4,325 research outputs found
Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel
Let be the tensor product of finite-dimensional polynomial evaluation
Yangian -modules. Consider the universal difference operator whose coefficients are the XXX transfer matrices associated with . We show that the
difference equation for an -valued function has a basis of
solutions consisting of quasi-exponentials. We prove the same for the universal
differential operator whose
coefficients are the Gaudin transfer matrices associated
with the tensor product of finite-dimensional polynomial evaluation
-modules.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte Type Inequalities for Convex Functions via Fractional Integrals
The aim of this paper is to establish Hermite-Hadamard,
Hermite-Hadamard-Fej\'er, Dragomir-Agarwal and Pachpatte type inequalities for
new fractional integral operators with exponential kernel. These results allow
us to obtain a new class of functional inequalities which generalizes known
inequalities involving convex functions. Furthermore, the obtained results may
act as a useful source of inspiration for future research in convex analysis
and related optimization fields.Comment: 14 pages, to appear in Journal of Computational and Applied
Mathematic
One-sided Cauchy-Stieltjes Kernel Families
This paper continues the study of a kernel family which uses the
Cauchy-Stieltjes kernel in place of the celebrated exponential kernel of the
exponential families theory. We extend the theory to cover generating measures
with support that is unbounded on one side. We illustrate the need for such an
extension by showing that cubic pseudo-variance functions correspond to
free-infinitely divisible laws without the first moment. We also determine the
domain of means, advancing the understanding of Cauchy-Stieltjes kernel
families also for compactly supported generating measures
Supervised Learning with Indefinite Topological Kernels
Topological Data Analysis (TDA) is a recent and growing branch of statistics
devoted to the study of the shape of the data. In this work we investigate the
predictive power of TDA in the context of supervised learning. Since
topological summaries, most noticeably the Persistence Diagram, are typically
defined in complex spaces, we adopt a kernel approach to translate them into
more familiar vector spaces. We define a topological exponential kernel, we
characterize it, and we show that, despite not being positive semi-definite, it
can be successfully used in regression and classification tasks
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