What Makes a Theory of Infinitesimals Useful? A View by Klein and Fraenkel

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein

Internetwork and intranetwork communications during bursting dynamics: Applications to seizure prediction

We use a simple dynamical model of two interacting networks of integrate-and-fire neurons to explain a seemingly paradoxical result observed in epileptic patients indicating that the level of phase synchrony declines below normal levels during the state preceding seizures (preictal state). We model the transition from the seizure free interval (interictal state) to the seizure (ictal state) as a slow increase in the mean depolarization of neurons in a network corresponding to the epileptic focus. We show that the transition from the interictal to preictal and then to the ictal state may be divided into separate dynamical regimes: the formation of slow oscillatory activity due to resonance between the two interacting networks observed during the interictal period, structureless activity during the preictal period when the two networks have different properties, and bursting dynamics driven by the network corresponding to the epileptic focus. Based on this result, we hypothesize that the beginning of the preictal period marks the beginning of the transition of the epileptic network from normal activity toward seizing

Betting on Climate Policy: Using Prediction Markets to Address Global Warming

Global warming, sea level rise, and extreme weather events have made climate change a top priority for policymakers across the globe. But which policies are best suited to tackle the enormous challenges presented by our changing climate? This Article proposes that policymakers turn to prediction markets to answer that crucial question. Prediction markets have a strong track record of outperforming other forecasting mechanisms across a wide range of contexts — from predicting election outcomes and economic trends to guessing Oscar winners. In the context of climate change, market participants could, for example, bet on important climate outcomes conditioned on the adoption of particular policies. These prediction markets would aggregate policy-relevant information from a variety of sources to improve upon existing decision-making methods, including expert deliberation, peer review, and cost-benefit analysis. Prediction markets also have the potential to overcome resistance to climate change mitigation efforts, particularly among market-oriented conservatives. We explain how both the federal and state governments could use prediction markets to help resolve high-profile controversies, such as how best to allocate subsidies to promote clean technology innovation and which policy strategy promises the greatest reduction in carbon emissions

Cauchy, infinitesimals and ghosts of departed quantifiers

Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz's distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson's framework, while Leibniz's law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz's infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson's framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler's own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson's framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy's procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson's framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. Keywords: historiography; infinitesimal; Latin model; butterfly modelComment: 45 pages, published in Mat. Stu

Identifying phase synchronization clusters in spatially extended dynamical systems

We investigate two recently proposed multivariate time series analysis techniques that aim at detecting phase synchronization clusters in spatially extended, nonstationary systems with regard to field applications. The starting point of both techniques is a matrix whose entries are the mean phase coherence values measured between pairs of time series. The first method is a mean field approach which allows to define the strength of participation of a subsystem in a single synchronization cluster. The second method is based on an eigenvalue decomposition from which a participation index is derived that characterizes the degree of involvement of a subsystem within multiple synchronization clusters. Simulating multiple clusters within a lattice of coupled Lorenz oscillators we explore the limitations and pitfalls of both methods and demonstrate (a) that the mean field approach is relatively robust even in configurations where the single cluster assumption is not entirely fulfilled, and (b) that the eigenvalue decomposition approach correctly identifies the simulated clusters even for low coupling strengths. Using the eigenvalue decomposition approach we studied spatiotemporal synchronization clusters in long-lasting multichannel EEG recordings from epilepsy patients and obtained results that fully confirm findings from well established neurophysiological examination techniques. Multivariate time series analysis methods such as synchronization cluster analysis that account for nonlinearities in the data are expected to provide complementary information which allows to gain deeper insights into the collective dynamics of spatially extended complex systems

MHSP in reversed-biased operation mode for ion blocking in gas-avalanche multipliers

We present recent results on the operation of gas-avalanche detectors comprising a cascade of gas electron multipliers (GEMs) and Micro-Hole and Strip Plates (MHSPs) multiplier operated in reversed-bias (R-MHSP) mode. The operation mechanism of the R-MHSP is explained and its potential contribution to ion-backflow (IBF) reduction is demonstrated. IBF values of 4E-3 were obtained in cascaded R-MHSP and GEM multipliers at gains of about 1E+4, though at the expense of reduced effective gain in the first R- MHSP multiplier in the cascade.Comment: 23 pages, 8 figure

Topological Aspects of Epistemology and Metaphysics

The aim of this paper is to show that (elementary) topology may be useful for dealing with problems of epistemology and metaphysics. More precisely, I want to show that the introduction of topological structures may elucidate the role of the spatial structures (in a broad sense) that underly logic and cognition. In some detail I’ll deal with “Cassirer’s problem” that may be characterized as an early forrunner of Goodman’s “grue-bleen” problem. On a larger scale, topology turns out to be useful in elaborating the approach of conceptual spaces that in the last twenty years or so has found quite a few applications in cognitive science, psychology, and linguistics. In particular, topology may help distinguish “natural” from “not-so-natural” concepts. This classical problem that up to now has withstood all efforts to solve (or dissolve) it by purely logical methods. Finally, in order to show that a topological perspective may also offer a fresh look on classical metaphysical problems, it is shown that Leibniz’s famous principle of the identity of indiscernibles is closely related to some well-known topological separation axioms. More precisely, the topological perspective gives rise in a natural way to some novel variations of Leibniz’s principle

Physico-chemical characteristics and primary structure of an affinity-purified α-D-galactose-specific, jacalin-related lectin from the latex of mulberry (Morus indica)

An α-D-galactose specific lectin belonging to the family of jacalin-related lectins (JRL) has been purified by affinity chromatography on cross-linked guar-gum. Mass spectrometric data revealed that the protein harbors two chains like all the members of galactose-specific jacalin-related lectins (gJRL). De novo sequencing of proteolytic peptides demonstrated that the heavier chain consists of 133 amino acids and the lighter chain comprises of 21 or 24 amino acids. The heavier chain contains one N-glycosylation site (Asn47) occupied with either pauci-mannose type [GlcNAc2(Fuc)Man3(Xyl)] or complex type [GlcNAc2(Fuc)Man3(Xyl)GlcNAc(Fuc)Gal] N-glycans. Circular dichroism spectroscopy indicated that the secondary structure of the lectin is predominantly made up of β-sheets, and differential scanning calorimetry revealed a thermal denaturation temperature of 77.6 °C. MTT (3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide) cell viability assays on MCF-7 and MDCK cells showed that the lectin is highly cytotoxic towards both cell lines when dosed at micromolar concentrations, suggesting that it may play a role in the defense mechanism of the plant

A category-specific response to animals in the right human amygdala

The amygdala is important in emotion, but it remains unknown whether it is specialized for certain stimulus categories. We analyzed responses recorded from 489 single neurons in the amygdalae of 41 neurosurgical patients and found a categorical selectivity for pictures of animals in the right amygdala. This selectivity appeared to be independent of emotional valence or arousal and may reflect the importance that animals held throughout our evolutionary past

Neurons in the human amygdala encode face identity, but not gaze direction

The amygdala is important for face processing, and direction of eye gaze is one of the most socially salient facial signals. Recording from over 200 neurons in the amygdala of neurosurgical patients, we found robust encoding of the identity of neutral-expression faces, but not of their direction of gaze. Processing of gaze direction may rely on a predominantly cortical network rather than the amygdala