A modal logic framework for reasoning about comparative distances and topology

We propose and investigate a uniform modal logic framework for reasoning about topology and relative distance in metric and more general distance spaces, thus enabling the comparison and combination of logics from distinct research traditions such as Tarski’s S4 for topological closure and interior, conditional logics, and logics of comparative similarity. This framework is obtained by decomposing the underlying modal-like operators into first-order quantifier patterns. We then show that quite a powerful and natural fragment of the resulting first-order logic can be captured by one binary operator comparing distances between sets and one unary operator distinguishing between realised and limit distances (i.e., between minimum and infimum). Due to its greater expressive power, this logic turns out to behave quite differently from both S4 and conditional logics. We provide finite (Hilbert-style) axiomatisations and ExpTime-completeness proofs for the logics of various classes of distance spaces, in particular metric spaces. But we also show that the logic of the real line (and various other important metric spaces) is not recursively enumerable. This result is proved by an encoding of Diophantine equations.     

Towards an Account of Epistemic Luck for Necessary Truths

Modal epistemologists parse modal conditions on knowledge in terms of metaphysical possibilities or ways the world might have been. This is problematic. Understanding modal conditions on knowledge this way has made modal epistemology, as currently worked out, unable to account for epistemic luck in the case of necessary truths, and unable to characterise widely discussed issues such as the problem of religious diversity and the perceived epistemological problem with knowledge of abstract objects. Moreover, there is reason to think that this is a congenital defect of orthodox modal epistemology. This way of characterising modal epistemology is however optional. It is shown that one can non-circularly characterise modal conditions on knowledge in terms of epistemic possibilities, or ways the world might be for the target agent. Characterising the anti-luck condition in terms of epistemic possibilities removes the impediment to understanding epistemic luck in the case of necessary truths and opens the door to using these conditions to shed new light on some longstanding epistemological problems.     

On (strong) α-favorability of the Wijsman hyperspace

The Banach-Mazur game as well as the strong Choquet game are investigated on the Wijsman hyperspace from the nonempty player's (i.e. α's) perspective. For the strong Choquet game we show that if X is a locally separable metrizable space, then α has a (stationary) winning strategy on X iff it has a (stationary) winning strategy on the Wijsman hyperspace for each compatible metric on X. The analogous result for the Banach-Mazur game does not hold, not even if X is separable, as we show that α may have a (stationary) winning strategy on the Wijsman hyperspace for each compatible metric on X, and not have one on X. We also show that there exists a separable 1st category metric space such that α has a (stationary) winning strategy on its Wijsman hyperspace. This answers a question of Cao and Junnila (2010) [6].     

Lion and man—can both win?

This paper is concerned with continuous-time pursuit and evasion games. Typically, we have a lion and a man in a metric space: they have the same speed, and the lion wishes to catch the man while the man tries to evade capture. We are interested in questions of the following form: is it the case that exactly one of the man and the lion has a winning strategy?As we shall see, in a compact metric space at least one of the players has a winning strategy. We show that, perhaps surprisingly, there are examples in which both players have winning strategies. We also construct a metric space in which, for the game with two lions versus one man, neither player has a winning strategy. We prove various other (positive and negative) related results, and pose some open problems.     

A simple logic for reasoning about incomplete knowledge

The semantics of modal logics for reasoning about belief or knowledge is often described in terms of accessibility relations, which is too expressive to account for mere epistemic states of an agent. This paper proposes a simple logic whose atoms express epistemic attitudes about formulae expressed in another basic propositional language, and that allows for conjunctions, disjunctions and negations of belief or knowledge statements. It allows an agent to reason about what is known about the beliefs held by another agent. This simple epistemic logic borrows its syntax and axioms from the modal logic KD. It uses only a fragment of the S5 language, which makes it a two-tiered propositional logic rather than as an extension thereof. Its semantics is given in terms of epistemic states understood as subsets of mutually exclusive propositional interpretations. Our approach offers a logical grounding to uncertainty theories like possibility theory and belief functions. In fact, we define the most basic logic for possibility theory as shown by a completeness proof that does not rely on accessibility relations.     

Continuous modal control of linear multicoupled objects

A modal control method is considered in which the spectrum of the closed-loop system is continuously deformed in such a way that the spectrum of the open-loop object transforms into the desired spectrum. The algorithm of the continuous modal control is synthesized. The conditions for spectral control in the method are obtained. The approach is based on similar ideas to those in /1/, but a different class of controls is considered here. Moreover, by using the appratus of Lyapunov functions, specified in the one-parameter family of the deformed spectrum, the deviation between the required spectrum and the closed-loop system spectrum can be minimized in the Euclidean metric, in the case when the wanted spectrum cannot be obtained in the closed-loop system.     

Hypoelliptic heat kernel inequalities on Lie groups

This paper discusses the existence of gradient estimates for the heat kernel of a second order hypoelliptic operator on a manifold. For elliptic operators, it is now standard that such estimates (satisfying certain conditions on coefficients) are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated “Ricci curvature” takes on the value −∞$-\infty$ at points of degeneracy of the semi-Riemannian metric. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting.     

A model updating approach based on design points for unknown structural parameters

Finite element analysis has become an essential tool to estimate structural responses under static and dynamic loads. However, there are a lot of uncertainties in structural properties. For this reason, in many cases, the outcomes of the theoretical and experimental modal analyses do not match. Therefore, the analytical models of the structures need to be updated according to the experimental test results. The commonly used method to get parameters for model updating is experimental modal analysis which provides structural dynamic characteristic (natural frequencies, mode shapes and modal damping ratio). There are many methods available for the updating process. This study addresses an updating algorithm to modify the numerical models by using the design points for unknown structural properties. The proposed method aims to minimize the difference between the analytical and experimental natural frequencies by updating uncertain parameters for each mode and combine them to get an optimum solution. The algorithm is tested on a column and a 2D frame models. These models are investigated by taking the connection rigidity and elasticity modulus as unknown parameters. It is observed that the proposed algorithm gives better results for unknown structural properties compared to the initial values.     

A General Lindström Theorem for Some Normal Modal Logics

There are several known Lindström-style characterization results for basic modal logic. This paper proves a generic Lindström theorem that covers any normal modal logic corresponding to a class of Kripke frames definable by a set of formulas called strict universal Horn formulas. The result is a generalization of a recent characterization of modal logic with the global modality. A negative result is also proved in an appendix showing that the result cannot be strengthened to cover every first-order elementary class of frames. This is shown by constructing an explicit counterexample.     

Gromov–Wasserstein Distances and the Metric Approach to Object Matching

This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison. Objects are viewed as metric measure spaces, and based on ideas from mass transportation, a Gromov–Wasserstein type of distance between objects is defined. This reformulation yields a distance between objects which is more amenable to practical computations but retains all the desirable theoretical underpinnings. The theoretical properties of this new notion of distance are studied, and it is established that it provides a strict metric on the collection of isomorphism classes of metric measure spaces. Furthermore, the topology generated by this metric is studied, and sufficient conditions for the pre-compactness of families of metric measure spaces are identified. A second goal of this paper is to establish links to several other practical methods proposed in the literature for comparing/matching shapes in precise terms. This is done by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers. These lower bounds can be computed in polynomial time. The numerical implementations of the ideas are discussed and computational examples are presented.     

单峰函数最优化问题的进化策略

１．引言 考虑无约束优化问题ｆ（ｘ）,其中ｆ（ｘ）为单峰函数．这类优化问题,不仅包含具有某种凸性的函数的极小化问题,而且包含其它许多问题,例如相容的和不相容的线性方程组的求解,也都可以归结为这类优化问题． 如果函数ｆ（ｘ）的性态良好,各类以梯度为基础的算法无疑是求解问题（Ｐ）的首选方法．假若问题（Ｐ）不可微,或者虽然可微,但 ｆ（ｘ）的 Ｈｅｓｓｉａｎ阵高度病态,则应该另辟蹊径． 近年来颇受人们重视的进化类算法,由于不使用梯度,计算过程对函数的性态依赖性较小,具有适应范围广、鲁棒性强的优点,而且特别…     

A Priori knowledge contextualised and Benacerraf’s dilemma

In this article, I discuss Hawthorne’s contextualist solution to Benacerraf’s dilemma. He wants to find a satisfactory epistemology to go with realist ontology, namely with causally inaccessible mathematical and modal entities. I claim that he is unsuccessful. The contextualist theories of knowledge attributions were primarily developed as a response to the skeptical argument based on the deductive closure principle. Hawthorne uses the same strategy in his attempt to solve the epistemologist puzzle facing the proponents of mathematical and modal realism, but this problem is of a different nature than the skeptical one. The contextualist theory of knowledge attributions cannot help us with the question about the nature of mathematical and modal reality and how they can be known. I further argue that Hawthorne’s account does not say anything about a priori status of mathematical and modal knowledge. Later, Hawthorne adds to his account an implausible claim that in some contexts a gettierized belief counts as knowledge.     

Hoisting Manipulation for Flying Cranes

A control strategy for hoisting drives of crane systems is discussed. Based on modal coupling control, the desired hoisting velocity is manipulated by superposition of a suitably modulated motion in order to suppress the so–called spaghetti–problem of hoisting–induced pendulations. For a 3–dimensional multibody system featuring the flying–crane –concept the extension of this control strategy to more complicated systems is exemplified. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

可行策略对应的图像拓扑下广义博弈Nash平衡的稳定性

以往关于广义博弈Nash平衡的稳定性的研究,均利用可行策略映射之间的一致度量.现考虑在更弱的度量下,利用可行策略映射图像之间的Hausdorff距离定义度量.在此弱图像拓扑下,证明了广义博弈空间的完备性,以及Nash平衡映射的上半连续性和紧性,进而得到广义博弈Nash平衡的通有稳定性.即在Baire分类的意义下,大多数的广义博弈都是本质的.     

基于参与主体异质性条件下囚徒困境合作演化博弈模型研究

利用演化博弈理论,对参与主体异质性条件下的囚徒困境模型进行了探讨,求出了满足不同条件下的演化稳定策略,并对种群中个体异质性对演化稳定策略的影响进行了分析,得出种群中选择相同策略的个体异质性差异越大,参与个体选择合作行为作为演化稳定策略的可能性就越大.极端地,当个体的异质性趋向于无穷大时,合作成为唯一的演化稳定占优策略,为现实大多数合作系统中能保持长期的一种合作稳定状态提供了合理地解释.     

A model-adaptive coupling strategy of non-linear and model-reduced linear subsystems

This article introduces a new three-level discretisation strategy in which the entire structure is divided into subsystems according to the required modelling accuracy(continuum, structural and substituted model level). This note especially focuses on the transition between the structural and the substituted model level. The principle idea is to separate the structure strictly into linear and non-linear substructures by the concept of coupling forces. Additionally the linear parts are reduced by modal reduction. The method is verified by a transient analysis of a bridge arch having regard to the evolution of damage. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong metric subregularity of mappings in variational analysis and optimization

Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong (metric) regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping F acting between metric spaces is stable under perturbations of the form $f+F$, where f is a function with a small calmness constant. This result is parallel to the Lyusternik–Graves theorem for metric regularity and to the Robinson theorem for strong regularity, where the perturbations are represented by a function f with a small Lipschitz constant. Then we study perturbation stability of the same kind for mappings acting between Banach spaces, where f is not necessarily differentiable but admits a set-valued derivative-like approximation. Strong metric q-subregularity is also considered, where q is a positive real constant appearing as exponent in the definition. Rockafellar's criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite-dimensional spaces. A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Fréchet coderivative, and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton's method for solving generalized equations are considered including inexact and semismooth methods, for which superlinear convergence is shown under strong metric subregularity. As applications to optimization, a characterization of the strong metric subregularity of the KKT mapping is obtained, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.     

Computational experience with known variable metric updates

Variable metric methods from the Broyden family are well known and commonly used for unconstrained minimization. These methods have good theoretical and practical convergence properties which depend on a selection of free parameters. We demonstrate, using extensive computational experiments, the influence of both the Biggs stabilization parameter and Oren scaling parameter on 12 individual variable metric updates, two of which are new. This paper focuses on a class of variable metric updates belonging to the so-called preconvex part of the Broyden family. These methods outperform the more familiar BFGS method. We also experimentally demonstrate the efficiency of the controlled scaling strategy for problems of sufficient size and sparsity.     

Amenability,locally finite spaces,and bi-lipschitz embeddings

We define the isoperimetric constant for any locally finite metric space and we study the property of having isoperimetric constant equal to zero. This property, called Small Neighborhood property, clearly extends amenability to any locally finite space. Therefore, we start making a comparison between this property and other notions of amenability for locally finite metric spaces that have been proposed by Gromov, Lafontaine and Pansu, by Ceccherini-Silberstein, Grigorchuk and de la Harpe and by Block and Weinberger. We discuss possible applications of the property SN in the study of embedding a metric space into another one. In particular, we propose three results: we prove that a certain class of metric graphs that are isometrically embeddable into Hilbert spaces must have the property SN. We also show, by a simple example, that this result is not true replacing property SN with amenability. As a second result, we prove that many spaces with uniform bounded geometry having a bi-lipschitz embedding into Euclidean spaces must have the property SN. Finally, we prove a Bourgain-like theorem for metric trees: a metric tree with uniform bounded geometry and without property SN does not have bi-lipschitz embeddings into finite-dimensional Hilbert spaces.