Generalizations of Perfect, Semiperfect, and Semiregular Rings

For a ring R and a right R-module M, a submodule N of M is said to be -small in M if, whenever N + X = M with M/X singular, we have X = M. If there exists an epimorphism p: P M such that P is projective and Ker(p) is -small in P, then we say that P is a projective -cover of M. A ring R is called -perfect (resp., -semiperfect, -semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective -cover. The class of all -perfect (resp., -semiperfect, -semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of -perfect, -semiperfect, and -semiregular rings. We define (R) by (R)/Soc(R_R) = Jac(R/Soc(R_R)) and show, among others, the following results:

On a Particular Class of Minihypers and Its Applications. I. The Result for General q

This article is the first in a series of three articles that discuss a particular class of minihypers and its applications. Proving that for small and < N, a {v _{+ 1}, v ; N, q}-minihyper consists of a sum of -spaces, we show that the excess points of an s-cover with excess of PG(N, q), (s + 1)|(N + 1), form a sum of s-spaces, and that no maximal partial s-spreads with deficiency of PG(N, q), (s + 1)|(N + 1), exist. The case q square will be studied in greater detail in [7] and further applications of these classification results on this class of minihypers will be published in [8].     

Thermal stability of a reactive slab with time-periodic heating of one surface

Summary The stability of time-periodic thermal oscillations in a slab with chemical reaction is examined. One surface is at a fixed temperature and the temperature of the other varies with amplitude and frequency. The reaction, which is of zero-order, is characterised by the Frank-Kamenetskii parameter(,,), where(=RT ₀/E) is the activation energy parameter. Perturbation series in and are used to calculate the critical values _crit; for _crit the slab can support time-periodic temperature oscillations.

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Zusammenfassung Die Stabilität von zeitperiodischen thermischen Schwingungen in einem Stab mit chemischer Reaktion wird untersucht. Eine Oberfläche hat eine feste Temperatur und auf der anderen verändert sich die Temperatur mit der Amplitude und der Frequenz. Die Reaktion der Ordnung null, wird vom Frank-Kamenetskii Parameter(,,) bestimmt, wobei(=RT ₀/E) der Aktivierungs-Energie Parameter ist. Reihenentwicklungen in und werden für die Berechnung des kritischen Wertes _crit benützt; für _crit können im Stab zeitperiodische Temperaturschwingungen bestehen.

     

Amalgams of Weyl algebras and theA (V, δ, Γ) conjecture

Summary We prove a long standing conjecture ([5, Conjecture 5.6]) concerning the algebrasA (V, , ). Namely, two such algebrasA (V, , ),A (W, , ) are isomorphic if and only if there is an isomorphism between the triples (V, , ), (W, , ) from which they are constructed. As a consequence, to each primitive ideal in the enveloping algebra of a solvable Lie algebra there is associated a unique (V, , ).     

Weak solutions of the Stokes problem in weighted Sobolev spaces

For n2 we consider the Stokes problem in ⁿ, -u + p=f, -divu=g, in weighted Soboiev spaces H ₆ ^m,r , where the weights are proportional to (1+|x|). We prove the existence of weak solutions for any K, whereK is a discrete set of critical values. Furthermore, we characterize the solutions of the homogeneous problem.This research was supported by the DFG research group Equations of Hydrodynamics, Universities of Bayreuth and Paderborn.     

An inequality linking packing and covering densities of plane convex bodies

For every convex body K in R ², let (K) denote the packing density of K, i.e. the density of the tightest packing of congruent copies of K in R ², and let (K) denote the covering density of K, i.e. the density of the thinnest covering of R ² with congruent copies of K. It is shown here that 4(K)3(K) for every convex body K in R ². This inequality is the strongest possible, since if E is an ellipse, then the equality 4(E)=3(E) holds. Two corollaries are presented, and a summary of known bounds for packing and covering densities is given.     

On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics

We consider the approximation by piecewise-constant functions for classes of functions of many variables defined by moduli of continuity of the form (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ), where _i ( _i ) are ordinary moduli of continuity that depend on one variable. In the case where _i ( _i ) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric L ₂ for (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ); (ii) in the integral metric L _p (p 1) for (₁, ..., _n ) = c ₁₁ + ... + c _{n n} ; (iii) in the integral metric L _{(2, ..., 2, 2r)} (r = 2, 3, ...) for (₁, ..., _n ) = ₁(₁) + ... + _{n – 1}( _{n – 1}) + c _{n n} .     

Canonical contact forms on spherical CR manifolds

We construct the CR invariant canonical contact form can(J) on scalar positive spherical CR manifold (M,J), which is the CR analogue of canonical metric on locally conformally flat manifold constructed by Habermann and Jost. We also construct another canonical contact form on the Kleinian manifold ()/, where is a convex cocompact subgroup of Aut_CRS²ⁿ⁺¹=PU(n+1,1) and () is the discontinuity domain of . This contact form can be used to prove that ()/ is scalar positive (respectively, scalar negative, or scalar vanishing) if and only if the critical exponent ()<n (respectively, ()>n, or ()=n). This generalizes Nayatanis result for convex cocompact subgroups of SO(n+1,1). We also discuss the connected sum of spherical CR manifolds.     

Some remarks on factoriality of certain hypersurfaces in \mathbb{P}^4

Let V be a reduced and irreducible hypersurface of degree k 3. In this paper we prove that if the singular locus of V consists of ₂ ordinary double points, ₃ ordinary triple points and if ₂ + 4₃ < (k – 1)², then any smooth surface contained in V is a complete intersection on V.Received: 7 January 2004     

On upper bounds of Fourier-Walsh coefficients

An upper bound is established for the upper bounds of the Fourier-Walsh coefficients a_n(f) whose modulus of continuity (,f) does not exceed a given modulus of continuity (). In the case of convex majorants of (), these bounds are attained for individual ordinal numbers n.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 725–736, December, 1969.     

The Connes–Moscovici Approach to Cyclic Cohomology for Compact Quantum Groups

Consider the Hopf algebra (A, ) of regular functions on a compact quantum group. Let (A ^o ,) denote its maximal dual Hopf algebra. We show that the tensor product Hopf algebra (H ₂,₂) of (A ^o ,) and its opposite Hopf algebra is endowed with a modular pair (,) in involution; a notion introduced by A. Connes and J. Moscovici, who associate canonically a cocyclic object to such Hopf algebras. Denote the Hopf cyclic cohomology thus obtained by HC ^* _(,)(H ₂). Next we define an action of H ₂),₂ on A and show that the Haar state of (A, ) is a -invariant -trace on A with respect to this action. This gives us a canonical map from HC ^* _(,)(H ₂) to the ordinary cyclic cohomology of A.     

Problem of correctness of the best approximation in the space of continuous functions

Let W^rH _w be the subclass of those functions of C^r[a, b], for which (f ^(r),)(), where () is a given modulus of continuity, and P_n be the space of algebraic polynomials of degree at most n and _n(f) be the polynomial of best approximation for f(x) on [a, b]. Estimates for and moduli of continuity of the operators of best approximation on W^rH _w are established. For example, if ()=, then Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 351–360, March, 1978.The author thanks S. B. Stechkin for the formulation of the problem and assistance with the article.     

Finite-dimensional attainable sets for stochastic control systems

Letx _t ^u () be a stochastic control system on the probability space (, ,P) intoR ⁿ. We say that the pointxR ⁿ is (, ) attainable at timet if there exists an admissible controlu such thatP _xo{x _t ^u ()S (x)}, wherex ₀()=x ₀, 0, 10, andS (x) is the closed Euclidean -ball inR ⁿ centered atx. We define the attainable setA (t) to be the set of all pointsxR ⁿ which are (, ) attainable at timet. For a large class of stochastic control systems, it is shown thatA (t) is compact for eacht and continuous as a function oft in an appropriate metric. From this, the existence of stochastic time-optional controls is established for a large class of nonlinear stochastic differential equations.This research was supported by the National Research Council of Canada, Grant No. A-9072.     

Gaussian approximations of Brownian motion in a stochastic integral

Given a Brownian motionB, Gaussian approximationsB , >0, of the form B _t = ₀ ^t K (u, s)dB _s du, t O , including polygonal and mollifier approximations, are considered. A limit theorem is proved for the integrals _T ⁰ X _t dB ^t as O. In particular, in the case of symmetric kernelsK the limit is the Fisk-Stratonovich integral ₀ ^t X _t odB _t.Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Vilnius Pedagogical University, Studentu 39, 2034 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 33, No. 4, pp. 508–526, October–December, 1993. Translated by V. Mackeviius     

Uniformities for the convergence in law and in probability

The convergence (X _n, Y_n)0 is investigated and characterized for probability metrics which metrize convergence in distribution or in probability. Some related metrics are also considered.     

Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates

Let H be a Hilbert space and K be a nonempty closed convex subset of H. For f H, we consider the (ill-posed) problem of finding u K for which 0 for all v K, where A : H H is a monotone (not necessarily linear) operator. We study the approximation of the solutions of the variational inequality by using the following perturbed variational inequality: for f H, f – f , find u, K for which 0 for all v K, where , , and are positive parameters, and K, a perturbation of the set K, is a nonempty closed convex set in H. We establish convergence and a rate O(1 / 3) of convergence of the solutions of the regularized variational inequalities to a solution of the original variational inequality using the Mosco approximation of closed convex sets, where A is a weakly differentiable inverse-strongly-monotone operator.     

Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes

Let (X_t⁽⁾,t0) be the BESQ process starting at x. We are interested in large deviations as for the family {^–1X_t⁽⁾,tT}, – or, more generally, for the family of squared radial OU process. The main properties of this family allow us to develop three different approaches: an exponential martingale method, a Cramér–type theorem, thanks to a remarkable additivity property, and a Wentzell–Freidlin method, with the help of McKean results on the controlled equation. We also derive large deviations for Bessel bridges.Mathematics Subject Classification (2000): 60F10, 60J60     

Pyramids in the complex projective plane

We study the critical points of the diameter functional on the n-fold Cartesian product of the complex projective plane C P ² with the Fubini-Study metric. Such critical points arise in the calculation of a metric invariant called the filling radius, and are akin to the critical points of the distance function. We study a special family of such critical points, P _kC P ¹C P ², k=1,2... We show that P _k is a local minimum of by verifying the positivity of the Hessian of (a smooth approximation to) at P _k. For this purpose, we use Shirokov's law of cosines and the holonomy of the normal bundle of C P ¹C P ². We also exhibit a critical point of , given by a subset which is not contained in any totally geodesic submanifold of C P ².     

Superconvergence for galerkin methods for the two point boundary problem via local projections

The two point boundary problemy'-a(x)y–b(x)y=-f(x), o<x<1,y(0)=y(1)=0, is first solved approximately by the standard Galerkin method, (Y, ) + (aY+bY, )=(f, ), ₁ ⁰ (r, ), for a function Y ₁ ⁰ (r, ), the space ofC ¹-piecewise--degree-polynomials vanishing atx=0 andx=1 and having knots at {x ₀ ,x ₁ , ...,x _M }=. ThenY is projected locally into a polynomial of higher degree by means of one of several projections. It is then shown that higher-order convergence results locally, provided thaty is locally smooth and is quasi-uniform.This research was supported in part by the National Science Foundation.     

Adiabatic limits of eta and zeta functions of elliptic operators

We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator , constructed from an elliptic family of operators indexed by S ¹ . We show that the regularized values ( _t ,0) and t( _t ,0) are smooth functions of t at t=0, and we identify their values at t=0 with the holonomy of the determinant bundle, respectively with a residue trace. For invertible families of operators, the functions ( _t ,s) and t( _t ,s) are shown to extend smoothly to t=0 for all values of s. After normalizing with a Gamma factor, the zeta function satisfies in the adiabatic limit an identity reminiscent of the Riemann zeta function, while the eta function converges to the volume of the Bismut-Freed meromorphic family of connection 1-forms. Mathamatics Subject Classification (2000): 58J28, 58J52Partially supported by ANSTI (Romania), the European Commission RTN HPRN-CT-1999-00118 Geometric Analysis and by the IREX RTR project.