Planar nearrings, like most of the subjects in finite geometries, can be applied for the construction of block designs. In this article we introduce the ideas of constructing simple BIBDs (balanced incomplete block designs) from field-generated (or nearfield-generated) planar nearrings. Further investigation reveals that there are strong connections between these kinds of constructions and the action of a sharply 2-transitive group on a set. We next explore the structures of the constructions. The theory is derived from finite fields directly. The main point is on finding a subset $S$ (called generating block) of the field $F$ with respect to the given stabilizer $Stab_{F^*}(S)$. A big portion of simple BIBDs with various parameters can be obtained in this way; many simple BIBDs with the same parameters appear. We classify the constructed BIBDs according to the types of the respective generating blocks.

In this paper, we prove that the normally uniform-hyperbolic lower dimensional invariant tori of the un-perturbed system will persist under small perturbations. The proof is based on the theory of exponentially dichotomous linear systems and an improved KAM machinery adapted for the perturbations of angle dependent unperturbed parts.

In this paper we prove the existence of coincidence points and common fixed points for a large class of noncommuting discontinuous contractive type mappings in cone metric spaces. These results generalize, extend and unify several well-known recent related results in literature.

In this paper, we consider approximate controllability for nonlinear deterministic and stochastic systems with resolvent operators and unbounded delay. We study the problem of approximate controllability of deterministic nonlinear differential equations with impulsive terms, resolvent operators and unbounded delay. Next, approximate controllability results are being established for a class of nonlinear stochastic differential equations with resolvent operators in a real separable Hilbert spaces. By using the resolvent operators and fixed point technique, sufficient conditions have been formulated and proved. In this paper, we prove the approximate controllability of nonlinear deterministic and stochastic control systems under the assumption that the corresponding linear system is approximately controllable. Examples are presented to illustrate the utility and applicability of the proposed method.

In the present paper, we prove that the classical sets $\ell_{\infty}(F)$, $c(F)$, $c_0(F)$ and $\ell_p(F)$ of sequences of fuzzy numbers are normed quasilinear spaces and the $\beta−$, $\alpha−$duals of the set $\ell_1(F)$ is the set $\ell_{\infty}(F)$. Besides this, we show that $\ell_{\infty}(F)$ and $c(F)$ are normed quasialgebras and an operator defined by an infinite matrix belonging to the class $(\ell_{\infty}(F) : \ell_{\infty}(F))$ is bounded and quasilinear. Finally, as an application, we characterize the class $(\ell_1(F) : \ell_p(F))$ of infinite matrices of fuzzy numbers and establish the perfectness of the spaces $\ell_{\infty}(F)$ and $\ell_1(F)$.

We study a learning algorithm for regression. The algorithm is a regularization scheme with $\ell^1$ regularizer stated in a hypothesis space trained from data or samples by a nonsymmetric kernel. The data dependent nature of the algorithm leads to an extra error term called hypothesis error, which is essentially different from regularization schemes with data independent hypothesis spaces. By dealing with regularization error, sample error and hypothesis error, we estimate the total error in terms of properties of the kernel, the input space, the marginal distribution, and the regression function of the regression problem. Learning rates are derived by choosing suitable values of the regularization parameter. An improved error decomposition approach is used in our data dependent setting.

We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n \geq 2$ and $1 \lt p \lt \infty$, it is shown that $\ell^n_{\infty}$ is representable in a Banach space $X$ if and only if it is representable in the Lebesgue-Bochner $L_p(X)$. New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice $E$ is uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is uniformly convex and that a Köthe function space $E$ is upper locally uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is midpoint locally uniformly convex.

We study a reaction-diffusion system of activator-inhibitor type. Variational and ordered methods are used to obtain the existence of steady states and standing pulses. It will be seen that the diffusion rates seem to play important roles in the existence of standing pulses.

In this note we prove sharp weak type estimates for lacunary maximal operators of Riesz means associated with homogeneous functions on $H^p$ spaces, $0 \lt p \leq 1$.

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 an implicit viscosity approximation method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a pseudocontractive mapping in a Hilbert space. Then, we prove a strong convergence theorem which is the improvements and development of Takahashi and Takahashi’s (2008) corresponding result. Using this theorem, we prove three new strong convergence theorems in fixed point problems, variational inequalities and equilibrium problems.

Starting from a minimal system of the ideal of centro-affine covariants of a polynomial differential system, we develop an algorithmic method to reduce the polynomial decomposition of a given centro-affine covariant of this system to a linear decomposition by constructing a matrix whose size depends on the type of the given covariant. This method avoids the Aronhold symbolic calculation and offers new means to calculate syzygies and can be used to describe the algebra of the centro-affine covariants. We also give many examples in the case where the system is a planar polynomial quadratic differential system.

Necessary Lagrangian conditions for a constrained minimum become sufficient under generalized convex assumptions, in particular invex, and duality results follow. Many classes of vector functions with properties related to invex have been studied, but It has not been clear how far these classes are distinct. Various inclusions between these classes are now established Some modifications of invex can be regarded as perturbations of invex. There is a stability criterion for when the invex property is preserved under small perturbations. Some results extend to nondifferentiable (Lipschitz) functions.

Let $\mathbb{F}_q$ be a field with $q$ elements of characteristic $p$ and $\mathbb{F}_q((X^{−1}))$ be the field of formal power series over $\mathbb{F}_q$. Let $f$ be a quadratic formal power series of continued fraction expansion $[b_0; b_1, \ldots, b_s, \overline{a_1, \ldots, a_t}]$, we denote by $t = \operatorname{Per}(f)$ the period length of the partial quotients of $f$. The aim of this paper is to study the continued fraction expansion of $Af$ where $A$ is a polynomial $\in \mathbb{F}_q[X]$. In particular we study the asymptotic behavior of the functions \[ S(N,n) = \sup_{\operatorname{deg} A = N} \sup_{f \in \Lambda_{n}} \operatorname{Per}(Af) \quad \textrm{and} \quad R(N) = \sup_{n \geq 1} \frac{S(N,n)}{n}, \] where $\Lambda_{n}$ is the set of quadratic formal power series of period $n$ in $\mathbb{F}_q((X^{−1}))$.

We study hypersurfaces either in the sphere $\mathbb{S}^{n+1}$ or in the hyperbolic space $\mathbb{H}^{n+1}$ whose position vector $x$ satisfies the condition $L_k x = Ax + b$, where $L_k$ is the linearized operator of the $(k+1)$-th mean curvature of the hypersurface for a fixed $k = 0, \ldots, n−1$, $A \in \mathbb{R}^{(n+2) \times (n+2)}$ is a constant matrix and $b \in \mathbb{R}^{n+2}$ is a constant vector. For every $k$, we prove that when $A$ is self-adjoint and $b=0$, the only hypersurfaces satisfying that condition are hypersurfaces with zero $(k+1)$-th mean curvature and constant $k$-th mean curvature, and open pieces of standard Riemannian products of the form $\mathbb{S}^m(\sqrt{1-r^2}) \times \mathbb{S}^{n-m}(r) \subset \mathbb{S}^{n+1}$, with $0 \lt r \lt 1$, and $\mathbb{H}^m(−\sqrt{1+r^2}) \times \mathbb{S}^{n-m}(r) \subset \mathbb{H}^{n+1}$, with $r \gt 0$. If $H_k$ is constant, we also obtain a classification result for the case where $b \neq 0$.

Let $\alpha \geq 0$, and $C$ be a bounded linear injection on a complex Banach space $X$. We first show that if $A$ generates an exponentially bounded nondegenerate $\alpha$-times integrated $C$-semigroup $S_{\alpha}(\cdot)$ on $X$, $B$ is a bounded linear operator on $\overline{D(A)}$ such that $BC = CB$ on $\overline{D(A)}$ and $BA \subset AB$, then $A+B$ generates an exponentially bounded nondegenerate $\alpha$-times integrated $C$-semigroup $T_{\alpha}(\cdot)$ on $X$. Moreover, $T_{\alpha}(\cdot)$ is also exponentially Lipschitz continuous or norm continuous if $S_{\alpha}(\cdot)$ is. We show that the exponential boundedness of $T_{\alpha}(\cdot)$ can be deleted and $\alpha$-times integrated $C$-semigroups can be extended to the context of local $\alpha$-times integrated $C$-semigroups when $R(C) \subset \overline{D(A)}$ and $BS_{\alpha}(\cdot) = S_{\alpha}(\cdot)B$ on $\overline{D(A)}$ both are added. Moreover, $T_{\alpha}(\cdot)$ is also locally Lipschitz continuous or norm continuous if $S_{\alpha}(\cdot)$ is. We show that $A+B$ generates a nondegenerate local $\alpha$-times integrated $C$-semigroup $T_{\alpha}(\cdot)$ on $X$ if $A$ generates a nondegenerate local $\alpha$-times integrated $C$-semigroup $S_{\alpha}(\cdot)$ on $X$ and $B$ is a bounded linear operator on $X$ such that either $BC = CB$, $BS_{\alpha} = S_{\alpha}B$ on $X$; or $BC = CB$ on $\overline{D(A)}$ and $BA \subset AB$. Moreover, $T_{\alpha}(\cdot)$ is also locally Lipschitz continuous, (norm continuous, exponentially bounded or exponentially Lipschitz continuous) if $S_{\alpha}(\cdot)$ is.

We introduce and study $M$-outer derivations and automorphisms of graded rings. We prove a version of the Chevalley-Jacobson density theorem for graded rings with such derivations and automorphisms.

In this paper, we establish some fixed point theorems for the generalized $\Psi$-set contraction mapping on an abstract convex space, which need not to be a compact map.

Let $\alpha$ be a nonnegative number, $C : X \to X$ a bounded linear injection on a Banach space $X$ and $A : D(A) \subset X \to X$ a closed linear operator in $X$ which satisfies $C^{−1}AC = A$ and may not be densely defined. We prove some equivalence relations between the generation of a local $\alpha$-times integrated $C$-cosine function on $X$ with generator $A$ and the uniqueness existence of weak solutions of the abstract Cauchy problem: \[ \operatorname{ACP}_{2}(A,f,x,y) \qquad \begin{cases} u''(t) = Au(t) + f(t) \quad \textrm{for } t \in (0,T_0), \\ u(0) = x, \; u'(0) = y, \end{cases} \] where $x,y \in X$ are given and $f$ is an $X$-valued function defined on a subset of $\mathbb{R}$.

The initial boundary value problem for an integro-differential equation with nonlinear damping in a bounded domain is considered. The local existence and blow-up of solutions with positive initial energy are discussed under some conditions. Estimates of the lifespan of solutions are also given.

The aim of this paper is to establish the existence of three solutions for a two-point boundary value problem with the Sturm-Liouville equation having discontinuous nonlinearities. The approach is based on the critical point theory for non-differentiable functions.

In this paper, we introduce two algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Furthermore, we prove that the proposed algorithms converge strongly to a solution of the minimization problem of finding $x^* \in \Gamma$ such that $\left\| x^* \right\| = \min_{x \in \Gamma} \|x\|$ where $\Gamma$ stands for the intersection set of the solution set of the equilibrium problem and the fixed points set of a nonexpansive mapping.

In this paper, we show that the set of deterministic volatility [10] term structure models with finite dimensional realizations (fdrs) considered in [2] can be identified with an open subset of a Euclidean space, and hence be equipped with the topological and analytical properties of the latter. In particular, the notions of distance, and differentiability of functions defined on this set, can be defined which have important implications for parameter estimation and risk analysis. It is also shown that Lie algebras, which play a key role in the characterization of term structure models with fdrs in [3] and [7], do not separate, and are hence unable to parameterize, these models.

This paper is mainly devoted to applications of modern tools of variational analysis and generalized differentiation to the metric regularity of parametric generalized inequality systems in infinite-dimensional spaces. The basic tools of our analysis involve the Mordukhovich normal coderivatives of set-valued mappings, the limiting subgradient estimate for the marginal functions, and the Ekeland variational principle. Using these tools, we establish new sufficient conditions for the metric regularity of parametric generalized inequality systems. Our results extend the corresponding results in [23] which established some pointbased sufficient conditions for the metric regularity in the Robinson’s sense of implicit multifunctions in finite-dimensional setting.