When is an L(B, m, n, r, s, t, z, w) loop SRAR?

In [5, 6], the second author and D. A. ROBINSON initiated a study of non-Moufang Bol loops with the property that over a field, necessarily of characteristic 2, their loop rings satisfy the right, but not the left, Bol identity. They called such loops SRAR and showed that the family of SRAR loops includes those Bol loops which have a unique non-identity commutator/associator. In [4, 2], the current authors presented a construction for a new class of Bol loops denoted L(B,m,n,r,s,t,z,w) with initial data a given (possibly associative) Bol loop B, elements, r, s, t, z and w in the centre of B, and integers m and n.     

Quasi-symmetric designs,codes, quadrics,and hyperplane sections

It is proved that a quasi-symmetric design with theSymmetric Difference Property (SDP) is uniquely embeddable as a derived or a residual design into a symmetric SDP design. Alternatively, any quasi-symmetric SDP design is characterized as the design formed by the minimum weight vectors in a binary code spanned by the simplex code and the incidence vector of a point set in PG(2m-1, 2) that intersects every hyperplane in one of two prescribed numbers of points. Applications of these results for the classification of point sets in PG(2m-1, 2) with the same intersection properties as an elliptic or a hyperbolic quadric, as well as the classification of codes achieving the Grey-Rankin bound are discussed.     

Benchmarking curriculum-based course timetabling: formulations,data formats,instances, validation,visualization, and results

We propose a set of formulations for the Curriculum-Based Course Timetabling problem, with the aim of “capturing” many real-world formulations, and thus encouraging researchers to “reduce” their specific problems to one of them, gaining the opportunity to compare and assess their results. This work is accompanied by a web application that maintains all the necessary infrastructures for benchmarking: validators, data formats, instances, reference scores, lower bounds, solutions, and visualizers. All instances proposed here are based on real data from various universities and they represent a variety of possible situations.     

Woodhouse, babbage, peacock, and modern Algebra

In a recent article, J. M. Dubbey [Historia Mathematica 4 (1977), 295–302] showed that George Peacock's A Treatise on Algebra (1830) was similar to an unpublished work written by Charles Babbage in 1821. Evidently perplexed about the absence of a dispute over priority, Dubbey concluded that Peacock had unconsciously assimilated Babbage's ideas, and that Babbage was too busy with other activities to be concerned. The thesis of this article is that the innovative aspects of the work of both Babbage and Peacock are extensions of ideas put forth in 1803 by Robert Woodhouse, and that probably neither Babbage nor Peacock was overly concerned with acknowledgments because their approach to algebra was not unique at Cambridge.     

Discontinuous Solutions of Linearized, Steady-State, Viscous, Compressible Flows

In thIs paper we prove the existence and uniqueness of the solution to a linearized, two dimensional, steady state, viscous, compressible Navier-Stokes equations in a strip. We obtain a regularity result and a new a priori estimate of the solution. We establish the discontinuity of the solution when the boundary data of pressure have a jump discontinuity. We also derive a formula for the discontinuous part and show that the remainder is smooth in the strip.     

Обобщение регулярны х в круге функций, близких к выпуклым, на случай многих комплексных перемен ных

Extremal problems are treated for the classR _d, the latter being a two-dimensional generalization of the class of complex variable functions which are regular in a disc and close-to-convex.     

Entropy,Universal Coding,Approximation, and Bases Properties

We shall present here results concerning the metric entropy of spaces of linear and nonlinear approximation under very general conditions. Our first result computes the metric entropy of the linear and m-terms approximation classes according to a quasi-greedy basis verifying the Temlyakov property. This theorem shows that the second index r is not visible throughout the behavior of the metric entropy. However, metric entropy does discriminate between linear and nonlinear approximation. Our second result extends and refines a result obtained in a Hilbertian framework by Donoho, proving that under orthosymmetry conditions, m-terms approximation classes are characterized by the metric entropy. Since these theorems are given under the general context of quasi-greedy bases verifying the Temlyakov property, they have a large spectrum of applications. For instance, it is proved in the last section that they can be applied in the case of L _p norms for R ^d for 1 < p < \infty. We show that the lower bounds needed for this paper in fact follow from quite simple large deviation inequalities concerning hypergeometric or binomial distributions. To prove the upper bounds, we provide a very simple universal coding based on a thresholding-quantizing constructive procedure.     

Contractualism, Politics, and Morality

Rawls developed a contractualist theory of social justice and Scanlon attempted to extend the Rawlsian framework to develop a theory of rightness, or morality more generally. I argue that there are some good reasons to adopt a contractualist theory of social justice, but that it is a mistake to adopt a contractualist theory of rightness. I begin by illustrating the major shared features of Scanlon and Rawls’ theories. I then show that the justification for these features in Rawls’ theory, the centrality of cooperative fairness to social justice, cannot be used to defend their use in Scanlon’s. Finally, I argue that Scanlon has not provided an adequate alternative defense of these features, and show that they create problems when contractualists try to explain major features of our common-sense morality.     

Cayley compositions,partitions, polytopes,and geometric bijections

In 1857, Cayley showed that certain sequences, now called Cayley compositions, are equinumerous with certain partitions into powers of 2. In this paper we give a simple bijective proof of this result and a geometric generalization to equality of Ehrhart polynomials between two convex polytopes. We then apply our results to give a new proof of Braun?s conjecture proved recently by the authors [15].     

Characterization of vector valued,gaussian, stationary,markov processes

We extend an old result by Doob characterizing real-valued, Gaussian, stationary, Markov processes to the vector case. In this case a deterministic component appears that consists of a system of harmonic oscillators while the random part is a collection of independent oscillator processes, modulo linear changes of coordinates.     

Worms,gaps, and hydras

We define a direct translation from finite rooted trees to finite natural functions which shows that the Worm Principle introduced by Lev Beklemishev is equivalent to a very slight variant of the well‐known Kirby‐Paris' Hydra Game. We further show that the elements in a reduction sequence of the Worm Principle determine a bad sequence in the well‐quasi‐ordering of finite sequences of natural numbers with respect to Friedman's gapembeddability. A characterization of gap‐embeddability in terms of provability logic due to Lev Beklemishev is also presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Randomness,chaos, and structure

We show how a simple scheme of symbolic dynamics distinguishes a chaotic from a random time series and how it can be used to detect structural relationships in coupled dynamics. This is relevant for the question at which scale in complex dynamics regularities and patterns emerge. © 2009 Wiley Periodicals, Inc. Complexity 2009     

Toughness,trees, and walks

A graph is t‐tough if the number of components of G\S is at most |S|/t for every cutset S ⊆ V (G). A k‐walk in a graph is a spanning closed walk using each vertex at most k times. When k = 1, a 1‐walk is a Hamilton cycle, and a longstanding conjecture by Chvátal is that every sufficiently tough graph has a 1‐walk. When k ≥ 3, Jackson and Wormald used a result of Win to show that every sufficiently tough graph has a k‐walk. We fill in the gap between k = 1 and k ≥ 3 by showing that, when k = 2, every sufficiently tough (specifically, 4‐tough) graph has a 2‐walk. To do this we first provide a new proof for and generalize a result by Win on the existence of a k‐tree, a spanning tree with every vertex of degree at most k. We also provide new examples of tough graphs with no k‐walk for k ≥ 2. © 2000 John Wiley & Sons, Inc. J Graph Theory 33:125–137, 2000     

Hadamard matrices,Baumert-Hall units,four-symbol sequences,pulse compression,and surface wave encodings

If a Williamson matrix of order 4w exists and a special type of design, a set of Baumert-Hall units of order 4t, exists, then there exists a Hadamard matrix of order 4tw. A number of special Baumert-Hall sets of units, including an infinite class, are constructed here; these give the densest known classes of Hadamard matrices. The constructions relate to various topics such as pulse compression and image encodings.     

Right-angularity, flag complexes, asphericity

The ??polyhedral product functor?? produces a space from a simplicial complex L and a collection of pairs of spaces, {(A(i), B(i))}, where i ranges over the vertex set of L. We give necessary and sufficient conditions for the resulting space to be aspherical. There are two similar constructions, each of which starts with a space X and a collection of subspaces, {X _i }, where ${i \in \{0,1. . . . , n\}}$ , and then produces a new space. We also give conditions for the results of these constructions to be aspherical. All three techniques can be used to produce examples of closed aspherical manifolds.     

Continue,quit, restart probability model

We discuss a new applied probability model: there is a system whose evolution is described by a Markov chain (MC) with known transition matrix on a discrete state space and at each moment of a discrete time a decision maker can apply one of three possible actions: continue, quit, and restart MC in one of a finite number of fixed “restarting” points. Such a model is a generalization of a model due to Katehakis and Veinott (Math. Oper. Res. 12:262, 1987), where a restart to a unique point was allowed without any fee and quit action was absent. Both models are related to Gittins index and to another index defined in a Whittle family of stopping retirement problems. We propose a transparent recursive finite algorithm to solve our model by performing O(n³) operations.     

A finite algorithm to fit geometrically all midrange lines, circles, planes, spheres, hyperplanes, and hyperspheres

Summary. The algorithm proved here solves the problem of orthogonal distance regression for the maximum norm with hyperplanes and hyperspheres. For each finite set of points in a Euclidean space of any dimension, the algorithm determines – through finitely many arithmetic operations – all the hyperplanes and hyperspheres that minimize the maximum Euclidean distance measured perpendicularly from the data. The algorithm finds all the slabs (bounded by parallel hyperplanes) and all the spherical shells (bounded by concentric hyperspheres) that contain all the data and are “rigidly supported” by the data (for which there does not exist any other pair of parallel hypersurfaces of the same type that intersect the data at the same points.) The computational complexity of the algorithm increases as the number of data points raised to the dimension of the ambient space. The solutions are then the midrange hyperplanes in the thinnest slabs, and the midrange hyperspheres in the thinnest shells. Their sensitivity to perturbations of the data is of the order of a power of the reciprocal of the smallest angle between two median hyperplanes separating two pairs of data points. The methods of proof consist in showing that if a pair of parallel hyperplanes or hyperspheres is not rigidly supported but encompasses all the data, then there exists a projective shift of their common projective center producing a thinner slab or shell that still contains all the data. Received December 14, 1999 / Revised version received August 30, 2000 / Published online September 19, 2001