53 research outputs found
Thomas rotation and Thomas precession
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
For a set of points in the plane, not all collinear, we denote by the number of triangles in any triangulation of ; that is, where and are the numbers of points of in the
boundary and the interior of (we use to denote "convex hull of
"). We conjecture the following analogue of the Brunn-Minkowski inequality:
for any two point sets one has
We prove this conjecture in several cases: if , if ,
if , or if none of or has interior points.Comment: 30 page
Weakly nonlocal fluid mechanics - the Schrodinger equation
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
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
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
The aim of this paper is to prove that if a planar set has a difference
set satisfying for suitable than
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
is a set of exponentials mutually orthogonal with respect to any symmetric
convex set in the plane with a smooth boundary and everywhere non-vanishing
curvature, then # (A \cap {[-q,q]}^2) \leq C(K) q where is a constant
depending only on . 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 is a centrally symmetric convex body with a smooth
boundary and non-vanishing curvature, then does not possess an
orthogonal basis of exponentials
A superadditivity and submultiplicativity property for cardinalities of sumsets
For finite sets of integers A1, . . . ,An we study the cardinality of the n-fold
sumset A1 + · · · + An compared to those of (n â 1)-fold sumsets A1 + · · · + Aiâ1 +
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
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
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
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