Ball,sphere, and all,all, all

The relationship of multidimensional geometry with statistical thermodynamics and with laws of large numbers is described.     

Constructivism,Factoring, and Beliefs

The errors made by remedial intermediate algebra students in factoring polynomials were analyzed in the light of student definitions of factoring. Certain belief sets about factoring are found to logically entail many of the errors made. The vocabulary of epistemic semantics is used to describe how beliefs are changed. It is suggested that belief-based teaching can be successful in teaching factoring.     

Ceva, Menelaus, and Selftransversality

The purpose of this paper is to state and prove a theorem (the CMS Theorem) which generalizes the familiar Ceva's Theorem and Menelaus' Theorem of elementary Euclidean geometry. The theorem concernsn -acrons (generalizations of n-gons) in affine space of any number of dimensions and makes assertions about circular products of ratios of lengths, areas, volumes, etc. In particular it contains, as special cases, many results in this area proved by earlier authors.     

Noncomputability,unpredictability, undecidability,and unsolvability in economic and finance theories

We outline, briefly, the role that issues of the nexus between noncomputability and unpredictability, on the one hand, and between undecidability and unsolvability, on the other hand, have played in Computable Economics (CE). The mathematical underpinnings of CE are provided by (classical) recursion theory, varieties of computable and constructive analysis and aspects of combinatorial optimization. The inspiration for this outline was provided by Professor Graça's thought‐provoking recent article. © 2012 Wiley Periodicals, Inc. Complexity, 2012     

Domination, discretization, and scalarization

This is an overview of a few possibilities that are open by model theory in applied mathematics. The most attention is paid to the present state and frontiers of the Cauchy method of majorants, approximation of operator equations with finite-dimensional analogs, and the Lagrange multiplier principle in multiobjective decision making.     

Fractions,Decimals,and Percents

<正>When am I ever going to use this'?Surveys The graph shows the results of a survey in which teens were asked to name the most important invention of the 20th century.1.What percent of the teens said that the personal computer was the most important invention?2.How is this percent written as a ratio?     

Corporate taxes, changing risk, and wealth transfers between shareholders, lenders, and the taxman

This paper examines how risk and corporate taxes jointly affectthe value of the debt and equity of a firm. We view thetotalpretax value of the firm as being divided between the taxman,lenders, and shareholders, and examine how increasing the riskof a leveraged firm, with no change in its pretax value, affectsthese claimants. We show that shareholders may gain while thetaxman loses, or vice versa, or that both gain at the expenseof lenders. The outcome depends on several factors includingthe ranking of the claims of the taxman and lenders. The analysisis extended to investments which expand the firm; here our conclusionsare at variance with some frequently expressed views on possibletax advantages of merging. We obtain different conclusions becausewe make allowance for the diversification effects of mergingon the value of the debt claims. A careful examination of thisrequires a precise mathematical formulation of the values ofthe tax, debt, and equity claims using option-pricing techniques.     

Riffle shuffles,cycles, and descents

We derive closed form expressions and limiting formulae for a variety of functions of a permutation resulting from repeated riffle shuffles. The results allow new formulae and approximations for the number of permutations inS _n with given cycle type and number of descents. The theorems are derived from a bijection discovered by Gessel. A self-contained proof of Gessel's result is given.     

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.     

Terminality,normality, and transversality conditions

The maximum principle distinguishes between two phases of the optimal control problem. Some of the stated conditions are to be satisfied at points other than endpoints and some conditions are to be satisfied specifically at the endpoints. This paper utilizes the first set of conditions from the maximum principle to reexamine the second set. In the process, a new necessary condition to be satisfied at the endpoints is obtained. This condition is for certain cases easier to apply than the transversality conditions, yields additional information which may be of computational advantage, and lends itself quite naturally to an exposition of abnormal solutions. The relationship of the new condition to the transversality conditions and a discussion on normality are included. Several examples are given to illustrate the results.This research was supported in part by NASA under Grant NGR-03-002-224 and NSF Science Faculty Fellowship. The authors are indebted to Professors G. Leitmann, G. Basile, and E. Cliff for their helpful comments and suggestions.     

Order,genus, and diagram invariance

The genus of any ordered set equals the genus of its covering graph, and, therefore, the genus of an ordered set is a diagram invariant.     

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.     

Centroids,Representations, and Submodular Flows

The notion of centroid of a tree is generalized to apply to an arbitrary intersecting family of sets. Centroids are used to construct a compact representation for any intersecting family of sets, as well as any crossing family. The size of the representation for a family on n elements is O(n²), compared to size O(n³) for previous representations. Efficient algorithms to construct the representation are given. For example on a network of n vertices and m edges, the representation of all minimum cuts uses O(m log(n²/m)) space; it is constructed in O(nm log(n²/m)) time (this is the best-known time for finding one minimum cut). The representation is used to improve several submodular flow algorithms. For example a minimum-cost dijoin is found in time O(n²m); as a result a minimum-cost planar feedback are set is found in time O(n³). The previous best-known time bounds for these two problems are both a factor n larger.     

Configurations, braids, and homotopy groups

The main results of this article are certain connections between braid groups and the homotopy groups of the -sphere. The connections are given in terms of Brunnian braids over the disk and over the -sphere. The techniques arise from the natural structure of simplicial and -structures on fundamental groups of configuration spaces.

     

Semigroups, Rings, and Markov Chains

We analyze random walks on a class of semigroups called left-regular bands. These walks include the hyperplane chamber walks of Bidigare, Hanlon, and Rockmore. Using methods of ring theory, we show that the transition matrices are diagonalizable and we calculate the eigenvalues and multiplicities. The methods lead to explicit formulas for the projections onto the eigenspaces. As examples of these semigroup walks, we construct a random walk on the maximal chains of any distributive lattice, as well as two random walks associated with any matroid. The examples include a q-analogue of the Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are generalized derangement numbers, which may be of independent interest.     

Educational Models,Manpower, Planning and Control

The paper describes the setting up of models of the educational system and their use in decision-making. The methodological problems of planning and controlling a large socio-economic system, such as education, are discussed.     

Gradients,subgradients, and economic equilibria

Two concepts of differentiability are shown to underlie two concepts of equilibrium for nonatomic economies. The weaker concept of subdifferentiability corresponds to Walrasian equilibrium and the stronger concept of Fréchet differentiability to product-exhaustion. Product-exhaustion is shown to be equivalent to a definition of economic equilibrium in terms of the rationality of price-taking.     

Taxation,incentives, and risk sharing

The tendency for taxation to weaken incentives and induce inefficient behavior on the part of entrepreneurs is demonstrated with a simple example. Next the Mossin model is extended from a proportional to a progressive tax structure that closely approximates the one in place in the U.S.A. It is shown that increased progressivity causes the amount of wealth invested in the risky asset to decline rather than increase as is true with proportional taxes.     

Contextualism,art, and rigidity: Levinson,Currie and Davies

The topic of this paper is the role played by context in art. In this regard I examine three theories linked to the names of J. Levinson, G. Currie and D. Davies. Levinson’s arguments undermine the structural theory. He finds it objectionable because it makes the individuation of artworks independent of their histories. Secondly, such a consequence is unacceptable because it fails to recognise that works are created rather than discovered. But, if certain general features of provenance are always work-constitutive, as it seems that Levinson is willing to claim, these features must always be essential properties of works. On the other hand, consideration of our modal practice suggests that whether a given general feature of provenance is essential or non-essential depends upon the particular work in question or is “work relative”. D. Davies builds his performance theory on the basis of the critical evaluation of Currie’s action-type hypotheses (ATH). Performances, says Davies, are not to be identified with “basic actions” to which their times belong essentially, but with “doings” that permit of the sorts of variation in modal properties required by the work-relativity of modality. He is also a fierce critic of the contextualist account. Contextualism is in his view unable to reflect the fact that aspects of provenance bear upon our modal judgements with variable force.In the second part of the paper I consider Davies’s “modality principle”. Davies is inclined to defend the claim that labels used for designation of works are rigid designators. Such a view offers a ground for discussion about the historicity of art. What has been meant when people claim that art is an historical concept? I argue that any historical theory implies a two-dimensional notion of “art”. At the end of the paper I suggest that Davies should embrace the theory of contingent identity and not the colocationist view about the relationship that exists between a particular artwork and its physical bearer.