Modern Reformulation Research Articles

An infinite dimensional version of the Kronecker index and its relation with the Leray–Schauder degree

Let \mathfrak{f} be a compact vector field of class C^{1} on a real Hilbert space \mathbb{H} . Denote by \mathbb{B} the open unit ball of \mathbb{H} and by \mathbb{S}= \partial\mathbb{B} the unit sphere. Given a point q \notin \mathfrak{f}(\mathbb{S}) , consider the self-map of \mathbb{S} defined by \mathfrak{f}_{q}^{\partial}(p)= \frac{\mathfrak{f}(p)-q}{\|\mathfrak{f}(p)-q\|}, \quad p \in \mathbb{S}. If \mathbb{H} is finite dimensional, then \mathbb{S} is an orientable, connected, compact differentiable manifold. Therefore, the Brouwer degree, \deg_{\mathrm{Br}}(\mathfrak{f}_{q}^{\partial}) is well defined, no matter what orientation of \mathbb{S} is chosen, assuming it is the same for \mathbb{S} as domain and codomain of \mathfrak{f}_{q}^{\partial} . This degree may be considered as a modern reformulation of the Kronecker index of the map \mathfrak{f}_{q}^{\partial} . Let \deg_{\mathrm{Br}}(\mathfrak{f},\mathbb{B},q) denote the Brouwer degree of \mathfrak{f} on \mathbb{B} with target q . It is known that one has the equality \deg_{\mathrm{Br}}(\mathfrak{f},\mathbb{B},q) = \deg_{\mathrm{Br}}(\mathfrak{f}_{q}^{\partial}). Our purpose is an extension of this formula to the infinite dimensional context. Namely, we will prove that \deg_{\mathrm{LS}}(\mathfrak{f},\mathbb{B},q) = \deg_{\mathrm{bf}}(\mathfrak{f}_{q}^{\partial}), where \deg_{\mathrm{LS}}(\cdot) denotes the Leray–Schauder degree and \deg_{\mathrm{bf}}(\cdot) is the degree earlier introduced by M. Furi and the first author, which extends, to the infinite dimensional case, the Brouwer degree and the Kronecker index. In other words, here, we extend to the Leray–Schauder degree the boundary dependence property which holds for the Brouwer degree in the finite dimensional context.

Electromagnetic induction: How the “flux rule” has superseded Maxwell's general law

As documented by textbooks, the teaching of electromagnetic induction in university and high school courses is primarily based on what Feynman labeled as the “flux rule,” downgrading it from the status of physical law. However, Maxwell derived a “general law of electromagnetic induction” in which the vector potential plays a fundamental role. A modern reformulation of Maxwell's law can be easily obtained by defining the induced electromotive force as ∮l(E→+v→c×B→)·dl→, where v→c is the velocity of the positive charges which, by convention, are the current carriers. Maxwell did not possess a model for the electric current. Therefore, in his law, he took v→c to be the velocity of the circuit element containing the charges. This paper aims to show that the modern reformulation of Maxwell's law governs electromagnetic induction, and the “flux rule” is not a physical law but only a calculation shortcut that does not always yield the correct predictions. This paper also tries to understand why Maxwell's law has been ignored, and how the “flux rule” has taken root. Finally, a section is dedicated to teaching this modern reformulation of Maxwell's law in high schools and elementary physics courses.

Ontogeny, Phylotypic Periods, Paedomorphosis, and Ontogenetic Systematics

The key terms linking ontogeny and evolution are briefly reviewed. It is shown that their application and usage in the modern biology are often inconsistent and incorrectly understood even within the “evo-devo” field. For instance, the core modern reformulation that ontogeny not merely recapitulates, but produces phylogeny implies that ontogeny and phylogeny are closely interconnected. However, the vast modern phylogenetic and taxonomic fields largely omit ontogeny as a central concept. Instead, the common “clade-” and “tree-thinking” prevail, despite on the all achievements of the evo-devo. This is because the main conceptual basis of the modern biology is fundamentally ontogeny-free. In another words, in the Haeckel’s pair of “ontogeny and phylogeny,” ontogeny is still just a subsidiary for the evolutionary process (and hence, phylogeny), instead as in reality, its main driving force. The phylotypic periods is another important term of the evo-devo and represent a modern reformulation of Haeckel’s recapitulations and biogenetic law. However, surprisingly, this one of the most important biological evidence, based on the natural ontogenetic grounds, in the phylogenetic field that can be alleged as a “non-evolutionary concept.” All these observations clearly imply that a major revision of the main terms which are associated with the “ontogeny and phylogeny/evolution” field is urgently necessarily. Thus, “ontogenetic” is not just an endless addition to the term “systematics,” but instead a crucial term, without it neither systematics, nor biology have sense. To consistently employ the modern ontogenetic and epigenetic achievements, the concept of ontogenetic systematics is hereby refined. Ontogenetic systematics is not merely a “research program” but a key biological discipline which consistently links the enormous biological diversity with underlying fundamental process of ontogeny at both molecular and morphological levels. The paedomorphosis is another widespread ontogenetic-and-evolutionary process that is significantly underestimated or misinterpreted by the current phylogenetics and taxonomy. The term paedomorphosis is refined, as initially proposed to link ontogeny with evolution, whereas “neoteny” and “progenesis” are originally specific, narrow terms without evolutionary context, and should not be used as synonyms of paedomorphosis. Examples of application of the principles of ontogenetic systematics represented by such disparate animal groups as nudibranch molluscs and ophiuroid echinoderms clearly demonstrate that perseverance of the phylotypic periods is based not only on the classic examples in vertebrates, but it is a universal phenomenon in all organisms, including disparate animal phyla.

English

A new approach to etale homotopy theory is presented which applies to a much broader class of objects than previously existing approaches, namely it applies not only to all schemes (without any local Noetherian hypothesis), but also to arbitrary higher stacks on the etale site of such schemes, and in particular to all algebraic stacks. This approach also produces a more refined invariant, namely a pro-object in the infinity category of spaces, rather than in the homotopy category. We prove a profinite comparison theorem at this level of generality, which states that if $\mathcal{X}$ is an arbitrary higher stack on the etale site of affine schemes of finite type over $\mathbb{C},$ then the etale homotopy type of $\mathcal{X}$ agrees with the homotopy type of the underlying stack $\mathcal{X}_{top}$ on the topological site, after profinite completion. In particular, if $\mathcal{X}$ is an Artin stack locally of finite type over $\mathbb{C}$, our definition of the etale homotopy type of $\mathcal{X}$ agrees up to profinite completion with the homotopy type of the underlying topological stack $\mathcal{X}_{top}$ of $\mathcal{X}$ in the sense of Noohi. In order to prove our comparison theorem, we provide a modern reformulation of the theory of local systems and their cohomology using the language of $\infty$-categories which we believe to be of independent interest.

Rota's Fubini lectures: The first problem

In his 1998 Fubini Lectures, Rota discusses twelve problems in probability that “no one likes to bring up”. The first problem calls for a revision of the notion of a sample space, guided by the belief that mention of sample points in a probabilistic argument is bad form and that a “pointless” foundation of probability should be provided by algebras of random variables.In 1958 Chang introduced MV-algebras to prove the completeness theorem of Łukasiewicz logic Ł∞. The aim of this paper is to show that MV-algebras provide a solution of Rota's first problem.The adjunction between MV-algebras and unital commutative C*-algebras equips every MV-algebra A with a natural ring structure, as advocated by Nelson for algebras of random variables. The closed compact set S(A)⊆[0,1]A of finitely additive probability measures on A (the states of A) coincides with the set of [0,1]-valued functions on A whose finite restrictions are consistent in de Finetti's sense. MV-algebras and Ł∞ thus provide the framework for a generalization (known as ŁIPSAT) of Boole's probabilistic inference problem, and its modern reformulation known as probabilistic satisfiability, PSAT. We construct an affine homeomorphism γA of S(A) onto the weakly compact space of regular Borel probability measures on the maximal spectral space μ(A). The latter is the most general compact Hausdorff space. As a consequence, for every Kolmogorov probability space (Ω,FΩ,P), with FΩ the sigma-algebra of Borel sets of a compact Hausdorff space Ω, and P a regular probability measure on FΩ, there is an MV-algebra A and a state σ of A such that (Ω,FΩ,P)≅(μ(A),Fμ(A),γA(σ)).

Sraffa’s Constructive and Interpretive Work, and Marx

ABSTRACTThis article provides a summary account of Piero Sraffa’s constructive and interpretive work on the classical approach to the theory of value and distribution and its relationship with Marx’s contributions. It is shown that in the early phase of his constructive work Sraffa developed his equation systems by adopting a ‘physical real cost’ approach and a strictly objectivist point of view, and completely eschewed Marx’s labour-based approach and the related Marxian concepts. Only at a later stage did he explore systematically the relationship between his own modern re-formulation of the surplus approach to the theory of value and distribution and Marx’s contribution. He considered Marx’s most important analytical contribution to the further development of the surplus approach to consist of the re-integration of circular production relations, which allowed him to see the existence of a maximum rate of profits and its role in an analysis of accumulation and technical change.

Approach to large distribution network optimisation using modern implementation of benders decomposition

Long-term electricity distribution network expansion planning problems (EDNEPPs) are complex combinatorial problems due to the large number of variables involved. Therefore, exact mathematical methods can only be applied to solve EDNEPPs for small-scale systems. Due to this reason, in the literature, to solve EDNEPPs for real-size networks, a variety of solution techniques have been utilised, with varying results. Early research on solving EDNEPPs focused on simplification of the model, while later research used decomposition techniques to solve EDNEPPs for real-sized networks. However, with the rapid changes and increased complexities of today's EDNEPPs, it has been difficult or impossible to solve EDNEPPs for real-sized networks using traditional decomposition approaches. Therefore, many authors have recently applied heuristics or local optimisation techniques, but these methods do not guarantee exact, globally optimal solutions, and provide little indication of solution quality. Recently, many new problem reformulation and decomposition techniques have been reported to cope with large-scale optimisation problems. However, there is a major void in the literature surrounding applications of modern problem decomposition and reformulation techniques from operations research to solve EDNEPPs. This study presents a modern approach to convex outer approximation using lazy constraints and capabilities of modern solvers for solving EDNEPPs.

Religion som verdenshåndtering

The article discusses two questions: whether (and in whatsense) Christianity can be ‘naturalized’; and whether ancient Stoicismmay contribute to a modern reformulation of ‘Christianity naturalized’.To answer these questions, the article focuses on articulating an understandingof ‘religion’ in relation to ‘science’. Building on the accountgiven of the philosophical discipline of ‘ethics’ by Hilary Putnam inEthics without Ontology, the article attempts to construct a structurallysimilar understanding of ‘religion’ (and its philosophical counterpart,‘theology’) that will give it a legitimate position ‘in an age of science’(cf. Putnam, Philosophy in an Age of Science). ‘Religion’ is here seen asone particular way of ‘coping with the world’. The article concludesby sketching some ways in which ancient Stoicism (as a specimen of a‘natural philosophy and theology’) may help in reformulating an adequate,contemporary understanding of Christianity.

The classical ether-drift experiments: A modern re-interpretation

The condensation of elementary quanta and their macroscopic occupation of the same quantum state, say k=0 in some reference frame Sigma, is the essential ingredient of the degenerate vacuum of present-day elementary particle physics. This represents a sort of `quantum ether' which characterizes the physically realized form of relativity and could play the role of preferred reference frame in a modern re-formulation of the Lorentzian approach. In spite of this, the so called `null results' of the classical ether-drift experiments, traditionally interpreted as confirmations of Special Relativity, have so deeply influenced scientific thought as to prevent a critical discussion on the real reasons underlying its alleged supremacy. In this paper, we argue that this traditional null interpretation is far from obvious. In fact, by using Lorentz transformations to connect the Earth's frame to Sigma, the small observed effects point to an average Earth's velocity of about 300 km/s, as in most cosmic motions. A common feature is the irregular behaviour of the data. While this has motivated, so far, their standard interpretation as instrumental artifacts, our new re-analysis of the very accurate Joos experiment gives clear indications for the type of Earth's motion associated with the CMB anisotropy and leaves little space for this traditional interpretation. The new explanation requires instead a view of the vacuum as a stochastic medium, similar to a fluid in a turbulent state of motion, in agreement with basic foundational aspects of both quantum physics and relativity. The overall consistency of this picture with the present experiments with vacuum optical resonators and the need for a new generation of dedicated ether-drift experiments are also emphasized.

Le parole di Orfeo: Dante, Petrarca, Leopardi, e gli archetipi di un genere

Starting from the idea of 'canzonieri di morte', collections of poems written by poets for their dead lovers, which is a genre or subgenre at the very origin of the Western poetical tradition, and taking Orpheus' role and specific poetical position within the genre as a key to understanding it, the article focuses on collections or single works by three authors who are fundamental to this tradition: Dante's Vita nova, Petrarch's anzoniere, and Leopardi's 'Il sogno'. The article considers the treatment of loss, absence, desire, and love (both as private events and cultural ideas) in these works of lyric poetry, taking into account historical and cultural differences between the three poets. The main aim is to trace an outline of the tradition, with all its internal changes, in Dante and Petrarch and to identify strands which lead to its modern recurrence and reformulation in a poet such as Leopardi.

Topos and Enthymeme

This paper is a modern reformulation of Aristotle's concepts of topos and enthymeme and the relation between them. Briefly, a topos may be understood as a binary relation which replaces implication in the syllogism to yield an enthymeme. If a syllogism is an argument of the form, (1) If P, and P implies Q, then Q, then an enthymeme is an argument of the form, (2) If P, and T(P,Q), then Q. All of Aristotle's twenty-eight topoi in the Rhetoric may be shown to have the form (2), and the enthymeme can thus be understood as a generalized (or weakened) syllogism.

The Moral Basis of Humanitarian Intervention

This article discusses the moral principles underlying the idea of humanitarian intervention. The analysis is in two parts, one historical and the other philosophical. First, the article examines arguments made in late medieval and early modern Europe for using armed force to punish the violation of natural law and to defend communities from tyranny and oppression, regardless of where they occur. It seeks to understand how moralists writing before the emergence of modern international law conceived what we now call humanitarian intervention.In the context of international law, humanitarian intervention is usually understood to be an exception to the nonintervention principle. However, the natural law tradition regards international law as less important than the moral imperative to punish wrongs and protect the innocent.Second, the article considers how humanitarian intervention is justified within the reformulation of the natural law tradition displayed in recent efforts to theorize morality along Kantian lines. In this reformulation, humanitarian intervention is a product of the duty of beneficence and, more specifically, of the right to use force to protect the innocent. The article draws upon the biblical injunction “Thou shalt not stand idly by the blood of thy neighbor,” which has become a centerpiece of the modern reformulation, and briefly explores its application to humanitarian intervention in the context of international relations today. This reformulation of natural law explains why, despite modern efforts to make it illegal, humanitarian intervention remains, in principle, morally defensible.

Factor Prices and Technical Change: From Induced Innovations to Recent Debates

This paper revisits the induced innovation literature of the 1960s to which Phelps was a major contributor (Drandakis and Phelps, 1965). This literature was the first systematic study of the determinants of technical change and also the first investigation of the relationship between factor prices and technical change. I present a modern reformulation of this literature based on the tools developed by the endogenous growth literature. This reformulation confirms many of the insights of the induced innovations literature, but reveals a new force, which I refer to as the market size effect: there will be more technical change directed at more abundant factors. I use this modern reformulation to shed light on two recent debates: (1) why is technical change often skill biased, and why has it become more skill biased during recent decades? (2) what is the role of human capital differences in accounting for income differences across countries? Interestingly, an application of this modern reformulation to these debates also reiterates some of the insights of another important paper by Phelps, Nelson and Phelps (1966).