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We call a ring R NJ-semicommutative if wh ∈ N(R) implies wRh ⊆ J(R) for any w,h ∈ R. The class of NJ-semicommutative rings is large enough that it contains semicom- mutative rings, left (right) quasi-duo rings, J-clean rings, and J-quasipolar rings. We provide some conditions for NJ-semicommutative rings to be reduced. We also observe that if R/J(R) is reduced, then R is NJ-semicommutative, and therefore we provide some conditions for NJ- semicommutative ring R for which R/J(R) is reduced. We also study some extensions of NJ- semicommutative rings wherein, among other results, we prove that the polynomial ring over an NJ-semicommutative ring need not be NJ-semicommutative. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three new fractional operators, MRL fractional integral, MRL fractional derivative and MC fractional derivative operators, which including generalized M-series in their kernels and give some of their fundamental properties. Then we apply Laplace, Mellin and beta integral transformations to the new fractional operators and obtain conclusions involving classical gamma and beta and modified gamma and beta functions. As examples, we also obtain similar conclusions by applying new fractional operators to the functions zλ and (1 - az)-λ. Furthermore, we present the relationships of the new fractional operators with other fractional operators in the literature. Finally, we compare the behavior of the function z² in classical Riemann-Liouville and Caputo fractional operators and MRL and MC fractional operators for orders ε = 0:2;0:4;0:6;0:8. [ABSTRACT FROM AUTHOR]

We consider the integral type problem for linear and nonlinear abstract wave equations with operator coefficient in E-valued L p , 1 ≤ p ≤ ∞ spaces. The equation includes an abstract linear operator A that defined in a Banach space E. Here, the existence, uniqueness, uniform estimates of solution and quality properties of integral wave problem are derived. We also obtain the same properties for solutions of mixed integral problem for nonlocal and anisotropic classes, by choosing the space E and operator A. [ABSTRACT FROM AUTHOR]

Given an injective closed linear operator A defined in a Banach space X , and writing C F D t α the Caputo–Fabrizio fractional derivative of order α ∈ (0 , 1) , we show that the unique solution of the abstract Cauchy problem (∗) C F D t α u (t) = A u (t) + f (t) , t ≥ 0 , where f is continuously differentiable, is given by the unique solution of the first order abstract Cauchy problem u ′ (t) = B α u (t) + F α (t) , t ≥ 0 ; u (0) = − A − 1 f (0) , where the family of bounded linear operators B α constitutes a Yosida approximation of A and F α (t) → f (t) as α → 1. Moreover, if 1 1 − α ∈ ρ (A) and the spectrum of A is contained outside the closed disk of center and radius equal to 1 2 (1 − α) then the solution of (∗) converges to zero as t → ∞ , in the norm of X, provided f and f ′ have exponential decay. Finally, assuming a Lipchitz-type condition on f = f (t , x) (and its time-derivative) that depends on α , we prove the existence and uniqueness of mild solutions for the respective semilinear problem, for all initial conditions in the set S : = { x ∈ D (A) : x = A − 1 f (0 , x) }. [ABSTRACT FROM AUTHOR]

We investigate the relations between the (completely bounded) local Coulhon-Varopoulos dimension and the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) L p -spaces associated to finite measure spaces. More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension d exceeds the spectral dimension, i.e. that the associated Hodge-Dirac operator is d + -summable. We explore different settings to compare these two values: compact Riemannian manifolds, compact Lie groups, sublaplacians, metric measure spaces, noncommutative tori and quantum groups. Specifically, we prove that, while very often equal in smooth compact settings, these dimensions can diverge. Finally, we show that the existence of a symmetric sub-Markovian semigroup on a von Neumann algebra with finite completely bounded local Coulhon-Varopoulos dimension implies that the von Neumann algebra is necessarily injective. [ABSTRACT FROM AUTHOR]

We present the existence and uniqueness of strong solutions for the continuous coagulation-fragmentation equation with singular fragmentation and essentially bounded coagulation kernel using semigroup theory of operators. Initially, we reformulate the coupled coagulation-fragmentation problem into the semilinear abstract Cauchy problem (ACP) and consider it as the nonlinear perturbation of the linear fragmentation operator. The existence of the substochastic semigroup is proved for the pure fragmentation equation. Using the substochastic semigroup and some related results for the pure fragmentation equation, we prove the existence of global nonnegative, strong solution for the coagulation-fragmentation equation. [ABSTRACT FROM AUTHOR]

Differentiability of semigroups is useful for many applications. Here we focus on stochastic differential equations whose diffusion coefficient is the square root of a differentiable function but not differentiable itself. For every m ∈ {0, 1, 2} we establish an upper bound for a Cm-norm of the semigroup of such a diffusion in terms of the Cm-norms of the drift coefficient and of the squared diffusion coefficient. The constants in our upper bound are often bounded in the dimension. Our estimates are thus suitable for analyzing certain high-dimensional and infinitedimensional degenerate stochastic differential equations. [ABSTRACT FROM AUTHOR]

In this paper, regularity properties, Strichartz type estimates for solution of integral problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Banach space E , in which by choosing E and A we can obtain numerous classis of initial value problems for Schrödinger equations which occur in a wide variety of physical systems. [ABSTRACT FROM AUTHOR]

This paper presents a survey of maximal inequalities for stochastic convolutions in 2-smooth Banach spaces and their applications to stochastic evolution equations. This article is part of the theme issue 'Semigroup applications everywhere'. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the diskcyclicity and disk transitivity of a set of operators. We establish a diskcyclicity criterion and give the relationship between this criterion and the diskcyclicity. As applications, we study the diskcyclicty of C0-semigroups and C-regularized groups. We show that a diskcyclic C0-semigroup exists on a complex topological vector space X if and only if dim(X) = 1 or dim(X) = ∞ and we prove that diskcyclicity and disk transitivity of C0-semigroups (resp C-regularized groups) are equivalent. [ABSTRACT FROM AUTHOR]

We study a diffusion process on a finite graph with semipermeable membranes on vertices. We prove, in L 1 and L 2 -type spaces that for a large class of boundary conditions, describing communication between the edges of the graph, the process is governed by a strongly continuous semigroup of operators, and we describe asymptotic behaviour of the diffusion semigroup as the diffusions' speed increases at the same rate as the membranes' permeability decreases. Such a process, in which communication is based on the Fick law, was studied by Bobrowski (Ann. Henri Poincaré 13(6):1501–1510, 2012) in the space of continuous functions on the graph. His results were generalized by Banasiak et al. (Semigroup Forum 93(3):427–443, 2016). We improve, in a way that cannot be obtained using a very general tool developed recently by Engel and Kramar Fijavž (Evolut. Equ. Control Theory 8(3)3:633–661, 2019), the results of J. Banasiak et al. [ABSTRACT FROM AUTHOR]

Motivated by models of signaling pathways in B-lymphocytes, which have extremely large nuclei, we study the question of how reaction-diffusion equations in thin 2D domains may be approximated by diffusion equations in regions of smaller dimensions. In particular, we study how transmission conditions featuring in the approximating equations become integral parts of the limit master equation. We devise a scheme which, by appropriate rescaling of coefficients and finding a common reference space for all Feller semigroups involved, allows deriving the form of the limit equation formally. The results obtained, expressed as convergence theorems for the Feller semigroups, may also be interpreted as a weak convergence of underlying stochastic processes. [ABSTRACT FROM AUTHOR]

Product systems have been originally introduced to classify E0-semigroups on type I factors by Arveson. The idea of product systems is influenced the constructions of dilations of CP0-semigroups. In this paper, we will develop the dilation theory and the classification theory of E0-semigroups on a general von Neumann algebra in the view point of Connes correspondences. For this, we will provide a concept of product system whose components are Connes correspondences. There exists a one-to-one correspondence between a CP0-semigroup and a unit of a product system of Connes correspondences. The correspondence enables us to construct dilations and classify E0-semigroups by product systems of Connes correspondences up to cocycle equivalence. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of square-mean piecewise almost automorphic function. By using the theory of semigroups of operators and the contraction mapping principle, the existence of square-mean piecewise almost automorphic mild solutions for linear and nonlinear impulsive stochastic evolution equations is investigated. In addition, the exponential stability of square-mean piecewise almost automorphic mild solutions for nonlinear impulsive stochastic evolution equations is obtained by the generalized Gronwall–Bellman inequality. Finally, we provide an illustrative example to justify the results. [ABSTRACT FROM AUTHOR]

We consider a singular limit of diffusion equations in 3D domains of thickness converging to zero. In the 2D limit the resulting reaction-diffusion equation has a source term resulting from the Robin-type boundary conditions imposed on boundaries of the original 3D domain. The proposed approach can be applied to constructing approximate solutions of diffusion problems in thin planar, cylindrical, or spherical layers between two membranes. As an example we refer to the problem of activation of B lymphocytes, which typically have large nuclei and a thin cytoplasmic layer which can be considered as a spherical shell. For this example, assuming additionally axial symmetry we provide a rigorous convergence theorem in the language of semigroups of operators. [ABSTRACT FROM AUTHOR]

We consider the Dunkl differential-difference operator associated with a reflection group W and a nonnegative parameter function k. We exhibit and study a new family of sharp integral inequalities related to the Dunkl heat semigroup generated by the Dunkl-Laplacian . These estimations are a natural extension and refinement of the Poincaré inequality established in [1] and used to derive other applications. [ABSTRACT FROM AUTHOR]

In this paper, regularity properties and Strichartz type estimates for solutions of the Cauchy problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Banach space E, in which by choosing E and A, we can obtain numerous classes of initial value problems for Schrödinger equations, which occur in a wide variety of physical systems. [ABSTRACT FROM AUTHOR]

In this article, we study the strongly continuous semigroups of fuzzy-valued operators defined on the space of triangular fuzzy numbers. New notions of fuzzy infinitesimal generators and fuzzy resolvent operators of fuzzy semigroups are established in the sense of the generalized Hukuhara difference. A Hille–Yosida-like theorem in the framework of fuzzy metric space is investigated. We demonstrate the efficiency of theoretical results by studying the existence of mild solutions of fuzzy nonlinear fractional evolution equations. [ABSTRACT FROM AUTHOR]

We consider the optimal goodwill control problem in a segmented market where the length of the product life cycle is affected by unpredictable technological turbulence. In order to maximize profit over a random time horizon, a company controls the marketing efforts directed to each market segment. Assuming an exponential distribution for the product life cycle, we modify the optimal goodwill model into the infinite time horizon control problem. Based on the semigroup approach, we prove the existence and uniqueness of the optimal solution. We formulate optimality conditions for the problem and we prove the existence of a stationary long-run equilibrium. Next, we construct a numerical algorithm to find the optimal solution. Finally, we examine several scenarios of optimal marketing strategies. [ABSTRACT FROM AUTHOR]

The objective of this paper is to present some sufficient conditions for approximate and exact controllability of semilinear stochastic control system with finite delay. Sufficient conditions for approximate controllability are obtained by separating the given semilinear system into two systems namely a semilinear deterministic system and a linear stochastic system using Schauder fixed point theorem. For obtaining the sufficient conditions for exact controllability of assumed system, Banach fixed point theorem is applied. Instead of a C0-semigroup associated with the mild solution of the system we use the so-called fundamental solution. At the end, examples are given to illustrate the theory. [ABSTRACT FROM AUTHOR]

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering. [ABSTRACT FROM AUTHOR]

The paper is intended to show a new state space, non integer order model of an one-dimensional heat transfer process. The proposed model derives directly from time continuous, state space semigroup model. The fractional order derivative with respect to time is described by a new operator proposed by Caputo and Fabrizio, the non integer order spatial derivative is expressed by Riesz operator. The Caputo-Fabrizio operator can be directly implementated using MATLAB, because it does not require us to apply any approximation. Analytical formulae of step response are given, the system decomposition was discussed also. Main results from the paper show that the use of Caputo Fabrizio operator allows us to obtain the simple in implementation and analysis model of the considered heat transfer process. The accuracy of the proposed model in the sense of a MSE cost function is satisfying. [ABSTRACT FROM AUTHOR]

Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L1(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr. [ABSTRACT FROM AUTHOR]

The new class of functions almost periodic at infinity is defined using the subspace of functions with integrals decreasing at infinity. We obtain spectral criteria for almost periodicity at infinity of bounded solutions to differential equations with unbounded operator coefficients. For the new class of asymptotically finite operator semigroups we prove the almost periodicity at infinity of their orbits. [ABSTRACT FROM AUTHOR]

Deepening the study of a new approximation sequence of positive linear operators we introduced and studied in [12] , in this paper we disclose its relationship with the Markov semigroup (pre)generation problem for a class of degenerate second-order elliptic differential operators which naturally arise through an asymptotic formula, as well as with the approximation of the relevant Markov semigroups in terms of the approximating operators themselves. The analysis is carried out in the context of the space C ( K ) of all continuous functions defined on an arbitrary convex compact subset K of R d , d ≥ 1 , having non-empty interior and a not necessarily smooth boundary, as well as, in some particular cases, in L p ( K ) spaces, 1 ≤ p < + ∞ . The approximation formula also allows to infer some preservation properties of the semigroup such as the preservation of the Lipschitz-continuity as well as of the convexity. We finally apply the main results to some noteworthy particular settings such as balls and ellipsoids, the unit interval and multidimensional hypercubes and simplices. In these settings the relevant differential operators fall into the class of Fleming–Viot operators. [ABSTRACT FROM AUTHOR]

For a locally compact semigroup S, we study a general fixed point property in terms of Banach left S-modules. We then use this property to give our main result which is a new characterization for left amenability of a large class of locally compact semigroups; finally, we investigate several examples which lead us to the conjecture that the main result remains true for all locally compact semigroups. [ABSTRACT FROM AUTHOR]

This paper deals with the infinite dimensional Hamiltonian operator with unbounded entries. Using the core of its entries, we obtain the conditions under which the numerical range of such an operator is symmetric with respect to the imaginary axis. Based on the symmetry above, a necessary and sufficient condition for generating C0 semigroups is further given. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet boundary problem for semilinear fractional Schrödinger equation with subcritical nonlinear term. Local and global in time solvability and regularity properties of solutions are discussed. But our main task is to describe the connections of the fractional equation with the classical nonlinear Schrödinger equation, including convergence of the linear semigroups and continuity of the nonlinear semigroups when the fractional exponent α approaches 1. [ABSTRACT FROM AUTHOR]

Let $V$ be a vector space and let $T(V)$ denote the semigroup (under composition) of all linear transformations from $V$ into $V$. For a fixed subspace $W$ of $V$, let $T(V,W)$ be the semigroup consisting of all linear transformations from $V$ into $W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’, Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$ is the largest regular subsemigroup of $T(V,W)$ and characterized Green’s relations on $T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of $Q$ when $W$ is a finite-dimensional subspace of $V$ over a finite field. Moreover, we compute the rank and idempotent rank of $Q$ when $W$ is an $n$-dimensional subspace of an $m$-dimensional vector space $V$ over a finite field $F$. [ABSTRACT FROM PUBLISHER]

The recent remarkable result of Cassaigne et al. about the existence of an infinite word without three consecutive factors "of the same sum and size" shows the importance of the theory of "repetitive semigroups" in combinatorics on words. This seems to be an occasion to recall the main definitions and results in the theory of repetitivity (which had been presented in Chapter 4 of the Lothaire book). We recall the progress since that time and in the last section we present some applications of the result of Cassaigne et al. In particular, we recall a result we obtained with Stefano Varricchio and the techniques we pioneered with him. [ABSTRACT FROM AUTHOR]

We introduce the concept of intuitionistic fuzzy BCK-submodules of a BCK-module with respect to a t-norm and a s-norm and present some basic properties. [ABSTRACT FROM AUTHOR]

New methods for obtaining representations of solutions of the Cauchy problem for linear evolution equations, i.e., equations of the form u '( t, x) = Lu( t, x), where the operator L is linear and depends only on the spatial variable x and does not depend on time t, are proposed. A solution of the Cauchy problem, that is, the exponential of the operator tL, is found on the basis of constructions proposed by the author combined with Chernoff's theorem on strongly continuous operator semigroups. [ABSTRACT FROM AUTHOR]

This article is concerned with bending plate theory for thermoelastic diffusion materials under Green-Naghdi theory. First, we present the basic equations which characterize the bending of thin thermoelastic diffusion plates for type II and III models. The theory allows for the effect of transverse shear deformation without any shear correction factor, and permits the propagation of waves at a finite speed without energy dissipation for type II model and with energy dissipation for type III model. By the semigroup theory of linear operators, we prove the well-posedness of the both models and the asymptotic behavior of the solutions of type III model. For unbounded plate of type III model, we prove that a measure associated with the thermodynamic process decays faster than an exponential of a polynomial of second degree. Finally, we investigate the impossibility of the localization in time of solutions. The main idea to prove this result is to show the uniqueness of solutions for the backward in-time problem. [ABSTRACT FROM AUTHOR]

This article discusses necessary and sufficient conditions for the existence of global attractors for transformation semigroups on principal bundles. Since the global attractor is a compact set, the discussion involves the compactness of the fibres. A compact structure group is a necessary condition for the existence of the global attractor. In specific situations, the global attractor exists if the structure group is compact. [ABSTRACT FROM AUTHOR]

In this work, we extend the theory of quantum Markov processes on a single quantum state to a broader theory that covers Markovian evolution of an ensemble of quantum states, which generalizes Lindblad's formulation of quantum dynamical semigroups. Our results establish the equivalence between an exponential decrease of the matrix Φ-entropies and the Φ-Sobolev inequalities, which allows us to characterize the dynamical evolution of a quantum ensemble to its equilibrium. In particular, we study the convergence rates of two special semigroups, namely, the depolarizing channel and the phase-damping channel. In the former, since there exists a unique equilibrium state, we show that the matrix Φ-entropy of the resulting quantum ensemble decays exponentially as time goes on. Consequently, we obtain a stronger notion of monotonicity of the Holevo quantity-the Holevo quantity of the quantum ensemble decays exponentially in time and the convergence rate is determined by the modified log-Sobolev inequalities. However, in the latter, the matrix Φ-entropy of the quantum ensemble that undergoes the phase-damping Markovian evolution generally will not decay exponentially. There is no classical analogy for these different equilibrium situations. Finally, we also study a statistical mixing of Markov semigroups on matrix-valued functions. We can explicitly calculate the convergence rate of a Markovian jump process defined on Boolean hypercubes and provide upper bounds to the mixing time. [ABSTRACT FROM AUTHOR]

In this paper, we prove an asymptotic formula for generalized Kantorovich operators associated with the canonical Markov projection on a given Bauer simplexK. That formula involves an operator acting on the subalgebra of all products of affine functions onK. Moreover, we prove that such an operator is closable and its closure is the generator of a Markov semigroup which, in turn, may be represented in terms of iterates of the above-mentioned generalized Kantorovich operators. [ABSTRACT FROM AUTHOR]

In the present paper, a new family of integral transforms depending on two parameters and related to the Dunkl transform is introduced. Well-known transforms, such as the fractional Dunkl transform, Dunkl transform, linear canonical transform, canonical Hankel transform, Fresnel transform, etc., can be seen to be special cases of this general transform. Some useful properties of the considered transform such as Riemann–Lebesgue lemma, reversibility property, additivity property, operational formula, Plancherel formula, Bochner type identity and master formula are derived. The intimate connection that exists between this transformation and the quantum harmonic oscillator is developed. [ABSTRACT FROM AUTHOR]

In this paper, an equivalence relationon an ordered Γ-semigroupSis introduced. Using the relation0-minimal (m, n)-Γ-ideals ofSis studied. The results obtained generalize the results on semigroups(without order) and Γ-semigroups(without order). [ABSTRACT FROM AUTHOR]

This article is concerned with a nonlinear theory of thermodynamics for elastic materials in which the particles are subjected to classical displacement, temperature, and mass diffusion fields and whose microelements possess microtemperatures and microconcentrations. The equations of the linear theory are also obtained. It is shown that there exists coupling between temperature, chemical potential, microtemperatures, and microconcentrations even for isotropic bodies. With the help of the semigroup theory of linear operators, we prove the well-posedness of the linear anisotropic problem and we study the asymptotic behavior of the solutions. [ABSTRACT FROM PUBLISHER]

We prove an averaging principle which asserts convergence of diffusion processes on domains separated by semi-permeable membranes, when diffusion coefficients tend to infinity while the flux through the membranes remains constant. In the limit, points in each domain are lumped into a single state of a limit Markov chain. The limit chain's intensities are proportional to the membranes' permeability and inversely proportional to the domains' sizes. Analytically, the limit is an example of a singular perturbation in which boundary and transmission conditions play a crucial role. This averaging principle is strongly motivated by recent signaling pathways models of mathematical biology, which are discussed toward the end of the paper. [ABSTRACT FROM AUTHOR]

We state and, for the first time, prove a theorem in the theory of strongly continuous operator semigroups. This theorem, which has essentially been suggested by O.G. Smolyanov, in particular, enables one to reduce solving the Schr¨odinger equation to solving the heat equation. [ABSTRACT FROM AUTHOR]

The main motivation behind this paper is to study some structural properties of a non-associative structure (LAsemigroup) in terms of fuzzy sets as it hasn't attracted much attention compared to associative structures. An LA-semigroup can be referred to as a non-associative semigroup, as the main difference between a semigroup and an LA-semigroup is the switching of an associative law. In this paper, we introduce and characterize fuzzy (0, 2)-ideals, fuzzy (0, 2)-bi-ideals and fuzzy (1, 2)-ideals of an LA-semigroup. We give a new and alternate definition for a strongly regular element of a unitary LA-semigroup and shownovel conditions under which a strongly regular LA-semigroup becomes an LA**-semigroup. These characterizations do not exist in literature before this work. As an application of our results we get characterizations of a strongly regular LA-semigroup in terms of fuzzy one-sided ideals and fuzzy bi-ideals. Finally we introduce and investigate the properties of fuzzy inner-unitary LA-subsemigroups in an LA**-semigroup. These concepts will help in verifying the existing characterizations and will help in achieving new and generalized results in future works. [ABSTRACT FROM AUTHOR]

Generalizing the notion of character amenability for Banach algebras, we study the concept of φ-Connes amenability of a dual Banach algebra A with predual A*, where φ is a homomorphism from A onto C that lies in A*. Several characterizations of φ-Connes amenability are given. We also prove that the following are equivalent for a unital weakly cancellative semigroup algebra ℓ¹(S): (i) S is χ-amenable. (ii) ℓ¹(S) is ...-Connes amenable. (iii) ℓ¹(S) has a ...-normal, virtual diagonal. [ABSTRACT FROM AUTHOR]

The result that goes essentially back to Euler [15] says that for any element a of a unital Banach algebra A with unit u , the limit lim ε → 0 + ⁡ ( u + ε a ) [ ε − 1 t ] (where [ ⋅ ] denotes the integral part) exists for all t ∈ R and equals e t a . As developed by E. Hille [22, Thm. 12.2.1] , in the case where a is replaced by the generator A of a strongly continuous semigroup { e t A , t ≥ 0 } in a Banach space X , a proper counterpart of this formula is e t A = lim ε → 0 + ⁡ ( I X − ε A ) − [ ε − 1 t ] strongly in X . Motivated by an example from mathematical biology (related to Rotenberg's model of cell growth [40] ) we study convergence of a similar approximation in which u (resp. I X ) is replaced by j ∈ A (resp. J ∈ L ( X ) ) such that for some ℓ ≥ 2 , j ℓ = u (resp. J ℓ = I X ). [ABSTRACT FROM AUTHOR]

We find the solutions of the cosine addition law with an additional term, including the case of the cosine addition law on semigroups. They are expressed in terms of multiplicative functions and solutions of the sine addition law, apart from an obvious exception. On groups multiplicative and additive functions suffice. As an application we solve extensions of Butler's problem on semigroups. [ABSTRACT FROM AUTHOR]

Linear and nonlinear degenerate abstract parabolic equations with variable coefficients are studied. Here the equation and boundary conditions are degenerated on all boundary and contain some parameters. The linear problem is considered on the moving domain. The separability properties of elliptic and parabolic problems in mixed Lp spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering. [ABSTRACT FROM AUTHOR]