On differentiability of SRB states for partially hyperbolic systems

Consider a one parameter family of diffeomorphisms f such that f ₀ is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let be any u-Gibbs state for f . We prove (Theorem 1) that for any C function A the map (A) is differentiable at =0. This implies (Corollary 2.2) that the difference of Birkhoff averages of the perturbed and unperturbed systems is proportional to . We apply this result (Corollary 3.3) to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique SRB measure with good statistical properties.     

Uniform boundedness of a family of set functions

Let be a ring of sets, X a normed space, : X ( ) a bounded family of triangular functions. The following generalized Nikodym theorem is established: the family {} is uniformly bounded on if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a part of some set in . An analogous criterion is established also for semiadditive functions. In addition, it is shown that uniform boundedness of a family of triangular functions is preserved in passing from a ring to the -ring it generates.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 855–861, June, 1978.     

Developments on MTP2 properties of absolute value multinormal variables with nonzero means

LetX=(X ₁,...,X _n ) have a multivariate normal distribution with mean and covariance matrix. In the case=0, Karlin and Rinott^[6] obtained a necessary and sufficient condition on for |X|=(|X ₁|,...|X _n|) to be MTP₂. In this paper we consider the case0, and give some conditions under which |X| is MTP₂. A necessary and sufficient condition is given for |X| to be TP₂ whenn=2 and0. Some results about the TP₂ stochastic ordering are also given. The results are applied to obtain positive dependence and associated inequalities for multinormal and related distributions.This research is supported by the National Natural Science Foundation of China, the Doctoral Program Foundation of Institute of High Education and a grant of the Chinese Academy of Sciences.     

Nonlocal problems in the theory of the motion equations of Kelvin-Voight fluids

For the motion equations of Kelvin-Voight fluids one proves: 1) a global theorem for the existence and uniqueness of a solution (v;{u_e}) of the initial-boundary value problem on the semiaxis t R⁺ from the class W ¹ (R⁺); W ₂ ² () H()) with initial condition v_o(x) W ₂ ² () H() when the right-hand side f(x, t) L(R ⁺; L₂()); 2) a global theorem for the existence and uniqueness of a solution (v; {u_l}) on the entire axisR from the classW ¹ (R; W ₂ ² () H()) when the right-hand side f(x, t) L(R; L₂()); 3) a global theorem for the existence of at least one solution (v; {u_l}), periodic with respect to t with period , from the class W ¹ (R ⁺; W ₂ ² () H()) when the right-hand side f(x, t) L(R ⁺; L₂()) is periodic with respect to t with period , and a local uniqueness theorem for such a solution; 4) a theorem for the existence and uniqueness in the small of a solution (v; {u_l}), almost periodic with respect to t R, from V. V. Stepanov's class S ¹ (R; W ₂ ² ()H()) when the right-hand side f(x, t) S(R; L₂()) is almost periodic with respect to t; 5) the linearization principle (Lyapunov's first method) is justified in the theory of the exponential stability of the solutions of an initial-boundary value problem in the space H() and conditions are given for the exponential stability of a stationary and periodic solution, with respect to t R, of the system (1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 146–185, 1990.     

Homogenization of Some Elasticity Problems with Rapidly Oscillating Convex Constraints on Displacements

The problem of homogenization is considered for an elastic body occupying a perforated domain = obtained from a fixed domain and an -contraction of a 1-periodic domain .     

On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

Exchangeable random variables and the subsequence principle

Summary Call a sequence {X _n } of r.v.'s -exchangeable if on the same probability space there exists an exchangeable sequence {Y _n } such thatP(|X _n –Y _n |) for alln. We prove that any tight sequence {X _n } defined on a rich enough probability space contains -exchangeable subsequences for every >0. The distribution of the approximating exchangeable sequences is also described in terms of {X _n }. Our results give a convenient way to prove limit theorems for subsequences of general r.v. sequences. In particular, they provide a simplified way to prove the subsequence theorems of Aldous [1] and lead also to various extensions.     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

Generation of alternating groups by pairs of conjugates

Considering the conjugacy classes of the alternating group of degreen, those classes that contain a pair of generators are in the majority. In fact, the proportion of such classes is 1 –(n), and(n) 0 asn .     

On the optimality of a characterization theorem for the gamma function using the multiplication formula

Summary The following Artin type characterization of : _{+ +} is proved: Assume thatf: _{+ +} satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim _{x 0} xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.     

A correction theorem and the dyadic space H1, ∞

It is shown that for every l-function f and for every , >0, there exists a function g such that mes {t=g} <, while the partial sums of the Fourier and Fourier-Walsh series of the function g are uniformly bounded by the number C log (^–1)f. In the proof we make use of the characterization of the dyadic space H^1, in terms of atomic decompositions (it is, apparently, new).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 67–75, 1986.     

Effect of loading history on changes in the characteristic Angles and the length of the retardation trace

The local strains theory and the theory of small elastoplastic strains are used to determine the values of the components of the plastic strain vector {₁, ₂} for nonlinearity n=3 in five-dimensional Euclidean space in the case of complex loading along a two-segment broken line, when both tensile and compressive stresses are present in each loading stage. The relation between the vectors , and S and the tangents to the deformation and loading trajectories are examined. Values of the retardation trace are obtained in terms of the loading history. Numerical results have been derived with the aid of a BÉSM-3M computer.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 92–97, January–February, 1971.     

Theorem on a convergence condition in the spaces

For a given -function (u), a condition on a -function (u) is found such that it is necessary and sufficient for the following to hold: if f_n(x) f(x) and f _n (x)M (n=1, 2, ...) where M>0 is an absolute constant, then f _n (x)–f(x)0(n). An analogous condition for convergence in Orlicz spaces is obtained as a corollary.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 615–626, May, 1977.The author thanks V. A. Skvortsov for his constant attention and guidance on this paper.     

Application of a computer to the proof of a conjecture of Minkowski in the geometry of numbers

A computer-assisted proof is given of Minkowski's conjecture on the critical determinant of the region x^p+y^p<1 in the cases 1.03p 1.9745, p2.40, p2.577.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 163–180, 1977.     

Approximation by infinitely divisible distributions in the multidimensional case

Let be the collection of parallelepipeds in R with edges parallel with the coordinate axes and let be the collection of closed sets in R. Let (G, H)=inf {G{A}H{A}+, H{A}G{A}+ for any; L(G, H)= inf {G{A}H{A}+, H{A}G{A}+ for any, where G, H are distributions in . In the paper one gives the proofs of results announced earlier by the author (Dokl. Akad. Nauk SSSR,253, No. 2, 277–279 (1980)). One considers the problem of the approximation of the distributions of sums of independent random vectors with the aid of infinitely divisible distributions. One obtains estimates for the distances (·, ·), L(·, ·) and. It is proved that, where 0p_i1, ; E is the distribution concentrated at zero; V_i(i=1, ..., n) are arbitrary distributions; the products and the exponentials are understood in the sense of convolution.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 89–103, 1983.     

Parabolic subgroups of Chevalley groups over a semilocal ring

Let G be the Chevalley group over a commutative semilocal ring R which is associated with a root system . The parabolic subgroups of G are described in the work. A system =() of ideals in R ( runs through all roots of the system ) is called a net of ideals in the commutative ring R if ₊ for all those roots and for which + is also a root. A net is called parabolic if =R for >0. The main theorem: under minor additional assumptions all parabolic subgroups of G are in bijective correspondence with all parabolic nets . The paper is related to two works of K. Suzuki in which the parabolic subgroups of G are described under more stringent conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 43–58, 1978.     

Continuity of martingales in the Brownian excursion filtration

Summary Given a Brownian motionB, we consider the filtration ( _{x xR} ), where _x is defined as the -field generated by the excursions ofB belowx. In this paper we prove a conjecture of Walsh which says that all -martingales are continuous.     

Fano'sche Moufang-Ebenen mit Gauss-Figur sind pappos'sch

LetV be a quadrilateral in aMoufang-plane , in which theFano-proposition is valid. Take the pointsP,Q,R respectively in the diagonalsp,q,r ofV and construe the pointsP ^*,Q ^*,R ^* inp,q, r harmonic toP,Q,R with respect to pairs of edges ofV. IfP,Q,R are collinear, so areP ^*,Q ^*,R ^*, if and only if is aPappos-plane. Is V classical, the pointsP ₁ p,Qq,Rr and their harmonic conjugatesP ₁ ^* ,Q ^*,R ^* (construed as above mentioned) lay in a curve of 2^nd order.

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R. Artzy zum 70. Geburtstag zugeeignet     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

On flag-transitive affine planes of orderq 3

Let be a translation plane of orderq ³,q an odd prime power, whose kern GF(q). Letl be the line at infinity of . LetG be a solvable collineation group of in the linear translation complement, which acts transitively onl , and letH be a maximal normal cyclic subgroup ofG. Then the restriction ofH onl acts semiregularly onl and {1, 2, 3, 6}, where is the restriction ofG onl (ifq –1(mod 3), then {1, 2}). Ifq {3, 5} and {1, 2}, then is determined completely, using a computer.