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Articles 1 - 9 of 9
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Ideals Of Functions With Compact Support In The Integer-Valued Case, Themba Dube, Oghenetega Ighedo, Batsile Tlharesakgosi
Ideals Of Functions With Compact Support In The Integer-Valued Case, Themba Dube, Oghenetega Ighedo, Batsile Tlharesakgosi
Mathematics, Physics, and Computer Science Faculty Articles and Research
For a zero-dimensional Hausdorff space X, denote, as usual, by C(X, ℤ) the ring of continuous integer-valued functions on X. If f ∈ C(X, ℤ), denote by Z(f) the set of all points of X that are mapped to 0 by f. The set CK(X; ℤ) = {f ∈ C(X; ℤ) | clX(X \ Z(f)) is compact} is the integer-valued analogue of the ideal of functions with compact support in C(X). By first working in the category of locales and then interpreting …
An Approach To The Gaussian Rbf Kernels Via Fock Spaces, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini
An Approach To The Gaussian Rbf Kernels Via Fock Spaces, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
We use methods from the Fock space and Segal–Bargmann theories to prove several results on the Gaussian RBF kernel in complex analysis. The latter is one of the most used kernels in modern machine learning kernel methods and in support vector machine classification algorithms. Complex analysis techniques allow us to consider several notions linked to the radial basis function (RBF) kernels, such as the feature space and the feature map, using the so-called Segal–Bargmann transform. We also show how the RBF kernels can be related to some of the most used operators in quantum mechanics and time frequency analysis; specifically, …
An Uncountable Ergodic Roth Theorem And Applications, Polona Durcik, Rachel Greenfeld, Annina Iseli, Asgar Jamneshan, José Madrid
An Uncountable Ergodic Roth Theorem And Applications, Polona Durcik, Rachel Greenfeld, Annina Iseli, Asgar Jamneshan, José Madrid
Mathematics, Physics, and Computer Science Faculty Articles and Research
We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements a result of Zorin- Kranich. We establish the following two additional results: First, a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. Second, a uniformity aspect in the double recurrence theorem for Γ-systems for arbitrary uniformly amenable groups Γ. Our uncountable Roth …
A Question Of Fundamental Methodology: Reply To Mikhail Katz And His Coauthors, Tom Archibald, Richard T. W. Arthur, Giovanni Ferraro, Jeremy Gray, Douglas Jesseph, Jesper Lützen, Marco Panza, David Rabouin, Gert Schubring
A Question Of Fundamental Methodology: Reply To Mikhail Katz And His Coauthors, Tom Archibald, Richard T. W. Arthur, Giovanni Ferraro, Jeremy Gray, Douglas Jesseph, Jesper Lützen, Marco Panza, David Rabouin, Gert Schubring
Philosophy Faculty Articles and Research
This paper is a response by several historians of mathematics to a series of papers published from 2012 onwards by Mikhail Katz and various co-authors, the latest of which was recently published in the Mathematical Intelligencer, “Two-Track Depictions of Leibniz’s Fictions” (Katz, Kuhlemann, Sherry, Ugaglia, and van Atten, 2021). At issue is a question of fundamental methodology. These authors take for granted that non-standard analysis provides the correct framework for historical interpretation of the calculus, and castigate rival interpretations as having had a deleterious effect on the philosophy, practice, and applications of mathematics. Rather than make this case by reasoned …
On Superoscillations And Supershifts In Several Variables, Yakir Aharonov, Fabrizio Colombo, Andrew N. Jordan, Irene Sabadini, Tomer Shushi, Daniele C. Struppa, Jeff Tollaksen
On Superoscillations And Supershifts In Several Variables, Yakir Aharonov, Fabrizio Colombo, Andrew N. Jordan, Irene Sabadini, Tomer Shushi, Daniele C. Struppa, Jeff Tollaksen
Mathematics, Physics, and Computer Science Faculty Articles and Research
The aim of this paper is to study a class of superoscillatory functions in several variables, removing some restrictions on the functions that we introduced in a previous paper. Since the tools that we used with our approach are not common knowledge we will give detailed proof for the case of two variables. The results proved for superoscillatory functions in several variables can be further extended to supershifts in several variables.
Fock And Hardy Spaces: Clifford Appell Case, Daniel Alpay, Kamal Diki, Irene Sabadini
Fock And Hardy Spaces: Clifford Appell Case, Daniel Alpay, Kamal Diki, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper, we study a specific system of Clifford–Appell polynomials and, in particular, their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions. The construction is based on a general idea which allows us to obtain various function spaces by specifying a suitable sequence of real numbers. We focus on the Fock and Hardy cases in this setting, and we study the action of the Fueter mapping and its range.
Superoscillating Sequences And Supershifts For Families Of Generalized Functions, F. Colombo, I. Sabadini, Daniele Carlo Struppa, A. Yger
Superoscillating Sequences And Supershifts For Families Of Generalized Functions, F. Colombo, I. Sabadini, Daniele Carlo Struppa, A. Yger
Mathematics, Physics, and Computer Science Faculty Articles and Research
We construct a large class of superoscillating sequences, more generally of F-supershifts, where F is a family of smooth functions in (t, x) (resp. distributions in (t, x), or hyperfunctions in x depending on the parameter t) indexed by λ ∈ R. The frame in which we introduce such families is that of the evolution through Schrödinger equation (i∂/∂t−H (x))(ψ) = 0 (H (x) = −(∂2/∂x2)/2+V (x)), V being a suitable potential). If F = {(t, x) → ϕλ(t, x) ; λ ∈ R}, where ϕλ is evolved from the initial datum x → eiλx , F-supershifts will be of …
Hadamard’S Variational Formula In Terms Of Stress And Strain Tensors, Björn Gustafsson, Ahmed Sebbar
Hadamard’S Variational Formula In Terms Of Stress And Strain Tensors, Björn Gustafsson, Ahmed Sebbar
Mathematics, Physics, and Computer Science Faculty Articles and Research
Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.
Boundedness Of Some Multi-Parameter Fiber-Wise Multiplier Operators, Frédéric Bernicot, Polona Durcik
Boundedness Of Some Multi-Parameter Fiber-Wise Multiplier Operators, Frédéric Bernicot, Polona Durcik
Mathematics, Physics, and Computer Science Faculty Articles and Research
We prove Lp estimates for various multi-parameter bi- and trilinear operators with symbols acting on fibers of the two-dimensional functions. In particular, this yields estimates for the general bi-parameter form of the twisted paraproduct studied in [V. Kovač, Boundedness of the twisted paraproduct. Rev. Mat. Iberoam. 28 (2012), no. 4, 1143–1164.].