24,954 research outputs found
Source identities and kernel functions for deformed (quantum) Ruijsenaars models
We consider the relativistic generalization of the quantum
Calogero-Sutherland models due to Ruijsenaars, comprising the rational,
hyperbolic, trigonometric and elliptic cases. For each of these cases, we find
an exact common eigenfunction for a generalization of Ruijsenaars analytic
difference operators that gives, as special cases, many different kernel
functions; in particular, we find kernel functions for Chalykh-
Feigin-Veselov-Sergeev-type deformations of such difference operators which
generalize known kernel functions for the Ruijsenaars models. We also discuss
possible applications of our results.Comment: 24 page
Kernel functions and B\"acklund transformations for relativistic Calogero-Moser and Toda systems
We obtain kernel functions associated with the quantum relativistic Toda
systems, both for the periodic version and for the nonperiodic version with its
dual. This involves taking limits of previously known results concerning kernel
functions for the elliptic and hyperbolic relativistic Calogero-Moser systems.
We show that the special kernel functions at issue admit a limit that yields
generating functions of B\"acklund transformations for the classical
relativistic Calogero-Moser and Toda systems. We also obtain the
nonrelativistic counterparts of our results, which tie in with previous results
in the literature.Comment: 76 page
Invariant four-variable automorphic kernel functions
Let be a number field, let be its ring of adeles, and let
. Previously the author
provided an absolutely convergent geometric expression for the four variable
kernel function where the sum is over isomorphism classes of cuspidal
automorphic representations of . Here
is the typical kernel function representing the action of a test
function on the space of the cuspidal automorphic representation . In this
paper we show how to use ideas from the circle method to provide an alternate
expansion for the four variable kernel function that is visibly invariant under
the natural action of .Comment: The formula in this version is more explicit and simpler than the
previous versio
Kernel functions based on triplet comparisons
Given only information in the form of similarity triplets "Object A is more
similar to object B than to object C" about a data set, we propose two ways of
defining a kernel function on the data set. While previous approaches construct
a low-dimensional Euclidean embedding of the data set that reflects the given
similarity triplets, we aim at defining kernel functions that correspond to
high-dimensional embeddings. These kernel functions can subsequently be used to
apply any kernel method to the data set
The Harmonic Analysis of Kernel Functions
Kernel-based methods have been recently introduced for linear system
identification as an alternative to parametric prediction error methods.
Adopting the Bayesian perspective, the impulse response is modeled as a
non-stationary Gaussian process with zero mean and with a certain kernel (i.e.
covariance) function. Choosing the kernel is one of the most challenging and
important issues. In the present paper we introduce the harmonic analysis of
this non-stationary process, and argue that this is an important tool which
helps in designing such kernel. Furthermore, this analysis suggests also an
effective way to approximate the kernel, which allows to reduce the
computational burden of the identification procedure
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