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Functional Analysis and Semi-Groups
... If is weakly continuous on , then is strongly measurable and hence weakly measurable ( [9,12]. It is evident that in reflexive Banach spaces, if is weakly continuous function on [ , ], then is weakly Riemann integrable [12]. ...
... If is weakly continuous on , then is strongly measurable and hence weakly measurable ( [9,12]. It is evident that in reflexive Banach spaces, if is weakly continuous function on [ , ], then is weakly Riemann integrable [12]. ...
... Then ( , ) is said to be weakly-weakly continuous at ( 0 ; 0 ) if given > 0, ∈ * there exists > 0 and a weakly open set containing 0 such that | ( ( , ) − ( 0 − 0 )| < whenever | − 0 | < and ∈ . Now, we have the following fixed point theorem, due to O'Regan, in the reflexive Banach space [18] and some propositions which will be used in the sequel [12,19]. Theorem 1. ...
By reformulating fractional differential equations of the Riemann-Liouville type into the Volterra-Stieltjes integral framework, this study examines the presence of weak solutions. The contributions integrals and fractional derivatives contribute to the development of mathematical theory and its applications across a range of domains are highlighted in the study. Using reflexive Banach spaces, the paper establishes conditions for the existence of weak solutions, characterized by functions of bounded variation.
... For more details we refer, e.g., [21,46]. In particular, it will be said that a family of operators ...
... Given an unbounded linear operator A, we begin by recalling an equivalent characterization of when it serves as the infinitesimal generator of a C 0 -semigroup S(·) = {S(t) ∈ L(X ) | t ≥ 0} (e.g., cf. [21,39,46]). A natural question that arises in this context is whether the C 0 -semigroup or its infinitesimal generator comes first. ...
... Given y ∈ Y , the map t → d dt A(t)y ∈ X is strongly measurable (cf. [21]). Now, fixed T > 0, we shall ensure the existence of a constant C > 0 such that ...
In this paper it is studied the well-posedness in several senses for non-homogeneous Cauchy problems where the infinitesimal generator depends on the time parameter. More specifically, we analyze the existence of classical, mild and weak solutions and their relationships. Thanks to uniqueness arguments, mild solutions are proved to satisfy a classical variational formulation. Finally, these results are applied to a thermoelastic plate model where the thermal part is of Cattaneo type and all the physical coefficients depend on time.
... This means that βck 1 satisfies the same properties as e λ , establishing the first equality in (35). ...
... The following proposition is a key to all the results presented later. Since B is a generator, B − Γ is a generator also, by the Phillips Perturbation Theorem [12,24,35]. Moreover, e t B ≤ M and so for λ > Γ , ...
... , PO is also a generator by the Phillips Perturbation Theorem [12,24,35]. This, by Kurtz's theorem described above, implies (63). ...
We study a system of coupled bulk-surface partial differential equations (BS-PDEs), describing changes in concentration of
certain proteins (Rho GTPases) in a living cell. These proteins, when activated, are bound to the plasma membrane where they diffuse and react with the inactive species; inactivated species diffuse inside the cell cortex; these react with the activated species when they are close to the plasma membrane. For our case study, we model the cell cortex as an annulus, and the plasma membrane as its outer circle.
Mathematically, the aim of the paper is twofold: Firstly, we show the master equation for the changes in concentration of Rho GTPases is the Kolmogorov forward differential equation for an underlying Feller stochastic process, and, in particular, the related Cauchy problem is well-posed. Secondly, since the cell cortex is typically a rather thin domain, we study the situation where the thickness of the annulus modeling the cortex converges to 0. To this end, we note that letting the thickness of the annulus to 0 is equivalent to keeping it constant while increasing the rate of radial diffusion. As a result, {in the limit,} solutions to the master equation loose dependence on the radial coordinate and can be thought of as functions on the circle. Furthermore, the limit master equation can be seen as describing diffusion on two copies of the circle with jumps from one copy to the other.
... The following properties are valid for submultiplicative functions defined on the whole line [5,Theorem 7.6.2]: ...
... here |κ| stands for the total variation of κ. The collection S(ϕ) is a Banach algebra with norm · ϕ by the usual operations of addition and scalar multiplication of measures, the product of two elements ν and κ of S(ϕ) is defined as their convolution ν * κ [5,Section 4.16]. The unit element of S(ϕ) is the measure δ 0 of unit mass concentrated at zero. ...
... Combining the estimates for J 1 (x) and J 3 (x) and similar ones for J 2 (x) and J 4 (x), we get (5). It remains to establish (6). ...
... The uniform well-posedness of the Cauchy problem for first-order differential equations is traditionally studied in terms of semigroups of operators. Briefly, a resolving C 0 -continuous semigroup of operators {Z(t) : t ∈ R + } of the equation D 1 z(t) = Az(t) exists if and only if the respective Cauchy problem z(0) = z 0 is uniformly well posed [1][2][3][4]. In this case, every operator Z(t) maps the initial data z 0 to a value z(t) of a solution of the problem at time t ≥ 0. Such operators for a first-order equation form a semigroup, but for other equations, it is not so. ...
... where D α,β is the Hilfer derivative [16] of an order α ∈ (m − 1, m), m ∈ N, of a kind β ∈ [0, 1], and A is a linear closed operator in Z. The Cauchy-type problem ...
... is considered for Equation (1). Hereafter, D γ z is the Riemann-Liouville fractional derivative for γ > 0, and the Riemann-Liouville fractional integral for γ ≤ 0 (see below for details) is D γ z(0) := lim t→0+ D γ z(t), γ ∈ R. ...
In the qualitative theory of differential equations in Banach spaces, the resolving families of operators of such equations play an important role. We obtained necessary and sufficient conditions for the existence of strongly continuous resolving families of operators for a linear homogeneous equation resolved with respect to the Hilfer derivative. These conditions have the form of estimates on derivatives of the resolvent of a linear closed operator from the equation and generalize the Hille–Yosida conditions for infinitesimal generators of C0-semigroups of operators. Unique solvability theorems are proved for the corresponding inhomogeneous equations. Illustrative examples of the operators from the considered classes are constructed.
... See the write-up [12] on StackExchange. This utilizes the bounded Borel function calculus of the Spectral Theorem (see p. 225 of [11], or 5.2.8 in [13], or 1.51 in [8]). Corollary 2.20. ...
... Let H be a separable Hilbert space; as usual, we regard L 2 (X; H ) as the direct integral of the constant field of Hilbert spaces {H x ≡ H }, with measurable vector fields E given by (38). Since H is separable, it follows that L 2 (X; H ) is separable as well, since L 2 (X; H ) ∼ = L 2 (X) ⊗ Hsee [11], p. 52. Let A be a countable dense subset of L 2 (X; H ). ...
... An alternate way to describe the subspaces Q M is as follows. It can be shown (Theorem II.10 on p. 52 of [11]) that there is a unique unitary transformation Thus α = v a,δ M ∈ H M , where a = I M (α). 5 Reed and Simon express this with more concise notation. Specifically, the theorem cited states that, for a separable Hilbert space H , there is a unique isomorphism from L 2 (Ω) ⊗ H to L 2 (Ω; H ) such that a(ω) ⊗ ζ → a(ω)ζ. ...
It is well known that the subspaces of that are invariant under translations consist precisely of the sets of functions whose Fourier transforms are zero almost everywhere on a measurable set. More recently, Bownik and Ross \cite{BownikRoss} utilized range functions to give a characterization of subspaces of -- where G is a locally compact abelian group -- invariant under translation by elements of a closed co-compact subgroup . In this thesis we introduce a unitary representation on which acts by crystal symmetry shifts. We show that this representation is unitarily equivalent to a direct integral of factor representations, and use this to characterize the subspaces of invariant under crystal symmetry shifts. Finally, we show that our decomposition is -- up to an identification of the base space -- the \textit{central decomposition} guaranteed by direct integral theory.
... We will need the following Theorem; for completeness, we show how to make the simple modification needed to the result given in [HilP,Th. 7.6.1]. (We replace their additional blanket condition of measurability of S by local boundedness, and give more of the details, as they are needed later.) ...
... Proof. Following [HilP,Th.7.5.1], for a > 0 and ma ≤ t < (m + 1)a with m = 2, 3, ..., we note two inequalities, valid according as S(a) ≥ 0 or S(a) < 0 : ...
... To lighten the notation, we write S ± for S ± A . Local boundedness of S ± follows immediately from local boundedness of S. Subadditivity is routine, and follows much as in [HilP,§7.8]. As regards sublinearity of S ± , note that if a n → x for a n ∈ A with lim n→∞ S(a n ) = S ± (x), then, as ka n ∈ A for k ∈ N, by sublinearity of S kS + (x) = lim n→∞ kS(a n ) = lim n→∞ S(ka n ) ≤ S + (kx), kS − (x) = lim n→∞ kS(a n ) = lim n→∞ S(ka n ) ≥ S − (kx). ...
We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function -- results of quantifier-weakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature.
... In this Section, we clarify the terminology connected with Banach algebras [18,67,110] and recall some of their properties. ...
... The unit [18,67,110] of an algebra B is an element 1 ∈ B such that 1A = A1 for all A ∈ B. If an algebra B has a unit, it is called an algebra with a unit or unital. ...
... ϕ is called a morphism of unital algebras. If A and B are Banach algebras [18,67,110] and, in addition, the morphism ϕ is continuous, then ϕ is called a morphism of Banach algebras. ...
Properties of the mappings \begin{align*} C&\mapsto\frac1{(2\pi i)^2}\int_{\Gamma_1}\int_{\Gamma_2}f(\lambda,\mu)\,R_{1,\,\lambda}\,C\, R_{2,\,\mu}\,d\mu\,d\lambda, C&\mapsto\frac1{2\pi i}\int_{\Gamma}g(\lambda)R_{1,\,\lambda}\,C\, R_{2,\,\lambda}\,d\lambda \end{align*} are discussed; here and are pseudo-resolvents, i.~e., resolvents of bounded, unbounded, or multivalued linear operators, and f and g are analytic functions. Several applications are considered: a representation of the impulse response of a second order linear differential equation with operator coefficients, a representation of the solution of the Sylvester equation, and an exploration of properties of the differential of the ordinary functional calculus.
... Also, we refer to classical literature Hille and Phillips [15], and Yosida [30], for the semigroup theory and expesially for the Bochner integration. For the notions of mild, weak or generalized, strong and classical solution which may be local or global in time we refer to Engel and Nagel [11] and Pazy in [23]. ...
... Indeed, applying classical results about Bochner integral in Hille and Phillips [15] for (i) we have: ...
In this article we present strong global solutions for the initial value problem in a reflexive Banach space, when the linear part of the corresponding differential evolution equation is Bohl-Bohr or Stepanov almost periodic function, and the displayed operator is infinitesimal generator of (C 0)-semigroup. As an application, we consider a problem from magnetohydrodynamics.
... Note by (26) ...
... and, by the first part of Proposition 5.1, the fact u n ∈Jn Hence (see, e.g. Theorem 2.6.1 of [26] or Proposition 3.13 of [12] ) , there exists an operatorQ n on B n such that P ψ n,paths e γQn = P ψ n,paths A ψγ n , so we endeavor to arrive at a description ofQ n = d dγ γ=0 A ψγ n . This involves computing T ...
We introduce a two parameter () family of interacting particle systems with determinantal correlation kernels expressible in terms of Jacobi polynomials . The family includes previously discovered Plancherel measures for the infinite-dimensional orthogonal and symplectic groups. The construction uses certain BC-type orthogonal polynomials which generalize the characters of these groups. The local asymptotics near the hard edge where one expects distinguishing behavior yields the multi-time -dependent discrete Jacobi kernel and the multi-time -dependent hard-edge Pearcey kernel. For nonnegative integer values of , the hard-edge Pearcey kernel had previously appeared in the asymptotics of non-intersecting squared Bessel paths at the hard edge.
... for some positive constant C. Then, from (17) and (18), we obtain (16). ...
... Then, by (16) and the dominated convergence theorem, it follows that ...
The purpose of this paper is to construct the law of a L\'evy process conditioned to avoid zero, under mild technicals conditions, two of them being that the point zero is regular for itself and the L\'evy process is not a compound Poisson process. Two constructions are proposed, the first lies on the method of h-transformation, which requires a deep study of the associated excessive function; while in the second it is obtained by conditioning the underlying L\'evy process to avoid zero up to an independent exponential time whose parameter tends to 0. The former approach generalizes some of the results obtained by Yano \cite{Yano10} in the symmetric case and recovers some of main results in Yano's work \cite{Yano13}, while the latter is reminiscent of \cite{Chaumont-Doney05}. We give some properties of the resulting process and we describe in some detail two examples: alpha stable and spectrally negative L\'evy processes.
... The study of approximately subadditive and higher order subadditive functions, discovering the linkage of subadditive function with convex and periodic functions, several important limit theorems have been investigated in the last few decades. The detailed research can be found in the papers [1][2][3][4][5][6][7][8][9][10] and the reference therein. ...
In this paper, we present an alternative proof of Fekete’s Lemma. We demonstrate that for any subadditive sequence, it is possible to construct a subadditive function that exactly interpolates the sequence. Using this result, along with Hille’s theorem on subadditive functions, we naturally arrive at Fekete’s Lemma. Additionally, we provide an explicit formula for determining the largest subadditive minorant of a given sequence. We explore a sandwich-type result and derive a discrete version of the Hyers-Ulam type stability theorem. For approximately periodic sequences, we offer a decomposition result. In the final section, we propose two characterization theorems for ordinary periodic sequences. The motivation, research progress, and other important details are covered in the introduction.
... Bu bölümde tezimiz için gerekli olan temel bilgiler ayrıca referans belirtilmedikçe [4,[25][26][27][28][29][30][31][32] kaynaklarından alınmıştır. Bahsedecegimiz teoremler, önermeler ve sonuçlar iyi bilindigi için ispatsız olarak verilecektir. ...
In our thesis, we consider the set S as an m-system, and investigate the properties and relationships of various classes of ideals defined with respect to S, including S-maximal ideals, S-semiprime ideals, weakly S-semiprime ideals, weakly S-prime ideals, and S-rho-ideals defined via special radicals. We explore connections of these ideals with Jacobson and Nil ideals, as well as their relationships with S-prime ideals. Furthermore, we examine the behavior of these ideals in ring extensions. At the end of the thesis, several open problems and directions for future research are presented. The thesis is written in Turkish.
In the conclusion section of this thesis, we provided some possible avenues for further research and highlighted several open problems that remain unsolved:
7.1.3.1 Future Studies
* S-primary ideals in noncommutative rings.
* S-primary modules (submodules) in noncommutative rings.
* Noncommutative ρ-modules.
* S-2-absorbing ideals in noncommutative rings.
* Quasi S-prime ideals in noncommutative rings.
* Weakly S-1-absorbing (2-absorbing) ideals in noncommutative rings.
* S-maximal submodules in noncommutative rings.
* S-𝒥-prime ideals in noncommutative rings.
* Quasi S-𝒥-ideals in noncommutative rings.
* S-𝒥-submodules in noncommutative rings.
* S-𝒫-submodules in noncommutative rings.
7.1.3.2 Open Problems
* In a noncommutative ring, are there properties that influence the ring structure between a right weak S-prime ideal and the Jacobson radical of the ring?
* Can a property be obtained between a right weak S-prime ideal and the Jacobson radical, similar to what was achieved in Proposition 3.2.3 regarding the nil radical?
* Is there a special ring in which every ideal is a ρ-ideal?
* Is there a special ring in which every ideal is an S-ρ-ideal?
* Is there a special noncommutative ring or rings in which every ideal is an S-𝒥-ideal?
* Is there a special noncommutative ring in which every weak (ρ, S-ρ, S-𝒥)-ideal is an ideal?
* Suppose IS⁻¹ is an ideal of the ring RS⁻¹. Then, is I necessarily an S-𝒥-ideal of R?
* Is the set S used in ring localization the same as the set of S-𝒥-ideals defined in our thesis?
* How can rings be characterized where an ideal is an S-𝒥-ideal but not necessarily a 𝒥-ideal?
* How can rings be characterized where an ideal is an S-𝒫-ideal but not necessarily a 𝒫-ideal?
* How can rings be characterized where an ideal is an S-ρ-ideal but not necessarily a ρ-ideal?
The publications related to our thesis are as follows:
1. *S-weakly prime ideals in noncommutative rings*, Ukrainian Mathematical Journal, accepted for Vol 77, 2025.
2. *On weakly S-ρ-ideals in noncommutative rings*, Gulf Journal of Mathematics, published.
3. *Generalization of ρ-ideals associated with an m-system and a special radical class*, Ricerche di Matematica, published.
4. *S-J-Ideals: A Study in Commutative and Noncommutative Rings*, Journal of Mathematics, published.
... The class of semigroups over topological monoids provides a proper generalization of the class of one-parameter strongly continuous semigroups of bounded linear operators. The class of strongly continuous semigroups defined on the set [0, +∞) n , which are usually called multiparameter semigroups (this class of semigroups was introduced by E. Hille in 1944; see [3] and [4] for more details in this direction) is a special subclass of the general class of strongly continuous semigroups defined on topological monoids. The precise definition goes as follows: If (X, ∥ · ∥) is a Banach space and (M, +) is a topological monoid with the neutral element 0, then by a semigroup defined over a monoid M we mean any operator-valued function T : M → L(X), where L(X) denotes the Banach space of all bounded linear operators on X, such that T (0) = I and T (t+s) = T (t)T (s) for all t, s ∈ M. A semigroup (T (t)) t∈M is called strongly continuous if the mapping t → T (t)x, t ∈ M is strongly continuous at t = 0. ...
In this paper, we investigate some classes of n-parameter abstract fractional Cauchy inclusions and n-parameter abstract Volterra inclusions with multivalued linear operators. We work in the framework of sequentially complete locally convex spaces and we use only the Caputo partial fractional derivatives.
... We see that any word with total weight bounded above by R + R ′ + w max can be written as a concatenation of words with weights bounded above by R + w max and R ′ + w max , respectively. Therefore, we have |B w (R + R ′ + w max )| ≤ |B w (R + w max )||B w (R ′ + w max )|, and by the continuous version of Fekete's lemma [18,Theorem 7.6.1] the limit lim R→∞ ln |B w (R + w max )| R = lim R→∞ ln |B w (R)| R exists and is finite. We restrict G to be a torsion-free, non-elementary hyperbolic group, and we assume the weights to be normalized, meaning s∈S w(s) = 1. ...
We study the rigidity of the volume entropy for weighted word metrics on hyperbolic groups, building on a recent convexity result due to Cantrell-Tanaka. Using ideas from small cancellation theory, we give conditions under which a hyperbolic group admits a unique normalized weight minimizing the entropy. Moreover, we show that these conditions are generic for random groups at small densities, and that the unique minimizer of such a generic group is arbitrarily close to the uniform weight.
... The class of semigroups over topological monoids provides a proper generalization of the class of one-parameter strongly continuous semigroups of bounded linear operators. The class of strongly continuous semigroups defined on the set [0, +∞) n , which are usually called multiparameter semigroups (this class of semigroups was introduced by E. Hille in 1944; see [4] and [8] for more details in this direction) is a special subclass of the general class of strongly continuous semigroups defined on topological monoids. The precise definition goes as follows: If (X, ∥ · ∥) is a Banach space and (M, +) is a topological monoid with the neutral element 0, then by a semigroup defined over a monoid M we mean any operator-valued function T : M → L(X), where L(X) denotes the Banach space of all bounded linear operators on X, such that T (0) = I and T (t+s) = T (t)T (s) for all t, s ∈ M. A semigroup (T (t)) t∈M is called strongly continuous if the mapping t → T (t)x, t ∈ M is strongly continuous at t = 0. ...
In this paper, we investigate abstract Volterra integro-differential inclusions with multiple variables and abstract partial fractional differential inclusions with multiple variables. We also introduce and analyze several several new classes of multidimensional (F, G, C)-resolvent operator families in sequentially complete locally convex spaces and provide certain applications.
... For this purpose, the Noetherness of Riemann boundary value problems for analytic functions with respect to these classes was first developed, and then the obtained results were used to determine the basis properties of corresponding perturbed exponential systems. Their abstract generalizations are given by many authors in different senses (see, e.g., [9,11,12,17,19,22,23,27,28,30]). In general, many of these works consider the case of a Hilbert space. ...
... The class of semigroups over topological monoids generalizes the class of oneparameter strongly continuous semigroups of bounded linear operators. This broad class of semigroups includs the semigroups defined on the set [0, +∞) n , which are usually called multiparameter semigroups (this class of semigroups was introduced by E. Hille in 1944; see [3] and [5]). If (X, ∥ · ∥) is a Banach space and (M, +) is a topological monoid with the neutral element 0, then by a semigroup defined over a monoid M we mean any operator-valued function T : M → L(X) such that T (0) = I and T (t + s) = T (t)T (s) for all t, s ∈ M ; here, L(X) denotes the Banach space of all bounded linear operators on X. ...
In this paper, we introduce and analyze several new classes of multidimensional (a, k)-regularized C-resolvent solution operator families in sequentially complete locally convex spaces. We profile exponentially equicon-tinuous (a, k)-regularized C-resolvent solution operator families in terms of multidimensional vector-valued Laplace transform and provide some applications to abstract Volterra integro-differential inclusions with multiple variables. The introduced notion and established results seem to be new even in the setting of complex Banach spaces.
... □ Remark 5. In turn, the matrix e −sh T 2 (s) is the Laplace transform of an impulse h 2 (t) with support on [h, ∞] (otherwise, the Laplace transform is obviously not defined) [17,18]. In this situation, if one consider the transfer matrix from w to z, ...
The objective of this work is to study the equilibrium stability of a switched linear model with time-delayed control and additive disturbances, that in subsidiary represents the control of wing vibrations in the presence of the turbulence disturbances in an aerodynamic tunnel. The state system is modeled as a collection of subsystems, each corresponding to different levels of air velocity in the wind tunnel. The problem is closely related to the gain scheduling approach for stable control synthesis and to the design of stable, switched systems with time-delay control. A state-predictive feedback method is employed to compensate for actuator delay, resulting in closed-loop free delay switching systems both in presence and absence of disturbances. The main contribution of this study is a thorough analysis of system stability in the presence of disturbances. Finally, numerical simulation results are provided to support and complement the findings.
... When equipped with the inner product f, g = C n f (z), g(z) H e −2φ(z) dA(z), L 2 φ (H) becomes a Hilbert space. We say that f : C n → H is holomorphic if for every continuous linear functional φ ∈ H * , the scalar-valued function φ • f : C n → C is holomorphic in the usual sense (see, e.g., §3.10 in [10]). The vector-valued large Fock space F 2 φ (H) is defined by Our large Fock spaces F 2 φ (H) generalize the concept of doubling Fock spaces on C to higher dimensions and allow for vector-valued functions. ...
We characterize boundedness and compactness of Toeplitz operators on large vector-valued Fock spaces with Dall'Ara's weights [Adv.\ Math., 285 (2015) 1706--1740] in terms of generalized Berezin transforms, averaging functions, and Carleson measures. To determine Schatten class Toeplitz operators, we introduce the operator-valued Berezin transform and averaging functions.
... A. Example 1: To easily evaluate integration by parts requires prior knowledge of the derivative and anti-derivative of functions that form the product of functions. Even after identifying the two prerequisites -derivative and anti-derivative, undergraduate students are confused still when applying the general rule or formula of the integration by parts [4]. The proposed method in this paper is as followed, Where the arrows indicate the derivatives and the diagonal represents the anti-derivatives. ...
... where we use the Bochner integral [1,22,12]. Note that this is a version of Duhamel's principle, where {G(t)} t≥0 is the homogeneous part, see e.g. ...
In statistical physics, the Mori-Zwanzig projection operator formalism (also called Nakajima-Zwanzig projection operator formalism) is used to derive a linear integro-differential equation for observables in Hilbert space, the generalized Langevin equation (GLE). This technique relies on the splitting of the dynamics into a projected and an orthogonal part. We prove that the GLE together with the second fluctuation dissipation theorem (2FDT) uniquely define the fluctuating forces as well as the memory kernel. The GLE and 2FDT are an immediate consequence of the existence and uniqueness of solutions of linear Volterra equations. They neither rely on the Dyson identity nor on the concept of orthogonal dynamics. This holds true for autonomous as well as non-autonomous systems. Further results are obtained for the Mori projection for autonomous systems, for which the fluctuating forces are orthogonal to the observable of interest. In particular, we prove that the orthogonal dynamics is a strongly continuous semigroup generated by , where is the generator of the time evolution operator, and is the Mori projection operator. As a consequence, the corresponding orbit maps (e.g.~the fluctuating forces) are the unique mild solutions of the associated abstract Cauchy problem. Furthermore, we show that the orthogonal dynamics is a unitary group, if is skew-adjoint. In this case, the fluctuating forces are stationary. In addition, we present a proof of the GLE by means of semigroup theory, and we retrieve the commonly used definitions for the fluctuating forces, memory kernel, and orthogonal dynamics. Our results apply to general autonomous dynamical systems, whose time evolution is given by a strongly continuous semigroup. This includes large classes of systems in classical statistical mechanics.
... Under the assumption that L 2 (Ω, P) is separable, the group {U (x ′ ) : x ′ ∈ R 2 } is strongly continuous, that is U (x ′ )f → f strongly in L 2 (Ω, P), as x ′ → 0 (see Theorems 3.5.3 and 10.10.1 in [17]). ...
We present a derivation of a multidomain model for the electric potential in bundles of randomly distributed axons with different radii. The FitzHugh-Nagumo dynamics is assumed on the axons' membrane, and the conductivity depends nonlinearly on the electric field. Under ergodicity conditions, we study the asymptotic behavior of the potential in the bundle when the number of the axons in the bundle is sufficiently large and derive a macroscopic multidomain model describing the electrical activity of the bundle. Due to the randomness of geometry, the effective intracellular potential is not deterministic but is shown to be a stationary function with realizations that are constant on axons' cross sections. The technique combines the stochastic two-scale convergence and the method of monotone operators.
... For this, we need a notion of analyticity for functions that take values in B(H). There are a priori three obvious ways to interpret the derivative of such a function (in the norm topology, strong operator topology and weak operator topology) but we note that by [HP48,Sec. 3.9] they are all equivalent. In this paper we will therefore not make a distinction and simply write 'analytic'. ...
Let and be half-sided modular inclusions in a common standard subspace H. We prove that the inclusion holds if and only if we have an inclusion of spectral subspaces of the generators of the positive one-parameter groups associated to the half-sided modular inclusions and . From this we give a characterization of this situation in terms of (operator valued) symmetric inner functions. We illustrate these characterizations with some examples of non-trivial phenomena occurring in this setting.
... 3,7,8], [33,Ch. 4] and [19,Section 6.4]. The main goal of this section is to state basic definitions, notation, and further properties of Hardy classes of vector-valued holomorphic functions, i.e. with values in either Hilbert or Banach spaces. ...
We propose a reduced basis method to solve time-dependent partial differential equations based on the Laplace transform. Unlike traditional approaches, we start by applying said transform to the evolution problem, yielding a time-independent boundary value problem that depends on the complex Laplace parameter. First, in an offline stage, we appropriately sample the Laplace parameter and solve the collection of problems using the finite element method. Next, we apply a Proper Orthogonal Decomposition (POD) to this collection of solutions in order to obtain a reduced basis that is of dimension much smaller than that of the original solution space. This reduced basis, in turn, is then used to solve the evolution problem using any suitable time-stepping method. A key insight to justify the formulation of the method resorts to Hardy spaces of analytic functions. By applying the widely-known Paley-Wiener theorem we can then define an isometry between the solution of the time-dependent problem and its Laplace transform. As a consequence of this result, one may conclude that computing a POD with samples taken in the Laplace domain produces an exponentially accurate reduced basis for the time-dependent problem. Numerical experiments characterizing the performance of the method, in terms of accuracy and speed-up, are included for a variety of relevant time-evolution equations.
... Passing to the limit as λ → 0. By making the use of the Banach-Alaoglu theorem (see [38]) and denoting the subsequences again by y λ , we have the following convergences Case II of Table 2: Note that for r > 5, (C.3) does not hold true. In this case, the most difficult term to handle in (C.2) is T τ (| y + y e | r −1 ( y + y e ) − | y e | r −1 y e ), ϕ dt. ...
In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a d-dimensional torus: where , , , with . We prove the exponential stabilization of CBFeD system by finite- and infinite-dimensional feedback controllers. The solvability of the controlled problem is achieved by using the abstract theory of m-accretive operators and density arguments. As an application of the above solvability result, by using infinite-dimensional feedback controllers, we demonstrate exponential stability results such that the solution preserves an invariance condition for a given closed and convex set. By utilizing the unique continuation property of controllability for finite-dimensional systems, we construct a finite-dimensional feedback controller which exponentially stabilizes CBFeD system locally, where the control is localized in a smaller subdomain. Furthermore, we establish the local exponential stability of CBFeD system via proportional controllers.
... In this connection recall that a function with values in a Banach space is holomorphic if and only if it is "weakly holomorphic" (Dunford's theorem; see e.g. [20], p.93). ...
This note is an (exact) copy of the report of Jaak Peetre, "H-infinity and Complex Interpolation". Published as Technical Report, Lund (1981). Some more recent general references have been added, some references updated though (in italics) and some misprints corrected.
... Subadditivity is an important technique in pure and applied mathematics, extending its applications to mathematical physics and other scientific disciplines. The pioneering work on subadditive functions is primarily attributed to Hille and Phillips [11], whose research laid the groundwork for this area of study, which was further enhanced by Rosenbaum's [25] research on multivariable subadditive functions. These concepts find applications in diverse fields such as thermodynamics, electrical networks, quantum relative entropy, and dynamic systems. ...
This study advances Hermite--Hadamard inequalities for subadditive functions using conformable fractional integrals. It establishes and explores numerous versions of these inequalities, as well as fractional integral inequalities for the product of two subadditive functions via conformable fractional integrals. The findings indicate that these inequalities improve and extend prior results for convex and subadditive functions, significantly enhancing the theoretical framework of fractional calculus and inequality theory. Moreover, computational analysis is conducted on these inequalities for subadditive functions, and mathematical examples are given to validate the newly established results within the framework of conformable fractional calculus.
... If µ is the Lebesgue measure we use the notation L 2 (I, G). For the definition of these spaces see [27]. We denote by H j ((0, 1)), for j = 1, 2 the Sobolev space of all the complex-valued, square-integrable functions defined on (0, 1) whose derivates up to order j, in distribution sense, are given by square-integrable functions [1]. ...
We consider the gravitational Vlasov-Poisson system linearized around steady states that are extensively used in galaxy dynamics. Namely, polytropes and King steady states. We develop a complete stationary scattering theory for the selfadjoint, strictly positive, Antonov operator that governs the plane-symmetric linearized dynamics. We identify the absolutely continuous spectrum of the Antonov operator. Namely, we prove that the absolutely continuous spectrum of the Antonov operator coincides with its essential spectrum and with the spectrum of the unperturbed Antonov operator. Moreover, we prove that the part of the singular spectrum of the Antonov operator that is embedded in its absolutely continuous spectrum is contained in a closed set of measure zero, that we characterize. We construct the generalized Fourier maps and we prove that they are partially isometric with initial subspace the absolutely continuous subspace of the Antonov operator and that they are onto the Hilbert space where the Antonov operator is defined. Furthermore, we prove that the wave operators exist, are isometric, and are complete. Moreover, we obtain stationary formulae for the wave operators and we prove that Birman's invariance principle holds.Further, we prove that the gravitational Landau damping holds for the solutions to the linearized gravitational Vlasov-Poisson system with initial data in the absolutely continuous subspace of the Antonov operator. Namely, we prove that the gravitational force and its time derivative, as well as the gravitational potential and its time derivative, tend to zero as time tends to Furthermore, for these initial states the solutions to the linearized gravitational Vlasov-Poisson system are asymptotic, for large times, to the orbits of the gravitational potential of the steady state, in the sense that they are transported along these orbits.
... For the theory of Radon measures and their properties one may consult, e.g., [12], [19,Chapter 7], [22,Chapter 4], and [62,Chapter 4], which sometimes complement each other. A compact treatment (though without using the Radon measures terminology) in [31,Chapter 4.16] is also relevant. Being unable to go into finer details of the theory of Radon measures, we just remark that with certain care, all of the basic facts from the integration theory of positive (Borel) measures can be adapted to this quite general setting. ...
We identify measures arising in the representations of products of generalized Stieltjes transforms as generalized Stieltjes transforms, provide optimal estimates for the size of those measures, and address a similar issue for generalized Cauchy transforms. In the latter case, in two particular settings, we give criteria ensuring that the measures are positive. In this way, we also obtain new, applicable conditions for representability of functions as generalized Stieltjes transforms, thus providing a partial answer to a problem posed by Sokal and shedding a light on spectral multipliers emerged recently in probabilistic studies. As a byproduct of our approach, we improve several known results on Stieltjes and Hilbert transforms.
... Since the map that takes a linear invertible operator to its inverse is real analytic (cf. Hille and Phillips [17,Thms. 4.3.2 and 4.3.4]), and therefore of class C ∞ , we also deduce that the map from A 1,α ∂Ω,Q to L(C ...
In this paper, we review the construction of periodic fundamental solutions and periodic layer potentials for various differential operators. Specifically, we focus on the Laplace equation, the Helmholtz equation, the Lam\'e system, and the heat equation. We then describe how these layer potentials can be applied to analyze domain perturbation problems. In particular, we present applications to the asymptotic behavior of quasi-periodic solutions for a Dirichlet problem for the Helmholtz equation in an unbounded domain with small periodic perforations. Additionally, we investigate the dependence of spatially periodic solutions of an initial value Dirichlet problem for the heat equation on regular perturbations of the base of a parabolic cylinder.
... We recount here some of the essentials of the theory of contraction semigroups on Banach spaces, which can be found in [HP57], [DS88], [BR87], and [EN06]. ...
Dilations of completely positive semigroups to endomorphism semigroups have been studied by numerous authors. Most existing dilation theorems involve a non-unital embedding, corresponding to the embedding of B(H) as a corner of B(K) for Hilbert spaces . A 1986 paper of Jean-Luc Sauvageot shows how to achieve a unital dilation, but does not specify how to do so while also preserving continuity properties of the original semigroup. This thesis further develops Sauvageot's dilation theory in order to establish the existence of continuous unital dilations, and to explore connections with free probability.
... Our main reference on the subject of dissipative operators (sometimes called accretive in the literature) is the book by Hille and Phillips [20]. The proof of the following theorem can be found in [29,Theorem II.2]. ...
In this paper we introduce a new technique for proving norm inequalities in operator ideals with an unitarily invariant norm. Among the well known inequalities which can be proved with this technique are the L\"owner-Heinz inequality, inequalities relating various operator means and the Corach-Porta-Recht inequality. We prove two general inequalities and from them we derive several inequalities by specialization, many of them new. We also show how some inequalities, known to be valid for matrices or bounded operators, can be extended with this technique to normed ideals in -algebras, in particular to the noncommutative -spaces of a semi-finite von Neumann algebra.
... Re Ax, x * X,X * ≤ ω x 2 X . The necessary and sufficient condition for A generating the C 0 semigroup is shown by the Lumer-Phillips theorem [12], which is equivalent to the Hille-Yosida theorem [8,33]. ...
This paper provides a methodology of verified computing for solutions to 1-dimensional advection equations with variable coefficients. The advection equation is typical partial differential equations (PDEs) of hyperbolic type. There are few results of verified numerical computations to initial-boundary value problems of hyperbolic PDEs. Our methodology is based on the spectral method and semigroup theory. The provided method in this paper is regarded as an efficient application of semigroup theory in a sequence space associated with the Fourier series of unknown functions. This is a foundational approach of verified numerical computations for hyperbolic PDEs. Numerical examples show that the rigorous error estimate showing the well-posedness of the exact solution is given with high accuracy and high speed.
... We recall that some classical results on groups and semigroups of linear operators (see [26,28,31,37]) have been extended to the quaternionic setting in some recent papers. In [14] it has been proved the quaternionic Hille-Yosida theorem, in [5] has been studied the problem of generation by perturbations of the quaternionic infinitesimal generator and in [3] the natural functional calculus has been defined for the infinitesimal generator of quaternionic groups of operators. ...
In this paper we extend the functional calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called S-functional calculus. The S-functional calculus has two versions one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz-Dunford functional calculus based on slice hyperholomorphicity because it shares with it the most important properties. The S-functional calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the functional calculus based on the notion of S-spectrum for both quaternionic operators and for n-tuples of noncommuting operators. We remark that the functional calculus for (n+1)-tuples of operators applies, in particular, to the Dirac operator.
... This makes it possible to mimick the argument of Feldman [Fe], which is reproduced in Chernoff's book [Ch2,p. 90], see also [Fr], to conclude by means of the Vitali theorem (see, e.g., [HP,Thm 3.14.1]) that for Re ζ > 0 (I + S(ζ, τ )) −1 s −→ (I + ζH P ) −1 P as τ → 0 (4.6) holds uniformly on compact subsets of Re ζ > 0. At the boundary Re ζ = 0, or ζ = it with t real, (I + S(ζ, τ )) −1 still converges as τ → 0 but in a weaker sense only. ...
We prove a product formula which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection P on a separable Hilbert space \HH, with the convergence in L^2_\mathrm{loc}(\mathbb{R};\HH). It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace \hbox{}. The convergence in \HH is demonstrated in the case of a finite-dimensional P. The main result is illustrated in the example where the projection corresponds to a domain in and the unitary group is the free Schr\"odinger evolution.
In this paper, we review the construction of periodic fundamental solutions and periodic layer potentials for various differential operators. Specifically, we focus on the Laplace equation, the Helmholtz equation, the Lamé system, and the heat equation. We then describe how these layer potentials can be applied to analyze domain perturbation problems. In particular, we present applications to the asymptotic behavior of quasi-periodic solutions for a Dirichlet problem for the Helmholtz equation in an unbounded domain with small periodic perforations as the size of each hole tends to 0. Additionally, we investigate the dependence of spatially periodic solutions of an initial value Dirichlet problem for the heat equation on regular perturbations of the base of a parabolic cylinder.
The main purpose of this paper is to study an internal geometric structure of convex cones in infinite-dimensional vector spaces and to obtain an analytical description of those. We suppose that a convex cone has an elementary (one-piece) internal geometric structure if it is relatively algebraically open. It is proved that an arbitrary convex cone is the disjoint union of the partial ordered family of its relatively algebraically open convex subcones and, moreover, as an ordered set this family is an upper semilattice. We identify the structure of this upper semilattice with the internal geometric structure of the corresponding convex cone. Theorem 16 and Example 17 demonstrate that the internal geometric structure of a convex cone is related to its facial structure but in an infinite-dimensional setting these two structures may differ each other. Further, we study the internal geometric structure of conical halfspaces (convex cones whose complements also are convex cones). We show that every conical halfspace is the disjoint union of the linearly ordered family of relatively algebraically open convex subcones each of which is a conical halfspace in its linear hull. Using the internal geometric structure of conical halfspaces, each conical halfspace is associated with a linearly ordered family of linear functions, which generates in turn a real-valued function, called a step-linear one, analytically describing this conical halfspace. At last, we establish that an arbitrary asymmetric convex cone admits an analytical representation by the family of step-linear functions.
We investigate in this chapter the mathematical models for electromagnetic wave propagation in dispersive isotropic passive linear media for which the dielectric permittivity and magnetic permeability depend on the frequency. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notions of causality and passivity and its connection to the existence of Herglotz functions that determine the dispersion of the material. We consider successively the cases of the general passive media and the so-called local media for which and are rational functions of the frequency. This leads us to analyse the important class of non dissipative and dissipative generalized Lorentz models. In particular, we discuss the connection between mathematical and physical properties of models through the notions of stability, energy conservation, dispersion and modal analyses, group and phase velocities and energy decay in dissipative systems.
In this paper, we consider optimal control problems of stochastic Volterra equations (SVEs) with singular kernels, where the control domain is not necessarily convex. We establish a global maximum principle by means of the spike variation technique. To do so, we first show a Taylor type expansion of the controlled SVE with respect to the spike variation, where the convergence rates of the remainder terms are characterized by the singularity of the kernels. Next, assuming additional structure conditions for the kernels, we convert the variational SVEs appearing in the expansion to their infinite dimensional lifts. Then, we derive first and second order adjoint equations in form of infinite dimensional backward stochastic evolution equations (BSEEs) on weighted spaces. Moreover, we show the well-posedness of the new class of BSEEs on weighted spaces in a general setting.
Starting from a fixed measure space ( X , F , μ ) , with μ a positive sigma-finite measure defined on the sigma-algebra F , we continue here our study of a generalization W ( μ ) of Brownian motion, and introduce a corresponding white-noise process. In detail, the generalized Brownian motion is a centered Gaussian process W ( μ ) , indexed by the elements A in F of finite μ measure, and with covariance function μ ( A ∩ B ) . The purpose of our present paper is to make precise and study the corresponding white-noise process, i.e., a point-wise process which is indexed by X , and which arises as a generalized μ derivative of W ( μ ) . A key tool in our definition and analysis of this pair is a construction of three operators between the underlying Hilbert spaces. One of these operators is a stochastic integral, the second is a gradient associated with the measure μ , and the third is a mathematical expectation in the underlying probability space. We show that, with the setting of families of processes indexed by sets of measures μ , our results lead to new stochastic bundles. They serve in turn to extend the tool set for stochastic calculus.
Research on the mathematical modeling of infectious disease dynamics and their control has recently made remarkable progress. However, despite its growth, the current literature still exhibits scarcity and limitations, particularly in addressing the control of compartmental models governed by Partial Differential Equations (PDEs). Additionally, there are a few gaps in the research concerning the mathematical analysis and asymptotic behavior of compartmental models governed by Stochastic Delayed Differential Equations (SDDEs) and Stochastic Partial Differential Equations (SPDEs). This thesis aims to bridge some of these research gaps through the application of the theory of PDE-constrained optimization and stochastic dynamics. On the one hand, the focus is put on the control of some new compartmental models. Our first primary contribution in this direction involves extending previous findings, on optimal control of spatio-temporal compartmental models with a single group, to multi-group models, accounting for population heterogeneity. Our second primary contribution involves developing a novel practical approach based on the concept of cross-diffusion to mathematically model social distancing as an optimal control measure for disease outbreaks in the absence of extreme clinical interventions such as vaccination and treatment. On the other hand, we also devote our attention to addressing two new problems. The first problem resides in investigating the asymptotic behavior of a new compartmental model, enhancing the existing compartmental models in the context of SDDEs driven by L\'evy noise. Our proposed model considers additional new biological factors such as multiple vaccination stages, the time between vaccination stages, vaccination-related deaths, and sudden anti-vaccination movements in the population, which are exhibited by some recently-emerged epidemics. The second problem resides in the mathematical modeling of abrupt environmental changes in the spatio-temporal dynamics of epidemics by means of a new class of mathematical models, which is the first to be proposed in the context of SPDEs driven by pure-jump Lévy noise. By leveraging semi-group theory and the Banach fixed-point theorem, we establish various results pertaining to the mathematical well-posedness and biological feasibility. All the aforementioned contributions in the frameworks of PDEs, SDDEs and SPDEs are supported by various numerical simulations.
We prove that a locally bounded and differentiable in the sense of Gateaux function given in a finite-dimensional commutative Banach algebra over the complex field is also differentiable in the sense of Lorch.
Available on arxiv under {{{ https://doi.org/10.48550/arXiv.2501.16314 }}}. Commuting families of contractions or contractive -semigroups on Hilbert spaces often fail to admit power dilations resp. simultaneous unitary dilations which are themselves commutative (see [31, 11, 13]). In the \emph{non-commutative} setting, Sz.-Nagy [41] and Bożejko [4] provided means to dilate arbitrary families of contractions. The present work extends this discrete-time result to families of contractive -semigroups. We refer to these dilations as continuous-time \emph{free unitary dilations} and present three distinct approaches to obtain them: 1) An explicit derivation applicable to semigroups that arise as interpolations; 2) A full proof with an explicit construction, via the theory of co-generators à la Słociński [37, 38] as well as a version for topologised index sets via a reformulation of the Trotter–Kato theorem for co-generators, leading to applications to evolution families; and 3) A second full proof based on the abstract structure of semigroups, which admits a natural reformulation to semigroup defined over topological free products of and leads to various residuality results.
This article addresses a combinatorial problem with applications to algebraic geometry. To a convex lattice polytope P and each of its integer dilations kP one may associate the barycenter of its lattice points. This sequence of k-quantized barycenters converge to the (classical) barycenter of the polytope considered as a convex body. A basic question arises: is there a complete asymptotic expansion for this sequence? If so, what are its terms? This article initiates the study of this question. First, we establish the existence of such an expansion as well as determine the first two terms. Second, for Delzant lattice polytopes we use toric algebra to determine all terms using mixed volumes of virtual rooftop polytopes, or alternatively in terms of higher Donaldson–Futaki invariants. Third, for reflexive polytopes we show the quantized barycenters are colinear to first order, and actually colinear in the case of polygons. The proofs use Ehrhart theory, convexity arguments, and toric algebra. As applications we derive the complete asymptotic expansion of the Fujita–Odaka stability thresholds on arbitrary polarizations on (possibly singular) toric varieties. In fact, we show they are rational functions of k for sufficiently large k. This gives the first general result on Tian’s stabilization problem for -invariants for (possibly singular) toric Fanos: stabilize in k if and only if they are all equal to 1, and when smooth if and only if asymptotically Chow semistable (a condition stronger than existence of a Kähler–Einstein metric). We also relate the asymptotic expansions to higher Donaldson–Futaki invariants of test configurations motivated by Ehrhart theory, and unify in passing previous results of Donaldson, Futaki, Ono, Sano, and Rubinstein–Tian–Zhang on existence of canonical Kähler metrics, obstructions, and stability thresholds.
We consider the evolutionary Hamilton-Jacobi equation \begin{align*} w_t(x,t)+H(x,Dw(x,t),w(x,t))=0, \quad(x,t)\in M\times [0,+\infty), \end{align*} where M is a compact manifold, , H=H(x,p,u) satisfies Tonelli conditions in p and the Lipschitz condition in u. This work mainly concerns with the Lyapunov stability (including asymptotic stability, and instability) and uniqueness of stationary viscosity solutions of the equation. A criterion for stability and a criterion for instability are given. We do not utilize auxiliary functions and thus our method is different from the classical Lyapunov's direct method. We also prove several uniqueness results for stationary viscosity solutions. The Hamiltonian H has no concrete form and it may be non-monotonic in the argument u, where the situation is more complicated than the monotonic case. Several simple but nontrivial examples are provided, including the following equation on the unit circle where a, are parameters. We analyze the stability, and instability of the stationary solution w=0 when parameters vary, and show that w=0 is the unique stationary solution when a=0, and , . The sign of the integral of with respect to the Mather measure of the contact Hamiltonian system generated by H plays an essential role in the proofs of aforementioned results. For this reason, we first develop the Mather and weak KAM theories for contact Hamiltonian systems in this non-monotonic setting. A decomposition theorem of the Ma\~n\'e set is the main result of this part.
For a Cauchy problem associated with a controlled semilinear evolutionary equation with optionally bounded operator in a Hilbert space we obtain sufficient conditions for exact controllability to a given final state (and also to given intermediate states at intermediate time moments) on a arbitrarily fixed (without additional conditions) time interval. Here we use the Minty–Browder’s theorem and also a chain technology of successive continuation of the solution to a controlled system to intermediate states. As examples we consider a semilinear pseudoparabolic equation and a semilinear wave equation.
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