On the algebraic integrals of the dynamical equations

We bring to the reader’s attention a translation of Levi-Civita’s work “Sugli integrali algebrici delle equazioni dinamiche” (1896).     

Convolution equations and spaces of ultradifferentiable functions

We study spaces of “approximate solutions” to a given convolution equation. In particular we show how their quasianalyticity can be reduced to the study of a suitably constructed Cauchy problem for related spaces. We give a unicity theorem for this problem. This paper is dedicated to the author’s son, Alessandro. The author has been partially supported by Ministero P.I., and G.N.S.A.G.A. of Consiglio Nazionale delle Ricerche.     

Una caratterizzazione geometrica dei piani di rifrazione generalizzati

Si dà una caratterizzazione geometrica, che prescinde dall’ordine, dei piani di rifrazione generalizzati usufruendo delle nozioni di “metàtransitività” e di “transitività relativa” dovute, rispettivamente, a J. André e a F. Bartolozzi.     

Covering numbers of manifolds and critical points of a morse function

For a closed connected triangulatedn-manifoldM, we study some numerical invariants (namedcategory andcovering numbers) ofM which are strictly related to the topological structure ofM. We complete the classical results of 3-manifold topology and then we prove some characterization theorems in higher dimensions. Finally some applications are given about the minimal number of critical points (resp. values) of Morse functions defined on a closed connected smoothn-manifold. Work performed under the auspices of the G.N.S.A.G.A. of the C.N.R. and financially supported by the M.P.I. of Italy within the project “Geometria delle Varietà Differenziabili”.     

Special line bundles on curves with involution

Partially supported by the Bulgarian foundation “Scientific research” and by NSF under the US-Bulgarian joint project “Algebra and algebraic geometry”.     

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.     

Brill–Noether theory of curves on Enriques surfaces I: the positive cone and gonality

We study the existence of linear series on curves lying on an Enriques surface and general in their complete linear system. Using a method that works also below the Bogomolov–Reider range, we compute, in all cases, the gonality of such curves. We also give a new result about the positive cone of line bundles on an Enriques surface and we show how this relates to the gonality. Dedicated to the memory of Silvano Bispuri. The work of A. L. Knutsen is partially supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme. The work of A. F. Lopez is partially supported by the MIUR national project “Geometria delle varietà algebriche” COFIN 2002--2004.     

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.     

Statistics on wreath products and generalized binomial-stirling numbers

Various statistics on wreath products are defined via canonical words, “colored” right to left minima and “colored” descents. It is shown that refined counts with respect to these statistics have nice recurrence formulas of binomial-Stirling type. These extended Stirling numbers determine (via matrix inversion) dual systems, which are also shown to have combinatorial realizations within the wreath product. The above setting also gives rise to a MacMahon-type equi-distribution theorem over subsets with prescribed statistics. Partially supported by Minerva Grant No. 8441 and by EC's IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272. Partially supported by EC's IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272.     

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.     

Everywhere regularity of functionals with <Emphasis Type="Italic">φ</Emphasis>-growth

We prove C ^1,α -regularity for local minimizers of functionals with φ-growth, giving also the decay estimate. In particular, we present a unified approach in the case of power-type functions. Supported by PRIN Project: “Calcolo delle variazioni e Teoria Geometrica della Misura”.     

Some formulae arising in projective-differential geometry

We set up a framework in which some projective-differential intuitive concepts can be very easily formalized. Then we prove some general formulas. Among them, two “inversion formulas” about deformations of families of nonreduced subschemes are particularly remarkable. As applications, we give convenient definitions about some classic topics, such as higher order foci; finally we propose a new proof of the classic structure theorem about the first Gauss map.

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Sunto  Si stabilisce una terminologia di base che sembra essere particolarmente adatta a trattare alcune particolari questioni di geometria proiettivo-differenziale. Nell'ambito di questa si dimostrano alcuni risultati generali, tra cui delle formule “di inversione” riguardanti le deformazioni di famiglie di sottoschemi non ridotti. Come applicazione, si propongono nuove definizioni di alcuni concetti classici, tra cui quello di fuochi di ordine qualsiasi; infine si dà una nuova dimostrazione del teorema di struttura delle fibre della prima mappa di Gauss.

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Partially supported by M.U.R.S.T.     

Righting the early history of computing,or how sausage was made

The short answer to the question just posed seems to be, “Not much.” Since I have given “the long answer” elsewhere,¹² I can summarize it here. Berg could see no point in writing Bromley. What could he write to someone he believed guilty of plagiarism? What could such a letter accomplish? He did, however, write to New York University Press; to all the universities involved, and to the Works’ English publisher (Pickering and Chatto), who said they passed the letter on to Campbell-Kelly (30 June 1990); to a great many professional societies in Australia, England, and the United States; to a great many governmental agencies and some politicians in those countries; to some publications, both academic and popular; to the Pope and several cardinals; and to a miscellany of other individuals. Generally, those in the best position to do something—for example, the three universities involved —did not even answer Berg’s letter. Others often did answer, but their answer was generally that they were in no position to do anything. That was how matters stood when I published my first article on “the Berg Affair”.₁₂ Its publication finally roused those best positioned to answer. Late in 1993, Galler, Bromley, and Campbell-Kelly wrote letters to the editor of Accountability in Research criticizing me for not getting their side of the story before I published Berg’s. Campbell-Kelly threatened the journal’s publisher with a lawsuit if I (or it) did not retract. The three also provided some insight into what their explanation of events might be. Bromley, though listed prominently in ads for the Works, claimed to have had only a small part, merely advising Campbell-Kelly on selection and arrangement of the papers printed in Volumes 2 and 3. Campbell-Kelly confirmed that Bromley took no part in the detailed editing or in the provision of documents. That work was performed by one C.J.D. (“Jim”) Roberts, a “London-based independent scholar” who was “editorial consultant to the Works” (and, apparently, worked directly under Campbell-Kelly). Roberts seems to deserve more public credit than he has so far received. According to Campbell-Kelly, it was Roberts who, making a systematic search for unknown holdings of Babbage, turned up the original of the letter to Quetelet by writing the Royal Library (one “tiny triumph” among many). Campbell-Kelly also claimed that neither he nor Roberts knew of Berg’s prior discovery.     

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.     

Morse Index and Critical Groups for p-Laplace Equations with Critical Exponents

In this work we consider a class of Euler functionals defined in Banach spaces, associated to quasilinear elliptic problems involving the critical Sobolev exponent. We perform critical groups estimates via the Morse index. Dedicated to the memory of Professor Aldo Cossu The research of the authors was supported by the MIUR project “Variational and topological methods in the study of nonlinear phenomena” (PRIN 2005).     

A sensitivity analysis of a class of semi-coercive variational inequalities using recession tools

Using the recession analysis we study necessary and sufficient conditions for the existence and the stability of a finite semi-coercive variational inequality with respect to data perturbation. Some applications of the abstract results in mechanics and in electronic circuits involving devices like ideal diode and practical diode are discussed. The research of S. Adly has been supported by the “Fondation EADS” and the ANR project “Guidage”.     

The Auslander-Reiten quiver of a modular group algebra revisited

This work has been supported by the DFG project “Darstellungstheorie” at the University of Essen.     

<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 weight-decay technique in learning from data: an optimization point of view

The technique known as “weight decay” in the literature about learning from data is investigated using tools from regularization theory. Weight-decay regularization is compared with Tikhonov’s regularization of the learning problem and with a mixed regularized learning technique. The accuracies of suboptimal solutions to weight-decay learning are estimated for connectionistic models with a-priori fixed numbers of computational units. The Authors were partially supported by a PRIN grant from the Italian Ministry for University and Research, project “Models and Algorithms for Robust Network Optimization”.