Kostia Beidar published more than 120 research papers and solved many well-known problems. Our goal is to mention just some of his brilliant results in ring and nearring theory, and also a brief history of his life.

We introduce the notion of a pair of simultaneously asymptotically quasi-nonexpansive type mappings and prove a general strong convergence theorem of the iteration scheme with errors for a pair of simultaneously asymptotically quasi-nonexpansive type mappings in Banach spaces. The result of this paper is an extension and an improvement of the corresponding well known results.

Recently, Chen established a general sharp inequality for warped products in real space forms. As applications, he obtained obstructions to minimal isometric immersions of warped products into real space forms. Afterwards, Matsumoto and one of the present authors proved the Sasakian version of this inequality.

In the present paper, we obtain sharp estimates for the warping function in terms of the mean curvature for warped products isometrically immersed in Kenmotsu space forms. Some applications are derived.

Under the assumption that $\mu$ is a non-doubling measure on $\mathbb{R}^d$, the authors obtain the boundedness of commutators generated by Calderón-Zygmund operators or fractional integrals with Lipschitz functions in the Lebesgue space and the Hardy space.

In the case of the equally spaced fixed design nonparametric regression, the local constant $M$-smoother (LCM) with local maximizing is proposed by Chu, Glad, Godtliebsen, and Marron (1998) to correct for the effect of discontinuity on the kernel regression estimator. It has the interesting property of jump-preserving. However, in the jump region, it is inconsistent when the magnitude of the noise is larger than the size of the jump in the regression. To adjust for this drawback to the ordinary LCM, we propose to construct the LCM with global maximizing instead of local maximizing as well as with binned data instead of original data. Our proposed estimator is analyzed by the asymptotic mean square error. Both binning and global maximizing have no effect on the asymptotic mean square error of the ordinary LCM in the smooth region, but have an effect on improving the inconsistency of the ordinary LCM in the jump region. Simulation studies demonstrate that the regression function estimate produced by our modified LCM is better than those by alternatives, in the sense of yielding smaller sample mean integrated square error, showing more accurately the location of jump point, and having smoother appearance.

By using the Durfee rectangle and Durfee dissection we give combinatorial interpretations for Göllnitz-Gordon type identities. Also, we give a generalization of an identity due to G. E. Andrews.

In this paper, a generalized version of the famous KKM theorem is obtained by using the concept of generalized KKM mappings introduced by Chang and Zhang [5]. By employing our generalized KKM theorem, we obtain a generalized minimax inequality which includes several existing ones as special cases. Further, by applying our generalized minimax inequality we establish an existence result for the saddle-point problem under general setting. Finally, we also derive some existence results for generalized equilibrium problems and generalized variational inequalities.

In this paper we establish some results concerning the existence of nonzero nonnegative and nonzero nonpositive solutions for a Dirichlet problem via variational methods. In particular, a general variational principle of B.Ricceri is applied.

Explicit formula for the fundamental class of functions $\left( Z,\widetilde{Z},\Phi \right)$, introduced by J. Jorgenson and S. Lang, is given a new form valid for a more general fudge factor $\Phi$. This is done for a larger class of test functions of generalized bounded variation.

The notion of partial group $C^{*}$-algebra of a discrete group introduced by R. Exel in [3] is generalized to an idempotent unital inverse semigroup, and the partial inverse semigroup $C^{*}$-algebra is defined. By using the algebras of multipliers of ideals of an associative algebra, we can prove some theorem in the $C^{*}$-algebra context without using the approximate identity.

Let $G(V,E)$ be a graph of order $p$. Denote by $\sigma(G)$, $\sigma_1(G)$, $\alpha_T(G)$ and $\beta_T(G)$ the dominating number, the edge dominating number, the total covering number and the total independence number of $G(V,E)$, respectively. Let $\overline{G}$ denote the complement graph of $G$. This paper establishes some relations among $\sigma(G)$, $\beta_T(G)$, $\alpha_T(G)$, $\sigma_1(G)$, $\sigma(\overline{G})$ and $\sigma_1(\overline{G})$.

In this paper, the following fourth-order three-point boundary value problem with $p$-Laplacian operator is studied: $$ \left\{ \begin{array}{ll} (\phi_p(u''(t)))'' = a(t) f(u(t)), \quad t \in (0,1), \\ u(0) = \xi u(1), \; u'(1) = \eta u'(0), \\ u''(0) = \alpha_1 u''(\delta), \; u''(1) = \beta_1 u''(\delta), \end{array} \right. $$ where $\alpha_1, \; \beta_1 \ge 0$, $\xi \neq 1$, $\eta \neq 1$, $0 \lt \delta \lt 1$ and $\phi_p(z) = |z|^{p-2} z$ for $p \gt 1$. We impose growth conditions on $f$ which guarantee the existence of at least three positive solutions for the problem.

We consider the number of vertex independent sets $i(G)$. In general, the problem of determining the value of $i(G)$ is $NP$-complete. We present several upper and lower bounds for $i(G)$ in terms of order, size or independence number. We obtain improved bounds for $i(G)$ on restricted graph classes such as the bipartite graphs, unicyclic graphs, regular graphs and claw-free graphs.

Let $R$ be a ring and $M$ a right $R$-module. It is shown that: (1) $M$ is Artinian if and only if $M$ is a GAS-module and satisfies DCC on generalized supplement submodules and on small submodules; (2) if $M$ satisfies ACC on small submodules, then $M$ is a lifting module if and only if $M$ is a GAS-module and every generalized supplement submodule is a direct summand of $M$ if and only if $M$ satisfies $(P^{*})$; (3) $R$ is semilocal if and only if every cyclic module is a GWS-module.

Boundedness of components of the Fatou sets of iteration of transcendental entire or meromorphic functions are investigated in this paper.

In this paper, we introduce and study a new class of completely generalized nonlinear mixed quasi-variational inequalities. Using the resolvent operator technique for maximal monotone operators, we construct a perturbed three-step iterative algorithm with errors for solving this kind of completely generalized nonlinear mixed quasi-variational inequalities. Furthermore, we establish a few existence and uniqueness results of solutions for the completely generalized nonlinear mixed quasi-variational inequality involving relaxed Lipschitz, generalized pseudo-contractive and strongly monotone mappings and prove some convergence results of the iterative sequence generated by the perturbed three-step iterative algorithm with errors.

The complete elliptic integral of the first kind $K(k)$ is defined for $0 \lt k \lt 1$ by \[ K(k) := \int_{0}^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}. \] The real number $k$ is called the modulus of the elliptic integral. The complementary modulus is $k' = (1-k^2)^{\frac{1}{2}}$ ($0 \lt k' \lt 1$). Let $\lambda$ be a positive integer. The equation \[ K(k') = \sqrt{\lambda} K(k) \] defines a unique real number $k(\lambda)$ ($0 \lt k(\lambda) \lt 1$) called the singular modulus of $K(k)$. Let $H(D)$ denote the form class group of discriminant $D$. Let $d$ be the discriminant $-4 \lambda$. Using some recent results of the authors on values of the Dedekind eta function at quadratic irrationalities, a formula is given for the singular modulus $k(\lambda)$ in terms of quantities depending upon $H(4d)$ if $\lambda \equiv 0 \pmod{2}$; $H(d)$ and $H(4d)$ if $\lambda \equiv 1 \pmod{4}$; $H(d/4)$ and $H(4d)$ if $\lambda \equiv 3 \pmod{4}$. Similarly a formula is given for the complete elliptic integral $K[\sqrt{\lambda}] := K(k(\lambda))$ in terms of quantities depending upon $H(d)$ and $H(4d)$ if $\lambda \equiv 0 \pmod{2}$; $H(d)$ if $\lambda \equiv 1 \pmod{4}$; $H(d/4)$ and $H(d)$ if $\lambda \equiv 3 \pmod{4}$. As an example the complete elliptic integral $K[\sqrt{17}]$ is determined explicitly in terms of gamma values.

In this paper we give a new generalization of the Pell numbers in matrix representation. Also we extend the matrix representation and we show that the sums of the generalized order-$k$ Pell numbers could be derived directly using this representation. Further we present some identities, the generalized Binet formula and combinatorial representation of the generalized order-$k$ Pell numbers.

In this paper, we introduce a class of functions $(k,A,B,\alpha)-UCV$ which is convex in the unit disk. We give some results for the class $(k,A,B,\alpha)-UCV$, integral operators and radius of $k$-uniform convexity. Further, the proofs of distortion theorems for fractional calculus for functions $(k,A,B,\alpha)-UCV$ is given.

Convexity assumptions for fractional programming of variational type are relaxed to generalized invexity. The sufficient optimality conditions are employed to construct a mixed dual programming problem. It will involve the Wolfe type dual and Mond-Weir type dual as its special situations. Several duality theorems concerning weak, strong, and strict converse duality under the framework in mixed type dual form are proved.