53 research outputs found

    Thomas rotation and Thomas precession

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    Exact and simple calculation of Thomas rotation and Thomas precessions along a circular world line is presented in an absolute (coordinate-free) formulation of special relativity. Besides the simplicity of calculations the absolute treatment of spacetime allows us to gain a deeper insight into the phenomena of Thomas rotation and Thomas precession.Comment: 20 pages, to appear in Int. J. Theo. Phy

    Triangulations and a discrete Brunn-Minkowski inequality in the plane

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    For a set AA of points in the plane, not all collinear, we denote by tr(A){\rm tr}(A) the number of triangles in any triangulation of AA; that is, tr(A)=2i+b−2{\rm tr}(A) = 2i+b-2 where bb and ii are the numbers of points of AA in the boundary and the interior of [A][A] (we use [A][A] to denote "convex hull of AA"). We conjecture the following analogue of the Brunn-Minkowski inequality: for any two point sets A,B⊂R2A,B \subset {\mathbb R}^2 one has tr(A+B)12≄tr(A)12+tr(B)12. {\rm tr}(A+B)^{\frac12}\geq {\rm tr}(A)^{\frac12}+{\rm tr}(B)^{\frac12}. We prove this conjecture in several cases: if [A]=[B][A]=[B], if B=AâˆȘ{b}B=A\cup\{b\}, if ∣B∣=3|B|=3, or if none of AA or BB has interior points.Comment: 30 page

    Weakly nonlocal fluid mechanics - the Schrodinger equation

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    A weakly nonlocal extension of ideal fluid dynamics is derived from the Second Law of thermodynamics. It is proved that in the reversible limit the additional pressure term can be derived from a potential. The requirement of the additivity of the specific entropy function determines the quantum potential uniquely. The relation to other known derivations of Schr\"odinger equation (stochastic, Fisher information, exact uncertainty) is clarified.Comment: major extension and revisio

    Second order equation of motion for electromagnetic radiation back-reaction

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    We take the viewpoint that the physically acceptable solutions of the Lorentz--Dirac equation for radiation back-reaction are actually determined by a second order equation of motion, the self-force being given as a function of spacetime location and velocity. We propose three different methods to obtain this self-force function. For two example systems, we determine the second order equation of motion exactly in the nonrelativistic regime via each of these three methods, the three methods leading to the same result. We reveal that, for both systems considered, back-reaction induces a damping proportional to velocity and, in addition, it decreases the effect of the external force.Comment: 13 page

    A Robust Iterative Unfolding Method for Signal Processing

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    There is a well-known series expansion (Neumann series) in functional analysis for perturbative inversion of specific operators on Banach spaces. However, operators that appear in signal processing (e.g. folding and convolution of probability density functions), in general, do not satisfy the usual convergence condition of that series expansion. This article provides some theorems on the convergence criteria of a similar series expansion for this more general case, which is not covered yet by the literature. The main result is that a series expansion provides a robust unbiased unfolding and deconvolution method. For the case of the deconvolution, such a series expansion can always be applied, and the method always recovers the maximum possible information about the initial probability density function, thus the method is optimal in this sense. A very significant advantage of the presented method is that one does not have to introduce ad hoc frequency regulations etc., as in the case of usual naive deconvolution methods. For the case of general unfolding problems, we present a computer-testable sufficient condition for the convergence of the series expansion in question. Some test examples and physics applications are also given. The most important physics example shall be (which originally motivated our survey on this topic) the case of pi^0 --> gamma+gamma particle decay: we show that one can recover the initial pi^0 momentum density function form the measured single gamma momentum density function by our series expansion.Comment: 23 pages, 9 figure

    Distances sets that are a shift of the integers and Fourier basis for planar convex sets

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    The aim of this paper is to prove that if a planar set AA has a difference set Δ(A)\Delta(A) satisfying Δ(A)⊂Z++s\Delta(A)\subset \Z^++s for suitable ss than AA has at most 3 elements. This result is motivated by the conjecture that the disk has not more than 3 orthogonal exponentials. Further, we prove that if AA is a set of exponentials mutually orthogonal with respect to any symmetric convex set KK in the plane with a smooth boundary and everywhere non-vanishing curvature, then # (A \cap {[-q,q]}^2) \leq C(K) q where C(K)C(K) is a constant depending only on KK. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from \cite{IKP01} and \cite{IKT01} that if KK is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then L2(K)L^2(K) does not possess an orthogonal basis of exponentials

    A superadditivity and submultiplicativity property for cardinalities of sumsets

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    For finite sets of integers A1, . . . ,An we study the cardinality of the n-fold sumset A1 + · · · + An compared to those of (n − 1)-fold sumsets A1 + · · · + Ai−1 + Ai+1 + · · · + An. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets

    On quaternary complex Hadamard matrices of small orders

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    One of the main goals of design theory is to classify, characterize and count various combinatorial objects with some prescribed properties. In most cases, however, one quickly encounters a combinatorial explosion and even if the complete enumeration of the objects is possible, there is no apparent way how to study them in details, store them efficiently, or generate a particular one rapidly. In this paper we propose a novel method to deal with these difficulties, and illustrate it by presenting the classification of quaternary complex Hadamard matrices up to order 8. The obtained matrices are members of only a handful of parametric families, and each inequivalent matrix, up to transposition, can be identified through its fingerprint.Comment: 7 page

    Non-Equilibrium Evolution Thermodynamics Theory

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    Alternative approach for description of the non-equilibrium phenomena arising in solids at a severe external loading is analyzed. The approach is based on the new form of kinetic equations in terms of the internal and modified free energy. It is illustrated by a model example of a solid with vacancies, for which there is a complete statistical ground. The approach is applied to the description of important practical problem - the formation of fine-grained structure of metals during their treatment by methods of severe plastic deformation. In the framework of two-level two-mode effective internal energy potential model the strengthening curves unified for the whole of deformation range and containing the Hall-Petch and linear strengthening sections are calculated.Comment: 7 pages, 1 figur

    Long-term safety and efficacy of eculizumab in generalized myasthenia gravis

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