On totally geodesic affine immersions

In this paper, totally geodesic affine immersionsf: (M, ) are studied in the case when is an affine manifold of recurrent curvature. It is proved that(M, ) if flat or of recurrent curvature. And iff is additionally umbilical with the shape tensorA 0 and dimM 3, then(M, ) is locally projectively flat. Examples of such immersions are also stated.     

Extrapolation of transformations of random processes perturbed by white noise

We consider the problem of linear mean square optimal estimation of transformation of a stationary random process (t) in observations of process (t) + n(t) for t < – 0, where (t) is white noise uncorrelated with (t). We find least favorable spectral densities f₀() D and minimax (robust) spectral characteristics of an optimal estimator of transformation A for various classesD of densities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 216–223, February, 1991.     

The Gaussian Isoperimetric Inequality and Transportation

Any probability measure on ^d which satisfies the Gaussian or exponential isoperimetric inequality fulfils a transportation inequality for a suitable cost function. Suppose that W (x) dx satisfies the Gaussian isoperimetric inequality: then a probability density function f with respect to W (x) dx has finite entropy, provided that t . This strengthens the quadratic logarithmic Sobolev inequality of Gross (Amr. J. Math 97 (1975) 1061). Let (dx) = e ^–(x) dx be a probability measure on ^d, where is uniformly convex. Talagrand's technique extends to monotone rearrangements in several dimensions (Talagrand, Geometric Funct. Anal. 6 (1996) 587), yielding a direct proof that satisfies a quadratic transportation inequality. The class of probability measures that satisfy a quadratic transportation inequality is stable under multiplication by logarithmically bounded Lipschitz densities.     

On finite element approximation of the gradient for solution of Poisson equation

Summary A nonconforming mixed finite element method is presented for approximation of w with w=f,w| _r =0. Convergence of the order is proved, when linear finite elements are used. Only the standard regularity assumption on triangulations is needed.     

Elementary Abelian Covers of Graphs

Let _G (X) be the set of all (equivalence classes of) regular covering projections of a given connected graph X along which a given group G Aut X of automorphisms lifts. There is a natural lattice structure on _G (X), where _{1 2} whenever ₂ factors through ₁. The sublattice _G () of coverings which are below a given covering : X^~ X naturally corresponds to a lattice _G () of certain subgroups of the group of covering transformations. In order to study this correspondence, some general theorems regarding morphisms and decomposition of regular covering projections are proved. All theorems are stated and proved combinatorially in terms of voltage assignments, in order to facilitate computation in concrete applications.For a given prime p, let _G ^p (X) _G (X) denote the sublattice of all regular covering projections with an elementary abelian p-group of covering transformations. There is an algorithm which explicitly constructs _G ^p (X) in the sense that, for each member of _G ^p (X), a concrete voltage assignment on X which determines this covering up to equivalence, is generated. The algorithm uses the well known algebraic tools for finding invariant subspaces of a given linear representation of a group. To illustrate the method two nontrival examples are included.     

Asymptotic properties of local densities of measures generated by semimartingales

For families of probability measures (P , )) generated by semimartingales, we consider the local density)(y, )= _t (y, )) _t0 of a, measureP _y with respect to the measureP whose logarithm is the difference of a local martingale and a positive predictable increasing locally bounded process. Conditions are obtained under which the relations and hold, wherey _t depends in some way ont, while _t ast . Applications of these relations are exhibited and an example is given when the hypotheses of the theorems proved can be verified.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 48–55, 1986.     

über Wurzeln und Logarithmen beschrÄnkter Ma\e

Ohne ZusammenfassungBezeichnungen und Symbole G lokalkompakte topologische Gruppe - M(G)/R(G)/P(G)/ regulÄre komplexe/reelle/positive Ma\e/ - Q(G)/W(G) Ma\e mit · l/Wahrscheinlichkeitsma\e - _x Punktma\: _x(f)=f(x) - v Faltung, — Bekanntlich bildet M(G) bezüglich der Faltung eine Banachalgebra; - Involution in M(G), , wobei — die Komplexkonjugierte bezeichnet - × diskreter Anteil eines Ma\es, - T _gm Faltungsoperator auf L ² (G) (bezüglich des linken Haarschen Ma\es), f.ü. - p(·)/q(·)/u(·) - exp(·.) Exponentialfunktion, exp - normal/unitÄr/symmetrisch/positiv definit bezeichnet man ein Ma\ , wenn der Faltungsoperator T diese Eigenschaft besitzt - invertierbar hei\t M(G), wenn ein vM(G) existiert, so da\ v = v= _e - _1/n n-te Wurzel von ₁ hei\t wenn( _1/n)ⁿ= ₁ - ₁ hei\t unendlich teilbar wenn zu jedem natürlichen n eine n-te Wurzel _1/n von existiert - N Menge der natürlichen Zahlen     

Aperiodische Wahrscheinlichkeitsmaße auf topologischen Gruppen

Several conditions are shown to be equivalent to the aperiodicity of a regular probability measure on a locally compact, separated topological GroupG. In particular, is aperiodic if and only if the sequence ( ( ⁽ⁿ⁾ denoting then-th convolution power of ) is convergent for any nonvoid open subsetU ofG with compact closure. It is always assumed that the support of generatesG as a closed semigroup.     

Flat Affine Surfaces in \mathbb{R}^4 with Flat Normal Connection

In this paper we study nondegenerate affine surfaces in the 4-dimensional affine space . We assume that both the connection and the normal connection induced by the canonical equiaffine transversal bundle are flat. Surfaces with constant equiaffine transversal bundle are trivial examples of such surfaces. Here, we obtain a complete classification of all such surfaces which do not have constant equiaffine normal bundle.     

On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives

For 2-periodic functions and arbitrary q [1, ] and p (0, ], we obtain the new exact Kolmogorov-type inequality which takes into account the number of changes in the sign of the derivatives (x ^(k)) over the period. Here, = (r – k + 1/q)/(r + 1/p), _r is the Euler perfect spline of degree r, and . The inequality indicated turns into the equality for functions of the form x(t) = a _r (nt + b), a, b R, n N. We also obtain an analog of this inequality in the case where k = 0 and q = and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines.     

Signed Total Domination Nnumber of a Graph

The signed total domination number of a graph is a certain variant of the domination number. If is a vertex of a graph G, then N() is its oper neighbourhood, i.e. the set of all vertices adjacent to in G. A mapping f: V(G)-1, 1, where V(G) is the vertex set of G, is called a signed total dominating function (STDF) on G, if for each V(G). The minimum of values , taken over all STDF's of G, is called the signed total domination number of G and denoted by _st(G). A theorem stating lower bounds for _st(G) is stated for the case of regular graphs. The values of this number are found for complete graphs, circuits, complete bipartite graphs and graphs on n-side prisms. At the end it is proved that _st(G) is not bounded from below in general.     

G-closure of some particular sets of admissible material characteristics for the problem of bending of thin elastic plates

In Ref. 1, we considered theG-closure of some initially given arbitrary setU of the positive-definite, symmetrical plane tensorsD of the 2nd rank, connected with the differential operator ·D · in two dimensions. Here, theG-closure procedure is applied to the 4th-order operator ··D ·· in a plane, arising in the theory of plates and containing self-adjoint tensorsD of the 4th rank. The paper generalizes some results obtained earlier in Refs. 2 and 3. The complete solution of the general problem of regularization, which presupposes the arbitrary character of the initially given setU, is not yet obtained.     

On the Binomial Asymptotics of an Entire Dirichlet Series

Let M() be the maximum modulus and let () be the maximum term of an entire Dirichlet series with nonnegative exponents _n increasing to . We establish a condition for _n under which the relations

Homomorphisms of Edge-Colored Graphs and Coxeter Groups

Let be two edge-colored graphs (without multiple edges or loops). A homomorphism is a mapping : for which, for every pair of adjacent vertices u and v of G ₁, (u) and (v) are adjacent in G ₂ and the color of the edge (u)(v) is the same as that of the edge uv.We prove a number of results asserting the existence of a graphG , edge-colored from a set C, into which every member from a given class of graphs, also edge-colored from C, maps homomorphically.We apply one of these results to prove that every three-dimensional hyperbolic reflection group, having rotations of orders from the setM ={m₁, m₂,..., m_k}, has a torsion-free subgroup of index not exceeding some bound, which depends only on the setM .     

T-Periodic solutions of time dependent Hamiltonian systems with a potential vanishing at infinity

We are discussing existence and multiplicity of T-periodic solutions of the time dependent Hamiltonian system = U(x, t) under the assumption that U(x, t) approaches 0 for large |x| meaning that the force U is concentrated in a finite region. Our method is variational and it is shown how the difficulties with the Palais-Smale condition can be overcome in this case.     

Polynomial Filtrations and Lannes'' T Functor

This paper defines and studies the polynomial filtration [p_k ] of the shift functor : F , where F is the category of functors between F-vector spaces over a finite field F. The functors [p_k ] correspond to a system of functors (p_k T):U U, related to Lannes'T-functor on the category U of unstable modules over the Steenrod algebra. The main results concern the behaviour of the quotients ~ _s:=~/[p_s-1~ filtrations by ~_s-nilpotent functors are introduced and it is shown that the full subcategory of _s-nilpotent functors is thick.     

Riemann's Fragment on Limit Values of Elliptic Modular Functions

We give an interpretation different from that of Dedekind to work n° 28 of the complete works of Riemann Fragmente über die Grenzfälle der Elliptischen Modulfunctionen.We prove a theorem of inversion of radial limit and sum in a series of functions. This allows us to justify all of Riemann's reasoning in the fragment to obtain the limit values of modular elliptic functions. In particular we prove the statement of Riemann that for every rational number x we have where denotes the periodic function with period 1, such that (x) = x when |x| < 1/2, and (n + ) = 0 for every n Z.This assertion of Riemann was criticized by Dedekind. We also give the transformation formulae of the logarithms of the classical theta-function ₃(0), giving an alternative form to that obtained by B. C. Berndt [1].     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

Théorie du potentiel pour des opérateurs elliptiques non linéaires du second ordre à coefficients discontinus

Let a open subset of ⁿ , n3, and an open. Existence and unicity are proved for the Dirichlet problem