Green and Generalized Green’s Functionals of Linear Local and Nonlocal Problems for Ordinary Integro-differential Equations

A linear, completely nonhomogeneous, generally nonlocal, multipoint problem is investigated for a second-order ordinary integro-differential equation with generally nonsmooth coefficients, satisfying some general conditions like p-integrability and boundedness. A system of three integro-algebraic equations named the adjoint system is introduced for the solution. The solvability conditions are found by the solutions of the homogeneous adjoint system in an “alternative theorem”. A version of a Green’s functional is introduced as a special solution of the adjoint system. For the problem with a nontrivial kernel also a notion of a generalized Green’s functional is introduced by a projection operator defined on the space of solutions. It is also shown that the classical Green and Cauchy type functions are special forms of the Green’s functional. The author passed away in 2006 prior to publication of the article.     

Program explanations and causal relevance

Frank Jackson and Philip Pettit have defended a non-reductive account of causal relevance known as the ‘program explanation account’. Allegedly, irreducible mental properties can be causally relevant in virtue of figuring in non-redundant program explanations which convey information not conveyed by explanations in terms of the physical properties that actually do the ‘causal work’. I argue that none of the possible ways to spell out the intuitively plausible idea of a program explanation serves its purpose, viz., defends non-reductive physicalism against Jaegwon Kim’s Causal Exclusion Argument according to which non-reductive physicalism is committed to epiphenomenalism because irreducible mental properties are ‘screened off’ from causal relevance by their physical realizers. Jackson and Pettit’s most promising explication of a program explanation appeals to the idea of invariance of effect under variation of realization, but I show that invariance of effect under variation of realization is neither necessary nor sufficient for causal relevance.     

Getting on Top of Oneself: Comments on <Emphasis Type="Italic">Self-Expression</Emphasis>

This paper is a critical review of Mitchell Green’s Self-Expression. The principal focus is on Green’s contention that all expression is at route, a form of signalling by an agent or by some mechanism of the organism which has been evolutionary selected for signalling. Starting from the idea that in some but not all expression an agent seeks to express his or her self, I question the centrality of communication to the idea of expression.     

Metrics of Hyperbolic Type on Bounded Fractals

On the bounded Sierpinski gasket F we use the set of essential fixed points V ₀ as a boundary and consider the fractal Brownian motion on F killed in V ₀. The corresponding Dirichlet–Laplacian is described in terms of a kind of hyperbolic distance, a metric which explodes near the boundary. We consider Harnack inequalities, Green’s function estimates and (random) products of matrices defining the local energy of harmonic functions. Supported by the DFG research group ‘Spektrale Analysis, asymptotische Verteilungen und stochastische Dynamik.’     

‘Seeing is believing’ and ‘believing is seeing’

The principal concern of my paper is a distinction between two ways of appreciating works of art, characterised here in terms of the phrases ‘seeing is believing’ and ‘believing is seeing’. I examine this distinction in the light of an epistemological requirement at times at least grounded in what David Davies, in his Art as Performance, refers to as the ‘common sense theory of art appreciation’ in order to assess exactly what aspect of the philosophical approach generally known as aesthetic empiricism his account commits him to reject. I argue that the ‘experiential requirement’, if only conceived in a slightly broader way than is usual, might very well have an important role to play not only in the appropriate appreciation of works that do not demonstrate the need for such a requirement (primarily works of late modern and conceptual art), but also in the ontological account Davies himself favours.     

Bar-On on Self-Knowledge and Expression

I critically discuss the account of self-knowledge presented in Dorit Bar-On’s Speaking My Mind (OUP 2004), focusing on Bar-On’s understanding of what makes our capacity for self-knowledge puzzling and on her ‘neo-expressivist’ solution to the puzzle. I argue that there is an important aspect of the problem of self-knowledge that Bar-On’s account does not sufficiently address. A satisfying account of self-knowledge must explain not merely how we are able to make accurate avowals about our own present mental states, but how we can reasonably regard ourselves as entitled to claim self-knowledge. Addressing this aspect of the problem of self-knowledge requires confronting questions about the metaphysical nature of mental states, questions that Bar-On’s approach seeks to avoid.     

Self-expression,Expressiveness, and Sincerity

This paper examines some aspects of Mitchell Green’s account of self-expression. I argue that Green fails to address the distinction between success and evidential notions of expression properly, which prevents him from adequately discussing the relation between these notions. I then consider Green’s explanation of how a speech act shows what is within, i.e., because of the liabilities one incurs and argue that this is false. Rather, the norms governing speech acts and liabilities incurred give us reason to think that the speaker is in a particular state of mind. It thus supports an evidential rather than success notion. Finally, I suggest that it is because of the sincerity of what is said, rather than the liabilities incurred, that you show what is within.     

A new integral representation for quasi-periodic scattering problems in two dimensions

Boundary integral equations are an important class of methods for acoustic and electromagnetic scattering from periodic arrays of obstacles. For piecewise homogeneous materials, they discretize the interface alone and can achieve high order accuracy in complicated geometries. They also satisfy the radiation condition for the scattered field, avoiding the need for artificial boundary conditions on a truncated computational domain. By using the quasi-periodic Green’s function, appropriate boundary conditions are automatically satisfied on the boundary of the unit cell. There are two drawbacks to this approach: (i) the quasi-periodic Green’s function diverges for parameter families known as Wood’s anomalies, even though the scattering problem remains well-posed, and (ii) the lattice sum representation of the quasi-periodic Green’s function converges in a disc, becoming unwieldy when obstacles have high aspect ratio.     

Dancy Cartwright: Particularism in the philosophy of science

This paper aims to explore the space of possible particularistic approaches to Philosophy of Science by examining the differences and similarities between Jonathan Dancy’s moral particularism—as expressed in both his earlier writings (e.g., Moral Reasons, 1993), and, more explicitly defended in his book Ethics without Principles (2004)—and Nancy Cartwright’s particularism in the philosophy of science, as defended in her early collection of essays, How the Laws of Physics Lie (1983), and her later book, The Dappled World: A Study of the Boundaries of Science (1999). I shall argue that Dancy’s particularism is more radical, but also more plausible, than Cartwright’s, concluding that we have good reason to embrace a scientific particularism that is far closer to Dancy’s ethical particularism than any view defended by Nancy Cartwright, or any other philosopher from the ‘Stanford school’ of scientific theory.     

The Blackwell and Dubins Theorem and Rényi’s Amount of Information Measure: Some Applications

This survey paper provides first for an overview of how quantum-like concepts could be used in macroscopic environments like economics. The paper then argues for the use of the concept of a quantum mechanical wave function as an ‘information wave function’. A rationale is provided on why such interpretation is reasonable. After having defined the ‘information wave function’, Ψ(q), we argue how | Ψ(q)| ² can be interpreted as a Radon-Nikodym derivative. We consider how we can connect, using the | Ψ(q)| ², the Blackwell and Dubins (Ann. Math. Stat. 33:882–886, 1961) Theorem with Rényi’s (Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1961) measure of quantity of information. We also define ‘ambiguity of information’ and ‘multi-sourced information’.     

Hamiltonian systems as selfdual equations

Hamiltonian systems with various time boundary conditions are formulated as absolute minima of newly devised non-negative action functionals obtained by a generalization of Bogomolnyi’s trick of ‘dcompleting squares’. Reminiscent of the selfdual Yang-Mills equations, they are not derived from the fact that they are critical points (i.e., from the corresponding Euler-Lagrange equations) but from being zeroes of the corresponding non-negative Lagrangians. A general method for resolving such variational problems is also described and applied to the construction of periodic solutions for Hamiltonian systems, but also to study certain Lagrangian intersections.      

Addenda to “The entropy formula for linear heat equation”

We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li-Yau ’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to ℝ ⁿ .In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman ’s entropy in the case of Riemann surfaces.     

Modal Tense and the Absolutely Unrestricted Quantifier

In this paper, I examine Takashi Yagisawa’s response to van Inwagen’s ontic objection against David Lewis. Van Inwagen criticizes Lewis’s commitment to the absolutely unrestricted sense of ‘there is,’ and Yagisawa claims that by adopting modal tenses he avoids commitment to absolutely unrestricted quantification. I argue that Yagisawa faces a problem parallel to the one Lewis faces. Although Yagisawa officially rejects the absolutely unrestricted sense of a quantifying expression, he is still committed to the absolutely unrestricted sense of ‘is _a real.’     

Roof duality,complementation and persistency in quadratic 0–1 optimization

The paper is concerned with the ‘primal’ problem of maximizing a given quadratic pseudo-boolean function. Four equivalent problems are discussed—the primal, the ‘complementation’, the ‘discrete Rhys LP’ and the ‘weighted stability problem of a SAM graph’. Each of them has a relaxation—the ‘roof dual’, the ‘quadratic complementation,’ the ‘continuous Rhys LP’ and the ‘fractional weighted stability problem of a SAM graph’. The main result is that the four gaps associated with the four relaxations are equal. Furthermore, a solution to any of these problems leads at once to solutions of the other three equivalent ones. The four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem. The optimal solutions of the ‘roof-dual’ define ‘best’ linear majorantsp(x) off, having the following persistency property: if theith coefficient inp is positive (negative) thenx _i=1 (0) in every optimum of the primal problem. Several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal. On the other hand, a class of gap-free functions (properly including the supermodular ones) is exhibited.     

Expressing One’s Mind

Remarks such as ‘I am in pain’ and ‘I think that it’s raining’ are puzzling, since they seem to literally describe oneself as being in pain or having a particular thought, but their conditions of use tend to coincide with unequivocal expressions of pain or of that thought. This led Wittgenstein, among others, to treat such remarks as expressing, rather than as reporting, one’s mental states. Though such expressivism is widely recognized as untenable, Bar-On has recently advanced a neo-expressivist view, on which such remarks exhibit characteristics of both expressions of mental states and reports of those states. I argue against any attempt to see such remarks as both reporting and expressing the same mental states, and that a correct account rests on distinguishing the truth conditions of such remarks from their conditions of use.     

S-convexity revisited (fuzzy): long version

In this revisional article, we criticize (strongly) the use made by Medar et al., and those whose work they base themselves on, of the name ‘convexity’ in definitions which intend to relate to convex functions, or cones, or sets, but actually seem to be incompatible with the most basic consequences of having the name ‘convexity’ associated to them. We then believe to have fixed the ‘denominations’ associated with Medar’s (et al.) work, up to a point of having it all matching the existing literature in the field [which precedes their work (by long)]. We also expand his work scope by introducing s ₁-convexity concepts to his group of definitions, which encompasses only convex and its proper extension, s ₂-convex, so far. This article is a long version of our previous review of Medar’s work, published by FJMS (Pinheiro, M.R.: S-convexity revisited. FJMS, 26/3, 2007).     

Experiments with an interactive paired comparison simplex method for molp problems

In this paper, an interactive paired comparison simplex based method formultiple objective linear programming (MOLP) problems is developed and compared to other interactive MOLP methods. The decision maker (DM)’s utility function is assumed to be unknown, but is an additive function of his known linearized objective functions. A test for ‘utility efficiency’ for MOLP problems is developed to reduce the number of efficient extreme points generated and the number of questions posed to the DM. The notion of ‘strength of preference ’ is developed for the assessment of the DM’s unknown utility function where he can express his preference for a pair of extreme points as ‘strong ’, ‘weak ’, or ‘almost indifferent ’. The problem of ‘inconsistency of the DM’ is formalized and its resolution is discussed. An example of the method and detailed computational results comparing it with other interactive MOLP methods are presented. Several performance measures for comparative evaluations of interactive multiple objective programming methods are also discussed. All rights reserved. This study, or parts thereof, may not be reproduced in any form without written permission of the authors.     

Moral coherence,moral worth and explanations of moral motivation

Moral internalism and moral externalism compete over the best explanation of the link between judgment and relevant motivation but, it is argued, they differ at best only verbally. The internalist rational-conceptual nature of the link’ as accounted by M. Smith in The Moral Problem is contrasted to the externalist, also rational, link that requires in addition support from the agent’s psychological-dispositional profile; the internalist link, however, is found to depend crucially on a, similarly to the externalist, psychologically ‘loaded’ profile. It is also argued that the differentiation of the two competing explanations is insufficient partly because they both fail to consider crucial quantitative parameters of the judgment-motivation link. Such parameters become very important particularly in the light of Smith’s claim that this link is grounded on the observable “striking fact” where changes in the set of one’s moral beliefs systematically bring about changes in one’s moral behavior. Examples of algorithms measuring moral coherence and moral worth are provided to serve as evidence for what it comes down to, vis-à-vis the alleged fact, only a verbal dispute between the two camps. Finally, the ‘misfiring’ of these explanations is understood in connection to the irreducibility of concepts such as ‘moral worth’, and/or, ‘moral sensitivity’.     

Moore’s Paradox,Truth and Accuracy

G. E. Moore famously observed that to assert ‘I went to the pictures last Tuesday but I do not believe that I did’ would be ‘absurd’. Moore calls it a ‘paradox’ that this absurdity persists despite the fact that what I say about myself might be true. Krista Lawlor and John Perry have proposed an explanation of the absurdity that confines itself to semantic notions while eschewing pragmatic ones. We argue that this explanation faces four objections. We give a better explanation of the absurdity both in assertion and in belief that avoids our four objections.