Spin texture on the Fermi surface of tensile strained HgTe

We present ab initio and k.p calculations of the spin texture on the Fermi surface of tensile strained HgTe, which is obtained by stretching the zincblende lattice along the (111) axis. Tensile strained HgTe is a semimetal with pointlike accidental degeneracies between a mirror symmetry protected twofold degenerate band and two nondegenerate bands near the Fermi level. The Fermi surface consists of two ellipsoids which contact at the point where the Fermi level crosses the twofold degenerate band along the (111) axis. However, the spin texture of occupied states indicates that neither ellipsoid carries a compensating Chern number. Consequently, the spin texture is locked in the plane perpendicular to the (111) axis, exhibits a nonzero winding number in that plane, and changes winding number from one end of the Fermi ellipsoids to the other. The change in the winding of the spin texture suggests the existence of singular points. An ordered alloy of HgTe with ZnTe has the same effect as stretching the zincblende lattice in the (111) direction. We present ab initio calculations of ordered Hg_xZn_1-xTe that confirm the existence of a spin texture locked in a 2D plane on the Fermi surface with different winding numbers on either end.Comment: 8 pages, 8 figure

Comparative Study of Embedding Methods

Embedding experimental data is a common first step in many forms of dynamical analysis. The choice of appropriate embedding parameters (dimension and lag) is crucial to the success of the subsequent analysis. We argue here that the optimal embedding of a time series cannot be determined by criteria based solely on the time series itself. Therefore we base our analysis on an examination of systems that have explicit analytic representations. A comparison of analytically obtained results with those obtained by an examination of the corresponding time series provides a means of assessing the comparative success of different embedding criteria. The assessment also includes measures of robustness to noise. The limitations of this study are explicitly delineated. While bearing these limitations in mind, we conclude that for the examples considered here, the best identification of the embedding dimension was achieved with a global false nearest neighbors argument, and the best value of lag was identified by the mutual information function

Statistical Validation of Mutual Information Calculations: Comparison of Alternative Numerical Algorithms

Given two time series X and Y, their mutual information, I(X, Y)= I(Y, X), is the average number of bits of X that can be predicted by measuring Y and vice versa. In the analysis of observational data, calculation of mutual information occurs in three contexts: identification of nonlinear correlation, determination of an optimal sampling interval, particularly when embedding data, and in the investigation of causal relationships with directed mutual information. In this contribution a minimum description length argument is used to determine the optimal number of elements to use when characterizing the distributions of X and Y. However, even when using partitions of the X and Y axis indicated by minimum description length, mutual information calculations performed with a uniform partition of the XY plane can give misleading results. This motivated the construction of an algorithm for calculating mutual information that uses an adaptive partition. This algorithm also incorporates an explicit test of the statistical independence of X and Y in a calculation that returns an assessment of the corresponding null hypothesis. The previously published Fraser-Swinney algorithm for calculating mutual information includes a sophisticated procedure for local adaptive control of the partitioning process. When the Fraser and Swinney algorithm and the algorithm constructed here are compared, they give very similar numerical results (less than 4% difference in a typical application). Detailed comparisons are possible when X and Y are correlated jointly Gaussian distributed because an analytic expression for I(X, Y) can be derived for that case. Based on these tests, three conclusions can be drawn. First, the algorithm constructed here has an advantage over the Fraser-Swinney algorithm in providing an explicit calculation of the probability of the null hypothesis that X and Y are independent. Second, the Fraser-Swinney algorithm is marginally the more accurate of the two algorithms when large data sets are used. With smaller data sets, however, the Fraser-Swinney algorithm reports structures that disappear when more data are available. Third, the algorithm constructed here requires about 0.5% of the computation time required by the Fraser-Swinney algorithm

Comparative Study of Embedding Methods

Embedding experimental data is a common first step in many forms of dynamical analysis. The choice of appropriate embedding parameters (dimension and lag) is crucial to the success of the subsequent analysis. We argue here that the optimal embedding of a time series cannot be determined by criteria based solely on the time series itself. Therefore we base our analysis on an examination of systems that have explicit analytic representations. A comparison of analytically obtained results with those obtained by an examination of the corresponding time series provides a means of assessing the comparative success of different embedding criteria. The assessment also includes measures of robustness to noise. The limitations of this study are explicitly delineated. While bearing these limitations in mind, we conclude that for the examples considered here, the best identification of the embedding dimension was achieved with a global false nearest neighbors argument, and the best value of lag was identified by the mutual information function

Linear and Nonlinear Measures Predict Swimming in the Leech

Stimulation of a trigger interneuron of an isolated nerve cord preparation of the medicinal leech, Hirudo medicinalis, sometimes leads to swimming; sometimes it does not. We investigate signals transmitted in the ventral cord of the leech after stimulation and seek quantitative measures that would make it possible to distinguish signals that predict swimming from those that do not. We find that a number of linear as well as nonlinear measures provide statistically significant distinctions between the two kinds of signals. The linear measures are the time dependence of (i) the standard deviation and (ii) the autocorrelation function at a small time delay. The nonlinear measures are (i) a measure of nonlinear predictability and (ii) the time dependence of a measure of the size of the embedded signal trajectory. Calculations using surrogate data suggest that the differences between the two classes of signals are dynamical as well as statistical

Linear and Nonlinear Measures Predict Swimming in the Leech

Stimulation of a trigger interneuron of an isolated nerve cord preparation of the medicinal leech, Hirudo medicinalis, sometimes leads to swimming; sometimes it does not. We investigate signals transmitted in the ventral cord of the leech after stimulation and seek quantitative measures that would make it possible to distinguish signals that predict swimming from those that do not. We find that a number of linear as well as nonlinear measures provide statistically significant distinctions between the two kinds of signals. The linear measures are the time dependence of (i) the standard deviation and (ii) the autocorrelation function at a small time delay. The nonlinear measures are (i) a measure of nonlinear predictability and (ii) the time dependence of a measure of the size of the embedded signal trajectory. Calculations using surrogate data suggest that the differences between the two classes of signals are dynamical as well as statistical

Models, Brains, and Scientific Realism

Prediction Error Minimization theory (PEM) is one of the most promising attempts to model perception in current science of mind, and it has recently been advocated by some prominent philosophers as Andy Clark and Jakob Hohwy. Briefly, PEM maintains that “the brain is an organ that on average and over time continually minimizes the error between the sensory input it predicts on the basis of its model of the world and the actual sensory input” (Hohwy 2014, p. 2). An interesting debate has arisen with regard to which is the more adequate epistemological interpretation of PEM. Indeed, Hohwy maintains that given that PEM supports an inferential view of perception and cogni-tion, PEM has to be considered as conveying an internalist epistemological perspective. Contrary to this view, Clark maintains that it would be incorrect to interpret in such a way the indirectness of the link between the world and our inner model of it, and that PEM may well be combined with an externalist epistemological perspective. The aim of this paper is to assess those two opposite interpretations of PEM. Moreover, it will be suggested that Hohwy’s position may be considerably strengthened by adopting Carlo Cellucci’s view on knowledge (2013)

Mathematical practice, crowdsourcing, and social machines

The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. Mathematical practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question answering system {\it mathoverflow} contains around 40,000 mathematical conversations, and {\it polymath} collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of "soft" aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a "social machine", a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.Comment: To appear, Springer LNCS, Proceedings of Conferences on Intelligent Computer Mathematics, CICM 2013, July 2013 Bath, U

How to think about informal proofs

This document is the Accepted Manuscript version of the following article: Brendan Larvor, ‘How to think about informal proofs’, Synthese, Vol. 187(2): 715-730, first published online 9 September 2011. The final publication is available at Springer via doi:10.1007/s11229-011-0007-5It is argued in this study that (i) progress in the philosophy of mathematical practice requires a general positive account of informal proof; (ii) the best candidate is to think of informal proofs as arguments that depend on their matter as well as their logical form; (iii) articulating the dependency of informal inferences on their content requires a redefinition of logic as the general study of inferential actions; (iv) it is a decisive advantage of this conception of logic that it accommodates the many mathematical proofs that include actions on objects other than propositions; (v) this conception of logic permits the articulation of project-sized tasks for the philosophy of mathematical practice, thereby supplying a partial characterisation of normal research in the fieldPeer reviewedFinal Accepted Versio