Operator pencils on the algebra of densities

We continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role played by the geometry of the extended manifold where the algebra of densities is a special class of functions. Firstly we consider basic examples. We give a projective line of diff(M)-equivariant pencil liftings for first order operators and describe the canonical second order self-adjoint lifting. Secondly we study pencil liftings equivariant with respect to volume preserving transformations. This helps to understand the role of self-adjointness for the canonical pencils. Then we introduce the Duval-Lecomte-Ovsienko (DLO) pencil lifting which is derived from the full symbol calculus of projective quantisation. We use the DLO pencil lifting to describe all regular proj-equivariant pencil liftings. In particular, the comparison of these pencils with the canonical pencil for second order operators leads to objects related to the Schwarzian.     

Shape preserving polynomial curves

We introduce particular systems of functions and study the properties of the associated Bézier-type curve for families of data points in the real affine space. The systems of functions are defined with the help of some linear and positive operators, which have specific properties: total positivity, nullity diminishing property and which are similar to the Bernstein polynomial operator. When the operators are polynomial, the curves are polynomial and their degrees are independent of the number of data points. Examples built with classical polynomial operators give algebraic curves written with the Jacobi polynomials, and trigonometric curves if the first and the last data points are identical.     

Resolving sequences of operators for linear ordinary differential and difference systems of arbitrary order

We introduce the notion of a resolving sequence of (scalar) operators for a given differential or difference system with coefficients in some differential or difference field K. We propose an algorithm to construct, such a sequence, and give some examples of the use of this sequence as a suitable auxiliary tool for finding certain kinds of solutions of differential and difference systems of arbitrary order. Some experiments with our implementation of the algorithm are reported.     

Approximation by Durrmeyer type Jakimoski–Leviatan operators

In this study, we introduce the Durrmeyer type Jakimoski–Leviatan operators and examine their approximation properties. We study the local approximation properties of these operators. Further, we investigate the convergence of these operators in a weighted space of functions and obtain the approximation properties. Furthermore, we give a Voronovskaja type theorem for the our new operators. Copyright © 2015 John Wiley & Sons, Ltd.     

Explicit div-curl inequalities in bounded and unbounded domains of $${{\mathbb {R}}}^3$$

We give a constructive proof of some functional inequalities related to the div and curl operators in bounded and unbounded domains of \({{\mathbb {R}}}^3\). Our new innovation consists in giving explicit constants in several geometric configurations. These inequalities are of a first use in solving div-curl systems and vector potential problems arising in physics.     

Sémantique algébrique ďun système logique basé sur un ensemble ordonné fini

In order to modelize the reasoning of an intelligent agent represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems a set of constants constitutes a fundamental tool. In this papers, we consider logic systems called L′_T without this kind of constants but limited to the case where T is a finite poset. We prove a weak deduction theorem. We introduce also an algebraic semantics using Hey ting algebra with operators. To prove the completeness theorem of the L′_T system with respect to the algebraic semantics, we use the method of H. Rasiowa and R. Sikorski for first order logic. In the propositional case, a corollary allows us to assert that it is decidable to know “if a propositional formula is valid”. We study also certain relations between the L′_T logic and the intuitionistic and classical logics.     

Local maxima of solutions to some nonsymmetric reaction‐diffusion systems

In this study, we examine the solution profile of some reaction‐diffusion systems. The systems are derived after approximating the Arrhenius term in our system which models the sintering process and consists of two coupled equations in terms of two unknowns. The unknowns describe the temperature of the solid and the concentration of the fuel. We show the evolution over time of local temperature hot spots. The key idea is to use ‘microscopic scaling’ around the temperature hot spot at the initial time along with asymptotic analysis. We also provide some numerical results that support the efficiency of our analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Differential operators for systems of linear partial differential equations

The aim of this paper is the representation of solutions of systems of formally hyperbolic differential equations of second order. I. N.Vekua gave a representation of the solutions using the Riemann-matrix-function. Here we introduce special differential operators which map holomorphic functions into the set of solutions. An existence theorem for such operators is proved which gives a necessary and sufficient condition on the coefficients of a system. These operators are represented explicitly and several properties of them are investigated. We give different representations of the solutions of such systems and discuss the relation between the integral operator method and the method using differential operators which leads to an explicit representation of the Riemann-matrix-function by means of the differential operators. Two examples of special systems with differential operators are given.     

Large‐time behavior of a two‐scale semilinear reaction–diffusion system for concrete sulfatation

We study the large‐time behavior of (weak) solutions to a two‐scale reaction–diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (or cement)‐based materials with sulfates. We prove that as t → ∞ , the solution to the original two‐scale system converges to the corresponding two‐scale stationary system. To obtain the main result, we make use essentially of the theory of evolution equations governed by subdifferential operators of time‐dependent convex functions developed combined with a series of two‐scale energy‐like time‐independent estimates. Copyright © 2014 John Wiley & Sons, Ltd.     

Maximal regularity of evolution equations on discrete time scales

We consider the question of Lp-maximal regularity for inhomogeneous Cauchy problems in Banach spaces using operator-valued Fourier multipliers. This follows results by L. Weis in the continuous time setting and by S. Blunck for discrete time evolution equations. We generalize the later result to the case of some discrete time scales (discrete problems with nonconstant step size). First we introduce an adequate evolution family of operators to consider the general problem. Then we consider the case where the step size is a periodic sequence by rewriting the problem on a product space and using operator matrix valued Fourier multipliers. Finally we give a perturbation result allowing to consider a wider class of step sizes.     

Korovkin type theorem for non‐tensor Balázs type Bleimann,Butzer and Hahn operators

In this paper, we prove a certain Korovkin type approximation theorem by introducing new test functions. We introduce the non‐tensor Balázs type Bleimann, Butzer and Hahn operators and give the approximation property by using this new Korovkin theorem. Furthermore, we obtain the rate of convergence of these operators by means of modulus of continuity. Finally, we state the multivariate version of the abovementioned Korovkin type theorem. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic equivalence of nonlinear evolution equations in Banach spaces

We show how the approach of Yosida Approximation of the derivative serves to obtain new results for evolution systems. We give criteria for the asymptotic equivalence of two different evolution systems, i.e., $$\lim_{t \to \infty} \|U_A(t, s)x - U_B(t, s)x\| =0,$$ where the evolution systems are generated by two different families of nonlinear and multivalued time-dependent operators A(t), and B(t).     

Quantaloid-enriched 范畴上的闭包算子与闭包系统

In this paper, we introduce the fundamental notions of closure operator and closure system in the framework of quantaloid-enriched category. We mainly discuss the relationship between closure operators and adjunctions and establish the one-to-one correspondence between closure operators and closure systems on quantaloid-enriched categories.     

A lower bound for the stability radius of time-varying systems

We introduce and characterize the stability radius of systems whose state evolution is described by linear skew-product semiflows. We obtain a lower bound for the stability radius in terms of the Perron operators associated to the linear skew-product semiflow. We generalize a result due to Hinrichsen and Pritchard.

     

Evolution semigroups and time operators on Banach spaces

We present new general methods to obtain shift representation of evolution semigroups defined on Banach spaces. We introduce the notion of time operator associated with a generalized shift on a Banach space and give some conditions under which time operators can be defined on an arbitrary Banach space. We also tackle the problem of scaling of time operators and obtain a general result about the existence of time operators on Banach spaces satisfying some geometric conditions. The last part of the paper contains some examples of explicit constructions of time operators on function spaces.     

Spectralizable Operators

We introduce the notion of spectralizable operators. A closed operator A in a Hilbert space is called spectralizable if there exists a non-constant polynomial p such that the operator p(A) is a scalar spectral operator in the sense of Dunford. We show that such operators belongs to the class of generalized spectral operators and give some examples where spectralizable operators occur naturally. Vladimir Strauss gratefully acknowledges support by DFG, Grant No. TR 903/3-1.     

Duality in reconstruction systems

We consider reconstruction systems (RS’s), which are G-frames in a finite dimensional setting, and that includes the fusion frames as projective RS’s. We describe the spectral picture of the set of RS operators for the projective systems with fixed weights. We also introduce a functional defined on dual pairs of RS’s, called the joint potential, and study the structure of the minimizers of this functional. In the case of irreducible RS’s the minimizers are characterize as the tight systems. In the general case we give spectral and geometric characterizations of the minimizers of the joint potential. At the end of the paper we show several examples that illustrate our results.     

Combining evolutionary game theory and network theory to analyze human cooperation patterns

As natural systems continuously evolve, the human cooperation dilemma represents an increasingly more challenging question. Humans cooperate in natural and social systems, but how it happens and what are the mechanisms which rule the emergence of cooperation, represent an open and fascinating issue. In this work, we investigate the evolution of cooperation through the analysis of the evolutionary dynamics of behaviours within the social network, where nodes can choose to cooperate or defect following the classical social dilemmas represented by Prisoner’s Dilemma and Snowdrift games. To this aim, we introduce a sociological concept and statistical estimator, “Critical Mass”, to detect the minimum initial seed of cooperators able to trigger the diffusion process, and the centrality measure to select within the social network. Selecting different spatial configurations of the Critical Mass nodes, we highlight how the emergence of cooperation can be influenced by this spatial choice of the initial core in the network. Moreover, we target to shed light how the concept of homophily, a social shaping factor for which “birds of a feather flock together”, can affect the evolutionary process. Our findings show that homophily allows speeding up the diffusion process and make quicker the convergence towards human cooperation, while centrality measure and thus the Critical Mass selection, play a key role in the evolution showing how the spatial configurations can create some hidden patterns, partially counterbalancing the impact of homophily.