Effective elastic moduli and stress concentrator factors in random structure aligned fiber composites

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically uniform random set of aligned fibers. Effective elastic moduli as well as the stress concentrator factors in the components are estimated. The micromechanical approach is based on the Green’s function technique as well as on the generalization of the “multiparticle effective field method” (MEFM, see for references, Buryachenko [1]). The refined version of the MEFM takes into account the variation of the effective fields acting on each pair of fibers. The dependence of effective elastic moduli and stress concentrator factors on the radial distribution function of the fiber locations is analyzed. Received: October 20, 2004     

Towards a new theory of confirmation

Any sequence of events can be “explained” by any of an infinite number of hypotheses. Popper describes the “logic of discovery” as a process of choosing from a hierarchy of hypotheses the first hypothesis which is not at variance with the observed facts. Blum and Blum formalized these hierarchies of hypotheses as hierarchies of infinite binary sequences and imposed on them certain decidability conditions. In this paper we also consider hierarchies of infinite binary sequences but we impose only the most elementary Bayesian considerations. We use the structure of such hierarchies to define “confirmation”. We then suggest a definition of probability based on the amount of confirmation a particular hypothesis (i.e. pattern) has received. We show that hypothesis confirmation alone is a sound basis for determining probabilities and in particular that Carnap’s logical and empirical criteria for determining probabilities are consequences of the confirmation criterion in appropriate limiting cases.     

Flexible servers in tandem lines with setup costs

We study the dynamic assignment of flexible servers to stations in the presence of setup costs that are incurred when servers move between stations. The goal is to maximize the long-run average profit. We provide a general problem formulation and some structural results, and then concentrate on tandem lines with two stations, two servers, and a finite buffer between the stations. We investigate how the optimal server assignment policy for such systems depends on the magnitude of the setup costs, as well as on the homogeneity of servers and tasks. More specifically, for systems with either homogeneous servers or homogeneous tasks, small buffer sizes, and constant setup cost, we prove the optimality of “multiple threshold” policies (where servers’ movement between stations depends on both the number of jobs in the system and the locations of the servers) and determine the values of the thresholds. For systems with heterogeneous servers and tasks, small buffers, and constant setup cost, we provide results that partially characterize the optimal server assignment policy. Finally, for systems with larger buffer sizes and various service rate and setup cost configurations, we present structural results for the optimal policy and provide numerical results that strongly support the optimality of multiple threshold policies.     

A (2+1)-dimensional fermion in the Coulomb and magnetic field backgrounds

In the problem of a two-dimensional hydrogen-like atom in a magnetic field background, we construct quasi-classical solutions and the energy spectrum of the Dirac equation in a strong Coulomb field and in a weak constant homogeneous magnetic field in 2+1 dimensions. We find some “exact” solutions of the Dirac and Pauli equations describing the “spinless” fermions in strong Coulomb fields and in homogeneous magnetic fields in 2+1 dimensions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 105–118, April, 1999.     

Weyl-Eddington-Einstein affine gravity in the context of modern cosmology

We propose new models of the “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposed method for obtaining the geometry using the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein theory with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) meson, and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the Lagrangian determines further details of the theory, for example, the nature of the fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, dark energy must also arise. The basic parameters of the theory (cosmological constant, mass, possible dimensionless constants) are theoretically indeterminate, but in the framework of modern “multiverse” ideas, this is more a virtue than a defect. We consider further extensions of the affine models and in more detail discuss approximate effective (“physical”) Lagrangians that can be applied to the cosmology of the early Universe.     

Branes with asymmetric geometries in the bulk and localization of scalar fields

The model of a domain wall (“thick brane”) in a noncompact five-dimensional space-time with asymmetric geometries of AdS type aside the brane is proposed. This model is generated by fermion self-interaction in the presence of gravity. Asymmetric geometries in the bulk are provided by a space defect in the scalar field potential and the related defect of cosmological constant. The possibility of localization of scalar modes on such “thick branes” is studied. Bibliography: 21 titles.     

The Pepsi Putnam Challenge

Opening a copy of The Mathematical Intelligencer you may ask yourself uneasily, “What is this anyway—a mathematical journal, or what?” Or you may ask, “Where am I?” Or even “Who am I?” This sense of disorientation is at its most acute when you open to Cohn Adam’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be hannless.     

Potato kugel

LetP be a solid, homogeneous, compact, connected “potato” in space which attracts each point outside it (according to Newton’s law) as if all its mass were concentrated at a single point. Answering a question of Lee Rubel, we show thatP is a ball. The same conclusion is also obtained under substantially weakened hypotheses.     

A Probabilistic Approach to Carne’s Bound

Carne’s bound is a sharp inequality controlling the transition probabilities for a discrete reversible Markov chain (Section 1). Its ordinary proof uses spectral techniques which look as efficient as miraculous. Here we present a new proof, comparing a “drift” for ways “out” and “back”, to get the gaussian part of the bound (Section 2), and using a conditioning technique to get the flight factor (Section 4). Moreover we show how our proof is more “supple” than Carne’s one and may generalize (Section 3.2).      

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.     

Copulas: Tales and facts

“But he does not wear any clothes” said the little child in Hans Christian Andersen’s “The Emperor’s New Clothes.”Mikosch’s research is partially supported by the Danish Research Council (SNF) GrantNo 21-04-0400. This is a discussion paper which was initiated at the 4th InternationalConference on Extreme Value Analysis in Gothenburg, 15–19 August, 2005; see.     

Influence measures and robust estimators of dependence in multivariate extremes

We develop a simple influence measure to assess whether Bayesian estimators in multivariate extreme value problems are sensitive to outliers. The proposed measure is easy to compute by importance sampling and successfully captures two effects on the functional: the “data effect” and the “parameter uncertainty effect”. We also propose a new Bayesian estimator which is easy to implement and is robust. The methods are tested and illustrated using simulated data and then applied to stock market data.     

Sparse second moment analysis for elliptic problems in stochastic domains

We consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations. Assuming normal perturbations with small amplitude and known mean field and two-point correlation function, we derive, using a second order shape calculus, deterministic equations for the mean field and the two-point correlation function of the random solution for a model Dirichlet problem which are 3rd order accurate in the boundary perturbation size. Using a variational boundary integral equation formulation on the unperturbed, “nominal” boundary and a wavelet discretization, we present and analyze an algorithm to approximate the random solution’s mean and its two-point correlation function at essentially optimal order in essentially work and memory, where N denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain. This work was supported by the EEC Human Potential Programme under contract HPRN-CT-2002-00286, “Breaking Complexity.” Work initiated while HH visited the Seminar for Applied Mathematics at ETH Zürich in the Wintersemester 2005/06 and completed during the summer programme CEMRACS2006 “Modélisation de l’aléatoire et propagation d’incertitudes” in July and August 2006 at the C.I.R.M., Marseille, France.     

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).      

Centers of Mal’tsev and alternative algebras

Let ϕ be an associative commutative ring with unity containing 1/6. Let A and B be a free Mal’tsev and a free alternative ϕ-algebras on a set of k≥6 free generators, respectively. We construct nonzero homogeneous elements of degree 7 belonging to an annihilatorAnnA of A, and nonzero homogeneous elements of degree 7 belonging to the center Z(B) of B. It is shown that a nilpotent Mal’tsev algebra of index 8 on a set of 6 generators has no faithful representation. Supported by RFFR grant No. 96-01-01511, and by the Program “Universities of Russia: Fundamental Research.” Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 613–635, September–October, 1999.     

What Mary’s Aboutness Is About

The aim of this paper is to reinforce anti-physicalism by extending the “hard problem” to a specific kind of intentional states. For reaching this target, I investigate the mental content of the new intentional states of Jackson’s Mary. I proceed in the following way: I start analyzing the knowledge argument, which highlights the “hard problem” tied to phenomenal consciousness. In a second step, I investigate a powerful physicalist reply to this argument: the phenomenal concept strategy. In a third step, I propose a constitutional account of phenomenal concepts that captures the Mary scenario adequately, but implies anti-physicalist referents. In a last step, I point at the ramifications constitutional phenomenal concepts have on the constitution of Mary’s new intentional states. Therefore, by focusing the attention on phenomenal concepts, the so-called “hard problem” of consciousness will be carried over to the alleged “easy problem” of intentional states as well.     

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.     

Why vagueness is a mystery

This paper considers two “mysteries” having to do with vagueness. The first pertains to existence. An argument is presented for the following conclusion: there are possible cases in which ‘There exists something that is F’ is of indeterminate truth-value and with respect to which it is not assertable that there are borderline-cases of “being F.” It is contended that we have no conception of vagueness that makes this result intelligible. The second mystery has to do with “ordinary” vague predicates, such as ‘tall’. An argument is presented for the conclusion that although there are people who are “tall to degree 1”—definitely tall, tall without qualification—, no greatest lower bound can be assigned to the set of numbers n such that a man who is n centimeters tall is tall to degree 1. But, since this set is bounded from below, this result seems to contradict a well-known property of the real numbers.     

On equilibrium in pure strategies in games with many players

We demonstrate that, if there are sufficiently many players, any Bayesian equilibrium of an incomplete information game can be “ε-purified” . That is, close to any Bayesian equilibrium there is an approximate Bayesian equilibrium in pure strategies. Our main contribution is obtaining this result for games with a countable set of pure strategies. In order to do so we derive a mathematical result, in the spirit of the Shapley–Folkman Theorem, permitting countable strategy sets. Our main assumption is a “large game property,” dictating that the actions of relatively small subsets of players cannot have large affects on the payoffs of other players. E. Cartwright and M. Wooders are indebted to Phillip Reny, Frank Page and two anonymous referees for helpful comments.     

A simple analytic solution for tachyon condensation

We discuss a new, simple analytic solution for tachyon condensation in open string field theory. In particular, the new solution requires no regularization or “phantom term,” and the proof of Sen’s first conjecture requires only a few lines of calculation.