On the Borel-Cantelli lemma

A new variant of the “divergent” part of the Borel-Cantelli lemma for events derived from a Markov chain is given. Further two applications are considered. One of the applications refers to the denumerable Markov chain and the second is a new proof of the “strong” theorem corresponding to the “arc sine law”.     

Cogito ergo mundus talis est

This paper deals with one of the basic philosophical questions in modern cosmology: can the so-called “Anthropic Principle”, considered as an alternative to the classical teleology of creation, be an adequate explanation of the evidence that our universe is “fine-tuned” for the emergence of life and consciousness. The main problem with this principle is not its presumed teleology, as it is sometimes wrongly supposed, but quite the contrary: its intention to avoid teleological explanations by including the existence of many universes (“multiverse”) into extended cosmological models. After having compared logical and cosmological many-worlds concepts, this paper reaches the conclusion that the ontological reality of the “multiverse” is an even more problematic presupposition than some properly revised version of teleological causality. This in itself does not imply the classical theistic explanation of creation, since it also yields a pantheistic explanation of the emergence of life and consciousness in our universe.     

On the interaction of δ-shocks and contact vacuum states

We study the pressureless gas equations, with piecewise constant initial data. In the immediate solution, δ-shocks and contact vacuum states arise and even meet (interact) eventually. A solution beyond the “interaction” is constructed. It shows that the δ-shock will continue with the velocity it attained instantaneously before the time of interaction, and similarly, the contact vacuum state will move past the δ-shock with a velocity value prior to the interaction. We call this the “no-effect-from-interaction” solution. We prove that this solution satisfies a family of convex entropies (in the Lax’s sense). Next, we construct an infinitely large family of weak solutions to the “interaction”. Suppose further that any of these solutions satisfy a convex entropy, it is necessary and suffcient that these solutions reduce to only the “no-effect-from-interaction” solution. In [1], Bouchut constructed another entropy satisfying solution. As with other previous papers, it is obvious that it will not be sufficient that a “correct” solution satisfies a convex entropy, in a non-strictly hyperbolic conservation laws system.     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

Homotopy classification of equivariant regular sections

In his paper [2], Bierstone proves the equivariant Gromov theorem which is an integrability theorem for “open regularity condition” of equivariant sections of a smooth G-fibre bundle under the assumption that all orbit bundles of base manifold are non-closed. Here, we prove the result without his assumption under a nice “open regularity condition” which we call “G-extensible”. One of the examples of “G-extensible condition” is given by notions of Thom-Boardman singularities.     

On the interaction of δ-shocks and contact vacuum states

We study the pressureless gas equations, with piecewise constant initial data. In the immediate solution, δ-shocks and contact vacuum states arise and even meet (interact) eventually. A solution beyond the “interaction” is constructed. It shows that the δ-shock will continue with the velocity it attained instantaneously before the time of interaction, and similarly, the contact vacuum state will move past the δ-shock with a velocity value prior to the interaction. We call this the “no-effect-from-interaction” solution. We prove that this solution satisfies a family of convex entropies (in the Lax’s sense). Next, we construct an infinitely large family of weak solutions to the “interaction”. Suppose further that any of these solutions satisfy a convex entropy, it is necessary and suffcient that these solutions reduce to only the “no-effect-from-interaction” solution. In [1], Bouchut constructed another entropy satisfying solution. As with other previous papers, it is obvious that it will not be sufficient that a “correct” solution satisfies a convex entropy, in a non-strictly hyperbolic conservation laws system. Research done in the University of Michigan-Ann Arbor, submission from Temasek Laboratories, National University of Singapore.     

Game theoretical optimization inspired by information theory

Inspired by previous work on information theoretical optimization problems, the basics of an axiomatic theory of certain special two-person zero-sum games is developed. One of the players, “Observer”, is imagined to have a “mind”, the other, “Nature”, not. These ideas lead to un-symmetric modeling as the two players are treated quite differently. Basic concavity- and convexity results as well as a general minimax theorem are derived from the axioms.     

NP-completeness of some problems of choosing a vector subset

The NP-completeness is proved of some problems of choosing a Euclidean vector subset. One of the data analysis problems is reduced to these problems. The required subset is assumed to have a fixed cardinality and include the vectors that are “close” to each other by the criterium of the minimum sum of squares of distances.     

Supply chain organization and e-commerce: a model to analyze store-picking, warehouse-picking and drop-shipping

Across many industries, e-commerce generates substantial modifications in supply chain structures. The aim of this article is to assess different forms of existing organizations when a store-based sales network coexists with a web site order network. Three main organizational models can be detected: “store-picking”, “dedicated warehouse-picking” and “drop-shipping”. We use a “newsboy” order policy model to compare the advantages of these different models and to note the impact of some parameters on inventory and flow management policies throughout the supply chain. Several effects are presented, particularly those linked to the size of the Internet market in relation to traditional market size and market demand hazards.      

Junctions of singularly degenerating domains with different limit dimensions 1

The asymptotic series for solutions of the mixed boundary-value problem for the Poisson equation in a domain, which is a junction of singularly degenerating domains, are constructed. In this paper, which is the first part of the publication, the three-dimensional problem (“wheel hub with spokes”) and the analogous two-dimensional problems are considered. The methods of matched and compound asymptotic expansions are used. It is shown that a special self-adjoint extension of the operator of the limit problem in the “hub” supplied by the straight-line segments (“limits of spokes”) can be chosen as an asymptotical model of the problem in question; the extension parameters are to be some integral characteristics of the boundary-layer problems. Bibliography: 39 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo. No. 18, pp. 3–78, 1995.     

Human or Cylon? Group testing on ‘Battlestar Galactica’

As in the real world, statistics could have played a significant role in solving the sci-fi problem in “Battlestar Galactica.” However, the consequences for implementing these procedures might have prematurely stifled fans’ enthusiasm for the show because the humanoid Cylons would have been identified earlier. While the series ended in 2009, a prequel, “Caprica,” premieres in 2010. “Caprica” will investigate topics such as how Cylons were first developed by humans. Of course, this makes one wonder if the use of statistical science could have played a role in preventing the Cylon attack on the humans in the first place. One can only hope the producers will ask a statistician to serve as a consultant in the writing of the new show.     

Disagreement point axioms and the egalitarian bargaining solution

We provide new characterizations of the egalitarian bargaining solution on the class of strictly comprehensive n-person bargaining problems. The main axioms used in all of our results are Nash’s IIA and disagreement point monotonicity—an axiom which requires a player’s payoff to strictly increase in his disagreement payoff. For n = 2 these axioms, together with other standard requirements, uniquely characterize the egalitarian solution. For n > 2 we provide two extensions of our 2-person result, each of which is obtained by imposing an additional axiom on the solution. Dropping the axiom of anonymity, strengthening disagreement point monotonicity by requiring player i’s payoff to be a strictly decreasing function of the disagreement payoff of every other player j ≠ i, and adding a “weak convexity” axiom regarding changes of the disagreement point, we obtain a characterization of the class of weighted egalitarian solutions. This “weak convexity” axiom requires that a movement of the disagreement point in the direction of the solution point should not change the solution point. We also discuss the so-called “transfer paradox” and relate it to this axiom.     

A complexity measure for families of binary sequences

In earlier papers finite pseudorandom binary sequences were studied, quantitative measures of pseudorandomness of them were introduced and studied, and large families of “good” pseudorandom sequences were constructed. In certain applications (cryptography) it is not enough to know that a family of “good” pseudorandom binary sequences is large, it is a more important property if it has a “rich”, “complex” structure. Correspondingly, the notion of “f-complexity” of a family of binary sequences is introduced. It is shown that the family of “good” pseudorandom binary sequences constructed earlier is also of high f-complexity. Finally, the cardinality of the smallest family achieving a prescibed f-complexity and multiplicity is estimated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Compactness for a class of hit-and-miss hyperspaces

In the study of some kind of generalized Vietoris-type topologies for the hyperspace of all nonempty closed subsets of a topological space (X, τ), namely the so called Δ-hit-and-miss-topologies with Δ⊇Cl(X) (or Δ-topologies), which was initiated by the second author in 1965, it is obvious, that the non-compactness of such a hyperspace often depends on the non-compactness even in the lower-semifinite topology (induced by the “hit-sets”), which is contained in all hypertopologies of this type. Otherwise, compactness for these topologies is easily obtained from the compactness of (X, τ) by well-known theorems, if the “miss-sets” are induced either by compact or closed subsets. To obtain a similar result for topologies with “miss-sets” generated by subsets with a property which generalizes both, closedness and compactness especially in the non-Hausdorff case, we use consequently a quite set-theoretical lemma, stated at the beginning.     

Some linked subsets of posets

Recently Frank Tall has introduced the notion of a poset being “endowed” which is used in a forcing and reflection type proof to show that certain properties are preserved by forcing with such a poset. Subsequently Bill Fleissner introduced the property of a boolean algebra having a “lynx” and used this in an “Axiom and combinatorics” type proof of the same result. The two notions are basically the same thing. In this paper we prove that some of the usual Cohen algebras have lynxes, that it is consistent that nearly all of them have lynxes and investigate more generally the notion of lynxes. Research supported by NSERC of Canada.     

The semantics of knowledge attributions

The basic idea of conversational contextualism is that knowledge attributions are context sensitive in that a given knowledge attribution may be true if made in one context but false if made in another, owing to differences in the attributors’ conversational contexts. Moreover, the context sensitivity involved is traced back to the context sensitivity of the word “know,” which, in turn, is commonly modelled on the case either of genuine indexicals such as “I” or “here” or of comparative adjectives such as “tall” or “rich.” But contextualism faces various problems. I argue that in order to solve these problems we need to look for another account of the context sensitivity involved in knowledge attributions and I sketch an alternative proposal.     

Expressing and Describing Experiences. A Case of Showing Versus Saying

Experiences are interpreted as conscious mental occurrences that are of phenomenal character. There is already a kind of (weak) intentionality involved with this phenomenal interpretation. A stricter conception of experiences distinguishes between purely phenomenal experiences and intentional experiences in a narrow sense. Wittgenstein’s account of psychological (experiential) verbs is taken over: Usually, expressing mental states verbally is not describing them. According to this, “I believe” can be seen as an expression of one’s own belief, but not as an expression of a belief about one’s belief. Hence, the utterance “I believe it is raining” shows that I believe that it is raining, although it is not said by these words that I believe that it is raining. Thinking thoughts such as “I believe it is raining, but it is not raining” (a variant of Moore’s paradox) is an absurdity between what is already said by silently uttering “It is not raining” and what is shown by silently uttering “I believe it is raining.” The paper agrees with a main result of Wittgenstein’s considerations of Moore’s paradox, namely the view that logical structure, deducibility, and consistency cannot be reduced solely to propositions—besides a logic of propositions, there is, for example, a logic of assertions and of imperatives, respectively.     

Two Characterizations of Optimality in Dynamic Programming

It holds in great generality that a plan is optimal for a dynamic programming problem, if and only if it is “thrifty” and “equalizing.” An alternative characterization of an optimal plan, that applies in many economic models, is that the plan must satisfy an appropriate Euler equation and a transversality condition. Here we explore the connections between these two characterizations.     

The Betti reciprocity principle and the normal boundary component control problem for linear elastic systems

The background for this article is the question of modification of the geometric configuration of an elastic structure by means of “volume” type actuation. In this actuation mode stresses are applied to the elastic body by injection/extraction of a fluid into, or from, a large number of vacuoles in the elastic “matrix” material. Previous articles by the author, and others, have examined this process and studied its effectiveness in the context of a “naive” continuous model. The present paper continues along these lines, exploring “normal boundary component controllability” criterion for determining achievable configurations for the controlled system in the two-dimensional case. Connections with conformal mapping lead to affirmative results for approximate controllability in this sense and Fourier series techniques provide exact controllability results for the case wherein the domain of the uncontrolled system is a two-dimensional disk.      

Hypothetical knowledge and counterfactual reasoning

Samet introduced a notion of hypothetical knowledge and showed how it could be used to capture the type of counterfactual reasoning necessary to force the backwards induction solution in a game of perfect information. He argued that while hypothetical knowledge and the extended information structures used to model it bear some resemblance to the way philosophers have used conditional logic to model counterfactuals, hypothetical knowledge cannot be reduced to conditional logic together with epistemic logic. Here it is shown that in fact hypothetical knowledge can be captured using the standard counterfactual operator “>” and the knowledge operator “K”, provided that some assumptions are made regarding the interaction between the two. It is argued, however, that these assumptions are unreasonable in general, as are the axioms that follow from them. Some implications for game theory are discussed.