Pritchard’s angst

In this paper, I raise some questions about Pritchard’s (2005) internalist argument for scepticism. I argue that his internalism begs the question in support of scepticism. Correlatively I advance what I take to be a better internalist argument for scepticism, one that leaves open the possibility of empirically adjudicating sceptical hypotheses. I close by discussing what it means to be an internalist.     

Characterization of Transfer Functions of Pritchard–Salamon or Other Realizations with a Bounded Input or Output Operator

We show that the transfer functions that have a (continuoustime) well-posed realization with a bounded input operator are exactly those that are strong-H² (plus constant feedthrough) over some right half-plane. The dual condition holds iff the transfer function has a realization with a bounded output operator. Both conditions hold iff the transfer function has a Pritchard–Salamon (PS) realization. A state-space variant of the PS result was proved already in [3], under the additional assumption that the weighting pattern (or impulse response) is a function (whose values are bounded operators). We illustrate by an example that this does not cover all PS systems, not even if the input and output spaces are separable.     

Ball,sphere, and all,all, all

The relationship of multidimensional geometry with statistical thermodynamics and with laws of large numbers is described.     

Contextualism,art, and rigidity: Levinson,Currie and Davies

The topic of this paper is the role played by context in art. In this regard I examine three theories linked to the names of J. Levinson, G. Currie and D. Davies. Levinson’s arguments undermine the structural theory. He finds it objectionable because it makes the individuation of artworks independent of their histories. Secondly, such a consequence is unacceptable because it fails to recognise that works are created rather than discovered. But, if certain general features of provenance are always work-constitutive, as it seems that Levinson is willing to claim, these features must always be essential properties of works. On the other hand, consideration of our modal practice suggests that whether a given general feature of provenance is essential or non-essential depends upon the particular work in question or is “work relative”. D. Davies builds his performance theory on the basis of the critical evaluation of Currie’s action-type hypotheses (ATH). Performances, says Davies, are not to be identified with “basic actions” to which their times belong essentially, but with “doings” that permit of the sorts of variation in modal properties required by the work-relativity of modality. He is also a fierce critic of the contextualist account. Contextualism is in his view unable to reflect the fact that aspects of provenance bear upon our modal judgements with variable force.In the second part of the paper I consider Davies’s “modality principle”. Davies is inclined to defend the claim that labels used for designation of works are rigid designators. Such a view offers a ground for discussion about the historicity of art. What has been meant when people claim that art is an historical concept? I argue that any historical theory implies a two-dimensional notion of “art”. At the end of the paper I suggest that Davies should embrace the theory of contingent identity and not the colocationist view about the relationship that exists between a particular artwork and its physical bearer.     

Woodhouse, babbage, peacock, and modern Algebra

In a recent article, J. M. Dubbey [Historia Mathematica 4 (1977), 295–302] showed that George Peacock's A Treatise on Algebra (1830) was similar to an unpublished work written by Charles Babbage in 1821. Evidently perplexed about the absence of a dispute over priority, Dubbey concluded that Peacock had unconsciously assimilated Babbage's ideas, and that Babbage was too busy with other activities to be concerned. The thesis of this article is that the innovative aspects of the work of both Babbage and Peacock are extensions of ideas put forth in 1803 by Robert Woodhouse, and that probably neither Babbage nor Peacock was overly concerned with acknowledgments because their approach to algebra was not unique at Cambridge.     

Naturalness, Vagueness, and Sortals

In the past few years, deflationary positions in the debate on the nature of composite material objects have become prominent. According to Ted Sider these include the thesis of quantifier variance, against which he has defended ontological realism. Recently, Sider has considered the possibility of rejecting his arguments against the vagueness of the unrestricted quantifiers in terms of translation functions. Against this strategy, he has presented an intuitive complaint and has argued that it can only be resisted if quantifier variance is accepted. But this is false. In this paper I argue, against Sider, that there is a coherent way to combine the rejection of quantifier variance with the vagueness of the unrestricted quantifiers. I sketch a model to show this, and then I consider, on the basis of it, several versions of the indeterminacy argument against the vagueness of the unrestricted quantifiers that Sider has formulated over the years.     

Vortices,strings, and pseudoinstantons

Vladimir Polytechnic Institute, Kovrov Branch. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 83, No. 2, pp. 207–221, May, 1990     

Chip-Firing,Antimatroids, and Polyhedra

Starting from the chip-firing game of Björner and Lovász we consider a generalization to vector addition systems that still admit algebraic structures as sandpile group or sandpile monoid. Every such vector addition language yields an antimatroid. We show that conversely every antimatroid can be represented this way. The inclusion order on the feasible sets of an antimatroid is an upper locally distributive lattice. We characterize polyhedra, which carry an upper locally distributive structure and show that they can be modelled by chip-firing games with gains and losses. At the end we point out a connection to a membership problem discussed by Korte and Lovász.     

Weights,growth, and amenability

This paper is a collection of general results in geometric group theory, concerning, in particular, amenability and its relation with growth of groups, cohomology and weights. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.     

Rigidity,reduction, and ramification

In this paper we consider wildly ramified G-Galois covers of curves f:Y¹_k branched at exactly one point over an algebraically closed field k of characteristic p. For G equal to A_p or PSL₂(p), we prove Abhyankar's Inertia Conjecture that all possible inertia groups occur over infinity for such covers f. In addition, we prove that the set of conductors that can be realized depends on the group. The method we use is to compute the reduction of Galois covers of branched at 3 points. We observe that the existence of covers with given inertia in characteristic p is closely related to the arithmetic of covers in characteristic zero.Mathematics Subject Classification (2000): 14H30, 14G32     

Matching, matroids, and extensions

Perhaps the two most fundamental well-solved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. We review the basic results for both, and describe some more recent advances. Then we discuss extensions of these models, in particular, two recent ones—jump systems and path-matchings. Received: October 10, 2000 / Accepted: June 13, 2001?Published online December 6, 2001     

Tightness,torsion, and tubes

We give examples of spaces with 3-torsion in the homology admitting Z₃-tight polyhedral immersions into euclidean space. Polyhedral tubes are used to construct embedded hypersurfaces of this kind.     

Girth,valency, and excess

There is a simple lower bound for the number of vertices of a regular graph whose girth and valency are specified. If the graph is required to have certain additional properties, then the number of excess vertices needed may be unbounded.