Topological SL(2) Gauge Theory on Conifold

Using a two component $SL(2)$ isospinor formalism, we study the link between conifold $T^{\ast}\mathbb{S}^{3}$ and q-deformed non commutative holomorphic geometry in complex four dimensions. Then, thinking about conifold as a projective complex three dimension hypersurface embedded in non compact $WP^{5}(1,-1,1,-1,1,-1)$ space and using conifold local isometries, we study topological $SL(2)$ gauge theory on $T^{\ast}\mathbb{S}^{3}$ and its reductions to lower dimension sub-manifolds $T^{\ast}\mathbb{S}^{2}$, $T^{\ast}\mathbb{S}^{1}$ and their real slices. Projective symmetry is also used to build a supersymmetric QFT$$ realization of these backgrounds. Extensions for higher dimensions with conifold like properties are explored. \bigskip \textbf{Key words}: Conifold, q-deformation, non commutative complex geometry, topological gauge theory. Nambu like background.Comment: 42 page

Controllability of fractional stochastic neutral functional differential equations driven by fractional Brownian motion with infinite delay

In this paper we study the controllability of fractional neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:1602.0580

Controllability of neutral stochastic functional integro-differential equations driven by fractional Brownian motion

This paper focuses on controllability results of stochastic delay partial functional integro-differential equations perturbed by fractional Brownian motion. Sufficient conditions are established using the theory of resolvent operators combined with a fixed point approach for achieving the required result. An example is provided to illustrate the theory.Comment: 14 page

A covariance equation

Let $\S$ be a commutative semigroup with identity $e$ and let $\Gamma$ be a compact subset in the pointwise convergence topology of the space $\S'$ of all non-zero multiplicative functions on $\S.$ Given a continuous function $F: \Gamma \to \mathbb C$ and a complex regular Borel measure $\mu$ on $\Gamma$ such that $\mu(\Gamma) \not = 0.$ It is shown that $$\mu(\Gamma) \int_{\Gamma} \varrho(s) \overline{\varrho(t)} |F|^2(\varrho) d\mu(\varrho) = \int_{\Gamma} \varrho(s) F(\varrho) d\mu(\varrho) \int_{\Gamma} \overline{\varrho(t) F(\varrho)} d\mu(\varrho)$$ for all $(s, t) \in \S\times \S$ if and only if for some $\gamma \in \Gamma,$ the support of $\mu$ is contained is contained in $\{ F = 0 \} \cup \{\gamma\}$. Several applications of this characterization are derived. In particular, the reduction of our theorem to the semigroup of non-negative integers $(\mathbb N_{0}, +)$ solves a problem posed by El Fallah, Klaja, Kellay, Mashregui and Ransford in a more general context. More consequences of our results are given, some of them illustrate the probabilistic flavor behind the problem studied herein and others establish extremal properties of analytic kernels

Controllability of stochastic impulsive neutral functional differential equations driven by fractional Brownian motion with infinite delay

In this paper we study the controllability results of impulsive neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.Comment: 16 page

Hyperbolic Invariance in Type II Superstrings

We first review aspects of Kac Moody indefinite algebras with particular focus on their hyperbolic subset. Then we present two field theoretical systems where these structures appear as symmetries. The first deals with complete classification of $\mathcal{N}=2$ supersymmetric CFT$_{4}$s and the second concerns the building of hyperbolic quiver gauge theories embedded in type IIB superstring compactification of Calabi-Yau threefolds. We show, amongst others, that $\mathcal{N}=2$ CFT$_{4}$s are classified by Vinberg theorem and hyperbolic structure is carried by the axion modulus. Keywords: Classification of KM algebras, Indefinite KM sector and Hyperbolic subset, Quiver gauge theories embedded in type II superstrings.Comment: 18 pages, 6 figures, Talk given at IPM String School and Workshop, ISS2005, January 5-14, 2005, Qeshm Island, IRA

Tetrahedron in F-theory Compactification

Complex tetrahedral surface $\mathcal{T}$ is a non planar projective surface that is generated by four intersecting complex projective planes $CP^{2}$. In this paper, we study the family $\{\mathcal{T}_{m}\}$ of blow ups of $\mathcal{T}$ and exhibit the link of these $\mathcal{T}_{m}$s with the set of del Pezzo surfaces $dP_{n}$ obtained by blowing up n isolated points in the $CP^{2}$. The $\mathcal{T}_{m}$s are toric surfaces exhibiting a $U(1) \times U(1)$ symmetry that may be used to engineer gauge symmetry enhancements in the Beasley-Heckman-Vafa theory. The blown ups of the tetrahedron have toric graphs with faces, edges and vertices where may localize respectively fields in adjoint representations, chiral matter and Yukawa tri-fields couplings needed for the engineering of F- theory GUT models building.Comment: 27 pages, 9 figure

On the global attractivity and oscillations in a class of second order difference equations from macroeconomics

New global attractivity criteria are obtained for the second order difference equation $$x_{n+1}=cx_{n}+f(x_{n}-x_{n-1}),\quad n=1, 2, ...$$ via a Lyapunov-like method. Some of these results are sharp and support recent related conjectures. Also, a necessary and sufficient condition for the oscillation of this equation is obtained using comparison with a second order linear difference equation with positive coefficients.Comment: the final version of this paper will be published in Journal of Difference Equations and Application

On backward stochastic differential equations driven by a family of It\^o's processes

We propose to study a new type of Backward stochastic differential equations driven by a family of It\^o's processes. We prove existence and uniqueness of the solution, and investigate stability and comparison theorem

On the 1H\frac{1}{H}-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter H<1/2H < 1/2

In this paper, we study the $\frac{1}{H}$-variation of stochastic divergence integrals $X_t = \int_0^t u_s {\delta}B_s$ with respect to a fractional Brownian motion $B$ with Hurst parameter $H < \frac{1}{2}$. Under suitable assumptions on the process u, we prove that the $\frac{1}{H}$-variation of $X$ exists in $L^1({\Omega})$ and is equal to $e_H \int_0^T|u_s|^H ds$, where $e_H = \mathbb{E}|B_1|^H$. In the second part of the paper, we establish an integral representation for the fractional Bessel Process $\|B_t\|$, where $B_t$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H < \frac{1}{2}$. Using a multidimensional version of the result on the $\frac{1}{H}$-variation of divergence integrals, we prove that if $2dH^2 > 1$, then the divergence integral in the integral representation of the fractional Bessel process has a $\frac{1}{H}$-variation equals to a multiple of the Lebesgue measure.Comment: 29 page