Let $n{\geq}2$ be an even integer. We investigate that if an odd mapping f : X ${\rightarrow}$ Y satisfies the following equation $2_{n-2}C_{\frac{n}{2}-1}rf\(\sum\limits^n_{j=1}{\frac{x_j}{r}}\)\;+\;{\sum\limits_{i_k{\in}\{0,1\} \atop {{\sum}^n_{k=1}\;i_k={\frac{n}{2}}}}\;rf\(\sum\limits^n_{i=1}(-1)^{i_k}{\frac{x_i}{r}}\)=2_{n-2}C_{{\frac{n}{2}}-1}\sum\limits^n_{i=1}f(x_i),$ then f : X ${\rightarrow}$ Y is additive, where $r{\in}R$. We also prove the stability in normed group by using shadowing property and the Hyers-Ulam stability of the functional equation in Banach spaces and in Banach modules over unital C-algebras. As an application, we show that every almost linear bijection h : A ${\rightarrow}$ B of unital $C^*$-algebras A and B is a $C^*$-algebra isomorphism when $h(\frac{2^s}{r^s}uy)=h(\frac{2^s}{r^s}u)h(y)$ for all unitaries u ${\in}$ A, all y ${\in}$ A, and s = 0, 1, 2,....

The main object of this paper is to prove several inclusion relations associated with (j, ${\delta}$)-neighborhoods of various subclasses defined by Salagean operator by making use of the familiar concept of neighborhoods of analytic functions. Special cases of some of these inclusion relations are shown to yield known results.

By using the method of differential subordinations, we derive some sufficient conditions for strongly starlike functions associated with a linear operator. All these results presented here are sharp.

In this paper, we investigate the boundary control of semilinear neutral evolution systems with a nonlocal condition by using the Banach fixed point theorem.

We study the dynamics of transcendental entire functions with Siegel disks whose singular values are just two points. One of the two singular values is not only a superattracting fixed point with multiplicity more than two but also an asymptotic value. Another one is a critical value with free dynamics under iterations. We prove that if the multiplicity of the superattracting fixed point is large enough, then the restriction of the transcendental entire function near the Siegel point is a quadratic-like map. Therefore the Siegel disk and its boundary correspond to those of some quadratic polynomial at the level of quasiconformality. As its applications, the logarithmic lift of the above transcendental entire function has a wandering domain whose shape looks like a Siegel disk of a quadratic polynomial.

By using the Lyapunov functional method, stochastic analysis, and LMI (linear matrix inequality) approach, the mean square exponential stability of an equilibrium solution of uncertain stochastic Hopfield neural networks with delayed is presented. The proposed result generalizes and improves previous work. An illustrative example is also given to demonstrate the effectiveness of the proposed result.

This work is concerned with an optimal selling rule for a large position of stock in a market. Selling a large block of stock in a short period typically depresses the market, which would result in a poor filling price. In addition, the large selling intensity makes the regime more likely to be poor state in the market. In this paper, regime switching and depressing terms associated with selling intensity are considered on a set of geometric Brownian models to capture movements of underlying asset. We also consider the liquidation strategy to sell much smaller number of shares in a long period. The goal is to maximize the overall return under state constraints. The corresponding value function with the selling strategy is shown to be a unique viscosity solution to the associated HJB equations. Optimal liquidation rules are characterized by a finite difference method. A numerical example is given to illustrate the result.

An element of a ring is called strongly nil clean provided that it can be written as the sum of an idempotent and a nilpotent element that commute. A ring is strongly nil clean in case each of its elements is strongly nil clean. We investigate, in this article, the strongly nil cleanness of 2${\times}$2 matrices over local rings. For commutative local rings, we characterize strongly nil cleanness in terms of solvability of quadratic equations. The strongly nil cleanness of a single triangular matrix is studied as well.

It was obtained in [5] generalized Lichnerowicz and Obata theorems for Riemannian foliations, which reduce to the results on Riemannian manifolds for the point foliations. Recently in [3], they studied a generalized Obata theorem for Riemannian foliations admitting transversal conformal fields. Each transversal conformal field is a ${\lambda}$-automorphism with ${\lambda}=1-{\frac{2}{q}}$ in the sense of [8]. In the present paper, we extend certain results established in [3] and study Riemannian foliations admitting ${\lambda}$-automorphisms with ${\lambda}{\geq}1-{\frac{2}{q}}$.

In this paper we study the Banach space $L^1$(G) of real valued measurable functions which are integrable with respect to a vector measure G in the sense of D. R. Lewis. First, we investigate conditions for a scalarly integrable function f which guarantee $f{\in}L^1$(G). Next, we give a sufficient condition for a sequence to converge in $L^1$(G). Moreover, for two vector measures F and G with values in the same Banach space, when F can be written as the integral of a function $f{\in}L^1$(G), we show that certain properties of G are inherited to F; for instance, relative compactness or convexity of the range of vector measure. Finally, we give some examples of $L^1$(G) related to the approximation property.

This is a survey on American options. An American option allows its owner the privilege of early exercise, whereas a European option can be exercised only at expiration. Because of this early exercise privilege American option pricing involves an optimal stopping problem; the price of an American option is given as a free boundary value problem associated with a Black-Scholes type partial differential equation. Up until now there is no simple closed-form solution to the problem, but there have been a variety of approaches which contribute to the understanding of the properties of the price and the early exercise boundary. These approaches typically provide numerical or approximate analytic methods to find the price and the boundary. Topics included in this survey are early approaches(trees, finite difference schemes, and quasi-analytic methods), an analytic method of lines and randomization, a homotopy method, analytic approximation of early exercise boundaries, Monte Carlo methods, and relatively recent topics such as model uncertainty, backward stochastic differential equations, and real options. We also provide open problems whose answers are expected to contribute to American option pricing.

In this paper, we prove sufficient conditions for controllability of second order semi-linear neutral impulsive differential inclusions on unbounded domain with infinite delay in Banach spaces using the theory of strongly continuous Cosine families. We shall rely on a fixed point theorem due to Ma for multi-valued maps. The controllability results in infinite dimensional space has been proved without compactness on the family of Cosine operators.

In this paper, we consider the cusum of squares test for the dispersion parameter in stochastic differential equation models. It is shown that the test has a limiting distribution of the sup of a Brownian bridge, unaffected by the drift parameter estimation. A simulation result is provided for illustration.

In this paper we provide examples of pairs of conformally non-equivalent, but topologically equivalent, p-groups $H_1$, $H_2$ < Aut(S), where S is a closed Riemann surface of genus g ${\geq}$ 2, so that $S/H_j$ has genus zero and all its cone points are of order equal to p.

In this paper, we prove the Hyers-Ulam-Rassias stability of the following additive functional inequality: ${\parallel}f(2x)+f(2y)+2f(z){\parallel}\;{\leq}\;{\parallel}2f(x+y+z){\parallel}$ We investigate homomorphisms in proper $CQ^*$-algebras and derivations on proper $CQ^*$-algebras associated with the additive functional inequality (0.1).

The semi-discrete central scheme and central upwind scheme use Runge-Kutta (RK) time discretization. We do the Lax-Wendroff (LW) type time discretization for both schemes. We perform numerical experiments for various problems including two dimensional Riemann problems for Burgers' equation and Euler equations. The results show that the LW time discretization is more efficient in CPU time than the RK time discretization while maintaining the same order of accuracy.