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.     

Concentration of measure and isoperimetric inequalities in product spaces

The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product Ω^N of probability spaces has measure at least one half, “most” of the points of Ωn are “close” to A. We proceed to a systematic exploration of this phenomenon. The meaning of the word “most” is made rigorous by isoperimetrictype inequalities that bound the measure of the exceptional sets. The meaning of the work “close” is defined in three main ways, each of them giving rise to related, but different inequalities. The inequalities are all proved through a common scheme of proof. Remarkably, this simple approach not only yields qualitatively optimal results, but, in many cases, captures near optimal numerical constants. A large number of applications are given, in particular to Percolation, Geometric Probability, Probability in Banach Spaces, to demonstrate in concrete situations the extremely wide range of application of the abstract tools. Dedicated to Vitali Milman     

On duality gap in linear conic problems

In their paper “Duality of linear conic problems” Shapiro and Nemirovski considered two possible properties (A) and (B) for dual linear conic problems (P) and (D). The property (A) is “If either (P) or (D) is feasible, then there is no duality gap between (P) and (D)”, while property (B) is “If both (P) and (D) are feasible, then there is no duality gap between (P) and (D) and the optimal values val(P) and val(D) are finite”. They showed that (A) holds if and only if the cone K is polyhedral, and gave some partial results related to (B). Later Shapiro conjectured that (B) holds if and only if all the nontrivial faces of the cone K are polyhedral. In this note we mainly prove that both the “if” and “only if” parts of this conjecture are not true by providing examples of closed convex cone in \mathbbR⁴{\mathbb{R}^{4}} for which the corresponding implications are not valid. Moreover, we give alternative proofs for the results related to (B) established by Shapiro and Nemirovski.     

The Concavity Assumption on Felicities and Asymptotic Dynamics in the RSS Model

An analysis of the RSS model in mathematical economics involves the study of an infinite-horizon variational problem in discrete time. Under the assumption that the felicity function is upper semicontinuous and “supported” at the value of the maximally-sustainable level of a production good, we report a generalization of results on the equivalence, existence and asymptotic convergence of optimal trajectories in this model. We consider two parametric specifications, and under the second, identify a “symmetry” condition on the zeroes of a “discrepancy function” underlying the objective function that proves to be necessary and sufficient for the asymptotic convergence of good programs. With a concave objective function, as is standard in the antecedent literature, we show that the symmetry condition reduces to an equivalent “non-interiority” condition.     

The return of the linear search problem

The linear search problem concerns a search on the real line for a point selected at random according to a given probability distribution. The search begins at zero and is made by a continuous motion with constant speed, first in one direction and then the other. The problem is to determine when it is possible to devise a “best” search plan. In former papers the best plan has been selected according to the criterion of minimum expected path length. In this paper we consider a more general, nonlinear criterion for a “best” plan and show that the substantive requirements of the earlier results are not affected by these changes.     

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.     

Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L ²(L ²)”- and “DG”-norm formed by the “L ²(H ¹)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q.     

Dynamic Game Model of Endogenous Growth with Consumption Externalities

This paper introduces consumption externalities into an endogenous growth model of common capital accumulation and characterizes balanced growth equilibria. Contrary to the standard argument in previous studies, we show that the growth rate in a feedback Nash equilibrium can be higher than that in an open-loop Nash equilibrium if agents strongly admire the consumption of others. This result is irrelevant to whether preferences exhibit “keeping up with the Joneses” or “running away from the Joneses”.     

Reduction of hugoniot-maslov chains for trajectories of solitary vortices of the “shallow water” equations to the hill equation

According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only three types of singularities that are in general position and have the property of “structure self-similarity and stability.” Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations, we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to predict the trajectory of the vortex center if we know its observable part. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66.     

Converging Marriage in Honey-Bees Optimization and Application to Stochastic Dynamic Programming

In this paper, we first refine a recently proposed metaheuristic called “Marriage in Honey-Bees Optimization” (MBO) for solving combinatorial optimization problems with some modifications to formally show that MBO converges to the global optimum value. We then adapt MBO into an algorithm called “Honey-Bees Policy Iteration” (HBPI) for solving infinite horizon-discounted cost stochastic dynamic programming problems and show that HBPI also converges to the optimal value.     

Dual effect free stochastic controls

In stochastic optimal control, a key issue is the fact that “solutions” are searched for in terms of “closed-loop control laws” over available information and, as a consequence, a major potential difficulty is the fact that present control may affect future available information. This is known as the “dual effect” of control. Our main result consists in characterizing the maximal set of closed-loop control laws containing open-loop ones and for which the information provided by observations closed with such a feedback remains fixed. We give more specific results in the two following cases: multi-agent systems and discrete time stochastic input-output systems with dynamic information structure.     

Provability interpretations of modal logic

We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev ^* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive.     

Spectrum of operators in ideal spaces

One considers “weighted translation” operators in ideal Banach spaces. It is proved that if the translation is aperiodic (the set of periodic points has measure zero), then the spectrum of such an operator is rotationinvariant. This result can be extended (under certain additional restrictions) to “weighted translation” operators acting in regular subspaces of ideal spaces, in particular, to operators in Hardy spaces. In this note we prove the rotation-invariance of the spectrum of aperiodic operators of “weighted translation” in ideal spaces and uniform B-algebras. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 196–198, 1976.     

<Emphasis Type="Italic">g</Emphasis>-metrizable spaces and the images of semi-metric spaces

In this paper, we prove that a space X is a g-metrizable space if and only if X is a weak-open, π and σ-image of a semi-metric space, if and only if X is a strong sequence-covering, quotient, π and mssc-image of a semi-metric space, where “semi-metric” can not be replaced by “metric”. This project was supported by NNSF of China (No. 10571151 and 10671173).     

The sweet mystery of compatibilism

Any satisfactory account of freedom must capture, or at least permit, the mysteriousness of freedom—a “sweet” mystery involving a certain kind of ignorance rather than a “sour” mystery of unintelligibility, incoherence, or unjustifiedness. I argue that compatibilism can capture the sweet mystery of freedom. I argue first that an action is free if and only if a certain “rationality constraint” is satisfied, and that nothing in standard libertarian accounts of freedom entails its satisfaction. Satisfaction of this constraint is consistent with the universal causal predetermination of action (UCP). If UCP is true and the rationality constraint satisfied, there’s a sense in which our actions are explanatorily (though not necessarily causally) overdetermined. While it seems plausible (given UCP) that our actions are so overdetermined, it seems utterly mysterious why they should be so overdetermined. Compatibilism’s capacity to accommodate this mystery is a mark in its favor.     

Variation of diametrically symmetric or elliptically schlicht conformal mappings

A domain is called diametrically symmetric if it contains with each point its antipodal point on the Riemann sphere. We derive a variational formula for schlicht conformal mappings of such domains onto domains of the same type. This gives an analogue of a classical variational formula of Duren and Schiffer, which is in some sense an “elliptic analogue” of the “hyperbolic case” of Duren and Schiffer.     

Approximate equivalence in von Neumann algebras

One formulation of D. Voiculescu's theorem on approximate unitary equivalence is that two unital representations π and ρ of a separable C*-algebra are approximately unitarily equivalent if and only if rank οπ = rank ορ. We study the analog when the ranges of π and ρ are contained in a von Neumann algebra R, the unitaries inducing the approximate equivalence must come from R, and "rank" is replaced with "R -rank" (defined as the Murray-von Neumann equivalence of the range projection).     

A proximal method for identifying active manifolds

The minimization of an objective function over a constraint set can often be simplified if the “active manifold” of the constraints set can be correctly identified. In this work we present a simple subproblem, which can be used inside of any (convergent) optimization algorithm, that will identify the active manifold of a “prox-regular partly smooth” constraint set in a finite number of iterations.     

A homological spectral sequence of group products

In this paper the “residual complex” is defined when a group and its subgroups act on a complex. With its aid a homological spectral sequence of group products is given. And the author makes a concentrated study of the structure of the residual complex and proves that it becomes a clear “step complex” if the group can be expressed as an amalgamated free product of its subgroups. Project supported by the National Natural Science Foundation of China