Hilbert-Schmidt Theory of Symmetric Kernel

A classical theorem of Weyl [7] guarantees that the eigenvalues, ordered according to decreasing absolute values, of a symmetric kernel of class Cm (m ≥ 0) satisfy λn = o(n−m−½). Reade [5, 6] recently proved that if K is, in addition, positive definite, then λn = o(n−m−1;). He has also in [4] made similar improvements of classical spectral estimates for kernels of class Lip α. James Cochran pointed out to me that allied theorems for trigonometric Fourier coefficients seem to have been neglected in the literature. The trigonometric versions turn out to be elementary; nevertheless, in their conclusions concerning the decreasing rearrangement {f^*(n)} they generalize known results about the behaviour of monotone trigonometric transforms. Furthermore they suggest that the Cm hypothesis of Reade's theorem could be relaxed.

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Numerical Assessment of Symmetric and Non-Symmetric Kernel Functions on Second Order Non-Homogenous Volterra Integro-Differential Equations

Sakarya University Journal of Science ◽

10.16984/saufenbilder.668299 ◽

2021 ◽

Author(s):

Falade IYANDA ◽

Baoku ISMAEL ◽

Tiamiyu ABDULGAFAR

Keyword(s):

Differential Equations ◽

Kernel Functions ◽

Second Order ◽

Symmetric Kernel ◽

Numerical Assessment

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Capacitability of Analytic Sets

Nagoya Mathematical Journal ◽

10.1017/s0027763000007583 ◽

1960 ◽

Vol 16 ◽

pp. 91-109 ◽

Cited By ~ 1

Author(s):

Masanori Kishi

Keyword(s):

Metric Space ◽

Compact Set ◽

Locally Compact ◽

Symmetric Kernel ◽

Analytic Sets ◽

Separable Metric Space

Let Ω be a locally compact separable metric space and let Ф be a positive symmetric kernel. Then the inner and outer capacities of subsets of Ω are defined by means of Ф-potentials of positive measures in the following manner. We define the capacity c(K) of a compact set K in a certain manner by means of Ф-potentials.

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Introducing User-Prescribed Constraints in Markov Chains for Nonlinear Dimensionality Reduction

Neural Computation ◽

10.1162/neco_a_01184 ◽

2019 ◽

Vol 31 (5) ◽

pp. 980-997 ◽

Cited By ~ 1

Author(s):

Purushottam D. Dixit

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Dimensionality Reduction ◽

Markov Models ◽

Transition Probabilities ◽

Central Component ◽

Nonlinear Dimensionality Reduction ◽

Entropy Maximization ◽

Symmetric Kernel ◽

Data Points

Stochastic kernel-based dimensionality-reduction approaches have become popular in the past decade. The central component of many of these methods is a symmetric kernel that quantifies the vicinity between pairs of data points and a kernel-induced Markov chain on the data. Typically, the Markov chain is fully specified by the kernel through row normalization. However, in many cases, it is desirable to impose user-specified stationary-state and dynamical constraints on the Markov chain. Unfortunately, no systematic framework exists to impose such user-defined constraints. Here, based on our previous work on inference of Markov models, we introduce a path entropy maximization based approach to derive the transition probabilities of Markov chains using a kernel and additional user-specified constraints. We illustrate the usefulness of these Markov chains with examples.

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