POWERS OF REAL SYMMETRIC DIFFERENTIAL EXPRESSIONS WITHOUT SMOOTHNESS ASSUMPTIONS

Abstract

In recent years many authors have studied properties of powers of symmetric (formally self-adjoint) ordinary linear differential expressions. So that these powers can be formed in the classical way these authors have placed heavy smoothness assumptions on the coefficients. Here we show that no differentiability conditions whatsoever are needed on the coefficients in order to form powers of a given expression-provided these powers are formed in the quasi-differential sense rather than the classical one.     

A note on -operations in orthogonal K-theory

In Comment. Math. Helv. 55 (1980), 233-254, Kratzer defined Lambda operations on classical algebraic K-theory by using exterior powers of representations and a splitting principle (R. G. Swan, Proc. Sympos. in Pure Math. 21 (1971), 155-159). Because hyperbolic forms are not stable under exterior powers, we instead use a larger class of symmetric bilinear forms to define the operation of exterior powers on the classifying space of the orthogonal K-theory.

     

The Module Decomposition of Super-Symmetric Powers of Matrices

ABSTRACT

Let M be the k  ×  m matrices over ?. The GL ( k ) ×  GL ( m ) decompositions of the symmetric and of the exterior powers of M are described by two classical theorems. We describe a theorem for Lie superalgebras, which implies both of these classical theorems as special cases. The constructions of both the exterior and the symmetric algebras are generalized to a class of algebras defined by partitions. That superalgebra theorem is further generalized to these algebras.     

On the dimension of the algebra generated by a boolean matrix

We investigate the subspace of the space of all n × n Boolean (0,1)-matrices, spanned by the powers of an arbitrary matrix. We estimate the maximum dimension of such spaces as a function of n and show that their bases consists of consecutive integer powers of the matrix, starting at I. We also determine the maximum dimension of the space spanned by the powers of as symmetric matrix and characterise the matrices achieving that maximum.     

Centralizers of Rational Functions

We show that the elements of an open and dense subset of rational functions of the Riemann sphere have trivial centralizers; i.e, the rational functions commute only with their own powers.     

Extension and Application of Itô's Formula Under G-Framework

In continuing his study of the intrinsically nonlinear expectation and conditional expectation under the so-called G-framework, Peng introduced a nonlinear Itô calculus; here, the G refers to the generator of a nonlinear heat equation. There, he derived the corresponding Itô formula for C ²-functions with bounded Lipschiz derivatives. This restrictive class of functions limits its applicatory value to stochastic finances and cannot be applied to study the powers of the G-Brownian motion. We extend the Itô formula to a slightly more general class of functions (C ²-functions with uniformly continuous derivatives). This enables us to compute the G-expectations of the even powers of the G-Brownian motion. The G-expectation of odd powers behave differently; in particular, we show that the G-expectation of the cube of the G-Brownian motion is positive, which is qualitatively different from the classical Brownian motion case. We remark that we are not able to get a formula for the G-expectation of the general odd powers of the G-Brownian motion.     

Jacobi Elliptic Cliffordian Functions

The well-known Jacobi elliptic functions sn(z), cn(z), dn(z) are defined in higher dimensional spaces by the following method. Consider the Clifford algebra of the antieuclidean vector space of dimension 2m + 1. Let x be the identity mapping on the space of scalars + vectors. The holomorphic Cliffordian functions may be viewed roughly as generated by the powers of x, namely xⁿ , their derivatives, their sums, their limits (cf: zⁿ for classical holomorphic functions). In that context it is possible to define the same type of functions as Jacobi's.     

A Quantitative Central Limit Theorem for Linear Statistics of Random Matrix Eigenvalues

It is known that the fluctuations of suitable linear statistics of Haar distributed elements of the compact classical groups satisfy a central limit theorem. We show that if the corresponding test functions are sufficiently smooth, a rate of convergence of order almost 1/n can be obtained using a quantitative multivariate CLT for traces of powers that was recently proven using Stein’s method of exchangeable pairs.     

Infinite divisibility of sub-stable processes. II. Logarithm of probability measure

We show that in some cases it is possible to get the Lévy measure v for an infinitely divisible distribution μ as a limit of some signed measuresL _N(μ) based on the convolution powers of μ. In the paper we give a collection of results in this area and apply the method to the investigation of the infinite divisibility of sub-stable random vectors onR ⁿ. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russian, 1995, Part I.     

Zero cycles on homogeneous varieties

In this paper we study the group A₀(X) of zero-dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety X. To do this we translate rational equivalence of 0-cycles on a projective variety into R-equivalence on symmetric powers of the variety. For certain homogeneous varieties, we then relate these symmetric powers to moduli spaces of étale subalgebras of central simple algebras which we construct. This allows us to show A₀(X)=0 for certain classes of homogeneous varieties for groups of each of the classical types, extending previous results of Swan/Karpenko, of Merkurjev, and of Panin.     

The Waring Locus of Binary Forms

Abstract

For integers m, d, n, we study the locus of m-dimensional subspaces of degree d binary forms whose elements admit simultaneous decompositions as sums of powers of n linear forms. We show that this locus has rational singularities and it is arithmetically Cohen–Macaulay in its natural Plücker imbedding. We describe invariant theoretic formulations for the equations of these loci.     

The distribution of the characteristic roots ofS 1 S 2 −1 under violations in the complex case and power comparisons of four tests

Summary The joint density function of the latent roots ofS ₁ S ₂ ⁻¹ under violations is obtained whereS ₁ has a complex non-central Wishart distributionW _c (p,n ₁,Σ ₁,Ω) andS ₂, an independent complex central Wishart,W _c (p,n ₂,Σ ₂, 0). The density and moments of Hotelling's trace are also derived under violations. Further, the non-null distributions of the following four criteria in the two-roots case are studied for tests of three hypotheses: Hotelling's trace, Pillai's trace, Wilks' criterion and Roy's largest root. In addition, tabulations of powers are carried out and power comparisons for tests of each of three hypotheses based on the four criteria are made in the complex case extending such work of Pillai and Jayachandran in the classical Gaussian case. The findings in the complex Gaussian are generally similar to those in the classical.     

Global classical solutions of coagulation-fragmentation equations with unbounded coagulation rates

We study discrete fragmentation coagulation equations in spaces X_p, p>1, consisting of distributions having the pth moments finite. We show that for sufficiently regular fragmentation laws the fragmentation semigroup is analytic in X_p, and fully characterize the domain of its generator. This allows for explicit characterization of the domains of the fractional powers of the generator through real interpolation. Finally, we use the linear results to show the existence of global classical solutions to fragmentation coagulation equations for a class of unbounded coagulation kernels.     

CARTESIAN CLOSEDNESS,EXPONENTIALITY, AND FINAL HULLS IN PSEUDOTOPOLOGICAL SPACES

Abstract

We show that each non-trivial epireflective subcategory of the topological or pretopological spaces fails to be cartesian closed. Motivated by this “negative” result, we consider the supercategory of pseudotopological spaces and obtain: An epireflective subcategory of the pseudotopological spaces which contains a finite non-indiscrete space is cartesian closed iff it is closed with respect to powers in the pseudotopological spaces. Here the density property that every pseudotopological space is a final epi-sink of free ultraspaces is essential.     

A constructive approach to the finite congruence lattice representation problem

A finite lattice is representable if it is isomorphic to the congruence lattice of a finite algebra. In this paper, we develop methods by which we can construct new representable lattices from known ones. The techniques we employ are sufficient to show that every finite lattice which contains no three element antichains is representable. We then show that if an order polynomially complete lattice is representable then so is every one of its diagonal subdirect powers. Received August 30, 1999; accepted in final form November 29, 1999.     

A note on the Σ1 spectrum of a theory

Let T be a suitable system of classical set theory. We will show, that the Σ₁ spectrum of T, i.e. the set of ordinals having good Σ₁ definition in T is an initial segment of the ordinals. Received: 11 October 1999 / Published online: 12 December 2001     

Non-classical polar unitals in finite Figueroa planes

The finite Figueroa planes are non-Desarguesian projective planes of order q ³ for all prime powers q > 2. These planes were constructed algebraically in 1982 by Figueroa, and Hering and Schaeffer, and synthetically in 1986 by Grundh?fer. All Figueroa planes of finite square order are shown to possess a unitary polarity by de Resmini and Hamilton in 1998, and hence admit unitals. Using the result of O??Nan in 1971 on the non-existence of his configuration in a classical unital, and the intrinsic characterization by Taylor in 1974 of the notion of perpendicularity induced by a unitary polarity in the classical plane (introduced by Dembowski and Hughes in 1965), we show that these Figueroa polar unitals do not satisfy a necessary condition, introduced by Wilbrink in 1983, for a unitary block design to be classical, and hence they are not classical.     

Schur algebras of Brauer algebras,II

A classical problem of invariant theory and of Lie theory is to determine endomorphism rings of representations of classical groups, for instance of tensor powers of the natural module (Schur–Weyl duality) or of full direct sums of tensor products of exterior powers (Ringel duality). In this article, the endomorphism rings of full direct sums of tensor products of symmetric powers over symplectic and orthogonal groups are determined. These are shown to be isomorphic to Schur algebras of Brauer algebras as defined in Henke and Koenig (Math Z 272(3–4):729–759, 2012). This implies structural properties of the endomorphism rings, such as double centraliser properties, quasi-hereditary, and a universal property, as well as a classification of simple modules.     

PRODUCTS OF DIFFERENTIAL EXPRESSIONS WITHOUT SMOOTHNESS ASSUMPTIONS

Abstract

To form products of differential expressions in the classical way it is necessary to place heavy differentiability assumptions on the coefficients. Here we consider symmetric (formally self-adjoint) expressions defined, not in the classical way, but in terms of quasi-derivatives. With this very general notion of symmetry we show that products such as M1M2MI of symmetric expressions M1, Hp can be formed vithout any smoothness assumptions on the coefficients and such products are symmetric expressions.     

Generalized Bernstein–Durrmeyer Operators and the Associated Limit Semigroup

The aim of this paper is the study of a new sequence of positive linear approximation operators M_n, λ on C([0, 1]) which generalize the classical Bernstein–Durrmeyer operators. After proving a Voronovskaja-type result, we show that there exists a strongly continuous positive contraction semigroup on C([0, 1]) which may be expressed in terms of powers of these operators. As a direct consequence, we are able to represent explicitly the solutions of the Cauchy problems associated with a particular class of second order differential operators.