4,730 research outputs found
A nodal domain theorem for integrable billiards in two dimensions
Eigenfunctions of integrable planar billiards are studied - in particular,
the number of nodal domains, , of the eigenfunctions are considered. The
billiards for which the time-independent Schr\"odinger equation (Helmholtz
equation) is separable admit trivial expressions for the number of domains.
Here, we discover that for all separable and non-separable integrable
billiards, satisfies certain difference equations. This has been
possible because the eigenfunctions can be classified in families labelled by
the same value of , given a particular , for a set of quantum
numbers, . Further, we observe that the patterns in a family are similar
and the algebraic representation of the geometrical nodal patterns is found.
Instances of this representation are explained in detail to understand the
beauty of the patterns. This paper therefore presents a mathematical connection
between integrable systems and difference equations.Comment: 13 pages, 5 figure
Quantum Mechanics of a Rotating Billiard
Integrability of a square billiard is spontaneously broken as it rotates
about one of its corners. The system becomes quasi-integrable where the
invariant tori are broken with respect to a certain parameter, where E is the energy of the particle inside the billiard and
is the angular frequency of rotation of billiard. We study the system
classically and quantum mechanically in view of obtaining a correspondence in
the two descriptions. Classical phase space in Poincar\'{e} surface of section
shows transition from regular to chaotic motion as the parameter is
decreased. In the Quantum counterpart, the spectral statistics shows a
transition from Poisson to Wigner distribution as the system turns chaotic with
decrease in . The wavefunction statistics however show breakdown of
time-reversal symmetry as decreases
A Polymatroid Approach to Generalized Weights of Rank Metric Codes
We consider the notion of a -polymatroid, due to Shiromoto, and the
more general notion of -demi-polymatroid, and show how generalized
weights can be defined for them. Further, we establish a duality for these
weights analogous to Wei duality for generalized Hamming weights of linear
codes. The corresponding results of Ravagnani for Delsarte rank metric codes,
and Martinez-Penas and Matsumoto for relative generalized rank weights are
derived as a consequence.Comment: 22 pages; with minor revisions in the previous versio
Arithmetic Progressions in a Unique Factorization Domain
Pillai showed that any sequence of consecutive integers with at most 16 terms
possesses one term that is relatively prime to all the others. We give a new
proof of a slight generalization of this result to arithmetic progressions of
integers and further extend it to arithmetic progressions in unique
factorization domains of characteristic zero.Comment: Version 2 (to appear in Acta Arithmetica) with minor typos correcte
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