57 research outputs found
Solvable and/or integrable and/or linearizable N-body problems in ordinary (three-dimensional) space. I
Several N-body problems in ordinary (3-dimensional) space are introduced
which are characterized by Newtonian equations of motion (``acceleration equal
force;'' in most cases, the forces are velocity-dependent) and are amenable to
exact treatment (``solvable'' and/or ``integrable'' and/or ``linearizable'').
These equations of motion are always rotation-invariant, and sometimes
translation-invariant as well. In many cases they are Hamiltonian, but the
discussion of this aspect is postponed to a subsequent paper. We consider
``few-body problems'' (with, say, \textit{N}=1,2,3,4,6,8,12,16,...) as well as
``many-body problems'' (N an arbitrary positive integer). The main focus of
this paper is on various techniques to uncover such N-body problems. We do not
discuss the detailed behavior of the solutions of all these problems, but we do
identify several models whose motions are completely periodic or multiply
periodic, and we exhibit in rather explicit form the solutions in some cases
The Fourier Transform of Poisson Multinomial Distributions and its Algorithmic Applications
An -Poisson Multinomial Distribution (PMD) is a random variable of
the form , where the 's are independent random
vectors supported on the set of standard basis vectors in In
this paper, we obtain a refined structural understanding of PMDs by analyzing
their Fourier transform. As our core structural result, we prove that the
Fourier transform of PMDs is {\em approximately sparse}, i.e., roughly
speaking, its -norm is small outside a small set. By building on this
result, we obtain the following applications:
{\bf Learning Theory.} We design the first computationally efficient learning
algorithm for PMDs with respect to the total variation distance. Our algorithm
learns an arbitrary -PMD within variation distance using a
near-optimal sample size of and runs in time
Previously, no algorithm with a
runtime was known, even for
{\bf Game Theory.} We give the first efficient polynomial-time approximation
scheme (EPTAS) for computing Nash equilibria in anonymous games. For normalized
anonymous games with players and strategies, our algorithm computes a
well-supported -Nash equilibrium in time The best
previous algorithm for this problem had running time
where , for any
{\bf Statistics.} We prove a multivariate central limit theorem (CLT) that
relates an arbitrary PMD to a discretized multivariate Gaussian with the same
mean and covariance, in total variation distance. Our new CLT strengthens the
CLT of Valiant and Valiant by completely removing the dependence on in the
error bound.Comment: 68 pages, full version of STOC 2016 pape
Exact Analytic Solutions for the Rotation of an Axially Symmetric Rigid Body Subjected to a Constant Torque
New exact analytic solutions are introduced for the rotational motion of a
rigid body having two equal principal moments of inertia and subjected to an
external torque which is constant in magnitude. In particular, the solutions
are obtained for the following cases: (1) Torque parallel to the symmetry axis
and arbitrary initial angular velocity; (2) Torque perpendicular to the
symmetry axis and such that the torque is rotating at a constant rate about the
symmetry axis, and arbitrary initial angular velocity; (3) Torque and initial
angular velocity perpendicular to the symmetry axis, with the torque being
fixed with the body. In addition to the solutions for these three forced cases,
an original solution is introduced for the case of torque-free motion, which is
simpler than the classical solution as regards its derivation and uses the
rotation matrix in order to describe the body orientation. This paper builds
upon the recently discovered exact solution for the motion of a rigid body with
a spherical ellipsoid of inertia. In particular, by following Hestenes' theory,
the rotational motion of an axially symmetric rigid body is seen at any instant
in time as the combination of the motion of a "virtual" spherical body with
respect to the inertial frame and the motion of the axially symmetric body with
respect to this "virtual" body. The kinematic solutions are presented in terms
of the rotation matrix. The newly found exact analytic solutions are valid for
any motion time length and rotation amplitude. The present paper adds further
elements to the small set of special cases for which an exact solution of the
rotational motion of a rigid body exists.Comment: "Errata Corridge Postprint" version of the journal paper. The
following typos present in the Journal version are HERE corrected: 1)
Definition of \beta, before Eq. 18; 2) sign in the statement of Theorem 3; 3)
Sign in Eq. 53; 4)Item r_0 in Eq. 58; 5) Item R_{SN}(0) in Eq. 6
Zeta function method and repulsive Casimir forces for an unusual pair of plates at finite temperature
We apply the generalized zeta function method to compute the Casimir energy
and pressure between an unusual pair of parallel plates at finite temperature,
namely: a perfectly conducting plate and an infinitely permeable one. The high
and low temperature limits of these quantities are discussed; relationships
between high and low temperature limits are estabkished by means of a modified
version of the temperature inversion symmetry.Comment: latex file 9 pages, 3 figure
Calculating Casimir Energies in Renormalizable Quantum Field Theory
Quantum vacuum energy has been known to have observable consequences since
1948 when Casimir calculated the force of attraction between parallel uncharged
plates, a phenomenon confirmed experimentally with ever increasing precision.
Casimir himself suggested that a similar attractive self-stress existed for a
conducting spherical shell, but Boyer obtained a repulsive stress. Other
geometries and higher dimensions have been considered over the years. Local
effects, and divergences associated with surfaces and edges have been studied
by several authors. Quite recently, Graham et al. have re-examined such
calculations, using conventional techniques of perturbative quantum field
theory to remove divergences, and have suggested that previous self-stress
results may be suspect. Here we show that the examples considered in their work
are misleading; in particular, it is well-known that in two dimensions a
circular boundary has a divergence in the Casimir energy for massless fields,
while for general dimension not equal to an even integer the corresponding
Casimir energy arising from massless fields interior and exterior to a
hyperspherical shell is finite. It has also long been recognized that the
Casimir energy for massive fields is divergent for . These conclusions
are reinforced by a calculation of the relevant leading Feynman diagram in
and three dimensions. There is therefore no doubt of the validity of the
conventional finite Casimir calculations.Comment: 25 pages, REVTeX4, 1 ps figure. Revision includes new subsection 4B
and Appendix, and other minor correction
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