an application we show that every non-trivial ultrabornological lcs is representable as an inductive limit of isomorphic copies of [, ( , )], where is any nuclear power series space of infinite type with stable exponent sequence.     

Powers of the curvature operator of space forms and geodesics of the tangent bundle

It is well known that if is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold, then all geodesic curvatures of the projected curve = _1463-01 are constant. In this paper, we consider the case of the tangent (sphere) bundle over real, complex, and quaternionic space forms and give a unified proof of the following property: All geodesic curvatures of the projected curve are zero beginning with k ₃, k ₆, and k ₁₀ for the real, complex, and quaternionic space forms, respectively.__________Published in Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1231–1243, September, 2004.     

Compact interval spaces in which all closed subsets are homeomorphic to clopen ones,I

A topological space X whose topology is the order topology of some linear ordering on X, is called an interval space. A space in which every closed subspace is homeomorphic to a clopen subspace, is called a CO space. We regard linear orderings as topological spaces, by equipping them with their order topology. If L and K are linear orderings, then L ^*, L+K, L·K denote respectively the reverse orderings of L, the ordered sum of L and K and the lexicographic order on L×K (so ·2=+ and 2·=). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals , 0, let L(, )= + 1 + ^* . Main theorem. Let X be a compact interval space. Then X is a CO space if and only if X is homeomorphic to a space of the form + 1 + _i L( _i , _i ), where is any ordinal, n, for every ii, _i are regular cardinals and _{i i}, and if n>0, then max({ _i: i}) · . This first part is devoted to show the following result. Theorem: If X is a compact interval CO space, then X is a scattered space (that means that every subspace of X has an isolated point).Supported by the Université Claude-Bernard (Lyon-1), the Ben Gurion University of the Negev, and the C.N.R.S.: UPR 9016Supported by the City of Lyon     

On some problems of interpolation by ℒ-splines

_n (D) — ,s — _n (D), _v (v=1, 2, ...,s/2) — . _m={0x ₀<x ₁<...<x _2m–1<2,x _2m =x ₀+2} , x _j +1–x _j <(4s max _v )^–1,j=0, 1, ..., 2m –1, ( ) 2- - _n,m 2m , _m . , L _q - (1q) W ( _n )={f ₂ :f ^(n–1)AC ₂ , _n (D)f 1} 2- - (s _n f), _m . , - - _n,m .

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The author expresses his gratitude to Yu. N. Subbotin for a useful discussion on the results of this paper.     

Normalformen für Bewegungen in hyperbolischen Räumen

If K is an Euclidean field and f a symmetric bilinear form of index 1, there is for each element O₄ (K,f) an orthogonal decomposition of the corresponding metric vector space V₄(K,f) into — invariant subspaces U_i such that dim U_i 3. Thus it is possible to determine nomal form for and to decompose the characteristic polynomial X into irreducible factors.