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Although the fundamental breakthrough in the formulation of modern quantum mechanics came from Heisenberg, the robust and mathematically rigorous formulation of matrix mechanics was given by Max Born and Pascual Jordan. In the present article, the author discusses the contributions of Max Born in the context of the formulation of matrix mechanics. [ABSTRACT FROM AUTHOR]

We give an elementary proof of an inequality of Lin, Kim and Hsieh that implies strong subadditivity of the von Neumann entropy. [ABSTRACT FROM AUTHOR]

The information in the real world often contains many properties such as fuzziness, randomness, and approximation. Although existing linguistic collections attempt to solve these problems, with the emergence of more and more constraints and challenges, this information cannot fully express the problem, leading to an increasing demand for methods that can contain multiple uncertain information. In this paper, we comprehensively consider the various characteristics of information including membership degree, credibility and approximation based on rough sets, and propose the concept of χ -linguistic sets (χ LSs), which depend on original data rather than prior knowledge and effectively solve the problem of incomplete information representation. At the same time, the corresponding theories such as the comparison method and operational rules have also been proposed. Subsequently, we construct a new χ -linguistic VIKOR (χ LVIKOR) method for multi-attribute group decision making (MAGDM) problem with χ LSs, and apply it to the risk assessment of COVID-19. Through comparative analysis, we discuss the effectiveness and superiority of χ LSs. [ABSTRACT FROM AUTHOR]

The aim of the current article is to establish some coupled fixed-point theorems with different contractive conditions in complete complex valued b-metric space. We are extending the results of Kang et al. (Int J Math Anal 7(46):2269–2277, 2013). To show the significance of our result an example is also presented. [ABSTRACT FROM AUTHOR]

The Operator Argument against eternalism holds that having non-vacuous tense operators in the language is incompatible with the claim that every proposition has its truth-value eternally. Assuming that (1) there are non-vacuous tense operators, (2) tense operators operate on propositions and (3) tense operators which operate on eternal entities are vacuous, it may be argued that eternalism is false. In this paper, I examine the Operator Argument. The goal is threefold. First, I want to present some aspects of the debate in a more elaborate way, especially those concerning formal matters. Secondly, I will argue that eternalism can escape the Operator Argument. There are two main strategies for handling the Operator Argument. The first one is based on replacing temporal operators with object-language quantifiers. The second rejects the identification of compositional semantic value with assertoric content. My third goal is to show that none of them is as good as the strategy that adopts Timestamp Semantics (Fritz in Philosl Stud 176:2933–2959, 2019 ). I am going to argue that the quantificational treatment of tenses is compatible with temporalism and that the arguments for rejecting the identification of compositional semantic value with assertoric content provide, in fact, a motivation for the temporalist position. At the end, I will develop Timestamp Semantics by providing a novel formalization of it, and defend it against three potential counter-arguments. [ABSTRACT FROM AUTHOR]

In this work, we mainly study refinements and generalizations of the Young’s and its reverse inequalities. First, utilizing the famous weighted arithmetic and geometric mean inequality, we provide an alternative proof for the power version of the Young’s inequality given by Al-Manasrah and Kittaneh. In addition, some generalizations of the Al-Manasrah–Kittaneh’s inequalities are also given. Further, we also establish refinement of the Alzer–Fonseca–Kova c ˇ ec’s version of the Young’s inequality as follows. Let a, b, μ and v be positive real numbers with 0 ≤ μ < v ≤ 1 and m be a positive integer. Then [ μ a + (1 - μ) b ] m - (a μ b 1 - μ ) m + r [ a m 2 - (a v b 1 - v ) m 2 ] 2 ≤ ( 1 - μ 1 - v ) m [ (v a + (1 - v) b) m - (a v b 1 - v ) m ] and r 1 [ a m 2 - (a v b 1 - v ) m 2 ] 2 + (μ v ) m [ (v a + (1 - v) b) m - (a v b 1 - v ) m ] ≤ [ μ a + (1 - μ) b ] m - (a μ b 1 - μ ) m , where r is a constant with 0 < r ≤ ( 1 - μ 1 - v ) m v m - μ m and r 1 is a constant with 0 < r 1 ≤ min { (μ v ) m , μ m - (μ v ) m v m } . As an application, we present corresponding operator and matrix inequalities following from the established scalar inequalities. [ABSTRACT FROM AUTHOR]

Let P n be the class of all complex polynomials of degree at most n, and let T : P n → P n be a linear operator. We shall say that T is a B n -operator if for every P ∈ P n having all zeros in the closed unit disk, T[P] has all its zeros in the closed unit disk. Recently, Rather et al. [On the zeros of certain composite polynomials and an operator preserving inequalities, Ramanujan J., 54 (2021), 605–617] introduced and considered the generalized B n -operator N m : P n → P n , defined by N m [ P ] (z) : = ∑ k = 0 m λ k nz 2 k P (k) (z) k ! , m ≤ n , where λ k ∈ C , k = 0 , 1 , 2 , ... , m , are such that the polynomial ϕ (z) : = ∑ k = 0 m n k λ k z k has all its zeros in the half plane R e (z) ≤ n 4. They established several sharp Bernstein-type inequalities for this operator giving extensions and generalizations of various classical polynomial inequalities. In this paper, we establish sharp integral-norm estimates of Zygmund type for this operator and develop a unified method for getting various Bernstein-type inequalities and other related inequalities in the supremum-norm as special cases. [ABSTRACT FROM AUTHOR]

A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the form A = T (a) + E where T(a) is the Toeplitz matrix with entries (T (a)) i , j = a j - i , for a j - i ∈ C , i , j ≥ 1 , while E is a matrix representing a compact operator in ℓ 2 . The matrix A is finitely representable if a k = 0 for k < - m and for k > n , given m , n > 0 , and if E has a finite number of nonzero entries. The problem of numerically computing eigenpairs of a finitely representable QT matrix is investigated, i.e., pairs (λ , v) such that A v = λ v , with λ ∈ C , v = (v j) j ∈ Z + , v ≠ 0 , and ∑ j = 1 ∞ | v j | 2 < ∞ . It is shown that the problem is reduced to a finite nonlinear eigenvalue problem of the kind W U (λ) β = 0 , where W is a constant matrix and U depends on λ and can be given in terms of either a Vandermonde matrix or a companion matrix. Algorithms relying on Newton's method applied to the equation det W U (λ) = 0 are analyzed. Numerical experiments show the effectiveness of this approach. The algorithms have been included in the CQT-Toolbox [Numer. Algorithms 81 (2019), no. 2, 741–769]. [ABSTRACT FROM AUTHOR]

The fast evolution of drone technology as well as the ubiquitous adoption from private and commercial users lead to the necessity to introduce new regulations covering the operations of Unmanned Aircraft Systems (UAS) in European Member states. The regulations implemented rules for multiple scenarios of normal and complex UAS operations as well as the collection of registration data from remote pilots and Operators. This research project tried for the first time to determine the status quo of the adoption of the novel EASA regulations of drones within the European members states. Data from National Aviation Authorities has been collected, accumulated, and analyzed towards qualitative and quantitative features. [ABSTRACT FROM AUTHOR]

Probabilistic linguistic term set solves the problem of probabilistic distribution of linguistic terms. Due to the objective and subjective factors such as the decision makers'experience and preference, the credibility of the linguistic terms is different. However, current studies on PLTSs ignore this difference. In this paper, we first propose a novel concept called Z probabilistic linguistic term set (ZPLTS). As an extension of existing tools, it takes advantage of the fact that Z-number can represent both information and corresponding credibility. At the same time, we discuss the normalization, operational rules, ranking method and distance measure for ZPLTSs. Then, we propose a new weight calculation method, an aggregation-based method and an extended TOPSIS method, and apply them to multi-attribute group decision making in Z probabilistic linguistic environment. Finally, a numerical example and some comparisons with other methods illustrate the necessity and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of fixed points of some generalized contractible operators defined on modular G-metric spaces and also prove the modular G-continuity of such operators. Furthermore, we prove that some generalized weakly compatible contractive operators in modular G-metric spaces have a unique fixed point. Our results extend, generalize, complement and include several known results as special cases. [ABSTRACT FROM AUTHOR]

This paper considers problems of reconstructing from known solutions the parameters of differential equations describing anomalous diffusion processes. A variational interpolation method proposed and tested by the authors earlier is used. Reconstruction of the time dependence of the diffusion coefficient and determination of the order of fractional time and space derivatives in an anomalous diffusion equation are demonstrated by using some model problems. It is shown that sufficient accuracy can be obtained with insignificant computational costs. [ABSTRACT FROM AUTHOR]

Let A and A ~ be linear unbounded in general operators on a Hilbert space. We consider the following problem: let the spectrum of A lie in some angular sector. In what sector the spectrum of A ~ lies, if A and A ~ are sufficiently "close"? Illustrative examples are also presented. [ABSTRACT FROM AUTHOR]

For a given self-adjoint operator A with discrete spectrum, we completely characterise possible eigenvalues of its rank-one perturbations B and discuss the inverse problem of reconstructing B from its spectrum. [ABSTRACT FROM AUTHOR]

If all the zeros of nth degree polynomials f(z) and g (z) = ∑ k = 0 n λ k n k z k respectively lie in the cricular regions | z | ≤ r and | z | ≤ s | z - σ | , s > 0 , then it was proved by Marden (Geometry of polynomials, Math Surveys, No. 3, American Mathematical Society, Providence, 1949, p. 86) that all the zeros of the polynomial h (z) = ∑ k = 0 n λ k f (k) (z) (σ z) k k ! lie in the circle | z | ≤ r max (1 , s) . In this paper, we relax the condition that f(z) and g(z) are of the same degree and instead assume that f(z) and g(z) are polynomials of arbitrary degree n and m, respectively, m ≤ n , and obtain a generalization of this result. As an application, we also introduce a linear operator which preserves Bernstein type polynomial inequalities. [ABSTRACT FROM AUTHOR]

This essay argues that there are theoretical benefits to keeping distinct—more pervasively than the literature has done so far—the psychological states of imagining that p versus believing that in-the-story p, when it comes to cognition of fiction and other forms of narrative. Positing both in the minds of a story's audience helps explain the full range of reactions characteristic of story consumption. This distinction also has interesting conceptual and explanatory dimensions that haven't been carefully observed, and the two mental state types make distinct contributions to generating emotional responses to stories. Finally, the differences between the mental states illuminate how a given story can be both shared with others and at the same time experienced as personal. [ABSTRACT FROM AUTHOR]

In this paper, by means of congruence openings of multiplicative Green's relations on a semiring we define and study several varieties of semirings, obtain the relationship between these varieties and give Mal'cev product decompositions of some varieties of idempotent semirings. In particular, we establish order embeddings of the lattice of all subvarieties of the variety of multiplicatively idempotent semirings (resp. idempotent semirings) into the direct products of the lattices of open (resp. closed) varieties with respect to two opening (resp. closure) operators on this lattice what we introduced. [ABSTRACT FROM AUTHOR]

Over recent years, several studies regarding preference queries over data streams have been developed in database and artificial intelligence research fields. Preference queries are useful in many decision making application areas, such as e-commerce, financial analysis, and content personalization. In this article, we explore new aspects of temporal conditional preference queries (tcp-queries) for the StreamPref query language. Tcp-queries allow the user to express how past instants of a data stream can influence the preference of a user at a present instant. In order to increase the utility of the StreamPref query language, we propose herein new operators that allow dealing with subsequences and filtering of sequences by length. To validate our proposal we present a detailed complexity analysis and an extensive set of experiments with synthetic and real datasets, which corroborate the efficiency of the algorithms and the utility of the new operators. [ABSTRACT FROM AUTHOR]

The use of weaponized drones or "unmanned aerial vehicles" (UAVs) has become increasingly widespread and controversial over the past few decades. The current paper reviews the state of the research regarding the potential psychosocial impact of weaponized drones on operators and target populations and communities. It is concluded that research regarding the impact of drones in the psychological literature remains limited and most discussion of drones' impact has taken place in the public policy and legal/ethical spheres, often by entities invested in condoning or condemning the use of drones. The limited available data addresses potential new challenges to the well-being of drone operators, factors influencing decision making regarding the use of drones, and the impact on target communities. The current paper neither condones nor condemns the use of drones, but is advanced as a state of the research and a call for additional objective and empirical analysis on this relatively new form of warfare. [ABSTRACT FROM AUTHOR]

It is known that dual representation of problems (through the main function and its adjoint in the Lagrange sense) makes it possible to formulate an effective perturbation theory on which the successive approximation method in the inverse problem theory relies. Let us suppose that according to preliminary predictions, a solution to the inverse problem (for example, the structure of medium of interest) belongs to a certain set A. Then selecting a suitable (trial, reference) element a0 as an unperturbed one and applying the perturbation theory, one can approximately describe the behavior of a solution to the forward problem in this domain and find a subset A0 that matches the measurement data best. However, as the accuracy requirements increase, the domain of applicability of the first approximation A0 is rapidly narrowing, and its expansion via addition of higher terms of the expansion complicates the solving procedure. For this reason, a number of works have searched for unperturbed approaches, including the method of variational interpolation (VI method). In this method, not one but several reference problems a1, a2,...,an are selected, from which a linear superposition of the principal function and the adjoint one is constructed, followed by determination of coefficients from the condition of stationarity of the form of the desired functional representation. This paper demonstrates application of the VI method to solving inverse problems of cosmic rays astrophysics in the simplest statement. [ABSTRACT FROM AUTHOR]

Objectives. To assess the efficacy Cytoflavin in correcting functional status and work capacity among Navy operators after long work cycles. Results from studies of 25 Navy operators aged 27–42 years were analyzed after voyages (baseline) and after correctional measures using Cytoflavin. Operator status was evaluated using the Ruslov Questionnaire for the Formal Dynamic Properties of Individuality and EEG recordings. Results and conclusions. The data obtained here provide evidence that Cytoflavin has positive effects on the energy component of the functional state of Navy operators during the post-voyage period, increasing psychomotor ergicity and speed and optimizing spontaneous brain bioelectrical activity. Activation of measures of the energy potential, phasic activity, and emotional reactions by Cytoflavin were linked with processes expanding the reserve capacities of the CNS in Navy operators after prolonged work cycles. [ABSTRACT FROM AUTHOR]

The theory that structured propositions are complex act-types has been independently articulated by Peter Hanks and Scott Soames. The present paper argues that the role of the act in such theories is supererogatory, for the individuation conditions of the act-based propositions remain wholly at the level of concepts and their formal combination, features which the traditional structured proposition theorist endorses. Thus, it is shown that the traditional problems for structured propositions are only ameliorable on the act conception by appeal to the very resources of the traditional conception. It is also shown that the act theories have no act-based account of quantification and other operator-involving relations. An elementary account of propositions is sketched, after Fodor, which retains the virtues of the traditional structural conception without falling into ’Platonism’. [ABSTRACT FROM AUTHOR]

We assess how video lottery terminal (VLT) operators' self-perceive their ability to recognize a problem gambler, to what extent they are approached by problem gamblers seeking for assistance, how many detections and interventions they report, and the reasons they give for not intervening with clients who show signs of problem gambling. We also examine how these variables are related to the operators' gender and status in the establishment. 177 VLT operators anonymously completed a structured questionnaire at the beginning of a responsible gambling training class held in different French-speaking Swiss towns. The operators felt confident in their ability to detect problem gambling behaviors, were rarely approached by problem gamblers seeking assistance, and reported fewer interventions compared to the number of detections. This reluctance to intervene was mainly attributed to the fear of potential negative reactions from the client. Female staff were the most reluctant to intervene and the most fearful of potential negative reactions from the client. Responsible gambling training programs should include coping strategies for dealing with potential negative reactions from clients. Our findings suggest that staff gender and status are two individual characteristics that should be taken into account when planning responsible gambling trainings. [ABSTRACT FROM AUTHOR]

In recent years, multi-objective evolutionary algorithms (MOEAs) have been widely used to handle various water resources problems. One challenge within MOEAs' applications is a lack of understanding on how various operators alter a MOEA's behavior to achieve its final performance (i.e., MOEAs are black-boxes to practitioners), and hence it is difficult to select the most appropriate operators to ensure the MOEA's best performance for a given real-world problem. To address this issue, this study proposes the use of the run-time measure metrics to reveal the underlying searching behavior of the MOEA's operators. The proposed methodology is demonstrated by the non-dominated sorting genetic algorithm II (NSGA-II, a widely used MOEA in water resources) with five commonly used crossover operators applied to six water distribution system design problems. Results show that the simulated binary crossover (SBX) and the simplex crossover (SPX) operators possess great ability in extending the front and finding Pareto-front solutions, respectively, while the naive crossover (NVX) strategy exhibits the overall worst performance in identifying optimal fronts. The obtained understanding on the operators' searching behavior not only offers guidance for selecting appropriate operators for real-world water resources problems, but also builds fundamental knowledge for developing more advanced MOEAs in future. [ABSTRACT FROM AUTHOR]

This article is devoted to a deep study of the analytic Campanato space $$\mathcal {CA}_p$$ on the unit disk via not only exploring the first and second pre-duals of $$\mathcal {CA}_p$$ but also handling the boundedness of three operators: superposition $$\mathsf {S}^\phi $$ ; backward shift $$\mathsf {S}_{\mathsf {b}}$$ ; Schwarzian derivative $$\mathsf {S}$$ , acting on $$\mathcal {CA}_p$$ . [ABSTRACT FROM AUTHOR]

A nonhomogeneous Poisson process (NHPP) plays an important role in a variety of applications as reliability of repairable systems, software reliability and actuarial studies. An NHPP is characterized by its intensity function m( t), which provides information on the time-dependent nature of the reliability of the system. Various intensity functions, which describe different behavior (from reliability decay to reliability growth along with monotonicity, convexity or concavity), have been suggested for NHPP's for modeling repairable systems. Perhaps one of the most frequently utilized NHPP is the power lawprocess (PLP) in which m( t) is a power function of t. Inthis studywe present a general method for constructing new intensity functions for NHPP's yielding new classes of NHPP's. This method utilizes certain operators L , n ∈ N, acting on some suitable functions L = f (termed base functions). We call these classesOBIF's (operator-based intensity functions). These classes are represented in terms of three parameters of which one is an indexing parameter n ∈ N and two others are scale and shape parameters. The fact that n ∈ N is also a parameter provides a flexibility in the choice of the appropriate statistical model for NHPP's data. In particular, we consider the exponential operator acting on the PLP intensity function f and realize that L 's, n ≥ 2, inherit properties similar to those of L (convexity and concavity) and thus are suitable for modelling bathtub data. We also consider a more comprehensive treatment of OBIF classes where both, the operator and base functions, are general. All of the introduced operators are demonstrated with illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, by establishing free-probabilistic models on the Hecke algebras $$\mathcal {H}(G_{p})$$ , we construct canonical free probability spaces $$(\mathcal {H}(G_{p}), \psi _{p})$$ , where $$G_{p} = GL_{2}(\mathbb {Q} _{p})$$ , for primes $$p$$ . Dependent upon such free-probabilistic structures, we study corresponding representations of $$\mathcal {H}(G_{p})$$ , and consider spectral properties of operators realized under representations. [ABSTRACT FROM AUTHOR]

We investigate natural sufficient conditions for a space $$L^p(m)$$ of $$p$$ -integrable functions with respect to a positive vector measure to be smooth. Under some assumptions on the representation of the dual space of such a space, we prove that this is the case for instance if the Banach space where the vector measure takes its values is smooth. We give also some examples and show some applications of our results for determining norm attaining elements for operators between two spaces $$L^p(m_1)$$ and $$L^q(m_2)$$ of positive vector measures $$m_1$$ and $$m_2$$ . [ABSTRACT FROM AUTHOR]

In this paper, we compute $$K$$-groups $$\{K_{n}(C^{*}(x))\}_{n=0}^{\infty }$$ of the $$C^{*}$$-subalgebra $$C^{*}(x)$$ of $$B(H),$$ generated by a single operator $$x,$$ where $$H$$ is a separable infinite dimensional Hilbert space, and $$B(H)$$ is the operator algebra consisting of all (bounded linear) operators on $$H.$$ These computations not only provide nice examples in $$K$$-theory, but also characterize-and-classify projections in a $$C^{*}$$-algebra generated by a single operator. The main result of this paper shows that: the $$K$$-groups of $$C^{*}(x)$$ are completely characterized by those of $$C^{*}(q),$$ where $$q$$ is the positive-operator part of $$x$$ in the polar decomposition of $$x.$$ [ABSTRACT FROM AUTHOR]

For the adjudication of conflicting claims, we develop three general approaches to obtain Lorenz rankings of rules. Our first approach concerns a parameterized family that contains several important rules (Thomson in Soc Choice Welf 31:667-692, ). We give a condition that the parameters defining two members of the family should satisfy for one of them to Lorenz dominate the other. Our second approach exploits the concept of 'consistency' (Young in Math Oper Res 12:398-414, ). We derive a criterion to deduce Lorenz domination for arbitrarily many claimants from Lorenz domination in the two-claimant case. Our third approach is based on the notion of an 'operator' on the space of rules (Thomson and Yeh in J Econ Theory 143:177-198, ). We develop conditions under which operators preserve the Lorenz order, or reverse it. As corollaries of our general theorems, we obtain rankings of most of the rules that have been discussed in the literature. [ABSTRACT FROM AUTHOR]

Jeffrey King has recently argued: (i) that the semantic value of a sentence at a context is (or determines) a function from possible worlds to truth values, and (ii) that this undermines Jason Stanley's argument against the rigidity thesis, the claim that no rigid term has the same content as a non-rigid term. I show that King's main argument for (i) fails, and that Stanley's argument is consistent with the claim that the semantic value of a sentence at a context is (or determines) a function from worlds to truth values. [ABSTRACT FROM AUTHOR]

Let P( z) be s polynomial of degree n having all its zeros in | z| ≤ k, k ≥ 1. In this paper, we shall generalize a result of Aziz and Rather [2] which in turn provides an improvement of a recent result of Dewan et al. [4]. Also a refinement of a result of Shah [10] for the polar derivative of a polynomial has been obtained by using the location of the zeros of P( z). [ABSTRACT FROM AUTHOR]

We study a new kind of random processes with discrete time. In fact we study a certain class of operators called substitution operators whose iterations generate a process. Our operators act on probability measures on a configuration space ${\mathcal{A}}^{\mathbb{Z}}$, where $\mathcal{A}$ is a finite set called alphabet, elements of which are called letters. Elements of $\mathcal{A}^{\mathbb{Z}}$ are bi-infinite sequences of letters. Let us call any finite sequence of letters a word. Length of a word is the number of letters in it. Informally speaking, action of our operator consists in the following: any occurrence of a word G in a configuration may be substituted by another word H with a certain probability ρ. Many well-known operators fit into this description with G and H having equal lengths. Our main novelty is that the lengths of G and H may be different. This makes our operators non-linear and causes our main difficulties. Our main task is to define such operators rigorously (which is not trivial since our space is ℤ rather than its finite segment), prove some of their properties and use them to study invariant measures. [ABSTRACT FROM AUTHOR]

We investigate the implications and logical relations between progressivity (a principle of distributive justice) and merging-proofness (a strategic principle) in taxation. By means of two characterization results, we show that these two principles are intimately related, despite their different nature. In particular, we show that, in the presence of continuity and consistency (a widely accepted framework for taxation) progressivity implies merging-proofness and that the converse implication holds if we add an additional strategic principle extending the scope of merging-proofness to a multilateral setting. By considering operators on the space of taxation rules, we also show that progressivity is slightly more robust than merging-proofness. [ABSTRACT FROM AUTHOR]

The spectral properties of the operators inducing linear dynamical systems on Banach spaces determine the asymptotic properties of the systems. In this note an interaction between the spectral mapping properties of operators and stability properties of the corresponding dynamical systems is presented. [ABSTRACT FROM AUTHOR]

Quantum Neural Network (QNN) is a fledging science built upon the combination of classical neural network and quantum computing. After analyzing of traditional competitive neural network, this paper firstly presents a Quantum Competitive Neural Network (QCNN) that can recognize patterns and classify patterns via quantum competition. Contrasting to the conventional competitive neural network, the storage capacity or memory capacity of the QCNN is exponentially increased by a factor of 2 n, where n is the number of qubit. The QCNN has no weights, does not need to learn and update weights, which accelerates the learning process of the network. Besides, the case analysis validates the feasibility and validity of the QCNN in this paper. [ABSTRACT FROM AUTHOR]

In this paper we study some properties of a totally *-paranormal operator (defined below) on Hilbert space. In particular, we characterize a totally *-paranormal operator. Also we show that Weyl’s theorem and the spectral mapping theorem hold for totally *-paranormal operators through the local spectral theory. Finally, we show that every totally *-paranormal operator satisfies an analogue of the single valued extension property for W 2( D, H) and some of totally *-paranormal operators have scalar extensions. [ABSTRACT FROM AUTHOR]

For a locally compact group G, L1(G) is its group algebra and L∞(G) is the dual of L1(G). Lau has studied the bounded linear operators T : L∞(G) → L∞(G) which commute with convolutions and translations. For a subspace H of L∞(G), we know that M(L∞(G),H), the Banach algebra of all bounded linear operators on L∞(G) into H which commute with convolutions, has been studied by Pym and Lau. In this paper, we generalize these problems to L(K)*, the dual of a hypergroup algebra L(K) in a very general setting, i. e. we do not assume that K admits a Haar measure. It should be noted that these algebras include not only the group algebra L1(G) but also most of the semigroup algebras. Compact hypergroups have a Haar measure, however, in general it is not known that every hypergroup has a Haar measure. The lack of the Haar measure and involution presents many difficulties; however, we succeed in getting some interesting results. [ABSTRACT FROM AUTHOR]

In the paper interactions between Clifford analysis and differential geometry are presented. It is a review article on the results which I obtained together mainly with V. Souček and also with P. Van Lancken and F. Sommen. Two important applications of Clifford analysis in geometry, first for generalized Penrose transform and second for the description of the polynomial solutions of some conformally invariant differential equations on the flat space are given. [ABSTRACT FROM AUTHOR]

We associate a family of Hilbert spaces H q 2;λ( D) of analytic functions on the unit disk D= z∈ $$\mathbb{C}$$ :| z|<1 the q-continuous Gegenbauer polynomials C n λ( x; q) on the interval]−1;1[ and give a q-analogue of the unitary integral transform that Watanabe constructed from the Hilbert space L2(]−1;1[;(1− x2)λ− $$\frac{1}{2}$$ d x onto the weighted Hilbert space H2;λ( D). [ABSTRACT FROM AUTHOR]

We define and examine several probabilistic operators ranging over sets (i.e., operators of type 2), among them the formerly studied ALMOST-operator. We compare their power and prove that they all coincide for a wide variety of classes. As a consequence, we characterize the ALMOST-operator which ranges over infinite objects (sets) by a bounded-error probabilistic operator which ranges over strings, i.e., finite objects. This leads to a number of consequences about complexity classes of current interest. As applications, we obtain (a) a criterion for measure 1 inclusions of complexity classes, (b) a criterion for inclusions of complexity classes relative to a random oracle, (c) a new upper time bound for ALMOST-PSPACE, and (d) a characterization of ALMOST-PSPACE in terms of checking stack automata. Finally, a connection between the power of ALMOST-PSPACE and that of probabilistic $ NC^1 $ circuits is given. [ABSTRACT FROM AUTHOR]

Genetic Algorithms are a class of powerful, robust search techniques based on genetic inheritance and the Darwinian metaphor of “Natural Selection”. These algorithms maintain a finite memory of individual points on the search landscape known as the “population”. Members of the population are usually represented as strings written over some fixed alphabet, each of which has a scalar value attached to it reflecting its quality or “fitness”. The search may be seen as the iterative application of a number of operators, such as selection, recombination and mutation, to the population with the aim of producing progressively fitter individuals. These operators are usually static, that is to say that their mechanisms, parameters, and probability of application are fixed at the beginning and constant throughout the run of the algorithm. However, there is an increasing body of evidence that not only is there no single choice of operators which is optimal for all problems, but that in fact the optimal choice of operators for a given problem will be time-variant i.e. it will depend on such factors as the degree of convergence of the population. Based on theoretical and practical approaches, a number of authors have proposed methods of adaptively controlling one or more of the operators, usually invoking some kind of “meta-learning” algorithm, in order to try and improve the performance of the Genetic Algorithm as a function optimiser. In this paper we describe the background to these approaches, and suggest a framework for their classification, based on the learning strategy used to control them, and what facets of the algorithm are susceptible to adaptation. We then review a number of significant pieces of work within the context of this setting, and draw some conclusions about the relative merits of various approaches and promising directions for future work. [ABSTRACT FROM AUTHOR]

The convergence of the power sequence of ann ×n fuzzy matrix has been studied. Some theoretical necessary and sufficient conditions have been established for the power sequence to be convergent generally. Furthermore, as one of our main concerns, the convergence index was studied in detail, especially for some special types of Boolean matrices. Also it has been established that the convergence index is bounded by (n − 1)2 + 1 from above for an arbitraryn ×n fuzzy matrix if its power sequence converges. Our method is concentrated on the limit behavior of the power sequence. It helped us to make our proofs be simpler and more direct than those in pure algebraic methods. [ABSTRACT FROM AUTHOR]

The convergence of the power sequence of an n × n fuzzy matrix has been studied. Some theoretical necessary and sufficient conditions have been established for the power sequence to be convergent generally. Furthermore, as one of our main concerns, the convergence index was studied in detail, especially for some special types of Boolean matrices. Also it has been established that the convergence index is bounded by ( n − 1) + 1 from above for an arbitrary n × n fuzzy matrix if its power sequence converges. Our method is concentrated on the limit behavior of the power sequence. It helped us to make our proofs be simpler and more direct than those in pure algebraic methods. [ABSTRACT FROM AUTHOR]

The concept of equi-outer semicontinuity allows us to relate the pointwise and the graphical convergence of set-valued-mappings. One of the main results is a compactness criterion that extends the classical Arzelà-Ascolì theorem for continuous functions to this new setting; it also leads to the exploration of the notion of continuous convergence. Equi-lower semicontinuity of functions is related to the outer semicontinuity of epigraphical mappings. Finally, some examples involving set-valued mappings are re-examined in terms of the concepts introduced here. [ABSTRACT FROM AUTHOR]