Mohammad Abolghasemi

In this thesis, we focus on the Ekeland’s variational principle and some of its applications to equilibrium and quasi-equilibrium problems and existential results for equilibrium and Minty equilibrium problems. First, we state the Ekeland’s variational principle and then, using it, we present equilibrium forms of the Ekeland’s variational principle for cyclically antimonotone and cyclically monotone functions, and using these equilibrium forms, we present two existence theorems for equilibrium and Minty equilibrium problems, respectively. We also state an existence theorem for the quasiequilibrium problem using the Ekeland’s variational principle. Next, we study the existence of solutions for equilibrium and Minty equilibrium problems and then the relationship between them. It should be noted that with different assumptions such as the Minty lemma, cyclically monotonicity and cyclically antimonotonicity, we will study this relationship in topological spaces, topological vector spaces, and Banach spaces.   

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.

    ABSTRACT Two transformations of gradient-descent iterative methods for solving unconstrained optimization are proposed. The first transformation is called modification and it is defined using a small enlargement of the step size in various gradient-descent methods. The second transformation is termed as hybridization and it is defined as a composition of gradient-descent methods with the Picard–Mann hybrid iterative process. As a result, several accelerated gradient-descent methods for solving unconstrained optimization problems are presented, investigated theoretically and numerically compared. The proposed methods are globally convergent for uniformly convex functions satisfying certain condition under the assumption that the step size is determined by the backtracking line search. In addition, the convergence on strictly convex quadratic functions is discussed. Numerical comparisons show better behaviour of the proposed methods with respect to some existing methods in view of the Dolan and Moré’s performance profile with respect to all analysed characteristics: number of iterations, the CPU time, and the number of function evaluations.  KEYWORD : Unconstrained Optimization; Gradient-Descent methods; Muiti Step-Size; Convergence; line Search.

In this thesis a representation of continuous functions is presented and then the Reisz representation theorem by using equilibrium problem is investigated.

   دستگاه معادلات غيرخطي يكي از مسائل مهم و پركاربرد در رياضيات است. روش‌هاي متفاوتي براي حل اين مسائل تاكنون ارائه شده است. از ميان روش‌هاي تكراري براي حل اين مسائل، مي‌توان به روش نيوتون، روش‌هاي شبه نيوتن و نسخه‌هاي تغيير يافته آن‌ها اشاره كرد.يكي از نقاط ضعف مهم اين روش‌ها بخصوص براي مسائل با ابعاد بزرگ، نياز به محاسبه ماتريس ژاكوبي در هر تكرار و حل دستگاه معادلات خطي متناطر است. تلاش براي ارائه روش‌هاي بدون ژاكوبي براي حل دستگاه‌هاي معادلات غيرخطي در سال‌هاي اخير همواره مورد توجه محققان بوده است. در حالات خاص كه دستگاه معادلات داراي خواص ويژه مي‌باشد، الگوريتم‌هاي بسيار موثري معرفي شده‌اند. يكي از اين رده‌هاي خاص، دستگاه معادلات غيرخطي يكنوا مي‌باشد كه روش‌هاي حل متفاوتي براي آن ارائه شده است. يكي از مهمترين رده هاي موجود براي حل اين مسائل، الگوريتم‌هاي   مبتني بر تصوير است كه بواسطه نياز به حافظه كم، در حل دستگاه معادلات غيرخطي مقياس بزرگ يكنوا كاربردهاي زيادي دارند.   هدف اين پايان‌نامه، ارائه دو خانواده جديد از الگوريتم‌هاي بدون مشتق مبتني بر تصوير است كه از جهاتي شبيه جهات گراديان مزدوج سه‌جمله‌ا‌ي استفاده مي كنند جاييكه ثابت مي شود جهات تعريف شده در شرايط كاهش كافي صدق مي كنند. نتايج عددي به دست آمده نشان مي‌دهد كه اين روش‌ها براي حل اين نوع از مسائل موثر و كارا هستند.

In this thesis, we give a novel one parameter algebraic functional equation for fractional resolvent families. 

Estimation of the survival function by using the copula for the inverse Rayleigh distribution.

در اين پايان نامه وجود جواب معادلات ديفرانسيل كسري انتقال-انتشار با استفاده آناليز غير خطي، روش هاي تغييراتي و قضيه گذرگاه كوهستاني بررسي مي شود

اخيرا يكي از موضوعات مورد علاقه رياضي­دانان، مطالعه معادلات ديفرانسيل ضربه­اي به خصوص با عملگرهاي مشتق كسري( مشتق كسري ريمان- ليوويل و كاپاتو) بوده است به منابع [3-4] رجوع شود. با وجود اين وجود مطالعات جزئي روي جواب، يكتايي جواب و پايداري جواب ها براي معادله ديفرانسيل كسري با مشتق هيلفر انجام گرفته است، زيرا كه مشتق كسري هيلفر شكل كلي تري از مشتقات كسري ريمان- ليوويل را ارائه كرده است كه به صورت خلاصه ميتوان آن را يك درونيابي بين مشتق كاپاتو و مشتق ريمان- ليوويل فرض كرد. حال در اين پايان نامه، وجود جواب و پايداري معادلات ديفرانسيل كسري ضربه اي با عملگر مشتق كسري هيلفر مورد مطالعه و بررسي قرار خواهد گرفت.

In this thesis, we consider astrongly continuous cosine family {C(t)}t?0 on a Banach space, and prove that the condition lim t?0+ sup ? C(t) ? I ?< 2 implies that C(t) converges to I in the operator norm. we further prove that the stronger assumption supt?0 ? C(t) ? I ?< 2 yields that C(t) = I for all t ? 0. For discrete cosine functions, the assumption sup n?N ? C(n) ? I ? ? r < 3 2 yields that C(n) = I for all n ? N. Furtheremore, we find a discrete cosine family that shows for r ? 3 2 , this conclusion does no longer hold. Morevoer, from the estimate sup t?0 ? C(t) ? cos(at)I ?< 1 we conclude that C(t) equals cos(at)I.   

  ‎The ‎objective ‎of ‎this ‎thesis ‎is ‎to ‎obtain ‎sufficient ‎condition‎s for the dynamics of a vector field on the two dimensional center manifold to be topologically equivalent with the versal deformation of two dimensional Bognanov-Takens bifurcation. Next, using the results of this part, the bursting behaviours and the related bifurcations of the neurons in the Pre-Botzinger Complex is investigated.

This thesis examines of three chapters, which in the first chapter introduce the concepts that are needed, including KKM and generalized KKM which are tools for solving fixed-point problems.

We obtain ‎ relation between an exact   g -frame and a   g -Riesz basis under some conditions. We alsoprove that the stability of an exact   g -frame for a Hilbert space under perturbation is ‎di‎fferent to that of a   g -frame. These properties of exact   g -frames for Hilbert spaces are not similar to those of exact frames. then we mainly discuss the exact   K-g -frames in theHilbert spaces.  

در اين پايان نامه به بررسي مسئله ي تعادل عملگر تعميم يافته پرداخته شده است ودر ادامه مسئله ي شبه تعادل را مورد بررسي قرار داده وسپس كاربردهاي مسئله ي شبه تعادل در مسئله ي شبه بهينه سازي و مسئله ي نابرابري شبه تغييراتي ارايه شده است.

Firstly we explore some properties of K-frame  in terms of decompositions of operators. Then we  pay more attention to characterize the dual K-Bessel sequences of a given K-frames.                                                                                  

A J-frame for a Krein space H is in particular a frame for H (in the Hilbert space sense).But it is also compatible with the indefinite inner-product of H meaning that it determines a pair of maximal uniformly definite subspace.

  In this thesis, the existence of weak solutions to a franctional system of p-Laplacianequations and a singular p-Laplacian equation with sing-changing functions, is investigatedvia the Nehari manifold method. To do this, corresponding to critical pointsof fibering maps, the Nehari manifold is divided into three sets. Moreove is provedthat the energy functional is coercive and bounded from below on the Nehari manifoldand local minima on the Nehari manifold are critical points of the energy functional.Then by showing the existence of two minima on the subsets of Nehari manifoldcorrespanding to maxima and minima of fibering maps, the existence of two criticalpoints of the energy functional will be proved.

بررسي اختلال عملگر قاب هاي باناخ در فضاهاي باناخ

در اين پايان نامه بررسي ارتباط جواب هاي كارا و كاراي ضعيف از مساله بهينه سازي برداري با استفاده از توابع اسكالر ساز مي پردازد 

Banachs fixed point is one of the main fields of research in non-linear analysis of analysis, and is the first to be described in the Banach Rescort. This case has been considered by many researchers for its application and its simplicity, which has been generalized in various ways, such as weakening the contraction inequality, weakening the topology of space, and so on.‎‎‎‎‎This thesis consists of three chapters. In the first chapter, the definitions and the necessary theorems are expressed. The second chapter of this thesis is titled as the fixed point for a weak contraction, which deals with the articles of Chric, Zhang, and Sang, which are presented in full metric space. We also consider the Suzuki case, which is an extension of the Banach contraction theorem, and its generalization‎. We also provide examples and applications of these cases, which will make the results clearer. The third chapter includes  F ‎-‎weak contraction and some of its results. Then we will generalize the contraction of the Banach using the functions called the auxiliary functions introduced by Matkawski and known as the  \\ varphi ‎-‎contraction. We also consider a new type of weak mapping called  F ‎‎-‎‎‎contraction introduced by Imovsky. In this chapter we study the contraction of fixed point theorems for the mapping f using the  F ‎‎-contraction and then the applications of the theorems obtained in the fractal theory are presented.

  In this thesis, we study American option problem under di?erent methods and conditions. First, we consider American put option pricing under regime switching model (based on fron-?xing transformation and the calculation of optimal stopping boundary) and by calculating the optimal stopping boundary, we obtain a stable so- lution. In fact, this solution is the best price in the shortest possible time and has the better consistency with other methods. Then, by inserting rational parameter under regime switching model and employing a wieghted ?nite di?erence method, the problem would be discretized and we check the stability and positivity condition of American option problem, again. By having rational parameters and Thomas algorithm, we simplify the calculations and show that numerical analysis is e?ective in the stability and consistency of the solution. Finally, using a ?nite di?erence method for di?erential equation, we consider both time and space fractional deriva- tives. For this, introduce an implicit direction scheme and minimal residual method, we also propose a preconditioner and we calculate the accuracy of method up to the second order.

In this thesis, we study numerical schems for solving American put option pricing problem and for this purpose present efficient numerical methods. Unlike an European option, the value of an American option satisfies in a linear complementarity problem. We first approximate the linear complementarityproblemwithanonlinearfractionalpartialdifferentialequationbyapenaltyterm, thenweobtainsolutionsofthisequationbyFiniteDifferenceMethodandfinallywestudyanother linear complementarity problem by Laplace Transform Method and Finite Difference Method, and compare these methods by giving examples. So the purpose of this thesis is to provide methods for American put option pricing.   

In this article we study the duals of Grassmann codes, certain codes coming from the Grassmannian variety. Exploiting their structure, we are able to count and classify all their minimum weight codewords. In this classi?cation the lines lying on the Grassmannian variety play a central role. Related codes,namely the af?ne Grassmann codes. In this paper we also classify and count the minimum weight codewords of the dual af?ne Grassmann codes. Combining the above classi?cation results, we are able to show that the dual of a Grassmann code is generated by its minimum weight codewords.  

  In this thesis ‎we‎ generalize the notion of ‎$n$‎‎-weak module amenability of ‎$A$‎‎ which is a Banach module over another Banach ‎algebra‎ ‎$U$‎‎ ‎with‎ compatible actions to that ‎of‎ ‎‎‎‎‎$‎(\\sigma)$‎-‎‎‎$n$‎‎-‎weak‎ module amenability ‎for‎ ‎$n\\in‎ ‎‎\\mathbb{N}$‎‎‎ ‎and‎‎ ‎$\\sigma\\in‎ ‎‎Hom_{U}(A)‎$‎‏ ‎.‎We also investigate the relation between this new concept of amenability of ‎$A$‎‎ and the quotient Banach algebra ‎$‎‎A/‎J$‎‎ where ‎$J$‎‎ is the closed ideal of ‎$A$‎‎ generated by elements of the ‎form‎ ‎‎‎$‎‎(a.‎‎\\alpha‎)b -‎‎‎ a(\\alpha.‎‎b)$‎‎‎ ‎for‎‎ ‎$a‎,‎b\\in‎ ‎A$‎‎‎ ‎and‎‎ ‎$\\alpha‎‎ ‎\\in‎‎‎ ‎U$‎‎‎‎ ‎.‎As a consequence ,we show that the semigroup ‎algebra‎ ‎‎‎$‎l‎^{1}(S)‎$‎‎ is ‎$‎‎(\\sigma‎)‎$-‎‎‎‎$‎(‎2n+‎‎‎1)‎‎$-‎weakly module ‎amenable‎ as an ‎‎$‎l^{1}(E)‎$‎ -module for each $‎n\\in \\mathbb{N}‎$‎‎‏ ‎and‎ $‎‎‎‎\\sigma ‎\\in ‎‎‎‎Hom_{l^{1}(E)‎}(‎l‎^{1}(S)‎)‎$‎‎‎ where ‎‎‎$‎S‎$‎ is an inverse semigroup with the set of idempotents ‎‎‎$‎E‎$‎

بررسي كنترل در فضاهاي هيلبرت

  ‎In ‎this ‎theisi ‎we ‎invsestiyqte ‎on ‎concept ‎of ‎operator‎-‎valued frames.‎‎In ‎fact, ‎operator-‎‎valued ‎(or g‎-‎‎frames) ‎are ‎generalization‎s of frames and fusion frames and have been used in packets encoding, quantum computing, theory of coherent states and mor. In this article , we give a new formula for operator-valued frames for finite dimensional Hilbert spaces. As an application, we derive in a simple manner a recent result of A. Najati conceerning the approximation of g-frames by Parseval ones. we obtain also some results concerning the best approximation of operator-valued frames by its alternate dual, with optimal estimate.

  In this thesis we consier tree concepts of orthogonlity in a Hilbert C?-module V over a C?-algebra A : the Birkhof-James orthogonality ?B , the strong Birkhof-ames orthogonlity ?SB , and the orthogonality with respect to the A-valued inner product on V . We characterize the classes of Hilbert C?-modules in which any two of them coincide .