Chip-Firing,Antimatroids, and Polyhedra

Starting from the chip-firing game of Björner and Lovász we consider a generalization to vector addition systems that still admit algebraic structures as sandpile group or sandpile monoid. Every such vector addition language yields an antimatroid. We show that conversely every antimatroid can be represented this way. The inclusion order on the feasible sets of an antimatroid is an upper locally distributive lattice. We characterize polyhedra, which carry an upper locally distributive structure and show that they can be modelled by chip-firing games with gains and losses. At the end we point out a connection to a membership problem discussed by Korte and Lovász.     

Dispositions, Laws, and Categories

After a short sketch of Lowe’s account of his four basic categories, I discuss his theory of formal ontological relations and how Lowe wants to account for dispositional predications. I argue that on the ontic level Lowe is a pan-categoricalist, while he is a language dualist and an exemplification dualist with regard to the dispositional/categorical distinction. I argue that Lowe does not present an adequate account of disposition. From an Aristotelian point of view, Lowe conflates dispositional predication with hôs epi to poly statements about what is normally or mostly the case.     

Efficiency analysis, shortage functions, arbitrage, and martingales

This paper shows that standard tools of efficiency analysis, directional distance functions, can be used to characterize the investment-returns technology. That ability to characterize the investment-returns technology and fundamental duality relationships imply that directional distance functions can be used to detect the presence of an arbitrage, to value financial assets in the absence of an arbitrage lying in the span of the market and to place bounds on the no-arbitrage values of assets lying outside the span of the market.     

Optimality,identifiability, and sensitivity

Around a solution of an optimization problem, an “identifiable” subset of the feasible region is one containing all nearby solutions after small perturbations to the problem. A quest for only the most essential ingredients of sensitivity analysis leads us to consider identifiable sets that are “minimal”. This new notion lays a broad and intuitive variational-analytic foundation for optimality conditions, sensitivity, and active set methods.     

Spreads,flocks, and generalized quadrangles

A spread of PG(3,q), q an odd prime, recently constructed by R. Baker and G. Ebert, when generalized for q an odd prime power is isomorphic to a spread derived by J. A. Thas from a flock of a quadratic cone discovered by J. C. Fisher. The associated generalized quadrangle has an unusual colllneation.     

Embeddings, Operator Ranges, and Dirac Operators

We present a generalized operator range construction associated to an indefinite unbounded selfadjoint operator that yields closed embeddings of Kreĭn spaces. As an application we obtain an energy space representation, in the sense of Friedrichs, of a general free Dirac operator.     

Sub-Riemannian (2, 3, 5, 6)-Structures

Doklady Mathematics - We describe all Carnot algebras with growth vector (2, 3, 5, 6), their normal forms, an invariant that separates them, and a change of basis that transforms such an algebra...     

Doubling measures,monotonicity, and quasiconformality

We construct quasiconformal mappings in Euclidean spaces by integration of a discontinuous kernel against doubling measures with suitable decay. The differentials of mappings that arise in this way satisfy an isotropic form of the doubling condition. We prove that this isotropic doubling condition is satisfied by the distance functions of certain fractal sets. Finally, we construct an isotropic doubling measure that is not absolutely continuous with respect to the Lebesgue measure. L.V.K. was supported by an NSF Young Investigator award under grant DMS 0601926. J.-M.W. was supported by the NSF grant DMS 0400810.     

Noncommutative Topologies,Localization, and Sheaves

In this note, we introduce and study the notion of sheaf on a noncommutative topology and construct an associated sheafification functor in this noncommutative context. It appears that abstract localization in the category of presheaves on a partially ordered set provides an elegant way to define this sheafification functor, generalizing similar constructions in the commutative case.     

David G. Luenberger,Linear and nonlinear programming, 2nd edition,Addison-Wesley,Reading (1984), xv + 492 pages, $35.50

This paper considers a stochastic version of the linear continuous type knapsack problem in which the cost coefficients are random variables. The problem is to find an optimal solution and an optimal probability level of the chance constraint. This problem P₀ is first transformed into a deterministic equivalent problem P. Then a subproblem with a positive parameter is introduced and a close relation between P and its subproblem is shown. Further, an auxiliary problem of the subproblem is introduced and a direct relation between P and the auxiliary problem is derived through a relation connecting the subproblem and its auxiliary problem. Fully utilizing these relations, an efficient algorithm is proposed that finds an optimal solution of P in at most O(n⁴) computational time where n is the number of decision variables. Finally, further research problems are discussed.     

Berlekamp—Massey Algorithm, Continued Fractions, Pade Approximations, and Orthogonal Polynomials

The Berlekamp—Massey algorithm (further, the BMA) is interpreted as an algorithm for constructing Pade approximations to the Laurent series over an arbitrary field with singularity at infinity. It is shown that the BMA is an iterative procedure for constructing the sequence of polynomials orthogonal to the corresponding space of polynomials with respect to the inner product determined by the given series. The BMA is used to expand the exponential in continued fractions and calculate its Pade approximations.     

A theory,a story,its analysis,and some implications

We present a refined form of a personal theory of learning which examines two distinct parts of the complex interaction between a learner's internal mental structures and his or her world of experience: what we call natural, conflicting, and alien experiences and the corresponding responses to them.In developing the theory we have come to realize that there are analogies between the experiences and the response of pupils, teachers, and researchers, and we suggest, therefore, that using the theory to analyze the learning of one group of learners can provide implications for the learning of others.The majority of the article concentrates on an example of analyzing for implications by describing the story of an incident in the authors' shared learning, an analysis of it, and some of the implications it provides for learning in general.     

Non-Symmetrical Relations,O-Roles,and Modes

I examine and discuss in this paper Orilia’s theory of external, non-symmetrical relations, that is based on ontological roles (O-Roles). I explore several attempts to interpret O-Roles from an ontological viewpoint and I reject them because of two problems concerning the status of asymmetrical relations (to be distinguished from non-symmetrical relations simpliciter) and of exemplification as an external, non-symmetrical relation. Finally, following Heil’s and Lowe’s characterization of modes as particular properties that ontologically depend on their “bearers”, I introduce relational modes in order to define a new solution to the problems of the ontological status of both external, non-symmetrical relations and O-Roles. I also deal with five objections raised by Fraser MacBride against relational modes and O-Roles and I elaborate an analysis of the relations of being to the left of and being to the right of.     

Factorizations,M-separation,and extremal-epireflective subcategories

Every factorization structure on certain concrete categories induces an extremal-epireflective subcategory. We study these subcategories, give conditions under which an extremal-epireflective subcategory can be so induced, and then show that there are extremal-epireflective subcategories that cannot be induced in this manner.     

Constrained Consumptions, Lipschitzian Demands, and Regular Economies

We consider an exchange economy where the consumers face linear inequality constraints on consumption. We parametrize the economy with the initial endowments and constraints. We exhibit sufficient conditions on the constraints implying that the demand is locally Lipschitzian and continuously differentiable on an open dense subset of full Lebesgue measure. Using this property, we show that the equilibrium manifold is lipeomorphic to an open, connected subset of an Euclidean space and that the lipeomorphism is almost everywhere continuously differentiable. We prove that regular economies are generic and that they have a finite odd number of equilibrium prices and local differentiable selections of the equilibrium prices.Communicated by J. P. CrouzeixThis work was partially supported by CCE, ECOS, and ICM Sistemas Complejos de Ingeniería.     

Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions

We study finite-dimensional representations of current algebras, loop algebras and their quantized versions. For the current algebra of a simple Lie algebra of type ADE, we show that Kirillov-Reshetikhin modules and Weyl modules are in fact all Demazure modules. As a consequence one obtains an elementary proof of the dimension formula for Weyl modules for the current and the loop algebra. Further, we show that the crystals of the Weyl and the Demazure module are the same up to some additional label zero arrows for the Weyl module.For the current algebra Cg of an arbitrary simple Lie algebra, the fusion product of Demazure modules of the same level turns out to be again a Demazure module. As an application we construct the Cg-module structure of the Kac-Moody algebra -module V(?Λ₀) as a semi-infinite fusion product of finite-dimensional Cg-modules.     

一般形式下斜正态随机向量的矩

本文给出一般形式下斜正态随机向量及其平方型的矩公式. 作为应用, 计算出了斜正态随机向量的多元偏度和峰度.     

Frequent hypercyclicity,chaos, and unconditional Schauder decompositions

We prove that if X is any complex separable infinite-dimensional Banach space with an unconditional Schauder decomposition, X supports an operator T which is chaotic and frequently hypercyclic. In contrast with the complex case, we observe that there are real Banach spaces with an unconditional basis which support no chaotic operator.     

Convexifactors, Generalized Convexity, and Optimality Conditions

The recently introduced notion of a convexifactor is further studied, and quasiconvex and pseudoconvex functions are characterized in terms of convexifactors. As an application to a chain rule, a necessary optimality condition is deduced for an inequality constrained mathematical programming problem.     

Stability,prediction-correction,and dynamic programming

It is proven here that a bounded perturbation of the discrete dynamic programming functional equation arising from the Bolza problem yields a bounded change in its solution. This stability property encourages the development of approximation techniques for solving such equations. One such technique, involving the backward solution of an approximate functional equation as a prediction step, followed by a forward reconstruction using true equations as a correction step, is then discussed. Bounds for the errors arising from such an approximation procedure are derived. Successive approximations is suggested, in conclusion, as a means for obtaining improved solutions.