Constitutive equations involving functional dependence of one vector on another

Zusammenfassung Die vorliegende Arbeit befasst sich mit einer Klasse von Stoffgleichungen, die so beschaffen ist, dass der momentane Wert eines Vektors vom gleichzeitigen und von allen vorhergehenden Werten eines zweiten Vektors abhängt. Es wird gezeigt, wie man diejenigen Einschränkungen der Form solcher Stoffgleichungen, die von materiellen Symmetrieeigenschaften stammen, herleiten kann. Die Herleitung wird völlig durchgeführt für den Sonderfall eines isotropen Materials mit oder ohne einen Symmetriemittelpunkt.

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The work reported in this paper was carried out under a grant from the National Science Foundation.     

On the approximation of one Markov chain by another

Motivated by applications in Markov chain Monte Carlo, we discuss what it means for one Markov chain to be an approximation to another. Specifically included in that discussion are situations in which a Markov chain with continuous state space is approximated by one with finite state space. A simple sufficient condition for close approximation is derived, which indicates the existence of three distinct approximation regimes. Counterexamples are presented to show that these regimes are real and not artifacts of the proof technique. An application to the ``ball walk' of Lovász and Simonovits is provided as an illustrative example. Partially supported by EPSRC grant ``Sharper Analysis of Randomized Algorithms: a Computational Approach' and the IST Programme of the EU under contract IST-1999-14036 (RAND-APX). The work described here was partially carried out while the author was visiting the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK. Postal address: School of Informatics, University of Edinburgh, The King's Buildings, Edinburgh EH9 3JZ, United Kingdom.     

On the number of copies of one hypergraph in another

Given two hypergraphsH andG, letN(G, H) denote the number of subhypergraphs ofG isomorphic toH. LetN(l, H) denote the maximum ofN(G, H), taken over allG with exactlyl edges. In [1] Noga Alon analyzes the asymptotic behaviour ofN(l, H) forH a graph. We generalize this to hypergraphs: Theorem:For a hypergraph H with fractional cover number ρ*,N(G,H).=θ(lρ*) The interesting part of this is the upper bound, which is shown to be a simple consequence of an entropy lemma of J. Shearer. In a special case which includes graphs, we also provide a different proof using a hypercontractive estimate. Research supported in part by NSF.     

On the optimal coding of one class of vector functions

We consider the problem of coding of parametrically defined vector functions by continuous functionals which act on their coordinate functions. We obtain the optimal method for coding some class of vector functions. In the linear case, we show the relation between this method and the informativeness of functionals with respect to the class of coordinate functions. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1638–1645, December, 1997.     

On the product of rotations

Using the elementary tools of matrix theory, we show that the product of two rotations in the three-dimensional Euclidean space is a rotation again. For this purpose, three types of rotation matrices are identified which are of simple structure. One of them is the identity matrix, and each of the other two types can be uniquely characterized by exactly one vector. The resulting products are investigated by using the basic properties of the vector cross product.     

On conditions of existence of permanent rotations of connected rigid bodies system about vertical vector

In contrast to the classical problem of motion of a heavy rigid body about a fixed point where the permanent rotations are well known and completely investigated [7, 3] as the most simple and good visually demonstrated type of motions, in multibody mechanics under an increasing of quantity of the system bodies, mechanical parameters and the order of differential motion equations the study of such motions is more complicated problem. The problem on permanent rotations of two connected rigid bodies under influence of gravity force was investigated in [2, 4]. In this paper a system consisting of arbitrary constant quantity, n, of heavy rigid bodies which are sequentially jointed in a chain is considered. The conditions of existence of motions when each body permanently rotates about the vertical vector are determined. These conditions are analyzed in a general case when the bodies angular velocities are different.     

On the adjunction mapping of very ample vector bundles of corank one

Let be a very ample vector bundle of rank over a smooth complex projective variety of dimension . The structure of being known when , we investigate the structure of the adjunction mapping when .

     

On vector fields with simply connected trajectories and one invariant line

We study polynomial vector fields X on ${\mathbb{C}}^2$ which have simply connected trajectories and satisfy $dP(X)=a?P$, for a constant $a\in {\mathbb{C}}^{?}$ and a primitive polynomial $P\in \mathbb{C}[x,y]$. We determine X, up to an algebraic change of coordinates. In particular, we obtain that X is complete.     

On the CLT for rotations and BV functions

Let $x?x+\alpha$ be a rotation on the circle and let φ be a step function. Denote by ${\phi }_n(x)$ the ergodic sums ${\sum }_{j=0}^{n?1}\phi (x+j\alpha )$. For α in a class containing the rotations with bounded partial quotients and under a Diophantine condition on the discontinuities of φ, we show that ${\phi }_n/{6{\phi }_{n}6}_2$ is asymptotically Gaussian for n in a set of density 1.     

On Yau's problem of evolving one curve to another: Convex case

A revised Yau's Curvature Difference Flow is considered to deform one convex curve $X_0$ to another one $\stackrel{?}{X}$. It is proved that this flow exists globally on time interval $[0,+\infty )$ and the evolving curve, preserving its convexity and bounded area A, converges to a fixed limiting curve $X_{\infty }$ (congruent to $\sqrt{A/\stackrel{?}{A}}\stackrel{?}{X}$) as time tends to infinity, where $\stackrel{?}{A}$ is the area bounded by the target curve $\stackrel{?}{X}$.     

On invariant probabilities for random rotations

It is shown that a spacially dependent random rotation process on the circle always possesses an invariant probability equivalent to Lebesgue measure. To the memory of Shlomo Horowitz     

On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives

We obtain new sharp Kolmogorov-type inequalities, in particular the following sharp inequality for 2π-periodic functions x ∈ L _∞ ^r (T):

Rank one quotients of vector groups

ABSTRACT

In this paper we study the possible torsion in even-dimensional higher class groups Cl _2n (Λ)(n ≥ 1) of an order Λ in a semisimple algebra A over a number field F with a ring of integers 𝒪 _F . We show that for certain orders, called generalized Eichler orders, bip-torsion in Cl _2n (Λ) can only occur for primes p dividing prime ideals ? of 𝒪 _F , at which Λ is not maximal. In particular, the results apply to Eichler orders in quaternion algebras and to hereditary orders.     

Three-manifolds of constant vector curvature one

A Riemannian manifold has $\mathrm{CVC}(?)$ if its sectional curvatures satisfy $\sec \le \epsilon$ or $\sec \ge \epsilon$ pointwise, and if every tangent vector lies in a tangent plane of curvature ε. We present a construction of an infinite-dimensional family of compact $\mathrm{CVC}(1)$ three-manifolds.     

On the distribution of partial sums of irrational rotations

Periodica Mathematica Hungarica - For an irrational $$\alpha $$ , we investigate the sums $$\sum _{i=1}^n \left( \{i \alpha \} - \frac{1}{2} \right) $$ and $$\sum _{i=1}^n \left\{ \left( \{i \alpha...