Let $\mathcal{𝒜}$ be a separable Banach algebra, $G$ be a locally compact group and . We first provide a necessary and sufficient condition for which $L^p(G,\mathcal{𝒜})$ is a Banach algebra, under convolution product. Then we characterize the character space of $L^p(G,\mathcal{𝒜})$, in the case where $\mathcal{𝒜}$ is commutative and $G$ is abelian. Moreover, we investigate the BSE-property for $L^p(G,\mathcal{𝒜})$ and prove that $L^p(G,\mathcal{𝒜})$ is a BSE-algebra if and only if $\mathcal{𝒜}$ is a BSE-algebra and $G$ is finite. Finally, we study the BSE-norm property for $L^p(G,\mathcal{𝒜})$ and show that if $L^p(G,\mathcal{𝒜})$ is a BSE-norm algebra then $\mathcal{𝒜}$ is so. We prove the converse of this statement for the case where $G$ is finite and $\mathcal{𝒜}$ is a unital BSE-algebra.

It is well known that certain topological concepts such as the sum of topologies and the product of topologies are defined by particular classes of other topologies. In such topologies and others, it is crucial to look into the connections between these classes and the topology generated by them. One of the topologies that creates a family of general topologies is soft topology, which is defined over a family of soft sets satisfying the basic axioms of general topology. It is interesting to investigate how topological notions behave when applied to soft topologies because it enables us to study topological qualities by comparing them to their counterparts in other topologies. In this paper, we benefit from the fruitful variety via soft topology to introduce the concept of weakly soft somewhat open sets (briefly, weakly soft $so$-sets), which is constructed using its corresponding notion via parametric topologies. We demonstrate that the class of soft somewhat open sets is a stronger generalization of soft open sets than this type, and we also offer a condition that guarantees their equality. Then, we present new soft operators and discuss their essential characteristics by utilizing the notion of weakly soft $so$-sets and their complements. One of the unique characteristics that we prove of these operators is that the $so$-interior of a soft subset is the soft subset itself or the null soft set, whereas $so$-closure of a soft subset is the soft subset itself or the absolute soft set. Finally, we provide new categories of soft mappings and explore their main characteristics. We provide a few examples and counterexamples to clarify the findings and relationships obtained herein.

This paper is concerned with the rigidity of proper holomorphic self-mappings of some unbounded weakly pseudoconvex domain, called a generalized Fock–Bargmann–Hartogs domain, which is defined as a Hartogs domain fibered over ${ℂ}^n$ with the fiber being a generalized complex ellipsoid. We develop a new technique to show that any proper holomorphic self-mapping of a generalized Fock–Bargmann–Hartogs domain must be an automorphism without the restriction of the dimension of each fiber. As a main contribution, we partly solve the rigidity problems for proper holomorphic self-mappings of generalized Fock–Bargmann–Hartogs domains.

We investigate the problem of finite-time stabilization of a class of nonlinear systems with dilations. We develop a recursive design procedure to design a state feedback which stabilizes the nonlinear system in finite time. The proof is based on the finite-time Lyapunov stability theorem.

We introduce the notion of curvature-like tensors on Kähler manifolds satisfying the second Bianchi equation. As examples, we mention concircular curvature tensor, projective curvature tensor, conformal curvature tensor and Bochner curvature tensor which are all closed curvature-like tensors. In particular, we show that all of them, except the concircular curvature tensor, are harmonic curvature-like tensors and give some conditions for them to be harmonic.

A quadrilateral is said to be rational if its four sides, the two diagonals and the area are all expressible by rational numbers. The problem of constructing rational quadrilaterals dates back to the seventh century when Brahmagupta gave an elegant solution of the problem. In 1848 Kummer gave a method of generating all rational quadrilaterals. In this paper we present an alternative method of generating all rational quadrilaterals. For rational cyclic quadrilaterals, we obtain a complete parametrization and for rational noncyclic quadrilaterals, we give several parametrizations in terms of quadratic and quartic polynomials. The parametrizations obtained in this paper are simpler than the known parametrizations of rational quadrilaterals. We also describe how further parametrizations of rational quadrilaterals may be obtained.

We formulate sufficient conditions under which a nonautonomous dynamics exhibits Hyers–Ulam–Rassias stability. These conditions require that the linear part is exponentially stable and that the nonlinear part is Lipschitz small. We consider both the case of continuous and discrete time dynamics.

This paper focuses on the observer design for a class of nonlinear systems involving bounded disturbances. This class of systems is a larger class of nonlinearities than the Lipschitz. We provide sufficient conditions for the observer design based on the one-sided Lipschitz and quadratically inner-bounded ones. In addition, we show the practical stability of the closed-loop system. Furthermore, we establish a separation principle for a class of nonlinear systems with bounded uncertain part. We also give a numerical example to demonstrate the effectiveness and applicability of the proposed controller.

Using an infinite-dimensional nuclear space, we introduce the quantum fractional number operator (QFNO) and the associated quantum fractional Ornstein–Uhlenbeck (O–U) semigroups. Then, we solve the Cauchy problems associated with the QFNO and show that its solutions can be expressed in terms of the aforementioned semigroups. Besides, we prove that the quantum fractional O–U semigroups satisfy the Feller property. Finally, using an adequate definition of the quantum fractional potentials, we give the solutions of the quantum fractional Poisson equations.

We derive a Hartman–Wintner type inequality for a fractional boundary value problem depending on the Caputo derivative and with Dirichlet boundary conditions. We explicitly show how this inequality strengthens previously known results in the literature, in particular the Lyapunov inequality.

Let $\mathcal{ℰ}$ be an injectively resolving subcategory of left $R$-modules. We study a particular case of $\mathcal{ℰ}$-Gorenstein injective and flat modules, called strongly $\mathcal{ℰ}$-Gorenstein injective and flat modules, respectively. We prove that a module is $\mathcal{ℰ}$-Gorenstein injective if and only if it is a direct summand of a strongly $\mathcal{ℰ}$-Gorenstein injective module, and every $\mathcal{ℰ}$-Gorenstein flat module is a direct summand of a strongly $\mathcal{ℰ}$-Gorenstein flat module. Then we show the property of being a strongly $\mathcal{ℰ}$-Gorenstein injective (resp. flat) module can be inherited by its direct summands under certain condition. The connections between (strongly) $\mathcal{ℰ}$-Gorenstein injective and flat modules are also discussed. Finally, we investigate FC rings in terms of strongly $\mathcal{ℰ}$-Gorenstein injective and flat modules.

We introduce generalized Reynolds operators on Leibniz algebras as a generalization of twisted Poisson structures. We define the cohomology of a generalized Reynolds operator $K$ as the Loday–Pirashvili cohomology of a certain Leibniz algebra induced by $K$ with coefficients in a suitable representation. Then we consider formal deformations of generalized Reynolds operators from cohomological points of view. Finally, we introduce and study NS-Leibniz algebras as the underlying structure of generalized Reynolds operators.

The Fibonacci sequence modulo $m$, which we denote ${({\mathcal{ℱ}}_{m,n})}_{n=0}^{\infty }$ where ${\mathcal{ℱ}}_{m,n}$ is the Fibonacci number $F_n$ modulo $m$, has been a well-studied object in mathematics since the seminal paper by D. D. Wall in 1960 exploring a myriad of properties related to the periods of these sequences. Since the time of Lagrange it has been known that ${({\mathcal{ℱ}}_{m,n})}_{n=0}^{\infty }$ is periodic for each $m$. We examine this sequence when $m=10$, yielding a sequence of period length 60. In particular, we explore its subsequences composed of every $r$-th term of ${({\mathcal{ℱ}}_{10,n})}_{n=0}^{\infty }$ starting from the term ${\mathcal{ℱ}}_{10,k}$ for some $0\le k\le 59$. More precisely we consider the subsequences ${({\mathcal{ℱ}}_{10,k+rj})}_{j=0}^{\infty }$, which we show are themselves periodic and whose lengths divide 60. Many intriguing properties reveal themselves as we alter the $k$ and $r$ values. For example, for certain $r$ values the corresponding subsequences surprisingly obey the Fibonacci recurrence relation, that is, any two consecutive subsequence terms sum to the next term modulo 10. Moreover, for all $r$ values relatively prime to 60, the subsequence ${({\mathcal{ℱ}}_{10,k+rj})}_{j=0}^{\infty }$ coincides exactly with the original parent sequence ${({\mathcal{ℱ}}_{10,n})}_{n=0}^{\infty }$ (or a cyclic shift of it) running either forward or reverse. We demystify these phenomena and explore many other tantalizing properties of these subsequences. Lastly, we end with a few open questions, some of which ask whether results in this paper generalize to arbitrary moduli, but we provide evidence that $m=10$ may indeed be a very special case.

We introduce and study the following special type of differential subordination implication:

$$p(z)Q(z)+\frac{z{p}^{\prime }(z)}{\beta p(z)+\alpha }\prec h(z)\Rightarrow p(z)\prec h(z),$$

which generalizes the Briot–Bouquet differential subordination, where $Q(z)$ is analytic and $0\ne \beta ,\alpha \in ℂ$. Further, we discuss some special cases of our result. Finally, we obtain analogues of the open door lemma and the integral existence theorem with applications to univalent functions.

Let $G$ be a simple graph of order $n$. For $\alpha \in [0,1]$, the $A_{\alpha }$-matrix of $G$ is defined as $A_{\alpha }=\alpha D(G)+(1-\alpha )A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. The largest eigenvalue of $A_{\alpha }(G)$, denoted by ${\rho }_{\alpha }(G)$, is called the $A_{\alpha }$ spectral radius of $G$. We give an upper bound on ${\rho }_{\alpha }(G)$ for $k$-connected irregular graphs. Moreover, we also derive an upper bound on ${\rho }_{\alpha }(G)$ when $G$ is a subgraph of a $k$-connected regular graph. Our results improve or extend the existing results, respectively.

Let $R$ be a commutative ring with identity element, $U(R)$ its group of units and $J(R)$ its Jacobson radical. Recall that a ring $R$ is a $JU$-ring (resp. a weakly $JU$-ring) if $U(R)=1+J(R)$ (resp. $U(R)=\pm 1+J(R)$). In this note we investigate the properties of these notions in different contexts of commutative rings. Precisely, we study the transfer of the notions of $JU$-rings, $WJU$-rings, $UU$-rings, $WUU$-rings, and more to trivial ring extensions, amalgamations of rings and pullbacks. Our aim is to provide new classes of commutative rings satisfying these properties. Examples illustrating the limits and scopes of our results are provided.

The structured singular value is a well-known mathematical quantity which plays a vital role in the investigation of the stability, instability, robustness and performance of linear input and output systems in control. We present an iterative method for the approximation of the lower bounds of structured singular values. The iterative method is based on low rank ordinary differential equations and is restricted to performing when pure complex full block uncertainties are under consideration. Numerical experimentation shows the behavior of lower and upper bounds of structured singular values. Furthermore, we investigate and analyze graphically the behavior of eigenvalues and singular values. The pseudospectrum of three-dimensional real valued matrices is inspected with the help of EigTool.

Following the recent investigation of Somashekara and Thulasi on vanishing coefficients in a certain infinite $q$-product, we further study this phenomenon in a family of powers of theta functions with odd moduli, and prove that some arithmetic progressions on vanishing coefficients could be classified into a unified family, which significantly extends a result obtained by Somashekara and Thulasi.

This paper is concerned with the Carleson type maximal operator ${\mathcal{𝒯}}^{\ast }$ defined by

$${\mathcal{𝒯}}^{\ast }f(x)=\underset{\lambda }{\text{sup }}\left|{\int }_{{ℝ}^n}{e}^{i{P}_{\lambda }(y)}K(y)f(x-y)dy\right|,$$

where $P_{\lambda }(y)={\sum }_{2\le \left|\alpha \right|\le d}{\lambda }_{\alpha }{y}^{\alpha }$ is the polynomial in ${ℝ}^n$ with real coefficients $\lambda :={({\lambda }_{\alpha })}_{2\le \left|\alpha \right|\le d}$. Under the assumption that the kernel function $K$ satisfies an $L^r$-Hörmander condition with , the authors show that ${\mathcal{𝒯}}^{\ast }$ is bounded on the weighted Lebesgue spaces $L^p(\omega )$ for and $\omega \in A_{p∕r^{\prime }}$, which improves and generalizes the previous results obtained by Stein and Wainger (Math. Res. Lett. 8:5-6 (2001), 789–800), and Ding and Liu (Proc. Amer. Math. Soc. 140:8 (2012), 2739–2751).

The heat transform $H_t$, for positive time , is the convolution on the complex plane with the heat kernel. In the field of analytic function spaces and related operator theory, $H_t$ coincides with the Berezin transform for the Fock space $F_t^2$ induced by the Gaussian measure $e^{-|z{|}^2∕t}dA(z)$. We study fixed-points of $H_t$ and the limit behavior of $H_{t}f$ as $t\to 0^{+}$. Fixed-points of $H_t$ are shown to be closely related to eigenfunctions of the Laplacian corresponding to certain special eigenvalues, while the limit behavior of $H_{t}f$ as $t\to 0^{+}$ depends on certain continuity and oscillation properties of $f$.

The paper is expository, although it contains a few new results. In particular, the main results about fixed-points of $H_t$ are known, but we present a completely new proof here.

We consider the degenerate and singular oscillatory integral operator with a singular kernel which is not a Calderón–Zygmund kernel and satisfies suitable size and derivative conditions related to a real parameter $\mu$. For any given homogeneous polynomial phases, except monomial phases, of degree $n$, we give the range of $p$ for which the sharp decay rate $-\frac{1-\mu }{n}$ on $L^2$ spaces can be preserved on $L^p$ spaces.

This paper presents several families of anisotropic interpolatory subdivisions satisfying the optimal reproduction property with the dilation matrix $\text{diag}(3,2)$. Such schemes are obtained based on the discussion of the relationship between sum rules and polynomial generation for anisotropic schemes. The corresponding masks are symmetric about the origin and imply that, with a relaxed requirement on symmetry, different families of similar schemes with the optimal reproduction property can be obtained. We also show that, given the unimodular matrices $\mathrm{\Theta },{\mathrm{\Theta }}^{\prime }$, the interpolatory schemes with the dilation matrix $\mathrm{\Theta }\text{diag}(3,2){\mathrm{\Theta }}^{\prime }$ and the ones with the dilation matrix $\text{diag}(3,2)$ have the same reproduction property. The convergence of these new schemes is analyzed based on the convergence analysis in the isotropic case. Some numerical examples are given to illustrate the performance of these new schemes.