Let $R$ be a ring with identity and let $\mathcal{M}_R$ be the category of right $R$-modules. In this article, we study some relations between torsion theories and cotorsion theories in $\mathcal{M}_R$. As applications, we give some new characterizations of IF rings with essential flat envelopes.

Inverses of some new inequalities similar to Hilbert’s inequality are established. Our results provide new estimates on these types of inequalities.

By using $q$-calculus, in the present paper we construct Szász type operators in King sense, this meaning the operators preserve the first and the third test function of Bohman-Korovkin theorem. Rate of local and global convergence is obtained in the frame of weighted spaces. The statistical approximation property of our operators is also revealed.

Ruled surfaces with the Gauss map satisfying a partial differential equation which is similar to an eigenvalue problem in a 3-dimensional Euclidean space are studied. Such a Gauss map is said to be of pointwise 1-type, namely, the Gauss map $G$ satisfies $\Delta G = f(G+C)$, where $\Delta$ is the Laplacian operator, $f$ is a non-zero function and $C$ is a constant vector. As a result, such ruled surfaces are completely determined by the function $f$ and the vector $C$ when their Gauss map is of pointwise 1-type. New examples of ruled surfaces called cylinders of an infinite type and rotational ruled surfaces are introduced in this regard.

In this paper we establish the general solution of mixed additive and quadratic functional equation \begin{equation*} f(x+2y) + f(x-2y) + 8f(y) = 2f(x) + 4f(2y) \end{equation*} and investigate the generalized Hyers-Ulam-Rassias stability of this equation in non-Archimedean Banach modules over a unital Banach algebra.

We give a classification of real hypersurfaces in a non-flat complex space form, whose almost contact structure operator, induced from the complex structure of the complex space form, commutes with the Ricci operator and at the same time commutes with the structural Jacobi operator. In particular, we classify real hypersurfaces in 2-dimensional complex projective and hyperbolic spaces satisfying the first commutativity condition.

In this paper, we study the discounted penalty in a perturbed compound Poisson model with two sided jumps. We prove second-order regularity of this function and investigate its asymptotic behavior at infinity. Next, based on Boyarchenko and Levendorskii (2002), we justify an integro-differential equation for the discounted penalty.

Let $X$ be a Banach space and $Z$ a relatively weakly compact subset of $X$. Let $J: Z \rightarrow \mathbb{R}$ be a upper semicontinuous function bounded from above and $p \geq 1$. This paper is concerned with the perturbed optimization problem of finding $z_0 \in Z$ such that $\|x-z_0\|^p + J(z_0) = \sup_{z \in Z} \{\|x-z\|^p + J(z)\}$, which is denoted by $\max_J(x,Z)$. We prove in the present paper that if $X$ is Kadec w.r.t. $Z$, then the set of all $x \in X$ such that the problem $\max_J(x,Z)$ is generalized well-posed is a dense $G_\delta$-subset of $X$. If $X$ is additionally $J$-strictly convex w.r.t. $Z$ and $p \gt 1$, we prove that the set of all $x\in X$ such that the problem $\max_J(x,Z)$ is well-posed is a dense $G_\delta$-subset of $X$.

In this note, we extend and improve the corresponding result of R. E. Bruck [Nonexpansive retracts of Banach spaces, Bull. Amer. Math. Soc. 76 (1970), 384-386].

In the present paper, we study the hypercyclicity of the adjoint of a weighted composition operator acting on some holomorphic function spaces.

We are interested in the Julia set of a semigroup of rational functions with coefficients in $\mathbb{C}_p$ where the semigroup operation is composition. We prove that if a semigroup $G$ is generated by a finite number of rational functions of degree at least two with coefficients in a finite extension of $\mathbb{Q}_p$, and has a nonempty Julia set $\mathcal{J}(G)$, then $\mathcal{J}(G)$ is perfect and has an empty interior.

In the Trotter-Kato approximation theorem for $C_0$-semigroups on Banach spaces, we replace the strong by the weak operator topology and discuss the validity of the relevant implications.

The class $S_{H}(p)$, $0 \leq p \lt 1$, of complex valued, meromorphic harmonic univalent sense-preserving functions in the unit disk $U \backslash \{p\}$ is studied. The functions belong to $S_{H}(p)$ have the expansion $f(z) = \frac{\alpha}{z-p} + \sum_{n=0}^{\infty} c_{n} z^{n} + \overline{\sum_{n=1}^{\infty} d_{n} z^{n}} + A\log |z-p|$ and $\lim_{z \rightarrow p} f(z) = \infty$. Some coefficient estimates, distortion and area theorems are obtained. Sufficient coefficient conditions for a class of meromorphic harmonic univalent sense-preserving functions that are starlike and convex are given.

We consider the $n$-dimensional vectorial Sturm-Liouville problem under some coupled boundary conditions. In some special cases the potential can be reconstructed from one spectrum. We prove that if the spectrum is the same as the spectrum belonging to the zero potential, under periodic boundary conditions or semi-periodic boundary conditions in which case an additional condition on the potential is imposed, then the potential is actually zero.

In this paper, we consider a Mann-type iterative algorithm for nonself strict pseudo-contractions. Weak and strong convergence theorems are established in the framework of Hilbert spaces. The results presented in this paper improve and extend the results announced by many others.

In this paper, we show that the predual spaces of some Lorentz sequence spaces are strictly convex renormable such that any point of the unit sphere is not an extreme point of the unit ball of the dibual. This will partially answer the Question 2. in the paper of Peter Morris.

In this paper we prove Chen inequalities for submanifolds of real space forms endowed with a semi-symmetric metric connection, i.e., relations between the mean curvature associated with the semi-symmetric metric connection, scalar and sectional curvatures, Ricci curvatures and the sectional curvature of the ambient space. The equality cases are considered.

For a fixed analytic univalent function $\phi$, the class of meromorphic univalent $\alpha$-convex functions with respect to $\phi$ is introduced. A representation theorem for functions in the class, as well as a necessary and sufficient condition for functions to belong to the class are obtained. Also we obtain a sharp growth theorem and estimate on a certain coefficient functional for meromorphic starlike functions with respect to $\phi$. Differential subordination and superordination conditions are also obtained for the subclass of meromorphic starlike functions with respect to $\phi$.

Using the modulus of smoothness with Jacobi weights $\omega_{\varphi^\lambda}^2(f,t)_\omega$, the relationship between the derivatives Bernstein operators and the smoothness of the function its approximated in the weighted approximation is characterized, an equivalent theorem between Bernstein operators and the modulus of smoothness with Jacobi weights is established. The corresponding results without weights are generalized. In addition, we obtain the direct theorem in the approximation with Jacobi weights by Bernstein operators.

Let $\mathcal{A}$ be the abelian von Neumann subalgebra $\{M_{f}: f \in L^{\infty}(\mathbb{H},\mu_{r})\}$ of $\mathcal{B}(L^{2}(\mathbb{H},\mu_{r}))$, where $\mathbb{H}$ is the upper half plane and the measure $d\mu_{r} = dx dy/y^{2-r}$. For any integers $r \gt 1$, the linear fractional action of $SL_{2}(\mathbb{R})$ on $\mathbb{H}$ induces a continuous action $\alpha$ of $SL_{2}(\mathbb{R})$ on $\mathcal{A}$. It is shown that the crossed product $\mathcal{R}(\mathcal{A}, \alpha)$ of $\mathcal{A}$ under the action $\alpha$ of $SL_{2}(\mathbb{R})$ is *-isomorphic to $\mathcal{B}(L^{2}(P, 2dx dy/y^{3-2r})) \overline{\otimes} \mathcal{L}_{K}$, where $SL_{2}(\mathbb{R}) = PK$ is the Iwasawa decomposition of $SL_{2}(\mathbb{R})$. Thus $\mathcal{R}(\mathcal{A}, \alpha)$ is of type $\mathrm{I}$.

In this paper we study the existence, uniqueness and iterative approximation of solutions for a few classes of functional equations arising in dynamic programming of multistage decision processes. By using of monotone iterative technique, we also establish the existence and iterative approximation of coincidence solutions for certain kinds of system of functional equations. The results presented in this paper extend, improve and unify the results due to Bhakta and Mitra [7], Chang [10] and Liu [12] and others.

The domination number $\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$, and the independence number $\alpha(G)$ of $G$ is the maximum cardinality among all independent sets of $G$. For any graph $G$, it is easy to see that $\gamma(G) \leq \alpha(G)$. Jou [6] has characterized trees with equal domination numbers and independence numbers. In this paper, we extend the result and present a characterization of connected unicyclic graphs with equal domination numbers and independence numbers.

In this work, we consider the following nonlinear problem \begin{gather*} u_{tt} - M(\| \nabla u(t) \|^{2}_2) \Delta u - \frac{\partial}{\partial t} \Delta u = f(u), \\ u = 0 \text{ \ in } \Gamma_{0} \times (0,T), \\ M(\| \nabla u(t) \|^{2}_2) \frac{\partial u}{\partial \nu} + \frac{\partial}{\partial t} (\frac{\partial u}{\partial \nu}) = -u_{t} \text{ \ in } \Gamma_{1} \times (0,T), \\ u(x,0) = u_{0}(x), \text{ \ } u_{t}(x,0) = u_{1}(x), \end{gather*} in a bounded domain $\Omega$. The existence, asymptotic behavior and nonexistence of solutions are discussed under some conditions.

In this paper we first prove that the forgetful functor from the category of partial entwined modules to that of modules has a right adjoint functor. Then we give a Maschke-type theorem for a partial entwined module.

This paper is concerned with periodic boundary value problems for a kind of first order impulsive differential equations. Some new results related to the existence of solutions are obtained by the ideas involve differential inequalities and fixed point theorems.

Using the fixed point method, we prove the generalized Hyers-Ulam stability of a cubic and quartic functional equation and of an additive and quartic functional equation in Banach spaces.

In this paper, we will show that certain Lozi-like maps have the orbit-shifted shadowing property.

In this paper, sufficient conditions are found for a function involving the gamma function and its reciprocal to be logarithmically completely monotonic. Consequently, a decreasing monotonicity of the function is generalized and a known inequality is extended.

We say that a Banach space $Y$ embeds into a Banach space $X$ smoothly if there is a linear isometry $T$ from $Y$ into $X$ such that every subspace of $TY$ is Hahn-Banach smooth in $X$ (i.e., the ones with unique extension property). In this note, we show that they are exactly (isometric copies of) those subspaces of $X$ having the half-space property.

In this paper, we give a unified generalization of Aczél, Popoviciu and Bellman’s inequalities. The result is then applied to deriving a refinement of Aczél’s inequality and Bellman’s inequality. As consequences, several interesting integral inequalities of Aczél-Popoviciu-Bellman type are obtained.

Let $\mathbb{G}$ be a nilpotent Lie groups equipped with a Hörmander system of vector fields $X = (X_1,\ldots,X_m)$ and $\Delta = \sum_{i=1}^m X_i^2$ be the sub-Laplacians associated with $X$. Let $A = -\Delta + W$ be the Schrödinger operator with the potential function $W$ belongs to the reverse Hölder class $B_q$ for some $q \ge D/2$, where $D$ denote the dimension at infinity. In this paper, we prove that the Riesz transform $\nabla A^{-1/2}$ related to Schrödinger operator $A$ is bounded from the Hardy space $H^1(\mathbb{G})$ to itself.

Let $M$ be a regular Riemann surface with a metric which has constant scalar curvature $\rho$. We give the asymptotic expansion of the sum of the square norm of the sections of the pluricanonical bundles $K_{M}^{m}$. That is, \[ \sum_{i=0}^{d_{m}-1} \|S_{i}(x_{0})\|_{h_{m}}^{2} \sim m(1+\frac{\rho}{2m}) + O\left( e^{-\frac{(\log m)^{2}}{8}} \right), \] where $\{S_{0},\cdots,S_{d_{m}-1}\}$ is an orthonormal basis for $H^{0}(M, K_{M}^{m})$ for sufficiently large $m$.

We extend some major theorems in commutative algebra to the class of modules that are not necessarily finitely generated. The novelty of our extension is that the hypothesis of finite generation over $R$ is replaced by one over $S$, where $R$ and $S$ are commutative Noetherian local rings and there is a local homomorphism $\varphi: R \rightarrow S$. Among the results that we extend are: Intersection Theorem and Intersection dimension formula.

Very recently, Takahashi and Takahashi [S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Analysis 69 (2008) 1025-1033] suggested and analyzed an iterative method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. In this paper, we introduce a general system of generalized equilibria with inverse-strongly monotone mappings in a real Hilbert space. First, this system of generalized equilibria is proven to be equivalent to a fixed point problem of nonexpansive mapping. Second, by using the demi-closedness principle for nonexpansive mappings, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient-like method converges strongly to a solution of this system of generalized equilibria. In addition, utilizing this result, we provide some applications of the considered problem not just giving a pure extension of existing mathematical problems.