Finitely axiomatizable quasivarieties of graphs

Apart from four trivial quasivarieties of bipartite graphs, any finitely axiomatizable universal Horn class of graphs must contain graphs of arbitrarily large chromatic number. Hence, no finitely generated universal Horn class of graphs is finitely axiomatizable, except these four. On the other hand, there is a continuum of universal Horn classes of graphs.Presented by J. Mycielski.     

Which linear-fractional composition operators are essentially normal?

We characterize the essentially normal composition operators induced on the Hardy space H² by linear-fractional maps; they are either compact, normal, or (the nontrivial case) induced by parabolic nonautomorphisms. These parabolic maps induce the first known examples of nontrivially essentially normal composition operators. In addition, we characterize those linear-fractionally induced composition operators on H² that are essentially self-adjoint, and present a number of results for composition operators induced by maps that are not linear-fractional.     

Projections of transnormal manifolds

Transnormal manifolds are generalizations of convex hypersurfaces of constant width (see [7]). For such hypersurfaces it is known that every orthogonal projection onto a hyperplane has an outline which is of constant width, too. Orthogonal projections of transnormal manifolds have been studied by F.J. Craveiro de Carvalho [1] in the case when the projection is an immersion onto a convex hypersurface of constant width in a suitable affine subspace of the ambient Euclidean space.Here it will be shown that transnormality is not preserved under orthogonal projection onto hyperplanes without assuming that the manifold has the topological type of a sphere. This implies another general version of the transnormal graph theorem (see [2]). Furthermore, in the case of closed transnormal curves in 3-space also non-orthogonal parallel projection onto planes cannot preserve transnormality as is shown in the last section.     

Quasivarieties of orthomodular lattices determined by conditions on states

In this paper we carry on the research initiated in [13] and [14]. We consider classes of orthomodular lattices which satisfy certain state and polynomial conditions. We show that these classes form quasivarieties. We then exhibit basic examples of these quasivarieties (some of these examples originated in the quantum logic theory). We finally show how the quasivarieties in question can be described in terms of implicative equations. (It should be noted that in some cases we have not been able to clarify whether or not a class shown to be a quasivariety is a variety, see Section 2.) Received May 26, 1998; accepted in final form May 19, 1999.     

Higher order bending and membrane responses of thin linearly elastic plates

The limit behaviors of three-dimensional displacements in thin linearly elastic plates, as the half-thickness ε tends to zero, is now known for various lateral boundary conditions (see [1], [5]). In the generic case one obtains that the leading term of the asymptotic series u⁰ + ge¹+ε²u² +… of the scaled displacement is a Kirchhoff-Love field. In this Note we investigate the case where this leading term vanishes, giving the structure of the first non-vanishing term u^k and an error estimate for its deviation from the scaled solution u(ε) multiplied by ε^−k. There are essentially only three new cases (uncoupling in membrane and bending). Finally, in these situations a boundary layer term of the same order as the actual leading term appears in a generic way.     

Henstock–Kurzweil–Pettis integrability of compact valued multifunctions with values in an arbitrary Banach space

The aim of this paper is to describe Henstock–Kurzweil–Pettis (HKP) integrable compact valued multifunctions. Such characterizations are known in case of functions (see Di Piazza and Musia? (2006)  [16]). It is also known (see Di Piazza and Musia? (2010)  [19]) that each HKP-integrable compact valued multifunction can be represented as a sum of a Pettis integrable multifunction and of an HKP-integrable function. Invoking to that decomposition, we present a pure topological characterization of integrability. Having applied the above results, we obtain two convergence theorems, that generalize results known for HKP-integrable functions. We emphasize also the special role played in the theory by weakly sequentially complete Banach spaces and by spaces possessing the Schur property.     

On the Group of Symplectic Matrices Over a Free Associative Algebra

Symplectic groups are well known as the groups of isometriesof a vector space with a non-singular bilinear alternating form.These notions can be extended by replacing the vector spaceby a module over a ring R, but if R is non-commutative, it willalso have to have an involution. We shall here be concernedwith symplectic groups over free associative algebras (witha suitably defined involution). It is known that the generallinear group GL_n over the free algebra is generated by the setof all elementary and diagonal matrices (see [1, Proposition2.8.2, p. 124]). Our object here is to prove that the symplecticgroup over the free algebra is generated by the set of all elementarysymplectic matrices. For the lowest order this result was obtainedin [4]; the general case is rather more involved. It makes useof the notion of transduction (see [1, 2.4, p. 105]). When thereis only a single variable over a field, the free algebra reducesto the polynomial ring and the weak algorithm becomes the familiardivision algorithm. In that case the result has been provedin [3, Anhang 5].     

Stability theorems for Fourier frames and wavelet Riesz bases

In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see [3]). In the case of an orthonormal basis, our estimate reduces to Kadec’ optimal 1/4 result. The second application proves that a phenomenon discovered by Daubechies and Tchamitchian [4] for the orthonormal Meyer wavelet basis (stability of the Riesz basis property under small changes of the translation parameter) actually holds for a large class of wavelet Riesz bases.     

Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability

In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential Riccati equation (BDRE) associated with these problems (see [2], for finite dimensional stochastic equations or [21], for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see [10], [18]). Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive, we establish that BDRE has a unique, uniformly positive, bounded on ℝ ₊ and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known [18] that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see [10]).      

Orthomodular lattices with state-separated noncompatible pairs

In the logico-algebraic foundation of quantum mechanics one often deals with the orthomodular lattices (OML) which enjoy state-separating properties of noncompatible pairs (see e.g. [18], [9] and [15]). These properties usually guarantee reasonable richness of the state space—an assumption needed in developing the theory of quantum logics. In this note we consider these classes of OMLs from the universal algebra standpoint, showing, as the main result, that these classes form quasivarieties. We also illustrate by examples that these classes may (and need not) be varieties. The results supplement the research carried on in [1], [3], [4], [5], [6], [11], [12], [13] and [16].     

The matricial Schur problem in both nondegenerate and degenerate cases

The principal object of this paper is to present a new approach simultaneously to both nondegenerate and degenerate cases of the matricial Schur problem. This approach is based on an analysis of the central matrixvalued Schur functions which was started in [24]–[26] and then continued in [27]. In the nondegenerate situation we will see that the parametrization of the solution set obtained here coincides with the well‐known formula of D. Z. Arov and M. G. Kre?n for that case (see [1]). Furthermore, we give some characterizations of the situation that the matricial Schur problem has a unique solution (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A partial model of NF with ZF

The theory New Foundations (NF) of Quine was introduced in [14]. This theory is finitely axiomatizable as it has been proved in [9]. A similar result is shown in [8] using a system called K. Particular subsystems of NF, inspired by [8] and [9], have models in ZF. Very little is known about subsystems of NF satisfying typical properties of ZF; for example in [11] it is shown that the existence of some sets which appear naturally in ZF is an axiom independent from NF (see also [12]). Here we discuss a model of subsystems of NF in which there is a set which is a model of ZF. MSC: 03E70.     

A characterization of Hering's plane of order 27

Hering's translation plane of order 27 has been characterized by its order and the fact that it admits SL(2, 13) in its translation complement (see [1]). We show that, aside from the Desarguian plane and a Generalized, André plane, it is the only plane of order 27 which admits a subgroup of SL(2, 13) of order 13×12.Partially supported by FONDECYT 0343 and ANDES FOUNDATION.     

Right-hand solutions of the differential equations of dynamics for mechanical systems with sliding friction

As a further development of Painlevé's theory [1], the existence, continuability and uniqueness of righ-hand solutions of the differential equations of dynamics, and, under certain additional conditions, of the equations of motion of holonomic mechanical systems with sliding friction [2] are considered. In classical mechanics, acceleration is essentially defined as the right-hand derivative of velocity (see [3, 4]). Hence the most meaningful definition of the “solution of a differential equation” in problems of the dynamics of mechanical systems with sliding friction is that using the concept of right derivative [5].     

Congruence properties of lattices of quasivarieties

The congruence properties close to being lower boundedness in the sense of McKenzie are treated. In particular, the affirmative answer is obtained to a known question as to whether finite lattices of quasivarieties are lower bounded in the case where quasivarieties are congruence-Noetherian and locally finite. Namely, we state that for every congruence-Noetherian or finitely generated locally finite quasivariety K, the lattice L_q(K) possesses the Day-Pudlak-Tuma property. But if a quasivariety is locally finite without the condition of being finitely generated), that lattice satisfies only the Pudlak-Tuma property. Translated fromAlgebra i Logika, Vol. 36, No. 6, pp. 605–620, Noember, 1997.     

Fourier-Laplace级数的缺项算术平均在Lebesgue点处的收敛性

设f(x)为定义于n-维欧氏空间R~n中的单位球面∑(n-1)上的Lebesgue可积函数,σ_N~δ(f)表示f的Fourier-Laplace级数的Cesaro平均.众所周知,λ:=(n-2)/2是Cesaro平均的临界阶.本文就n是偶数的情形证明了,使得1/N∑_(k=1)~Nσ_(n_k)~λ(f)(x)→f(x),N→∞,在每个满足一定对极条件的Lebesgue点成立的具有一定“缺项程度”的数列{n_k}的存在性。     

A NOTE ON POLYNOMIAL RINGS OVER VON NEUMANN REGULAR RINGS

Abstract

Let R be an associative ring with 1. It is well known (see [1], [2]) that if R is commutative, then R is Yon Neumann regular (VNR) <=> the polynomial ring S = R[x] is semihereditary. While one of these implications is true in the general case, it is known that a polynomial ring over a regular ring need not be semihereditary (see [3]). In [4] we showed that a ring R is VNR <=> aS + xS is projective for each a ε R. In this note we sharpen this result and use it to show that if c is the ring epimorphism from R[x] to R that maps each polynomial onto its constant term, then R is Yon Neumann regular <=> the inverse image (under c) of each principal (right, left) ideal of R. is a principal (right. left) ideal of R[x] generated by a regular element. (Here an element is regular if and only if it is a non zero-divisor).     

Cohomology of the moduli of parabolic vector bundles

The purpose of this paper is to compute the Betti numbers of the moduli space ofparabolic vector bundles on a curve (see Seshadri [7], [8] and Mehta & Seshadri [4]), in the case where every semi-stable parabolic bundle is necessarily stable. We do this by generalizing the method of Atiyah and Bott [1] in the case of moduli of ordinary vector bundles. Recall that (see Seshadri [7]) the underlying topological space of the moduli of parabolic vector bundles is the space of equivalence classes of certain unitary representations of a discrete subgroup Γ which is a lattice in PSL (2,R). (The lattice Γ need not necessarily be co-compact). While the structure of the proof is essentially the same as that of Atiyah and Bott, there are some difficulties of a technical nature in the parabolic case. For instance the Harder-Narasimhan stratification has to be further refined in order to get the connected strata. These connected strata turn out to have different codimensions even when they are part of the same Harder-Narasimhan strata. If in addition to ‘stable = semistable’ the rank and degree are coprime, then the moduli space turns out to be torsion-free in its cohomology. The arrangement of the paper is as follows. In § 1 we prove the necessary basic results about algebraic families of parabolic bundles. These are generalizations of the corresponding results proved by Shatz [9]. Following this, in § 2 we generalize the analytical part of the argument of Atiyah and Bott (§ 14 of [1]). Finally in § 3 we show how to obtain an inductive formula for the Betti numbers of the moduli space. We illustrate our method by computing explicitly the Betti numbers in the special case of rank = 2, and one parabolic point.     

对流扩散方程的本质非振荡特征差分方法

本文把特征差分法[1]和本质非振荡插值[3]相结合,提出了对流扩散方程的本质非荡性征差分格式,避免了基于Lagrange插值特征差分格式在求解解具有大梯度问题时所产生的非物理振荡,并给出了格式的严格误差估计及数值算例。