Searcharxiv⌕ Search

arXiv subjects

D. C. Struppa

Publications and source records attributed to D. C. Struppa.

At least 19 recordsLinked to original sources
arXiv2024

Analyticity, superoscillations and supershifts in several variables

Superoscillations have roots in various scientific disciplines, including optics, signal processing, radar theory, and quantum mechanics. This intriguing mathematical phenomenon permits specific functions to oscillate at a rate surpassing their highest Fourier component. A more encompassing concept, supershifts, extends the idea of superoscillations to functions that are not sum of exponential functions. This broader notion is linked to Bernstein and Lagrange approximation of analytic functions in $\mathbb{C}^n$. Recent advancements in the theory of superoscillations and supershifts in one variable have focused on their time evolution. This paper takes a step further by expanding the notion of supershifts to include the case of several variables. We provide specific examples related to harmonic analysis where the variables vary in multi-dimensional frequency (space, or scale) domains.

math.CV↗
arXiv2023

Analyticity and supershift with irregular sampling

The notion of supershift generalizes that one of superoscillation and expresses the fact that the sampling of a function in an interval allows to compute the values of the function outside the interval. In a previous paper we discussed the case in which the sampling of the function is regular and we are considering supershift in a bounded set, while here we investigate how irregularity in the sampling may affect the answer to the question of whether there is any relation between supershift and real analyticity on the whole real line. We show that the restriction to $\mathbb{R}$ of any entire function displays supershift, whereas the converse is, in general, not true. We conjecture that the converse is true as long as the sampling is regular, we discuss examples in support and we prove that the conjecture is indeed true for periodic functions.

math.CV↗
arXiv2023

Analyticity and supershift with regular sampling

The notion of supershift (in itself a generalization of the notion of superoscillation arising in quantum mechanics) expresses the fact that the sampling of a function in an interval allows to compute the values of the function far from the interval. In this paper we study the relation between supershift and real analyticity. We use a classical result due to Serge Bernstein to show that real analyticity for a complex valued function implies a strong form of supershift. On the other hand, we use a parametric version of a result by Leonid Kantorovitch to show that the converse is not true. We additionally study the stability of functions that satisfy suitable supershift requirements under multiplication by complex numbers or primitivization.

math.CV↗
arXiv2021

A new method to generate superoscillating functions and supershifts

Superoscillations are band-limited functions that can oscillate faster than their fastest Fourier component. These functions (or sequences) appear in weak values in quantum mechanics and in many fields of science and technology such as optics, signal processing and antenna theory. In this paper we introduce a new method to generate superoscillatory functions that allows us to construct explicitly a very large class of superoscillatory functions.

math-ph↗
arXiv2020

Gauss sums, superoscillations and the Talbot carpet

We consider the evolution, for a time-dependent Schrödinger equation, of the so called Dirac comb. We show how this evolution allows us to recover explicitly (indeed optically) the values of the quadratic generalized Gauss sums. Moreover we use the phenomenon of superoscillatory sequences to prove that such Gauss sums can be asymptotically recovered from the values of the spectrum of any sufficiently regular function compactly supported on $\R$. The fundamental tool we use is the so called Galilean transform that was introduced and studied in the context on non-linear time dependent Schrödinger equations. Furthermore, we utilize this tool to understand in detail the evolution of an exponential $e^{iωx}$ in the case of a Schrödinger equation with time-independent periodic potential.

math-ph↗
arXiv2016

A Zeta Function for Multicomplex Algebra

In this paper we define and study a Dedekind-like zeta function for the algebra of multicomplex numbers. By using the idempotent representations for such numbers, we are able to identify this zeta function with the one associated to a product of copies of the field of Gaussian rationals. The approach we use is substantially different from the one previously introduced by Rochon (for the bicomplex case) and by Reid and Van Gorder (for the multicomplex case).

math.NT↗
arXiv2015

The mathematics of superoscillations

In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, we obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces. In particular, we will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrödinger equation and other equations.

math-ph↗
arXiv2014

Quantum harmonic oscillator with superoscillating initial datum

In this paper we study the evolution of superoscillating initial data for the quantum driven harmonic oscillator. Our main result shows that superoscillations are amplified by the harmonic potential and that the analytic solution develops a singularity in finite time. We also show that for a large class of solutions of the Schrödinger equation, superoscillating behavior at any given time implies superoscillating behavior at any other time.

quant-ph↗
arXiv2014

The quantum pigeonhole principle and the nature of quantum correlations

The pigeonhole principle: "If you put three pigeons in two pigeonholes at least two of the pigeons end up in the same hole" is an obvious yet fundamental principle of Nature as it captures the very essence of counting. Here however we show that in quantum mechanics this is not true! We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box. Furthermore, we show that the above "quantum pigeonhole principle" is only one of a host of related quantum effects, and points to a very interesting structure of quantum mechanics that was hitherto unnoticed. Our results shed new light on the very notions of separability and correlations in quantum mechanics and on the nature of interactions. It also presents a new role for entanglement, complementary to the usual one. Finally, interferometric experiments that illustrate our effects are proposed.

quant-ph↗
arXiv2011

Geometric Path Integrals. A Language for Multiscale Biology and Systems Robustness

In this paper we suggest that, under suitable conditions, supervised learning can provide the basis to formulate at the microscopic level quantitative questions on the phenotype structure of multicellular organisms. The problem of explaining the robustness of the phenotype structure is rephrased as a real geometrical problem on a fixed domain. We further suggest a generalization of path integrals that reduces the problem of deciding whether a given molecular network can generate specific phenotypes to a numerical property of a robustness function with complex output, for which we give heuristic justification. Finally, we use our formalism to interpret a pointedly quantitative developmental biology problem on the allowed number of pairs of legs in centipedes.

q-bio.MN↗
arXiv2009

A Phragmén - Lindelöf principle for slice regular functions

The celebrated 100-year old Phragmen-Lindelof principle is a far reaching extension of the maximum modulus theorem for holomorphic functions of one complex variable. In some recent papers there has been a resurgence of interest in principles of this type for functions of a hypercomplex variable and for solutions of suitable partial differential equations. In the present article we obtain a Phragmen-Lindelof principle for slice regular functions, a class of quaternion-valued functions of a quaternionic variable which has been recently introduced.

math.CV↗
arXiv2008

Slice monogenic functions

In this paper we offer a definition of monogenicity for functions defined on $\rr^{n+1}$ with values in the Clifford algebra $\rr_n$ following an idea inspired by the recent papers \cite{gs}, \cite{advances}. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra $\rr_n$. We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series.

math.CV↗
arXiv2008

Non commutative functional calculus: unbounded operators

In a recent work, \cite{cgss}, we developed a functional calculus for bounded operators defined on quaternionic Banach spaces. In this paper we show how the results from \cite{cgss} can be extended to the unbounded case, and we highlight the crucial differences between the two cases. In particular, we deduce a new eigenvalue equation, suitable for the construction of a functional calculus for operators whose spectrum is not necessarily real.

math.SP↗
arXiv2008

A new functional calculus for non-commuting operators

In this paper we use the notion of slice monogenic functions \cite{slicecss} to define a new functional calculus for an $n$-tuple $T$ of not necessarily commuting operators. This calculus is different from the one discussed in \cite{jefferies} and it allows the explicit construction of the eigenvalue equation for the $n$-tuple $T$ based on a new notion of spectrum for $T$. Our functional calculus is consistent with the Riesz-Dunford calculus in the case of a single operator.

math.SP↗
arXiv2007

Augmented Sparse Reconstruction of Protein Signaling Networks

The problem of reconstructing and identifying intracellular protein signaling and biochemical networks is of critical importance in biology today. We sought to develop a mathematical approach to this problem using, as a test case, one of the most well-studied and clinically important signaling networks in biology today, the epidermal growth factor receptor (EGFR) driven signaling cascade. More specifically, we suggest a method, augmented sparse reconstruction, for the identification of links among nodes of ordinary differential equation (ODE) networks from a small set of trajectories with different initial conditions. Our method builds a system of representation by using a collection of integrals of all given trajectories and by attenuating block of terms in the representation itself. The system of representation is then augmented with random vectors, and minimization of the 1-norm is used to find sparse representations for the dynamical interactions of each node. Augmentation by random vectors is crucial, since sparsity alone is not able to handle the large error-in-variables in the representation. Augmented sparse reconstruction allows to consider potentially very large spaces of models and it is able to detect with high accuracy the few relevant links among nodes, even when moderate noise is added to the measured trajectories. After showing the performance of our method on a model of the EGFR protein network, we sketch briefly the potential future therapeutic applications of this approach.

physics.data-an↗
arXiv2007

Non commutative functional calculus: bounded operators

In this paper we develop a functional calculus for bounded operators defined on quaternionic Banach spaces. This calculus is based on the notion of slice-regularity, see \cite{gs}, and the key tools are a new resolvent operator and a new eigenvalue problem.

math.SP↗
arXiv2007

Functional dissipation microarrays for classification

In this article, we describe a new method of extracting information from signals, called functional dissipation, that proves to be very effective for enhancing classification of high resolution, texture-rich data. Our algorithm bypasses to some extent the need to have very specialized feature extraction techniques, and can potentially be used as an intermediate, feature enhancement step in any classification scheme. Functional dissipation is based on signal transforms, but uses the transforms recursively to uncover new features. We generate a variety of masking functions and `extract' features with several generalized matching pursuit iterations. In each iteration, the recursive process modifies several coefficients of the transformed signal with the largest absolute values according to the specific masking function; in this way the greedy pursuit is turned into a slow, controlled, dissipation of the structure of the signal that, for some masking functions, enhances separation among classes. Our case study in this paper is the classification of crystallization patterns of amino acids solutions affected by the addition of small quantities of proteins.

physics.data-an↗