Division algebra counterexamples of degree 8

An example is given of a non-crossed product of degree 8 and exponent 4. On the other hand, every division algebra of degree 8 (arbitrary exponent) has a solvable splitting field; other positive results are also given. The research of the author is supported by the Anshel Pfeffer Chair.     

Ergodic theorem in variable Lebesgue spaces

We prove an Ergodic Theorem in variable exponent Lebesgue spaces, whenever the exponent is invariant under the transformation. Moreover, a counterexample is provided which shows that the norm convergence fails to hold for an arbitrary exponent.     

Temperature distribution in a laminar plane wall jet due to variably heated wall

The heat transfer in a laminar incompressible plane wall jet due to a variably heated wall has been studied. It is assumed that the difference of temperatures between the wall and the issuing jet is inversely proportional to an arbitrary exponent of the distance from the slit. A similar solution of the energy equation is possible. The solutions, for arbitrary values of the Prandtl number and of the exponent are obtained. It is found that in some cases the heat transfer at the wall may become zero or negative.     

Effective formulas for the ?ojasiewicz exponent at infinity

We give effective formulas for the ?ojasiewicz exponent at infinity of an arbitrary complex polynomial mapping.     

Existence and Uniqueness of Solutions of Differential Equations Weakly Controlled by Rough Paths with an Arbitrary Positive Hölder Exponent

Differential Equations - A functional version of the theory of rough paths with an arbitrary Hölder exponent is developed. It is used to prove a theorem on the existence and uniqueness of a...     

Computation of uniform integral means and some similar characteristics of piecewise continuous functions

We obtain formulas for the computation of a certain asymptotic characteristic of piecewise continuous functions defined on the half-line and use it to state the well-known necessary and sufficient condition on a linear homogeneous differential system for the corresponding inhomogeneous system with arbitrary inhomogeneity whose characteristic exponent is nonpositive to have a solution with nonpositive characteristic exponent. We give a new form of this necessary and sufficient condition similar to the Perron-Maizel condition for the exponential dichotomy of the system.     

Stability of Solutions of Stochastic Differential Equations Weakly Controlled by Rough Paths with Arbitrary Positive Hölder Exponent

Differential Equations - We prove theorems on the stability of solutions of one-dimensional stochastic differential equations controlled by rough paths with arbitrary positive Hölder exponent.     

Hybrid feedback stabilization of quasilinear systems in the plane

We apply Artstein's hybrid feedback algorithm to stabilize quasilinear dynamical systems with complex multipliers in the plane. We study only the case of incomplete observation when ordinary feedback controls do not work. The main results of the paper state that Artstein's procedure provides an arbitrary rate of asymptotic convergence/divergence of solutions. In other words, we prove the complete controllability from below of the upper Lyapunov exponent and the uniform upper Lyapunov exponent for the quasilinear systems in question.     

Decomposability for division algebras of exponent two and associated forms

This article investigates the structure of quadratic forms and of division algebras of exponent two over fields of characteristic different from two with the property that the third power of the fundamental ideal in the associated Witt ring is torsion free.      

Zur Struktur der vierdimensionalen quadratischen Algebren

Assuming properties, which are essential for division algebras, but mostly invariant to extensions of the ground field, we investigate the structure of quadratic division algebras of dimension four over an arbitrary field of characteristic not two. We relate the size of the group of automorphisms of such an algebra A to algebraic laws valid in A, characterize Lie-admissibility by means of the skew-commutative vector algebra of A and outline the possibilities of describing A by irreducible identities of degree 3. Some results of the last chapter apply to arbitrary dimensions. We show, that a simple quadratic algebra with the right (left) inverse property for invertible elements is a composition algebra. Finally we conclude, that a quadratic division algebra of dimension four with a right (left) nucleus different from the center is associative.     

Harmonic Calculus on Fractals - A Measure Geometric Approach I

Differentiation of functions w.r.t. finite atomless measures with compact support on the real line is introduced. The related harmonic calculus is similar to that of the classical Lebesgue case. As an application we obtain the Weyl exponent for the spectral asymptotics of the Laplacians w.r.t. linear Cantor-type measures with arbitrary weights.     

The exponent of discrepancy of sparse grids is at least 2.1933

We study bounds on the exponents of sparse grids for L ₂‐discrepancy and average case d‐dimensional integration with respect to the Wiener sheet measure. Our main result is that the minimal exponent of sparse grids for these problems is bounded from below by 2.1933. This shows that sparse grids provide a rather poor exponent since, due to Wasilkowski and Woźniakowski [16], the minimal exponent of L ₂‐discrepancy of arbitrary point sets is at most 1.4778. The proof of the latter, however, is non‐constructive. The best known constructive upper bound is still obtained by a particular sparse grid and equal to 2.4526.... This revised version was published online in June 2006 with corrections to the Cover Date.     

On solutions of anisotropic elliptic equations with variable exponent and measure data

ABSTRACT

The Dirichlet problem in arbitrary domains for a wide class of anisotropic elliptic equations of the second order with variable exponent nonlinearities and the right-hand side as a measure is considered. The existence of an entropy solution in anisotropic Sobolev spaces with variable exponents is established. It is proved that the obtained entropy solution is a renormalized solution of the considered problem.     

Łojasiewicz exponents and resolution of singularities

We show an effective method to compute the Łojasiewicz exponent of an arbitrary sheaf of ideals of O_X{\mathcal{O}_X} , where X is a non-singular scheme. This method is based on the algorithm of resolution of singularities.     

Identities in the varieties generated by the algebras of upper triangular matrices

Consider the algebra UT _s of upper triangular matrices of size s over an arbitrary field. Petrogradsky proved that the exponent of an arbitrary subvariety in var(UT _s ) exists and is an integer. We strengthen the estimates for the growth of these varieties and provide equivalent conditions for finding these exponents. Kemer showed that in the case of a ground field of characteristic zero there exists no varieties of associative algebras with growth intermediate between polynomial and exponential. We prove that this property extends to the case of the fields of arbitrary characteristic distinct from 2.     

STABLE AND UNSTABLE IDEAL PLANE FLOWS

IntroductionWe COnsider SOllltlollS     

Counting the order of the group of points of an elliptic curve over a finite field based on Shanks’s algorithm

Shanks’s baby-step-giant-step algorithm to count the order of the group of points of an elliptic curve over a finite prime field is described. The algorithm’s improvements are also discussed. The first one is based on Mestre’s theorem. The second improvement is based on the Sutherland’s probabilistic result allowing to calculate the exponent of an arbitrary finite abelian group.     

开平面C上的亚纯函数与完备极小曲面的Gauss映射

设■:D→R~3确定了以等温参数表示的极小曲面M,其中D是全平面R~2的开子区域,那么极小曲面的Gauss映射g(z)是D上的亚纯函数.Xavier与Chao提出了一个尚未解决的问题:任意给定区域■上的亚纯函数g(z),它是否是某完备极小曲面的Gauss映射?本文证明了若开平面C上的亚纯函数g(z)的零点列或极点列的收敛指数小于1/2,则g(z)—定是某完备极小曲面的Gauss映射.     

Separation of algebraic sets and the Łojasiewicz exponent of polynomial mappings

We prove a sharp regular separation condition for arbitrary projective algebraic sets. As a corollary an extension of Kollár's results on the Łojasiewicz exponent of polynomial mappings is obtained. Oblatum 9-II-1998 & 3-VI-1998 / Published online: 14 January 1999