We classify N($\kappa$)-contact metric manifolds which satisfy $Z(\xi,\;X)\cdotZ\;=\;0,\;Z(\xi,\;X)\cdotR\;=\;0\;or\;R(\xi,\;X)\cdotZ\;=\;0$.

We discuss a certain geometric property $X_{{\theta},{\gamma}}$ of dual algebras generated by a polynomially bounded operator and property ($\mathbb{A}_{N_0,N_0}$; these are central to the study of $N_{0}\timesN_{0}$-systems of simultaneous equations of weak$^{*}$-continuous linear functionals on a dual algebra. In particular, we prove that if T $\in$ $\mathbb{A}$$^{M}$ satisfies a certain sequential property, then T $\in$ $\mathbb{A}^{M}_{N_0}(H) if and only if the algebra $A_{T}$ has property $X_{0, 1/M}$, which is an improvement of Li-Pearcy theorem in [8].

Contact Homogeneous 3-manifolds are pseudo-symmetric spaces of constant type. All Sasakian 3-manifolds are pseudo-symmetric spaces of constant type.

Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = 1M. Let M be a non-zero multiplication R-module. Then we prove the following: (1) there exists a bijection: N(M)$\bigcap$V(ann$\_{R}$(M))$\rightarrow$Spec$\_{R}$(M) and in particular, there exists a bijection: N(M)$\bigcap$Max(R)$\rightarrow$Max$\_{R}$(M), (2) N(M) $\bigcap$ V(ann$\_{R}$(M)) = Supp(M) $\bigcap$ V(ann$\_{R}$(M)), and (3) for every ideal I of R, The ideal $\theta$(M) = $\sum$$\_{m(Rm :R M) of R has proved useful in studying multiplication modules. We generalize this ideal to prove the following result: Let R be a commutative ring with identity, P $\in$ Spec(R), and M a non-zero R-module satisfying (1) M is a finitely generated multiplication module, (2) PM is a multiplication module, and (3) P$^{n}$M$\neq$P$^{n+1}$ for every positive integer n, then $\bigcap$$^{$\_{n=1}$(P$^{n}$ + ann$\_{R}$(M)) $\in$ V(ann$\_{R}$(M)) = Supp(M) $\subseteq$ N(M).

For weighted sum of a sequence {X, X$\_{n}$, n $\geq$ 1} of identically distributed, negatively orthant dependent random variables such that |r| > 0, has a finite moment generating function, a strong law of large numbers is established.

In this paper, based on large deviation probabilities on Gaussian random vectors, we obtain Strassen's functional LIL for d-dimensional self-similar Gaussian process in Holder norm via estimating large deviation probabilities for d-dimensional self-similar Gaussian process in Holder norm.

On the setting of the upper half-space H of the Eu­clidean n-space, we show the boundedness of weighted Bergman projection for 1 < p < $\infty$ and nonorthogonal projections for 1 $\leq$ p < $\infty$ . Using these results, we show that Bergman norm is equiva­ lent to the normal derivative norms on weighted harmonic Bergman spaces. Finally, we find the dual of b$\_{$^{1}$.

Klein proved that polynomial rings over nil rings of bounded index are also nil of bounded index; while Puczylowski and Smoktunowicz described the nilradical of a power series ring with an indeterminate. We extend these results to those with any set of commuting indeterminates. We also study prime radicals of power series rings over some class of rings containing the case of bounded index, finding some examples which elaborate our arguments; and we prove that R is a PI ring of bounded index then the power series ring R[[X]], with X any set of indeterminates over R, is also a PI ring of bounded index, obtaining the Klein's result for polynomial rings as a corollary.

In this paper, we study averaging properties in Banach spaces using the Brunel-Sucheston's spreading model. We show that the Schlumprecht space S does not have the Banach­Saks property and II is finitely representable in the Schlumprecht space S using the spreading model properties.

In this paper, we define the conditional first variation over Wiener paths in abstract Wiener space and investigate its properties. Using these properties, we also investigate relationships among first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transforms of functions in a Banach algebra which is equivalent to the Fresnel class. Finally, we provide another method evaluating the Fourier-Feynman transform for the product of a function in the Banach algebra with n linear factors.

In this article, we prove the existence of two embedded minimal annuli in a slab which are all bounded by a Jordan convex curve and a straight line.

In this paper, we introduce a discrete time to the model to describe the time between infection of a CD4$^{+}$ T-cells, and the emission of viral particles on a cellular level. We study the effect of the time delay on the stability of the endemically infected equilibrium, criteria are given to ensure that the infected equilibrium is asymptotically stable for all delay. We also obtain the condition for existence of an orbitally asymptotically stable periodic solution.

In this note we give the mapping properties of the Marcinkiewicz integral !-to. at some end spaces. More precisely, we first prove that !-to. is a bounded operator from H$^{1,($\mathbb{R) to H$^{1, ($\mathbb{R). As a corollary of the results above, we obtain again the weak type (1,1) boundedness of $\mu$$_{, but the condition assumed on n is weaker than Stein's condition. Finally, we show that !-to. is bounded from BMO($\mathbb{R) to BMO($\mathbb{R). The results in this note are the extensions of the results obtained by Lee and Rim recently.

Recently, Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. In the present paper, we obtain sharp estimates for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for contact CR-warped products isometrically immersed in Kenmotsu space forms. The equality case is considered. Some applications are derived.