Weaving frames in separable Hilbert spaces have been recently introduced by Bemrose et al. to deal with some problems in distributed signal processing and wireless sensor   networks. Inthiset we study the notion of excess for woven frames and prove that any two frames in a separable Hilbert space that are woven have the same excess. We also show that every frame with a large class of duals is woven provided that its redundant elements have small enough norm. Also, we try to transfer the woven property from frames to their duals and vice versa. Finally, we look at which perturbations of dual frames preserve the woven property

dimension frames.Keyword: Frame, Tight frame, Parseval frame, synthesis operator , Analysis operator, Frame operator, Dual frame, Span, Redundancy, Upper redundancy, Lower Redundancy, orthonormal basis

In this thesis, we investigate the invertibility of multipliers on the frame, especially single g-frame multipliers, modular frames and frame on the Hilbert C*-module. then using some theorems, it is determined when a multiplier is inverted, and most importantly its inverter is an operator in terms of g-frame.

  Let $G$ be a graph with adjacency matrix $A(G)$‎. ‎The nullity $\\eta(G)$ of $G$ is the multiplicity of zero as an eigenvalue of $A(G)$‎, ‎which has received a lot of attention because of its chemical importance‎. ‎Here‎, ‎some upper bounds for $\\eta(G)$ are given‎. ‎For example‎, ‎it is shown that $\\eta(G) \\leq \\frac{(\\Delta‎- ‎2)n‎ + ‎2}{\\Delta‎- ‎1}$ and the equality holds if and only if $G \\cong C_n$ ($n \\equiv 0({\\rm mod} 4)$) or $G \\cong K_{\\Delta‎, ‎\\Delta}$‎. ‎The multiplicity of an arbitrary eigenvalue $\\lambda$ of $A(G)$ is denoted as $m(G‎, ‎\\lambda)$‎. ‎Let $\\theta(G) =|E(G)|-|V(G)|‎ +‎1$ be the cyclomatic number of $G$ and $p(G)$ denote the number of pendant vertices of $G$‎. ‎In this thesis‎, ‎it is proved that for a connected graph $G$‎, ‎$m(G‎, ‎\\lambda) \\leq 2\\theta(G)‎ +‎p(G)$ and the equality holds if and only if $G$ is a cycle $C_n$ and $\\lambda=2\\cos \\frac{2k\\pi}{n}$ with $k=1,2,\\ldots \\lceil\\frac{n}{2}\\rceil-1$‎.

  artial differential equations(PDEs) can model Many physical phenomena.\\\\ The Black–Scholes model is one of the most essential models in finincial mathematics, particularly for American options and European options. Since we can not solve this equation analytically, it seems necessary to provide numerical methods. In this thesis, we consider a compact finite difference method for solving the Black-Scholes equation. Furthermore, we investigate its convergence and stability analysis.\\\\ A higher-order compact finite difference method is introduced for generalised Black–Scholes equation. Moreover, stability analysis, consistent analysis, and convergence analysis of the presented method are investigated. In addition, A consistent and stable numerical scheme for solving a nonlinear option pricing model in illiquid markets is introduced. Finally, some numerical experiments are carried out to illustrate the accuracy and efficiency of mentioned schemes. \\\\

   Let T be a triangular algebra over a commutative ring R, _ be an automorphism of T and Z_(T ) be the _-center of T . Suppose that q : T _ T ??! T is an R-bilinear mapping and that Tq : T ??! T is a trace of q. Our aim is to describe the form of Tq satisfying the commuting condition [Tq; x]_ = 0 (resp. the centralizing condition [Tq; x]_ 2 Z_(T ) for all x 2 T . More precisely, we will consider the question of when Tq satisfying the previouse condition has the so-called proper form. We provide sufficient conditions for each centralizing trace of aribitrary mappings on a triangular algebra to be proper and apply this result to describe the centralizing traces of bilinear mappings on certain 0 (resp. the centralizing condition [Tq; x]_ 2 Z_(T ) for all x 2 T . More precisely, we will consider the question of when Tq satisfying the previouse condition has the so-called proper form. We provide sufficient conditions for each centralizing trace of aribitrary mappings on a triangular algebra to be proper and apply this result to describe the centralizing traces of bilinear mappings on certain classical traingular algebras.

  AbstractIn this thesis we study a host-generalist parasitoid model with Holling II functional response where the generalistparasitoids are introduced to control the invasion of the hosts. It is shown that the model can undergo a sequence ofbifurcations including cusp, focus and elliptic types degenerate Bogdanov-Takens bifurcations of codimension three, and adegenerate Hopf bifurcation of codimension at most two as the parameters vary, and the model exhibits rich dynamics suchas the existence of multiple coexistent steady states, multiple coexistent periodic orbits, homoclinic orbits, etc. Moreover,there exists a critical value for the carrying capacity of generalist parasitoids such that: (i) when the carrying capacity ofthe generalist parasitoids is smaller than the critical value, the invading hosts can always persist despite of the predation bythe generalist parasitoids, i.e., the generalist parasitoids cannot control the invasion of hosts; (ii) when the carrying capacityof the generalist parasitoids is larger than the critical value, the invading hosts either tend to extinction or persist in theform of multiple coexistent steady states or multiple coexistent periodic orbits depending on the initial populations, i.e.,whether the invasion can be stopped and reversed by the generalist parasitoids depends on the initial populatio   (iii) inboth cases, the generalist parasitoids always persist. Numerical simulations are presented to illustrate the theoretical results.-parasitoid model; Bogdanov-Takens bifurcation; Hopf bifurcation; Invasion; PersistencKeywords:Hoste; Extinction.