In this paper, we show that in order for a proper compact subset $K$ of plane ${\mathrm{ℝ}}^2$ to be convex, it is necessary and sufficient that inverse norm function be subharmonic.

This paper presents a new technique for solving linear Volterra integro-differential equations with boundary conditions. The method is based on the blending of the Chebyshev spectral methods. The application of the proposed method leads the Volterra integro-differential equation to a system of algebraic equations that are easy to solve. Some examples are introduced and the obtained results are compared with exact solution as well as the methods that reported in the literature to illustrate the effectiveness and accuracy of the method. The results demonstrate that there is congruence between the numerical and the exact results to a high order of accuracy. Tables were generated to verify the accuracy convergence of the method and error. Figures are presented to show the excellent agreement between the results of this study and the results from literature.

In this paper, we introduce a proximal point algorithm for approximating a common solution of finite family of convex minimization problems and fixed point problems for $k$-demicontractive mappings in complete CAT(0) spaces. We prove a strong convergence result and obtain other consequence results which generalize and extend some recent results in the literature. We further provide a numerical example to illustrate the convergence behaviour of the sequence generated by our algorithm.

In this paper, we consider a JS metric space endowed with convexity structure, which will allow us to examine and study convergence of Mann iteration and Ishikawa iteration for Banach type and Chatterjea type contractions defined on JS metric space.

This paper proves a common fixed point theorem for single-valued and multivalued mappings by using ternary relation in $G$-metric spaces. Henceforth, results obtained will be verified with the help of illustrative examples. Also, we demonstrate the results with an application.

It is known that injective (complex or real) $W*$-algebras with particular factors have been studied well enough. In the arbitrary cases, i.e., in noninjective case, to investigate (up to isomorphism) $W*$-algebras is hard enough, in particular, there exist continuum pairwise nonisomorphic noninjective factors of type II. Therefore, it seems interesting to study maximal injective $W*$-subalgebras and subfactors. On the other hand, the study of maximal injective $W*$-subalgebras and subfactors is also related to the well-known von Neumann’s bicommutant theorem. In the complex case, such subalgebras were investigated by S. Popa, L. Ge, R. Kadison, J. Fang, and J. Shen. In recent years, studies have also begun in the real case. Let us briefly recall the relevance of considering the real case. It is known that in the works of D. Topping and E. Stormer, it was shown that the study of JW-algebras (nonassociative real analogues of von Neumann algebras) of types II and III is essentially reduced to the study of real $W*$-algebras of the corresponding type. It turned out that the structure of real $W*$-algebras, generally speaking, differs essentially in the complex case. For example, in the finite-dimensional case, in addition to complex and real matrix algebras, quaternions also arise, i.e., matrix algebras over quaternions. In the infinite-dimensional case, it is proved that there exist, up to isomorphism, two real injective factors of type ${\mathrm{I}\mathrm{I}\mathrm{I}}_{\lambda }(0<\lambda <1)$, and a countable number of pairwise nonisomorphic real injective factors of type ${\text{III}}_0$, whose enveloping (complex) $W*$-factors are isomorphic, is constructed. It follows from the above that the study of the real analogue of problems in the theory of operator algebras is topical. Moreover, the real analogue is a generalization of the complex case, since the class of real linear operators is much wider than the class of complex linear operators. In this paper, the maximal injective real $W*$-subalgebras of real $W*$-algebras or real factors are investigated. For real factors $Q\subset R$, it is proven that if $Q+iQ$ is a maximal injective $W*$-subalgebra in $R+iR$, then $Q$ also is a maximal injective real $W*$-subalgebra in $R$. The converse is proved in the case “ ${\text{II}}_1$”-factors, that is, it is shown that if $R$ is a real factor of type ${\text{II}}_1$, then the maximal injectivity of $Q$ implies the maximal injectivity of $Q+iQ$. Moreover, it is proven that a maximal injective real subfactor $Q$ of a real factor $R$ is a maximal injective real $W*$-subalgebra in $R$ if and only if $Q$ is irreducible in $R$, i.e., $Q\prime \cap R=ℝ𝕀$ where $𝕀$ is the unit. The “splitting theorem” of Ge-Kadison in the real case is also proven, namely, if $R_1$ is a finite real factor, $R_2$ is a finite real $W*$-algebra, and $R$ is a real $W*$-subalgebra of $R_1\overline{\otimes }{R}_2$ containing $R_1\overline{\otimes }ℝ𝕀$, then there is some real $W*$-subalgebra $Q_2\subset R_2$ such that $R=R_1\overline{\otimes }{Q}_2$. Moreover, it is given some affirmative answers to the question of S. Popa for the real case.

The ill-posed Helmholtz equation with inhomogeneous boundary deflection in a Hilbert space is regularized using the divergence regularization method (DRM). The DRM includes a positive integer scaler that homogenizes the inhomogeneous boundary deflection in the Helmholtz equation’s Cauchy issue. This guarantees the existence and uniqueness of the equation’s solution. To reestablish the stability of the regularized Helmholtz equation and regularized Cauchy boundary conditions, the DRM uses its regularization term $\left(1+{\alpha }^{2m}\right)e^m$, where $\alpha >0$ is the regularization parameter. As a result, DRM restores all three Hadamard requirements for well-posedness.

The finite element approach was utilized in this study to solve numerically the two-dimensional time-dependent heat transfer equation coupled with the Darcy flow. The Picard-Lindelöf Theorem was used to prove the existence and uniqueness of the solution. The prior and posterior error estimates are then derived for the numerical scheme. Numerical examples were provided to show the effectiveness of the theoretical results. The essential code development in this study was done using MATLAB computer simulation.

A two-generator Kleinian group can be naturally associated with a discrete group with the generator $\varphi$ of order two and where This is useful in studying the geometry of the Kleinian groups since will be discrete only if is, and the moduli space of groups is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of . The dimension reduction here is realised by a projection of principal characters of the two-generator Kleinian groups. In applications, it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen’s results on algebraic convergence.

In this paper, a discretization of a three-dimensional fractional-order prey-predator model has been investigated with Holling type III functional response. All its fixed points are determined; also, their local stability is investigated. We extend the discretized system to an optimal control problem to get the optimal harvesting amount. For this, the discrete-time Pontryagin’s maximum principle is used. Finally, numerical simulation results are given to confirm the theoretical outputs as well as to solve the optimality problem.

The Bishop frame or rotation minimizing frame (RMF) is an alternative approach to define a moving frame that is well defined even when the curve has vanished second derivative, and it has been widely used in the areas of computer graphics, engineering, and biology. The main aim of this paper is using the RMF for classification of singularity type of timelike sweeping surface and Bishop spherical Darboux image which is mightily concerning a unit speed spacelike curve with timelike binormal vector in ${\mathbb{E}}_1^3$.

Based on the concept of pseudocomplement, we introduce a new representation of preopenness of $L$-fuzzy sets in $RL$-fuzzy bitopological spaces. The concepts of pairwise $RL$-fuzzy precontinuous and pairwise $RL$-fuzzy preirresolute functions are extended and discussed based on the $\left(i,j\right)$-$RL$-preopen gradation. Further, we follow up with a study of pairwise $RL$-fuzzy precompactness in $RL$-fuzzy bitopological spaces of an $L$-fuzzy set. We find that our paper offers more general results since $RL$-fuzzy bitopology is a generalization of $L$-bitopology, $RL$-bitopology, and $L$-fuzzy topology.

In this work, we introduce a time-like ruled surface in one-parameter hyperbolic dual spherical motions. This provides the ability to derive some formulae of surface theory into line spaces. Then, a time-like Plücker conoid associated with the motion has been obtained, and its kinematic geometry is researched in detail. Consequently, a characterization for a time-like ruled surface to be a constant Disteli-axis is derived and investigated in detail. At last, we have discussed some special cases which lead to some special time-like ruled surfaces such as the time-like helicoids, Lorentzian sphere, and time-like cone.

In this paper, we introduce generalized $\left(\alpha ,\psi \right)$-contraction mappings in the setting of rectangular $b$-metric spaces and established existence and uniqueness of fixed points for the mappings introduced. Our results extend and generalize related fixed point results in the existing literature. We derive some consequences and corollaries from our obtained results. Also, we provide examples in support of our main findings. Furthermore, we determined a solution to an integral equation by applying our obtained results.

Nirenberg proposed a problem as to whether or not a continuous and expansive operator $T:X⟶X$(where $X$ is a Hilbert space) is surjective if $R\left(T\right)°\ne \varnothing$. I shall give a positive answer for the problem provided that $R\left(T\right)°$ is unbounded. For contents related to this paper, the reader is referred to the remarks and the study of Asfaw (2021). The present paper gives a complete answer for the problem that has been open for about 47 years.

In this paper, we first introduce two new notions of uniform convexity on a geodesic space, and we prove their properties. Moreover, we reintroduce a concept of the set-convergence in complete geodesic spaces, and we prove a relation between the metric projections and the convergence of a sequence of sets.

Brucellosis is one of the most serious diseases that wreaks havoc on the production of livestock. Despite various efforts made to curb the spread of brucellosis, the disease remains a major health concern to both humans and animals. In this work, a deterministic model is developed to investigate the transmission dynamics and control of bovine brucellosis in a herd of cattle. The disease-free equilibrium point of the model is shown to be locally asymptotically stable whenever basic reproduction number ${\mathcal{R}}_0\le 1$ and unstable if . Also, the endemic equilibrium point of the model is shown to be locally asymptotically stable whenever and unstable otherwise. Numerical simulations of the model suggest that vaccination is the most efficient single control intervention. Also, the most efficient pair of control interventions is vaccination and culling of seropositive cattle. However, the best way to control bovine brucellosis in cattle is the combination of the three control interventions (vaccination, culling of seropositive cattle, and observation of comprehensive biosecurity protocols).

The study introduced a generalized multiplier operator used as a tool to define and investigate a new class of function, $TS{Q}_{\sigma ,\xi ,\omega ,ϵ}^{\mu ,\lambda ,\eta }\left(\omega ,n\right)$ and its subclass $TS{Q}_{\sigma ,\xi ,\omega ,ϵ}^{\mu ,\lambda ,\eta }\left(\omega ,n;M_i\right)$. Various properties of the class of functions were investigated. The results extend some known results in literature.