In this article, we study about the escaping rate of the Sinai-Ruelle-Bowen (SRB) measure through holes of positive measure constructed in the Julia set of hyperbolic rational maps. An explicit formula for the escaping rate through holes of different sizes is obtained and the dependence of this rate on the position of the hole is discussed. Later, the Hausdorff dimension of the survivor set is also computed. For an easier and better understanding, the simple quadratic polynomial restricted on the unit circle is studied.

An attempt has been made to investigate the thermal convection of a heterogeneous Walters B' viscoelastic fluid layer through porous medium under linear stability theory. It is found that medium permeability and viscoelastic parameter have destabilizing effect while density distribution and Prandtl number have stabilizing effect on the fluid layer. The sufficient conditions depending upon the monotonic behaviour of $f\left(z\right)\; \left[\frac{df}{dz} \; \rangle \; 0\; or\; \lt \; 0\right]$ for the non-existence of overstability are also derived. It is shown that the sufficient conditions for the validity of principle of exchange of stabilities for the present problem are $\frac{df}{dz} \lt 0$ and $F\lt \frac{P_{1} }{\varepsilon } $. Further, it has been found that the complex growth rate of an arbitrary oscillatory modes lies inside a circle in the $\sigma _{r} -\sigma _{i} $ plane, whose centre is at the origin and radius is $\frac{a}{p_{1} } \sqrt{\frac{\left|R_{2} \right|P_{1} }{E} }$.

In this paper we deal with sequence spaces inclusion equations (SSIE), which are determined by an inclusion where each term is a sum or a sum of products of sets of the form $\chi_{a}\left( T\right) $ and $\chi_{f\left( x\right) }\left( T\right) $ where $f$ map $U^{+}$ to itself, and $\chi \in \left\{ \mathbf{s},\mathbf{s}^{0},\mathbf{s}^{\left( c\right) }\right\} $, the sequence $x$ is the unknown and $T$ is a given triangle. Here we give characterizations of the (SSIE) $\chi_{x}\left( B\left( r,s\right) \right) \subset \chi_{x}\left( B\left( r',s'\right) \right) $ and of the (SSE) $\chi_{x}\left( B\left( r,s\right) \right) =\chi_{x}\left( B\left( r's'\right) \right) $, where $\chi=s,s^{0}$, or $s^{\left( c\right) }$ and $B\left( r,s\right) $ is the generalized operator of first difference defined by $B\left( r,s\right) _{n}y=ry_{n}+sy_{n-1}$ for all $n\geq 2$ and $B\left( r,s\right) _{1}y_{1}=ry_{1}$. We give an application to the spectrum of $B\left( r,s\right) $ considered as an operator from $\chi_{x}$ to itself, where $\chi=\mathbf{s}$, or $\mathbf{s}^{0}$. Then we apply these results to the solvability of the sequence spaces equation $\chi_{a}+\mathbf{s}_{x}^{(c)}(B(r,s)) = \mathbf{s}_{x}^{(c)}$ where $\chi =\mathbf{s}, \mathbf{s}^{0}$, or $\mathbf{s}^{\left( c\right) }$ and $x$ is the unknown.

Almost periodic functions with values in Banach spaces, and even more generally in Fréchet spaces and $p$-Fréchet spaces, have been investigated by many authors. The purpose of this paper is to investigate the extent to which some results of these authors also hold in the setting of topological vector spaces, not necessarily locally convex or $p$-locally convex. In fact, most of our results (Theorems 3.3, 3.5-3.8) do not require even completeness or metrizability of the range space. We thus extend and unify several known results.

We prove estimates for certain square functions of solutions to divergence form linear parabolic equations. The estimates are related to singularity and mutual absolute continuity of caloric measure with respect to surface measure of non-cylindrical domains. The square functions and the results included here are adapted from works on elliptic equations.

A different application of the familiar integral representation for the modified Bessel function drives to a new Kontorovich-Lebedev-like integral transformation of a general complex index. Mapping and operational properties, a convolution operator and inversion formula are established. Solvability conditions and explicit solutions of the corresponding class of convolution integral equations are exhibited. Finally, as a valuable application it is shown, that the introduced transformation is a key ingredient for solving difference equations of the order $n \in \mathbb{N}$ with constant coefficients in a class of analytic functions in the right half-plane ${\rm Re} z > n.$

We consider an infinite order entire functions $f(z)$, whose zeros $z_1(f), z_2(f),\dots$ are enumerated in the increasing order. For a nondecreasing sequence $\{p_k\}$ of positive numbers, a bound for the sums $$ \sum_{k=1}^j \frac{1}{|z_k(f)|^{p_k}}\;\;(j=1, 2,\dots) $$ is suggested. That bound gives us conditions providing the convergence of the corresponding series.

In this paper we investigate the foundations for analysis in infinitely-many (independent) variables. We give a topological approach to the construction of the regular $\sigma$-finite Kirtadze-Pantsulaia measure on $\mathbb{R}^\infty$ (the usual completion of the Yamasaki-Kharazishvili measure), which is an infinite dimensional version of the classical method of constructing Lebesgue measure on $\mathbb{R}^n$ (see [YA1], [KH] and [KP2]). First we show that von Neumann's theory of infinite tensor product Hilbert spaces already implies that a natural version of Lebesgue measure must exist on $\mathbb{R}^{\infty}$. Using this insight, we define the canonical version of $L^2[\mathbb{R}^{\infty}, \lambda_{\infty}]$, which allows us to construct Lebesgue measure on $\R^{\infty}$ and analogues of Lebesgue and Gaussian measure for every separable Banach space with a Schauder basis. When $\mathcal{H}$ is a Hilbert space and $\lambda_{\mathcal{H}}$ is Lebesgue measure restricted to $\mathcal{H}$, we define sums and products of unbounded operators and the Gaussian density for $L^2[\mathcal{H}, \lambda_{\mathcal{H}}]$. We show that the Fourier transform induces two different versions of the Pontryagin duality theory. An interesting new result is that the character group changes on infinite dimensional spaces when the Fourier transform is treated as an operator. Since our construction provides a complete $\s$-finite measure space, the abstract version of Fubini's theorem allows us to extend Young's inequality to every separable Banach space with a Schauder basis. We also give constructive examples of partial differential operators in infinitely many variables and briefly discuss the famous partial differential equation derived by Phillip Duncan Thompson [PDT], on infinite-dimensional phase space to represent an ensemble of randomly forced two-dimensional viscous flows.