1,622 research outputs found
A direct proof of Kim's identities
As a by-product of a finite-size Bethe Ansatz calculation in statistical
mechanics, Doochul Kim has established, by an indirect route, three
mathematical identities rather similar to the conjugate modulus relations
satisfied by the elliptic theta constants. However, they contain factors like
and , instead of . We show here that
there is a fourth relation that naturally completes the set, in much the same
way that there are four relations for the four elliptic theta functions. We
derive all of them directly by proving and using a specialization of
Weierstrass' factorization theorem in complex variable theory.Comment: Latex, 6 pages, accepted by J. Physics
Distortions Of The Phase Space Behavior Of A Particle During one-third integral Resonance Extraction
Electromagnetic Oscillations in a Driven Nonlinear Resonator: A New Description of Complex Nonlinear Dynamics
Many intriguing properties of driven nonlinear resonators, including the
appearance of chaos, are very important for understanding the universal
features of nonlinear dynamical systems and can have great practical
significance. We consider a cylindrical cavity resonator driven by an
alternating voltage and filled with a nonlinear nondispersive medium. It is
assumed that the medium lacks a center of inversion and the dependence of the
electric displacement on the electric field can be approximated by an
exponential function. We show that the Maxwell equations are integrated exactly
in this case and the field components in the cavity are represented in terms of
implicit functions of special form. The driven electromagnetic oscillations in
the cavity are found to display very interesting temporal behavior and their
Fourier spectra contain singular continuous components. To the best of our
knowledge, this is the first demonstration of the existence of a singular
continuous (fractal) spectrum in an exactly integrable system.Comment: 5 pages, 3 figure
Shear-driven size segregation of granular materials: modeling and experiment
Granular materials segregate by size under shear, and the ability to
quantitatively predict the time required to achieve complete segregation is a
key test of our understanding of the segregation process. In this paper, we
apply the Gray-Thornton model of segregation (developed for linear shear
profiles) to a granular flow with an exponential profile, and evaluate its
ability to describe the observed segregation dynamics. Our experiment is
conducted in an annular Couette cell with a moving lower boundary. The granular
material is initially prepared in an unstable configuration with a layer of
small particles above a layer of large particles. Under shear, the sample mixes
and then re-segregates so that the large particles are located in the top half
of the system in the final state. During this segregation process, we measure
the velocity profile and use the resulting exponential fit as input parameters
to the model. To make a direct comparison between the continuum model and the
observed segregation dynamics, we locally map the measured height of the
experimental sample (which indicates the degree of segregation) to the local
packing density. We observe that the model successfully captures the presence
of a fast mixing process and relatively slower re-segregation process, but the
model predicts a finite re-segregation time, while in the experiment
re-segregation occurs only exponentially in time
Manuscript for the Proceedings of the VI International Conference on High Energy Accelerators: Improvement possibilities in the performance of the CERN Intersecting Storage Rings
Nodal domains on quantum graphs
We consider the real eigenfunctions of the Schr\"odinger operator on graphs,
and count their nodal domains. The number of nodal domains fluctuates within an
interval whose size equals the number of bonds . For well connected graphs,
with incommensurate bond lengths, the distribution of the number of nodal
domains in the interval mentioned above approaches a Gaussian distribution in
the limit when the number of vertices is large. The approach to this limit is
not simple, and we discuss it in detail. At the same time we define a random
wave model for graphs, and compare the predictions of this model with analytic
and numerical computations.Comment: 19 pages, uses IOP journal style file
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