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Articles 1 - 8 of 8
Full-Text Articles in Other Mathematics
Superoscillations And Fock Spaces, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa
Superoscillations And Fock Spaces, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we use techniques in Fock spaces theory and compute how the Segal-Bargmann transform acts on special wave functions obtained by multiplying superoscillating sequences with normalized Hermite functions. It turns out that these special wave functions can be constructed also by computing the approximating sequence of the normalized Hermite functions. First, we start by treating the case when a superoscillating sequence is multiplied by the Gaussian function. Then, we extend these calculations to the case of normalized Hermite functions leading to interesting relations with Weyl operators. In particular, we show that the Segal-Bargmann transform maps superoscillating sequences onto …
An Extension Of The Complex–Real (C–R) Calculus To The Bicomplex Setting, With Applications, Daniel Alpay, Kamal Diki, Mihaela Vajiac
An Extension Of The Complex–Real (C–R) Calculus To The Bicomplex Setting, With Applications, Daniel Alpay, Kamal Diki, Mihaela Vajiac
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper, we extend notions of complex ℂ−ℝ-calculus to the bicomplex setting and compare the bicomplex polyanalytic function theory to the classical complex case. Applications of this theory include two bicomplex least mean square algorithms, which extend classical real and complex least mean square algorithms.
Operators Induced By Certain Hypercomplex Systems, Daniel Alpay, Ilwoo Choo
Operators Induced By Certain Hypercomplex Systems, Daniel Alpay, Ilwoo Choo
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper, we consider a family {Ht}t∈R of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations {(C2, πt)}t∈R of the hypercomplex system {Ht}t∈R, and study the realizations πt(h) of hypercomplex numbers h ∈ Ht, as (2 × 2)-matrices acting on C2, for an arbitrarily fixed scale t ∈ R. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.
A Hörmander–Fock Space, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa
A Hörmander–Fock Space, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
In a recent paper we used a basic decomposition property of polyanalytic functions of order 2 in one complex variable to characterize solutions of the classical ∂-problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander-Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by η= (ηn)n≥0 leading to a special entire function E(z) that is used to express the kernel function of the Hörmander-Fock space. We …
Hörmander’S L2 -Method, ∂-Problem And Polyanalytic Function Theory In One Complex Variable, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa
Hörmander’S L2 -Method, ∂-Problem And Polyanalytic Function Theory In One Complex Variable, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we consider the classical ∂-problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of polyanalytic functions in order to construct particular solutions of this problem and obtain new Hörmander type estimates using suitable powers of the Cauchy-Riemann operator. We also compute particular solutions of the ∂-problem for specific polyanalytic data such as the Itô complex Hermite polynomials and polyanalytic Fock kernels.
The Structure Of Locally Integral Involutive Po-Monoids And Semirings, José Gil-Férez, Peter Jipsen, Siddhartha Lodhia
The Structure Of Locally Integral Involutive Po-Monoids And Semirings, José Gil-Férez, Peter Jipsen, Siddhartha Lodhia
Mathematics, Physics, and Computer Science Faculty Articles and Research
We show that every locally integral involutive partially ordered monoid (ipo-monoid) A = (A,⩽, ·, 1,∼,−), and in particular every locally integral involutive semiring, decomposes in a unique way into a family {Ap : p ∈ A+} of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are semirings. Moreover, we show that there is a family of monoid homomorphisms Φ = {φpq : Ap → Aq : p ⩽ q}, indexed on the positive cone (A+,⩽), so that the structure of A can be recovered as a glueing R ΦAp of its integral …
Music: Numbers In Motion, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa
Music: Numbers In Motion, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
Music develops and appears as we allow numbers to acquire a dynamical aspect and create, through their growth, the various keys that permit the richness of the musical texture. This idea was simply adumbrated in Plato’s work, but its importance to his philosophical worldview cannot be underestimated. In this paper we begin by discussing what is probably the first written record of an attempt to create a good temperament and then follow the Pythagoreans approach, whose problems forced musicians, over the next several centuries up to the Renaissance and early modern times, to come up with many different variations.
The Merchant And The Mathematician: Commerce And Accounting, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa
The Merchant And The Mathematician: Commerce And Accounting, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this article we describe the invention of double-entry bookkeeping (or partita doppiaas it was called in Italian), as a fertile intersection between mathematics and early commerce. We focus our attention on this seemingly simple technique that requires only minimal mathematical expertise, but whose discovery is clearly the result of a mathematical way of thinking, in order to make a conceptual point about the role of mathematics as the humus from which disciplines as different as operations research, computer science, and data science have evolved.