Difference Equations∶ Detecting Oscillations via Higher Order Characteristic Equations

In these notes we introduce an alternative procedure to detect the presence of a wide range of oscillatory solutions-the exponential oscillatory ones-in both continuous-time and discrete linear difference equations. The technique is addressed to the scalar case of the linear two-delay difference equation, but it can be extended to other types of evolution equations including the nonlinear ones, in higher dimensions. In particular, we give an illustration of its applicability to the case of a simple two dimensional difference equation. Oscillatory solutions are presented in their closed formulae.     

Sur une extension de la notion de dérivée

Sans résuméCe mémoire est le développement de deux notes que nous avons publiées, sous le même titre, dans les Comptes rendus des séances de l'Académie des Sciences de Paris. C. R.206, 1938 S. 1444 et207, 1938 S. 110.     

In Short

Abstract

The Great Transformation in Higher Education, 1960–1980 by Clark Kerr, Albany; State University of New York Press, 1991, xxii + 383 pages, notes, index.

Learning in Adulthood: A Comprehensive Guide by Sharon B. Merriam and Rosemary S. Caffarella, San Francisco, Oxford: Jossey-Bass, 1991, xx + 376 pages, references, indices.

Literacy in the United States: Readers and Reading Since 1880 by Carl F. Kaestle, et al. New Haven, London: Yale University Press, 1991, xxiii + 338 pages, appendix, bibliography, index.

Major American Higher Education Issues and Answers in the 1990s by Richard I. Miller, London: Jessica Kingsley, 1990, 219 pages, references, index.     

Computer-Assisted Statistics Education at Delft University of Technology

Abstract

At Delft University of Technology many students experience difficulties in mastering basic concepts of probability and statistics. In the past few years the lectures have undergone a radical change—the lecture notes now contain modern data analysis techniques, like kernel density estimation, simulation, and bootstrapping. In the TWI-Stat project, a computer-aided instruction course was developed to help students become more familiar with modern statistical analysis. The course presents itself as a dynamic, interactive, personal book. Highly interactive analysis tools are available. The software will be available for MS-Windows.     

NOTES ON HOMOTOPY PULL-BACKS

Abstract

In these notes we investigate the relationships between two different notions of homotopy pull-back, namely Mather's original concept for homotopy commutative squares (1976) and Baues' notion for commutative squares in his book on algebraic homotopy (1989) (which does not contain any reference to Mather's work. In fact, Baues' definition seems to be based on investigations by V. Puppe (1974)). We also provide new characterizations of the two concepts; in particular we introduce a homotopy pull-back property for Baues' homotopy pull-backs.     

Fractals Everywhere

Abstract

Like every other component of government these days, the U.S. federal statistical system is being reinvented, reengineered, and downsized. At this writing, several bills are before the U.S. Congress to restructure the system, relocate key agencies, and reallocate programs and budgets. Janet L. Norwood, Senior Fellow with The Urban Institute and former Commissioner of Labor Statistics, offers a trenchant commentary on the situation. Norwood identifies serious problems with the U.S. federal statistical system, as it is currently organized, that need to be addressed. At the same time, she notes the grave danger that solutions attempted by the U.S. Congress may make matters worse. She ends with a call to action for those who “care about data quality and who understand the important uses of the government's statistical output.”     

Notes on Gromov’s systolic estimate

These notes cover some of the main results of Gromov’s paper Filling Riemannian manifolds. The goal of these notes is to make the results and proofs accessible to more people. The main result is that if (M,g) is a Riemannian manifold of dimension n, then there is a non-contractible curve in (M,g) of length at most C _n Vol(M,g)^1/n .     

Discrete real time flows with quasi-discrete spectra and algebras generated by expq(t)

These notes (essentially unedited) were sent to W. Parry in 1964. The first two parts are complete and in a letter to Parry at that time Hahn indicated his intention to publish them. Evidently he did not manage to do this. The remainder of these notes represents an attempt to establish a theory of quasidiscrete spectra for discrete one-parameter flows. Hahn indicates the gaps and in a following note Parry clarifies his theory. The first part of these notes presents a characteristic example of a discrete one-parameter flow with quasidiscrete spectrum. Ergodicity, minimality and distality are established. The second part examines the Banach algebra of functions onR generated by {expq(t): q a real polynomial of degree <n+1} and shows that the shift isometries arise from a discrete one-parameter flow on its maximal ideal space Λ _n and that ifn is finite this flow is isomorphic to the example examined in the first part. Deceased     

Constructive struggle in geometry classrooms

The purpose of these notes is to provide examples of large-scale geometry problems that have the potential to promote constructive struggling in high-school geometry classrooms. These problems are suitable for being explored in a dynamic geometry environment. The notes also contain brief sketches of solutions to the two problems.     

CONNEXION PROPERTIES AND FACTORISATION THEOREMS

Abstract

With the introduction of several new factorisation theorems, this paper is intended to show that previous efforts of the authors [3] [5] and of Strecker [15] to describe the factorisations involving connectedness are incomplete. In Section 1 we give a purely topological construction of such a factorisation, in which the right factor is the class of spreads and the left factor has a certain property hereditarily: crucially, not all members of the left factor need be quotients. Section 2 shows that, given a left factor consisting of onto maps in the category T of topological spaces, then the class of mappings with the relevant properties hereditarily is also a left factor, and the result of section 1 is a particular case of this. Section 3 combines the material in [3] on intrinsic connexion properties with ideas of Preuss (see [1]) on disconnectednesses to yield another range of factorisations, for example, involving the maps with strongly connected fibres; and Section 4 notes some outstánding problems which our work has provoked.     

Pricing Fixed-Income Securities in an Information-Based Framework

Abstract

The purpose of this article is to introduce a class of information-based models for the pricing of fixed-income securities. We consider a set of continuous-time processes that describe the flow of information concerning market factors in a monetary economy. The nominal pricing kernel is assumed to be given at any specified time by a function of the values of information processes at that time. Using a change-of-measure technique, we derive explicit expressions for the prices of nominal discount bonds and deduce the associated dynamics of the short rate of interest and the market price of risk. The interest rate positivity condition is expressed as a differential inequality. An example that shows how the model can be calibrated to an arbitrary initial yield curve is presented. We proceed to model the price level, which is also taken at any specified time to be given by a function of the values of the information processes at that time. A simple model for a stochastic monetary economy is introduced in which the prices of the nominal discount bonds and inflation-linked notes can be expressed in terms of aggregate consumption and the liquidity benefit generated by the money supply.     

Equidistant hypersurfaces of the bidisk

The following are notes on the geometry of the bidisk, H ² × H ². In particular, we examine the properties of equidistant surfaces in the bidisk.     

Riemann Surface Laminations with Singularities

In these introductory notes we give the basics of the theory of holomorphic foliations and laminations. The emphasis is on the theory of harmonic currents and unique ergodicity for laminations transversally Lipschitz in ?² and for generic holomorphic foliations in ?².     

The Converse of Schur's Lemma in Noetherian Rings and Group Algebras

ABSTRACT

If M is a simple module over a ring R, then, by Schur's Lemma, its endomorphism ring is a division ring. However, the converse of this property, which we called the CSL property, does not hold in general. The object of this article is to study this converse for a few classes of rings: left Noetherian rings, V-rings and group algebras. First, we establish that a left Noetherian ring R is a CSL ring if and only if a ring R is left–artinian and primary decomposable. Secondly, we prove that a left semiartinian V-ring is CSL. At last, we study the CSL property in group algebra K [ G ] where K a field algebraically closed of characteristic p and G is a finite group of order divisible by p. Our main contribution is that K [ G ] is a CSL ring if and only if Gbf = HP where H is a normal p′-subgroup and bfP a Sylow bfp-subgroup of bfG. In this case, K [ G ] is primary decomposable.     

A hybrid algorithm for solving linear inequalities in a least squares sense

The need for solving a system of linear inequalities, A x ≥ b, arises in many applications. Yet in some cases the system to be solved turns out to be inconsistent due to measurement errors in the data vector b. In such a case it is often desired to find the smallest correction of b that recovers feasibility. That is, we are looking for a small nonnegative vector, y ≥ 0, for which the modified system A x ≥ b - y is solvable. The problem of calculating the smallest correction vector is called the least deviation problem. In this paper we present new algorithms for solving this problem. Numerical experiments illustrate the usefulness of the proposed methods.     

Orders in regular rings with minimal condition for principal right ideals

ABSTRACT

A ring R is called an n-clean (resp. Σ-clean) ring if every element in R is n-clean (resp. Σ-clean). Clean rings are 1-clean and hence are Σ-clean. An example shows that there exists a 2-clean ring that is not clean. This shows that Σ-clean rings are a proper generalization of clean rings. The group ring ?_(p) G with G a cyclic group of order 3 is proved to be Σ-clean. The m× m matrix ring M _m (R) over an n-clean ring is n-clean, and the m×m (m>1) matrix ring M _m (R) over any ring is Σ-clean. Additionally, rings satisfying a weakly unit 1-stable range were introduced. Rings satisfying weakly unit 1-stable range are left-right symmetric and are generalizations of abelian π-regular rings, abelian clean rings, and rings satisfying unit 1-stable range. A ring R satisfies a weakly unit 1-stable range if and only if whenever a ₁ R + ˙˙˙ a _m R = dR, with m ≥ 2, a ₁,…, a _m, d ∈ R, there exist u ₁ ∈ U(R) and u ₂,…, u _m ∈ W(R) such that a ₁ u ₁ + ? a _m u _m = Rd.     

Singular Archimedean lattice-ordered groups

W stands for the category of all archimedean l-groups with designated weak unit. The subcategory W _s of all groups with singular weak unit is analyzed as a full subcategory of W which is both epireflective and monocoreflective. A general technique for "contracting" monoreflections of a category A to a monocoreflective subcategory B is developed and then applied to W _s to show that: (i) the projectable hull in W _s is a monoreflection; (ii) essential hulls in W _s are formed by simply taking the lateral completion, and G is essentially closed in this category if and only if , where X is compact, Hausdorff and extremally disconnected; (iii) the maximum monoreflection on W _s , denoted , is obtained by contracting the maximum monoreflection on W, and G is epicomplete in W _s precisely when G is laterally -complete; (iv) the maximum essential reflection on W _s , denoted , is the contraction of the maximum essential reflection on W. Received January 22, 1997; accepted in final form June 11, 1998.     

Finite-to-finite universal quasivarieties are Q-universal

Let K be a quasivariety of algebraic systems of finite type. K is said to be universal if the category G of all directed graphs is isomorphic to a full subcategory of K. If an embedding of G may be effected by a functor F:G K which assigns a finite algebraic system to each finite graph, then K is said to be finite-to-finite universal. K is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.?We establish a connection between these two, apparently unrelated, notions by showing that if K is finite-to-finite universal, then K is Q-universal. Using this connection a number of quasivarieties are shown to be Q-universal. Received February 8, 2000; accepted in final form December 23, 2000.     

k-Eulerian Posets

A poset P is called k-Eulerian if every interval of rank k is Eulerian. The class of k-Eulerian posets interpolates between graded posets and Eulerian posets. It is a straightforward observation that a 2k-Eulerian poset is also (2k+1)-Eulerian. We prove that the ab-index of a (2k+1)-Eulerian poset can be expressed in terms of c=a+b, d=ab+ba and e ^2k+1=(a–b)^2k+1. The proof relies upon the algebraic approaches of Billera-Liu and Ehrenborg-Readdy. We extend the Billera-Liu flag algebra to a Newtonian coalgebra. This flag Newtonian coalgebra forms a Laplace pairing with the Newtonian coalgebra ka,b studied by Ehrenborg-Readdy. The ideal of flag operators that vanish on (2k+1)-Eulerian posets is also a coideal. Hence, the Laplace pairing implies that the dual of the coideal is the desired subalgebra of ka,b. As a corollary we obtain a proof of the existence of the cd-index which does not use induction.     

Rank formulae from the perspective of orthogonal projectors

For matrices F and G having the same number of rows and the orthogonal projectors P?=?FF ^? and Q?=?GG ^?, with F ^? and G ^? denoting the Moore–Penrose inverses of F and G, respectively, several formulae for ranks of various functions of F, G, P and Q are established. Besides a collection of original characterizations, many of which involve the ranks of F*G and (F?:?G) (which coincide with the ranks of PQ and P?+?Q, respectively), some properties known in the literature are reestablished in a generalized form. The variety of relationships considered shows that the approach utilized in the article, based on the partitioned representations of the projectors, provides a powerful tool of wide applicability.