On the torsion function with Robin or Dirichlet boundary conditions

For $p\in (1,+\infty)$ and $b \in (0, +\infty]$ the $p$-torsion function with Robin boundary conditions associated to an arbitrary open set \Om \subset \R^m satisfies formally the equation $-\Delta_p =1$ in \Om and $|\nabla u|^{p-2} \frac{\partial u}{\partial n} + b|u|^{p-2} u =0$ on \partial \Om. We obtain bounds of the $L^\infty$ norm of $u$ {\it only} in terms of the bottom of the spectrum (of the Robin $p$-Laplacian), $b$ and the dimension of the space in the following two extremal cases: the linear framework (corresponding to $p=2$) and arbitrary $b>0$, and the non-linear framework (corresponding to arbitrary $p>1$) and Dirichlet boundary conditions ($b=+\infty$). In the general case, $p\not=2, p \in (1, +\infty)$ and $b>0$ our bounds involve also the Lebesgue measure of \Om.Comment: 19 page

Biased statistics for traces of cyclic p-fold covers over finite fields

In this paper, we discuss in more detail some of the results on the statistics of the trace of the Frobenius endomorphism associated to cyclic p-fold covers of the projective line that were presented in [1]. We also show new findings regarding statistics associated to such curves where we fix the number of zeros in some of the factors of the equation in the affine model

On the minimization of Dirichlet eigenvalues of the Laplace operator

We study the variational problem \inf \{\lambda_k(\Omega): \Omega\ \textup{open in}\ \R^m,\ |\Omega| < \infty, \ \h(\partial \Omega) \le 1 \}, where $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\Omega)$, \h(\partial \Omega) is the $(m-1)$- dimensional Hausdorff measure of the boundary of $\Omega$, and $|\Omega|$ is the Lebesgue measure of $\Omega$. If $m=2$, and $k=2,3, \cdots$, then there exists a convex minimiser $\Omega_{2,k}$. If $m \ge 2$, and if $\Omega_{m,k}$ is a minimiser, then $\Omega_{m,k}^*:= \textup{int}(\overline{\Omega_{m,k}})$ is also a minimiser, and $\R^m\setminus \Omega_{m,k}^*$ is connected. Upper bounds are obtained for the number of components of $\Omega_{m,k}$. It is shown that if $m\ge 3$, and $k\le m+1$ then $\Omega_{m,k}$ has at most $4$ components. Furthermore $\Omega_{m,k}$ is connected in the following cases : (i) $m\ge 2, k=2,$ (ii) $m=3,4,5,$ and $k=3,4,$ (iii) $m=4,5,$ and $k=5,$ (iv) $m=5$ and $k=6$. Finally, upper bounds on the number of components are obtained for minimisers for other constraints such as the Lebesgue measure and the torsional rigidity.Comment: 16 page

Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity

We present some open problems and obtain some partial results for spectral optimization problems involving measure, torsional rigidity and first Dirichlet eigenvalue.Comment: 18 pages, 4 figure

The nature of semantic priming by subliminal spatial words: Embodied or disembodied?

Theories of embodied semantics (ES) suggest that a critical part of understanding what a word means consists of simulating the sensorimotor experience related to the word's referent. Some proponents of ES have suggested that sensorimotor activations are mandatory and highly automatic during semantic processing. Evidence supporting this claim comes from masked priming studies showing that unconsciously perceived spatial words (e.g., up, down) can directly modulate action performance on the basis of their meaning. However, a closer look reveals that such priming effects can be explained also in terms of symbolic (disembodied) semantic priming or nonsemantic mechanisms. In this study we sought to understand whether sensorimotor processing takes place during language understanding outside awareness. We used spatial words as a test bed and across 6 experiments we teased apart the possibility that action priming could be explained by: (a) nonsemantic mechanisms, (b) symbolic semantic priming, or (c) embodied semantic priming. The critical finding is that when symbolic and nonsemantic mechanisms were prevented, allowing only for a genuinely embodied semantic priming, no effect was found. Conversely, facilitation emerged in the same experimental paradigm when embodied priming was prevented and symbolic priming was allowed. Despite extensive testing, we found no evidence that unconsciously perceived words can activate sensorimotor processes, although these words are processed up to the semantic level. We thus conclude that sensorimotor activations might need conscious access to emerge during language understanding. (PsycINFO Database Recor

On efficiency and localisation for the torsion function

We consider the torsion function for the Dirichlet Laplacian $-\Delta$, and for the Schr\"odinger operator $- \Delta + V$ on an open set $\Omega\subset \R^m$, with Lebesgue measure $0<|\Omega|<\infty$, with a real-valued, non-negative, measurable potential $V.$ We investigate the phenomena of vanishing efficiency and localisation, and large efficiency for the torsion function and the first Dirichlet eigenfunction.Comment: 34 pages, minor corrections and additions to v1 (May 2020

Optimizing the secondary coil of a tesla transformer to improve spectral purity

This paper provides an overview of the response of the tuned secondary circuit of a Tesla transformer, following impulse excitation from the tuned primary circuit. Multiorder oscillatory voltages and currents are energized in the secondary circuit, and research is ongoing to determine the fundamental and higher order modes for various secondary winding configurations, with the aim of developing design techniques that can be used to suppress the generation of the higher order modes. It is anticipated that this will lead to generators which exhibit enhanced spectral purity and which will be better suited to use in electronic warfare applications than conventionally wound Tesla transformers. © 2013 IEEE

Significant practical features of Tesla transformers

Although a large number of publications dealing with Tesla transformers have appeared, many of these are confined to providing an analysis of the transformer performance based on a lumped equivalent circuit model. The present paper is concerned with more practical issues and begins by considering the often overlooked significance of the magnetic coupling between the primary and secondary windings for the range of potential applications of these transformers. It continues by discussing the benefits of using a solid-state primary switch and providing an insight into various other additions that may be made to the basic circuit