ON (p, q)-STANCU-SZÁSZ-BETA OPERATORS AND THEIR APPROXIMATION PROPERTIES

In the present paper, we introduce the generalized form of (p; q)-analogue of the Szász-Beta operators with Stancu type parameters. We derived the local approximation properties of these operators and obtained the convergence rate and weighted approximation

Inequalities

Inequalities appear in various fields of natural science and engineering. Classical inequalities are still being improved and/or generalized by many researchers. That is, inequalities have been actively studied by mathematicians. In this book, we selected the papers that were published as the Special Issue ‘’Inequalities’’ in the journal Mathematics (MDPI publisher). They were ordered by similar topics for readers’ convenience and to give new and interesting results in mathematical inequalities, such as the improvements in famous inequalities, the results of Frame theory, the coefficient inequalities of functions, and the kind of convex functions used for Hermite–Hadamard inequalities. The editor believes that the contents of this book will be useful to study the latest results for researchers of this field

Functional inequalities in quantum information theory

Functional inequalities constitute a very powerful toolkit in studying various problems arising in classical information theory, statistics and many-body systems. Extensions of these tools to the noncommutative setting have been introduced in the beginning of the 90's in order to study the asymptotic properties of certain quantum Markovian evolutions. In this thesis, we study various extensions and problems arising from the specific noncommutative nature of such processes. The first logarithmic Sobolev inequality to be proved, due to Gross, was for the Ornstein Uhlenbeck semigroup, that is the Brownian motion with friction on the real line. The generalization of this result to the quantum Ornstein Uhlenbeck semigroup was found very recently by Carlen and Maas, and de Palma and Huber by means of different techniques. The latter proof consists of a quantum generalization of the so-called entropy power inequality. Here, we consider another possible version of the entropy power inequality and use it to derive asymptotic properties of the frictionless quantum Brownian motion. The proof of Carlen and Maas discussed in the previous paragraph relies on their new quantum extension of the classical notion of displacement convexity. This is classically known to imply most of the usual functional inequalities such as the modified logarithmic Sobolev inequality and Poincaré's inequality. Here, we further study the framework introduced by Carlen and Maas. In particular, we show how displacement convexity implies quantum functional and transportation cost inequalities. The latter are then used to derive certain concentration inequalities of quantum states in the spirit of Bobkov and Goetze. These concentration inequalities are used in order to derive finite sample size bounds for the task of quantum parameter estimation. The main advantage of classical logarithmic Sobolev inequalities over other methods resides in their tensorization property: the strong log-Sobolev constant of the product of independent Markovian evolutions is equal to the maximum over the set of strong log-Sobolev constants of the individual evolutions. However, this property is strongly believed to fail in the non-commutative case, due to the non-multiplicativity of noncommutative Lp to Lq norms. In this thesis, we show tensorization of the logarithmic Sobolev constants for the simplest quantum Markov semigroup, namely the generalized depolarizing semigroup. Moreover, we consider a new general method to overcome the issue of tensorization for general primitive quantum Markov semigroups by looking at their contractivity properties under the completely bounded Lp to Lq norms. This method was first investigated in the restricted case of unital semigroups by Beigi and King. Noncommutative functional inequalities considered in the present literature only deal with primitive quantum Markovian semigroups which model memoryless irreversible dynamics converging to a specific faithful state. However, quantum Markov semigroups can in general display a much richer behavior referred to as decoherence: In particular, under some mild conditions, any such semigroup is known to converge to an algebra of observables which effectively evolve unitarily. Here, we introduce the concept of a decoherence-free logarithmic Sobolev inequality, and the related notion of hypercontractivity of the associated evolution, to study the decoherence rate of non-primitive quantum Markov semigroups. Moreover, we utilize the transference method recently introduced by Gao, Junge and LaRacuente, in order to find decoherence times associated to a class of decoherent Markovian evolutions of great importance in the field of quantum error protection, namely collective decoherence semigroups. Finally, we develop the notion of quantum reverse hypercontractivity, first introduced by Cubitt, Kastoryano, Montanaro and Temme in the unital case, and apply it in conjunction with the tensorization of the modified logarithmic Sobolev inequality for the generalized depolarizing semigroup in order to find strong converse rates in quantum hypothesis testing and for the classical capacity of classical-quantum channels. Moreover, the transference method also allows us to find strong converse bounds on the various capacities of quantum Markovian evolutions

Uncertainty Quantification

Uncertainty quantification (UQ) is concerned with including and characterising uncertainties in mathematical models. Major steps comprise proper description of system uncertainties, analysis and efficient quantification of uncertainties in predictions and design problems, and statistical inference on uncertain parameters starting from available measurements. Research in UQ addresses fundamental mathematical and statistical challenges, but has also wide applicability in areas such as engineering, environmental, physical and biological applications. This workshop focussed on mathematical challenges at the interface of applied mathematics, probability and statistics, numerical analysis, scientific computing and application domains. The workshop served to bring together experts from those disciplines in order to enhance their interaction, to exchange ideas and to develop new, powerful methods for UQ

Classical and Quantum Mechanical Models of Many-Particle Systems

The topic of this meeting were non-linear partial differential and integro-differential equations (in particular kinetic equations and their macroscopic/fluid-dynamical limits) modeling the dynamics of many-particle systems with applications in physics, engineering, and mathematical biology. Typical questions of interest were the derivation of macro-models from micro-models, the mathematical analysis (well-posedness, stability, asymptotic behavior of solutions), and –to a lesser extend– numerical aspects of such equations

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

Integral Transformation, Operational Calculus and Their Applications

The importance and usefulness of subjects and topics involving integral transformations and operational calculus are becoming widely recognized, not only in the mathematical sciences but also in the physical, biological, engineering and statistical sciences. This book contains invited reviews and expository and original research articles dealing with and presenting state-of-the-art accounts of the recent advances in these important and potentially useful subjects

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282