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AbstractLet M(S;Λ;P) denote a Rees I×Λ matrix semigroup over a semigroup S in which I is a singleton. In this paper we describe the right regular representation of M(S;Λ;P), and find connection between properties of S and M(S;Λ;P). We also prove embedding theorems on Rees matrix semigroups M(S;Λ;P). [ABSTRACT FROM AUTHOR]

The theories of rough sets (RSs) and soft sets (SSs) are practical mathematical techniques to accommodate data uncertainty. On the other hand, fuzzy bipolar soft sets (FBSSs) can address uncertainty and bipolarity in various situations. The key objective of this study is to establish the notions of rough fuzzy bipolar soft ideals in semigroup (SG), which is an extension of the idea of rough fuzzy bipolar soft sets in a SG. Also, we have analyzed the roughness in the bipolar fuzzy subsemigroup (BF-SSG) by employing a congruence relation (CR) defined on the SG and investigating several related characteristics. Further, the idea is expanded to the rough fuzzy bipolar soft ideal, rough fuzzy bipolar soft interior ideal, and rough fuzzy bipolar soft bi-ideal in SGs. Moreover, it is observed that CR and complete CR (CCR) are critical in developing rough approximations of fuzzy bipolar soft ideals. Therefore, their related characteristics are studied via CRs and CCRs. [ABSTRACT FROM AUTHOR]

In this paper we study the complex symmetry in the several variable Fock space by using the techniques of weighted composition operators and semigroups. We characterize unbounded weighted composition operators that are (real) complex symmetric with respect to a concrete conjugation. Using this characterization, we study complex symmetric semigroups and their generators. We realize such generators as first-order differential operators. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations ut(x,t)=Δu(x,t)+ψ(t)f(u(x,t)),(x,t)∈Ω×(0,∞),u(·,0)=u0(x),x∈Ω,u(x,t)=0,(x,t)∈∂Ω×(0,∞),$$ \left\{\begin{array}{ll}{u}_t\left(x,t\right)=\Delta u\left(x,t\right)+\psi (t)f\left(u\left(x,t\right)\right),& \left(x,t\right)\in \Omega \times \left(0,\infty \right),\\ {}u\left(\cdotp, 0\right)={u}_0(x),& x\in \Omega, \\ {}u\left(x,t\right)=0,& \left(x,t\right)\in \mathrm{\partial \Omega}\times \left(0,\infty \right),\end{array}\right. $$where Ω$$ \Omega $$ is the half‐space ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$, ψ$$ \psi $$ is a nonnegative continuous function, and f$$ f $$ is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data u0∈C0(Ω)$$ {u}_0\in {C}_0\left(\Omega \right) $$ if and only if ∫1∞ψ(t)tN+K2fϵt−N+K2dt=∞$$ {\int}_1^{\infty}\psi (t){t}^{\frac{N+K}{2}}f\left(\epsilon \kern0.1em {t}^{-\frac{N+K}{2}}\right) dt=\infty $$ for every ϵ>0$$ \epsilon >0 $$. In fact, we introduce a very special curve in ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$x^(t):=t,⋯,t⏟K‐times,xK+1,⋯,xN,t>0,$$ \hat{x}(t):= \left(\underset{K\hbox{-} \mathrm{times}}{\underbrace{\sqrt{t},\cdots, \sqrt{t}}},{x}_{K+1},\cdots, {x}_N\right),t>0, $$to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result. [ABSTRACT FROM AUTHOR]

We study an asymptotic problem of a semilinear partial differential equation (PDE) with Neumann boundary condition, periodic coefficients and highly oscillating drift and nonlinear terms. Our analysis focuses on the double limiting behavior of the PDE-solution perturbed by ε (viscosity parameter) and δ (scaling coefficient) both tending to zero. To do so, we state basic properties of the large deviations principle (LDP) and we express the logarithmic asymptotic of the PDE-solution. Particularly, we provide it for the case when ε converges more quickly than δ. [ABSTRACT FROM AUTHOR]

We consider linear isotopes of commutative groups, i.e., central quasigroups and study the invertibility and orthogonality conditions. It turns out that it is sufficient to study these conditions solely for the unitary isotopes, i.e., for isotopes, which have an idempotent. We establish criteria for the possession of each invertibility property (inverse property, crossed inverse property, and mirroring) for unitary central and matrix quasigroups. In particular, for matrices of the second order, we describe the corresponding matrix quasigroups over the fields of characteristics 2 and 3. The orthogonality criteria are obtained for matrix quasigroups with the indicated invertibility properties. [ABSTRACT FROM AUTHOR]

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely related. To our knowledge, this is the first example of a finitely related semigroup where adjoining an identity yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related. [ABSTRACT FROM AUTHOR]

In this paper, we establish strong convergence of the Halpern's type iteration for a semigroup of α-nonexpansive mappings in Banach spaces. Using the concept of quasilinearization, we extend this result to CAT(0) spaces, an important subclass of metric spaces. Our work generalizes and complements several comparable results existing in the literature. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of liner fuzzy real numbers, and show that the set of all positive (or negative) symmetric linear fuzzy real numbers forms a semiring. Moreover, we discuss a complex transform of linear fuzzy real numbers. [ABSTRACT FROM AUTHOR]

This paper explores the relationship between fuzzy logic algebra and non associative groupoid. As a groupoid which can satisfy type-2 cyclic associative (T2CA) law, T2CA-groupoid is characterized by generalized symmetry. Fuzzy logic algebra is a major direction in the study of fuzzy logic. Residuated lattices are a class of fuzzy logic algebras with widespread applications. The inflationary pseudo general residuated lattice (IPGRL), a generalization of the residuated lattice, does not need to satisfy the associative law and commutative law. Moreover, the greatest element of IPGRL is no longer the identity element. In this paper, the notion of T2CA-IPGRL (IPGRL in T2CA-groupoid) is proposed and its properties are investigated in combination with the study of IPGRL and T2CA-groupoid. In addition, the generalized symmetry and regularity of T2CA-groupoid are investigated based on the characteristics of commutative elements. Meanwhile, the decomposition of T2CA-root of band with T2CA-unipotent radical is studied as well. The result shows that every T2CA-root of band is the disjoint union of T2CA-unipotent radicals. [ABSTRACT FROM AUTHOR]

The work is devoted to integro-differential equations related to stochastic processes. We study the relationship between differential equations with random perturbations — stochastic differential equations (SDEs) — and deterministic equations for the probability characteristics of processes determined by random perturbations. The resulting deterministic pseudodifferential equations are investigated by semigroup methods and Fourier transform methods. [ABSTRACT FROM AUTHOR]

We study two variants of the cosine subtraction law on a semigroup S. The main objective is to solve g (x y ∗) = g (x) g (y) + f (x) f (y) for unknown functions g , f : S → C , where x ↦ x ∗ is an anti-homomorphic involution. Until now this equation has not been solved on non-commutative semigroups, nor even on non-Abelian groups with x ∗ : = x - 1 . We solve this equation on semigroups under the assumption that g is central, and on groups generated by their squares under the assumption that x ∗ : = x - 1 . In addition we give a new proof for the solution of the variant g (x σ (y)) = g (x) g (y) + f (x) f (y) , where σ : S → S is a homomorphic involution. The continuous solutions on topological semigroups are also found. [ABSTRACT FROM AUTHOR]

The present article establishes two contrasting results: a finite involution semigroup is embeddable in some semigroup of complex matrices with conjugate transposition if and only if it is an inverse semigroup, while every finite involution semigroup is embeddable in some semigroup of binary matrices with the skew transposition across the secondary diagonal. [ABSTRACT FROM AUTHOR]

This paper presents new findings on varieties generated by small semigroups and groups, and offers a survey of existing results. A companion website is provided which hosts a computational system integrating automated reasoning tools, finite model builders, SAT solvers, and GAP. This platform is a living guide to the literature. In addition, the first complete and justified list of identity bases for all varieties generated by a semigroup of order up to 4 is provided as supplementary material. The paper concludes with an extensive list of open problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider a 2D stochastic quasi-geostrophic equation driven by jump noise in a smooth bounded domain. We prove the local existence and uniqueness of mild  L p (D) -solutions for the dissipative quasi-geostrophic equation with a full range of subcritical powers  α ∈ (1 2 , 1 ]  by using the semigroup theory and fixed point theorem. Our approach, based on the Yosida approximation argument and Itô formula for the Banach space valued processes, allows for establishing some uniform bounds for the mild solutions and we prove the global existence of mild solutions in  L ∞ (0 , T ; L p (D))  space for all  p > 2 2 α − 1 , which is consistent with the deterministic case. [ABSTRACT FROM AUTHOR]

By proving existence, regularity and uniqueness of solutions to Cauchy problems governed by abstract unbounded operators with finite pseudo-spectral bounds as an alternative and a serious enhancement of results by Melnikova and Filinkov, we establish a new generation criterion theorem for C 0 -semigroups in general Banach spaces. A generalization of Kaiser–Weis–Batty's perturbation generation theorem in reflexive Banach spaces is therefore derived. We apply our last theoretical result to a singular transport model in L p -spaces, p ∈ ] 1 , + ∞ [ , where the streaming (unperturbed) semigroup cannot be explicit. [ABSTRACT FROM AUTHOR]

Under suitable conditions on the unknown functions, solutions of Levi-Civita functional equations on commutative monoids with no prime ideals are exponential polynomials. This is not generally the case on commutative monoids with prime ideals. Here we describe the solutions of Levi-Civita equations on commutative monoids in which every prime ideal is tractable. Monoids with this property include those which are regular or generated by their squares, as well as many others. Our results also give the continuous solutions on topological commutative monoids with tractable prime ideals. [ABSTRACT FROM AUTHOR]

We present a novel approach to address integro-differential systems incorporating state, input, and output delays. Our approach leverages product spaces and employs a boundary perturbation technique. Initially, we focus on state-delay equations, wherein we introduce a variation of constants formula for the mild solution. Additionally, we establish spectral properties using a characteristic equation. Subsequently, we extend our analysis to integro-differential systems affected by state, input, and output delays. Notably, we demonstrate the equivalence between a such delay system and a regular free-delay system within the Salamon–Weiss framework. This equivalence sheds valuable insights on the nature of the integro-differential system under consideration. [ABSTRACT FROM AUTHOR]

We consider a binary mixture system with local Kelvin–Voigt damping in one of the components of the mixture. The well‐posedness and the polynomial stability are proved by the semigroup theory and the frequency domain approach. [ABSTRACT FROM AUTHOR]

This paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the chosen type of unbounded perturbations. To overcome this problem, we first proved a regularity property of the stochastic convolution with respect to the domain of ‘admissible’ unbounded linear operators (not necessarily closed or closable). This is done using Yosida extensions of such unbounded linear operators. After proving the well-posedness of these equations, we also establish the Feller property for the corresponding transition semigroups. Several examples like heat equations and Schrödinger equations with nonlocal perturbations terms are given. Finally, we give an application to a general class of semilinear neutral stochastic equations. [ABSTRACT FROM AUTHOR]

We investigate a non-conservative semigroup ( S t ) t ≥ 0 determined by a branching process tracing the evolution of particles moving in a domain in R d . When a particle is killed at the boundary, a new generation of particles with mean number K ¯ is born at a random point in the domain. Between branching, the particles are driven by a diffusion process with Dirichlet boundary conditions. According to the sign of K ¯ − 1 , we distinguish super/sub-critical regimes and determine the exact exponential rate for the total number of particles n ( t ) ∼ exp ⁡ ( α ∗ t ) , with α ∗ depending explicitly on K ¯ . We prove the Yaglom limit S t / n ( t ) → ν , where the quasi-stationary distribution ν is determined by the resolvent of the Dirichlet kernel at the point α ∗ . The main application is in particle systems, where the normalization of the semigroup by its total mass gives the hydrodynamic limit of the Bak-Sneppen branching diffusions (BSBD). Since ν is the asymptotic profile under equilibrium, and the family of quasi-stationary distributions ν is indexed by K ¯ , the model provides an explicit example of self-organized criticality. [ABSTRACT FROM AUTHOR]

We describe a practical algorithm for computing normal forms for semigroups and monoids with finite presentations satisfying so-called small overlap conditions. Small overlap conditions are natural conditions on the relations in a presentation, which were introduced by J. H. Remmers and subsequently studied extensively by M. Kambites. Presentations satisfying these conditions are ubiquitous; Kambites showed that a randomly chosen finite presentation satisfies the C (4) condition with probability tending to 1 as the sum of the lengths of relation words tends to infinity. Kambites also showed that several key problems for finitely presented semigroups and monoids are tractable in C (4) monoids: the word problem is solvable in O (min ⁡ { | u | , | v | }) time in the size of the input words u and v ; the uniform word problem for 〈 A | R 〉 is solvable in O (N 2 min ⁡ { | u | , | v | }) where N is the sum of the lengths of the words in R ; and a normal form for any given word u can be found in O (| u |) time. Although Kambites' algorithm for solving the word problem in C (4) monoids is highly practical, it appears that the coefficients in the linear time algorithm for computing normal forms are too large in practice. In this paper, we present an algorithm for computing normal forms in C (4) monoids that has time complexity O (| u | 2) for input word u , but where the coefficients are sufficiently small to allow for practical computation. Additionally, we show that the uniform word problem for small overlap monoids can be solved in O (N min ⁡ { | u | , | v | }) time. [ABSTRACT FROM AUTHOR]

For a semigroup S, the covering number of S with respect to semigroups, σs (S), is the minimum number of proper subsemigroups of S whose union is S. This article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. Our three main theorems give a complete description of the covering number of finite semigroups, finite inverse semigroups, and monoids (modulo groups and infinite semigroups). For a finite semigroup that is neither monogenic nor a group, its covering number is two. For all n ≥ 2, there exists an inverse semigroup with covering number n, similar to the case of loops. Finally, a monoid that is neither a group nor a semigroup with an identity adjoined has covering number two as well. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the null controllability of an abstract boundary linear control systems. We then transform the problem of the null controllability, where the control operator is unbounded, to a problem of the null controllability, where the control operator is bounded. An application to flows in networks controlled in a single vertex is applied, where a characterization of the null controllability is given by a matrix equality. Moreover, the construction of the control function is specified. Illustrative numerical examples are further provided. [ABSTRACT FROM AUTHOR]

In this paper, well-posedness and global boundary exponential stabilization problems are studied for the one-dimensional linearized Korteweg-de Vries equation (KdV) with state delay, which is posed in bounded interval |$[0,2\pi ]$| and actuated at the left boundary by Dirichlet condition. Based on the infinite-dimensional backstepping method for the delay-free case, a linear Volterra-type integral transformation maps the system into another homogeneous target system, and an explicit feedback control law is obtained. Under this feedback, we prove the well-posedness of the considered system in an appropriate Banach space and its exponential stabilization in the topology of |$L^{2}(0,2\pi)$| -norm by the use of an appropriate Lyapunov–Razumikhin functional. Moreover, under the same feedback law, we get the local exponential stability for the non-linear KdV equation. A numerical example is provided to illustrate the result. [ABSTRACT FROM AUTHOR]

An n-tuple semigroup is an algebra defined on a set with n binary associative operations. This notion play a prominent role in the theory of n-tuple algebras of associative type. Our paper is devoted to the development of the variety theory of n-tuple semigroups. We construct a free k-nilpotent n-tuple semigroup and characterize the least k-nilpotent congruence on a free n-tuple semigroup. [ABSTRACT FROM AUTHOR]

The inclusion ideal graph I n (S) of a semigroup S is an undirected simple graph whose vertices are all the nontrivial left ideals of S and two distinct left ideals I , J are adjacent if and only if either I ⊂ J or J ⊂ I. The purpose of this paper is to study algebraic properties of the semigroup S as well as graph theoretic properties of I n (S). We investigate the connectedness of I n (S) and show that the diameter of I n (S) is at most 3 if it is connected. We also obtain a necessary and sufficient condition of S such that the clique number of I n (S) is the number of minimal left ideals of S. Further, various graph invariants of I n (S) , viz. perfectness, planarity, girth, etc., are discussed. For a completely simple semigroup S , we investigate properties of I n (S) including its independence number and matching number. Finally, we obtain the automorphism group of I n (S). [ABSTRACT FROM AUTHOR]

The algebraic extension B Z F of an extended bicyclic semigroup is introduced for an arbitrary ω -closed family ℱ of subsets of ω. It is proved that B Z F is a combinatorial inverse semigroup. We describe Green's relations, the natural partial ordering on the semigroup B Z F , and its set of idempotents. We also establish the criteria of simplicity, 0-simplicity, bisimplicity, and 0-bisimplicity of the semigroup B Z F and a criterion for B Z F to be isomorphic either to the extended bicyclic semigroup or to the countable semigroup of matrix units. It is proved that, in the case where the family ℱ consists of all singletons of ω and the empty set, the semigroup B Z F is isomorphic to the Brandt λ-extension of the semilattice (ω,min). [ABSTRACT FROM AUTHOR]

The notion of neutral element generalizes to a pair of elements in ternary algebras. Biunit pairs are introduced as pairs of elements in a semiheap that generalize the notion of Mal'cev element. In order to generalize the known correspondences between semiheaps and certain kinds of semigroups, families of functions generalizing involutions and conjugations, called switches and warps, are investigated. The main theorem establishes that there is a one-to-one correspondence between monoids equipped with a particular switch and semiheaps with a fixed biunit pair. This generalizes the celebrated result in semiheap theory that gives a one-to-one correspondence between involuted monoids and semiheaps with a fixed biunit element. A novel, previously undocumented, algebra is motivated by this result: diheaps are introduced as semiheaps whose elements belong to biunit pairs, which generalize the well-known case of heaps. Diheaps are of great interest since they are shown to be isomorphic to heaps only when they are heaps themselves and explicit non-heap examples are constructed from abelian groups and hypermatrices. [ABSTRACT FROM AUTHOR]

In this paper, we investigate properties, characterizations and compactifications of almost periodic functions with values in a topological vector space. The techniques applied are based essentially on an analogue of a representation in Pettis-integration. Applications of the results to flows are indicated. [ABSTRACT FROM AUTHOR]

Our main result is that we describe the solutions g , f : S → C of the functional equation g (x σ (y)) = g (x) g (y) - f (x) f (y) + α f (x σ (y)) , x , y ∈ S , where S is a semigroup, α ∈ C is a fixed constant and σ : S → S an involutive automorphism. [ABSTRACT FROM AUTHOR]

For any semigroup S and any a , b , c ∈ S , the author (Drazin (2012) [3]) defined a as being " (b , c) -invertible", with " (b , c) -inverse" y , if there exists y ∈ S such that y ∈ b S y ∩ y S c (say with b h y = y = y g c), y a b = b and c a y = c. This article is concerned with pairwise implications among y a = a y and other "EP-like" properties of a , b , c , g , h , y such as y c = b y , a h = g a , h y = y g , c a = a b and b h = g c. Each implication is either proved to be always true, or shown by an example to be false. The important special case where b = c and g = h is considered separately, again with a counter-example for every false implication. [ABSTRACT FROM AUTHOR]

In this article, we consider the inclusion ideal graph In(S) of nontrivial right ideals of a semigroup S with zero element. We characterize a semigroup S for which the graph In(S) is complete, connected and also find various graph parameters of In(S). We determine the values of n for which the graph In(Zn) is complete, triangulated, split, unicyclic, thresold and also study minimal embedding of In(Zn) into compact orientable (resp. non-orientable) surface. We give both upper and lower bouds for metric and partition dimension of inclusion ideal graph of a completely 0-simple semigroup. Finally, we compute some graph parameters of the cartesian product of inclusion ideal graph of two monoids. [ABSTRACT FROM AUTHOR]

In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures. [ABSTRACT FROM AUTHOR]

We call a semigroup S right noetherian if it satisfies the ascending chain condition on right ideals, and we say that S satisfies ACCPR if it satisfies the ascending chain condition on principal right ideals. We investigate the behavior of these two conditions with respect to ideals and ideal extensions, with a particular focus on minimal and 0-minimal one-sided ideals. In particular, we show that the property of satisfying ACCPR is inherited by right and left ideals. On the other hand, we exhibit an example of a right noetherian semigroup with a minimal ideal that is not right noetherian. [ABSTRACT FROM AUTHOR]

We show that a unitary surjective Morita context connecting two semigroups yields Galois connections between certain lattices of compatible relations whenever either semigroup has common weak local units. In the event both semigroups have common weak local units, we obtain mutually inverse lattice isomorphisms that preserve reflexivity, symmetricity and transitivity between the lattices of compatible relations on the semigroups. [ABSTRACT FROM AUTHOR]

We study the functional equation for unknown functions f , g , h , k , ℓ : S → C , where S is a semigroup and m 1 , m 2 : S → C are multiplicative functions. The study is divided into two main parts: m 1 = m 2 and m 1 ≠ m 2 . In some cases we assume that one or more of the unknown functions is central and/or that S is a monoid. The solutions are found in all cases under the umbrella assumption that S is a commutative monoid. The continuous solutions on topological semigroups are also found. [ABSTRACT FROM AUTHOR]

Let S be a semigroup, let H be a 2 -torsion free abelian group, and let φ , ψ : S → S be two endomorphisms that need not be involutive. We express the solutions f : S → H of generalized variant of Jensen's functional equation \begin{equation*} f(x\varphi(y))+f(\psi(y)x)=2f(x), \quad x,y\in S, \end{equation*} in terms of additive maps. In all results it is supposed that at least one of the endomorphisms φ and ψ is surjective. Many consequences of this result are presented. [ABSTRACT FROM AUTHOR]

We study semigroups satisfying the identity a b a = a b . [ABSTRACT FROM AUTHOR]

It is shown that, in any semigroup with involution, both the DMP inverse of Malik and Thome [Appl Math Comp. 2014;226:575–580.] and the CMP inverse of Mehdipour and Salemi [Linear Multilinear Alg. 2018;66:1046–1053.] are special cases of the Bott–Duffin (e , f) -inverse introduced in [Linear Algebra Appl. 2012;436:1909-1923.] It is also shown that the core inverse of O.M. Baksalary and Trenkler [Linear Multilinear Alg. 2010;58:681–697.] can (even for other pairs of generalized inverses) also be alternatively described, in three other equivalent ways, as the (b , c) -inverse y of a for three other choices of (b , c) besides (b , c) = (a a † , a a †). [ABSTRACT FROM AUTHOR]

We review various constructions of wreath products of groups, semigroups, Lie algebras and associative algebras and discuss their realizations in matrix wreath products of associative algebras. As an application we prove a new version of Evans's embedding theorem [T. Evans, Embedding theorems for multiplicative systems and projective geometries, Proc. American Math. Soc. 3 (1952) 614–620]. [ABSTRACT FROM AUTHOR]

This paper studies a class of the fractional stochastic evolution equation with delay. With the aid of the compact semigroup theory and Schauder fixed point theorem, the existence of square-mean S-asymptotically periodic mild solutions is obtained under the situation that the nonlinear functions satisfy certain growth conditions. Moreover, by establishing a new Grönwall integral inequality corresponding to fractional differential equation with delay, the global asymptotic stability of the square-mean S-asymptotically periodic mild solutions are discussed. Finally, an example is given to illustrate our abstract results. [ABSTRACT FROM AUTHOR]

In this paper, the focal point is to study the stabilization problem of distributed systems via bilinear boundary control. Hence, we give some sufficient conditions to ensure weak, strong, and exponential stabilization of the considered system. Finally, an example of application is devoted, and the attained results are illustrated through numerical simulations. [ABSTRACT FROM AUTHOR]

Dai, Hom, Stoffregen and Truong defined a family of concordance invariants φ j . The example of a knot with zero Upsilon invariant but nonzero epsilon invariant previously given by Hom also has nonzero phi invariant. We show there are infinitely many such knots that are linearly independent in the smooth concordance group. In the opposite direction, we build infinite families of linearly independent knots with zero phi invariant but nonzero Upsilon invariant. We also give a recursive formula for the phi invariant of torus knots. [ABSTRACT FROM AUTHOR]

One proves herein that the flow S (t) , generated by the nonlinear Fokker–Planck equation ρ t − Δ β (ρ) + div (a (ρ) ρ) = 0 in (0 , ∞) × R d , is expressed by the Trotter product formula S (t) ρ 0 = lim n → ∞ ⁡ (S A 1 (t n) S A 2 (t n)) n ρ 0 in L 1 (R d) , where S A 1 (t) is the flow (continuous semigroup) generated in L 1 (R d) by the nonlinear diffusion operator A 1 (ρ) = − Δ β (ρ) , while S A 2 (t) is that generated in L 1 (R d) by the conservation law operator A 2 (ρ) = div (a (ρ) ρ) defined in the entropy sense. As an application, one obtains a split-product formula for the McKean–Vlasov stochastic differential equation associated with the Fokker–Planck equation. [ABSTRACT FROM AUTHOR]

The ℛ -height of a semigroup S is the height of the poset of ℛ -classes of S. Given a semigroup S with finite ℛ -height, we establish bounds on the ℛ -height of bi-ideals, one-sided ideals and two-sided ideals; in particular, these substructures inherit the property of having finite ℛ -height. We then investigate whether these bounds can be attained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stabilization of the generalized Rao‐Nakra beam equation, which consists of four wave equations for the longitudinal displacements and the shear angle of the top and bottom layers and one Euler–Bernoulli beam equation for the transversal displacement. Dissipative mechanism are provided through viscous damping for two displacements. The location of the viscous damping are divided into two groups, characterized by whether both of the top and bottom layers are directly damped or otherwise. Each group consists of three cases. We obtain the necessary and sufficient conditions for the cases in group 2 to be strongly stable. Furthermore, polynomial stability of certain orders are proved. The cases in group 1 are left for future study. [ABSTRACT FROM AUTHOR]

In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of interior filters, quasi-interior filters and weak-interior filters in a quasi-ordered semigroup. Additionally, we study the properties of these types of filters of quasi-ordered semigroups and their interrelationships. [ABSTRACT FROM AUTHOR]

The intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if their intersection is nontrivial. In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected, then the diameter of Γ(S) is at most two. Further, we classify the semigroups S in terms of their ideals such that the diameter of Γ(S) is two. We obtain the domination number, independence number, girth and the strong metric dimension of Γ(S). We have also investigated the completeness, planarity and perfectness of Γ(S). We show that if S is a completely simple semigroup, then Γ(S) is weakly perfect. Moreover, in this article, we give an upper bound of the chromatic number of Γ(S). Finally, if S is the union of n minimal left ideals, then we obtain the metric dimension and the automorphism group of Γ(S). [ABSTRACT FROM AUTHOR]