237,385 research outputs found
Several Methods on the Limit Problem of Integration
If the expression of a limit contains definite integral, we call this kind of limit as the limit of integration. This paper takes the graduate entrance exams of some famous universities in China as examples, and gives several methods to solve the limit problem of integration by comprehensively utilizing the characteristics and operational properties of integral
A Brief History of Elliptic Integral Addition Theorems
The discovery of elliptic functions emerged from investigations of integral addition theorems. An addition theorem for a function f is a formula expressing f(u+v) in terms of f(u) and f(v). For a function defined as a definite integral with a variable upper limit, an addition theorem takes the form of an equation between the sum of two such integrals, with upper limits u and v, and an integral whose upper limit is a certain function of u and v.In this paper, we briefly sketch the role which the investigation of such addition theorems has played in the development of the theory of elliptic intgrals and elliptic functions
Principles of Discrete Time Mechanics: IV. The Dirac Equation, Particles and Oscillons
We apply the principles of discrete time mechanics discussed in earlier
papers to the first and second quantised Dirac equation. We use the Schwinger
action principle to find the anticommutation relations of the Dirac field and
of the particle creation operators in the theory. We find new solutions to the
discrete time Dirac equation, referred to as oscillons on account of their
extraordinary behaviour. Their principal characteristic is that they oscillate
with a period twice that of the fundamental time interval T of our theory.
Although these solutions can be associated with definite charge, linear
momentum and spin, such objects should not be observable as particles in the
continuous time limit. We find that for non-zero T they correspond to states
with negative squared norm in Hilbert space. However they are an integral part
of the discrete time Dirac field and should play a role in particle
interactions analogous to the role of longitudinal photons in conventional
quantum electrodynamics.Comment: 27 pages LateX; published versio
Hierarchical Structure of Azbel-Hofstader Problem: Strings and loose ends of Bethe Ansatz
We present numerical evidence that solutions of the Bethe Ansatz equations
for a Bloch particle in an incommensurate magnetic field (Azbel-Hofstadter or
AH model), consist of complexes-"strings". String solutions are well-known from
integrable field theories. They become asymptotically exact in the
thermodynamic limit. The string solutions for the AH model are exact in the
incommensurate limit, where the flux through the unit cell is an irrational
number in units of the elementary flux quantum.
We introduce the notion of the integral spectral flow and conjecture a
hierarchical tree for the problem. The hierarchical tree describes the topology
of the singular continuous spectrum of the problem. We show that the string
content of a state is determined uniquely by the rate of the spectral flow
(Hall conductance) along the tree. We identify the Hall conductances with the
set of Takahashi-Suzuki numbers (the set of dimensions of the irreducible
representations of with definite parity).
In this paper we consider the approximation of noninteracting strings. It
provides the gap distribution function, the mean scaling dimension for the
bandwidths and gives a very good approximation for some wave functions which
even captures their multifractal properties. However, it misses the
multifractal character of the spectrum.Comment: revtex, 30 pages, 6 Figs, 8 postscript files are enclosed, important
references are adde
Olshanski spherical functions for infinite dimensional motion groups of fixed rank
Consider the Gelfand pairs associated
with motion groups over the fields
with and fixed as well as the inductive limit ,the
Olshanski spherical pair . We classify all Olshanski
spherical functions of as functions on the cone
of positive semidefinite -matrices and show that they appear as
(locally) uniform limits of spherical functions of as .
The latter are given by Bessel functions on . Moreover, we determine all
positive definite Olshanski spherical functions and discuss related positive
integral representations for matrix Bessel functions. We also extend the
results to the pairs which
are related to the Cartan motion groups of non-compact Grassmannians. Here
Dunkl-Bessel functions of type B (for finite ) and of type A (for
) appear as spherical functions
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