فهرست مطالب

  • سال پنجم شماره 2 (پاییز و زمستان 1398)
  • تاریخ انتشار: 1398/10/11
  • تعداد عناوین: 12
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  • الناز بابایی، الهام سادات هاشمی زاده* صفحات 121-136
    دستگاهی از معادلات انتگرال می تواند مسائل مختلفی در علوم و مهندسی را توصیف کند. روش های عددی مختلفی برای تقریب جواب های دستگاه معادلات انتگرال خطی و غیرخطی وجود دارد. در این مقاله، یک روش عددی بر اساس توابع کلاهی توسعه یافته برای تقریب جواب های دستگاه معادلات انتگرال فردهلم- همرشتاین ارائه شده است. روش ارائه شده دستگاه معادلات انتگرال را به دستگاهی از معادلات جبری غیرخطی کاهش می دهد که به راحتی با روش های معمول عددی حل می شود. برای اثبات درستی و کارایی روش پیشنهادی، چند مثال عددی همراه با مقایسه با سایر روش های مشابه ارائه شده است که کارایی روش جدید و برتری آن نسبت به سایر روش های موجود را نشان می دهد.
    کلیدواژگان: معادلات انتگرال فردهلم - همرشتاین، توابع کلاهی توسعه یافته، ماتریس عملیاتی، دستگاه معادلات انتگرال
  • میکیل بنزی، فاطمه پنجه علی بیک*، سید حسن عزیزی چپرپردی، زهره روی گر صفحات 137-150
    در این مقاله، به تعمیم برخی از روش های تکراری ایستا در شکل بلوکی برای حل مسائل نقطه زینی مضاعف می پردازیم. برای این منظور ابتدا روش ژاکوبی را تعمیم داده و تحت شرایط خاص همگرایی آن را بررسی می کنیم. هم چنین با اضافه کردن پارامتر تخفیف، شکل برونیابی شده روش ژاکوبی تعمیم یافته و همگرایی آن را نیز در نظر می گیریم. سپس به بررسی تعمیمی از روش گاوس- سیدل و آنالیز همگرایی آن تحت قید مناسبی می پردازیم. هم چنین در روش مذکور تخفیف متوالی تعمیم یافته به همراه شرایط کافی همگرایی آن بررسی شده است. برای نشان دادن کارایی روش های ارائه شده به گزارش نتایج عددی برای حل مسئله نقطه زینی مضاعف، دارای کاربرد در مدل سازی هدایت گرهای کریستال مایع می پردازیم.
    کلیدواژگان: مسئله نقطه زینی مضاعف، روش بلوکی گاوس- سیدل، روش بلوکی فوق تخفیف متوالی، همگرایی، کریستال مایع
  • مسعود سبزواری* صفحات 151-156
    در این مقاله، با به کارگیری نتایج موجود در نظریه تاناکا اثباتی کوتاه برای حدس بیشینه بلوشاپکا در بعد کوشی-ریمان یک ارائه می کنیم. به عبارت دیگر، ثابت می کنیم که هر مدل کاملا ناتباهیده بلوشاپکا از بعد کوشی-ریمان یک و طول بزرگ تر از سه صلبیت دارد. به عنوان نتیجه ، هم چنین خواهیم دید که گروه خودریختی های کوشی-ریمان متناظر با هر یک از مدل های مذکور تنها شامل نگاشت های خطی  است.
    کلیدواژگان: مدل های کاملا ناتباهیده کوشی-ریمان، حدس بیشینه، توسیع تاناکا
  • رضا سزیده*، فاطمه ساوجی صفحات 157-164
    فرض کنیم (R,m) حلقه ای موضعی، نوتری و جابه جایی باشد. در این مقاله وجود مدول های متناهی مولد از بعد انژکتیو گورنشتاین متناهی روی حلقه های کوهن-مکالی را بررسی می کنیم. در ابتدا بعدهای انژکتیو گورنشتاین کوهمولوژی موضعی هم بافت ها را بررسی می کنیم.
    کلیدواژگان: انژکتیو گورنشتاین، حلقه های کوهن-مکالی، مدول های کوهمولوژی موضعی
  • ایلدار صادقی*، علی حسن زاده صفحات 165-174
    در این مقاله مفهوم نقاط اتکاء مجموعه های محدب در مخروط های نرم دار معرفی شده و نشان داده می شود که در یک مخروط نرم دار پیوسته، تحت شرط های مناسب، مجموعه نقاط اتکاء مجموعه ای محدب اسکات بسته کران دار، ناتهی است. هم چنین قضیه بیشاب- فلپس را برای مخروط های نرم دار بیان و اثبات می کنیم.
    کلیدواژگان: نقطه اتکاء، مخروط نرم دار، قضیه - بیشاب فلپس
  • محمد طباطبایی، بنت الهدی سادات حسینی* صفحات 175-186
    در این مقاله به بررسی بیش تر مفهوم طیف های آروسون بر ابرگروه های موضعا فشرده می پردازیم و خواص اساسی آن را به گروه های آبلی موضعا فشرده و رده مهمی از ابرگروه ها را توسعه می دهیم که فشرده و شمارای نامتناهی هستند. به ویژه ثابت می کنیم ، که در آن  یک - سیستم و  و  زیرمجموعه های بسته از  هستند.
    کلیدواژگان: ابرگروه، طیف آروسون، زیرفضای طیفی، - جبر
  • کامل عبداله نژاد*، علی اکبر جعفری، نسرین طاطاری صفحات 187-196
    در این مقاله به مسئله مقایسه میانگین های چندین جامعه لگ نرمال چندمتغیره پرداخته می شود و یک روش مفید به نام روش متغیر تعمیم یافته ارائه می گردد. بررسی های شبیه سازی نشان می دهد که آزمون فرض روش پیشنهادی صرف نظر از حجم نمونه، اندازه و توان آزمون مناسبی دارد. برای ارزیابی این روش، آن را با روش مرسوم آنالیز واریانس چندمتغیره مقایسه می کنیم که اندازه واقعی هر دو آزمون به یک دیگر نزدیک هستند ولی توان آزمون و احتمال پوشش روش پیشنهادی در بسیاری از موارد به ویژه در اندازه نمونه های کوچک بهتر از روش آنالیز واریانس چندمتغیره است. بنابراین می توان این روش را برای حالتی که ماتریس های واریانس-کوواریانس برابر نیستند و روش دیگری برای مقایسه میانگین ها وجود ندارد به کار برد.
    کلیدواژگان: متغیر تعمیم یافته، توزیع لگ نرمال چند متغیره، اندازه و توان آزمون، احتمال پوشش، تجزیه چولسکی
  • رضا عرفی* صفحات 197-204
    فرض کنیم   یک گروه ناآبلی از مرتبه  باشد در این مقاله یک ساختار برای زیرگروه سیلوی گروه خودریختی های   معرفی می شود.
    کلیدواژگان: گروه خودریختی، خودریختی های مرکزی و -pگروه متناهی
  • پیام مختاری* صفحات 205-220
    در این مقاله روش هم محلی مبتنی بر چند جمله ای های ژاکوبی انتقال یافته به عنوان توابع پایه ای را برای تقریب مناسب جواب های معادلات دیفرانسیل کسری غیرخطی تک مرتبه ای معرفی می کنیم. با استفاده از قضایای وجود و یکتایی نتیجه می گیریم که برخی از مشتقات جواب های این دسته از معادلات در مبدا ناپیوسته اند که این به نوبه خود موجب می شود که پیاده سازی روش هم محلی به شکل معمول، مرتبه همگرایی پایینی داشته باشد. به منظور رفع این مشکل با استفاده از یک تغییر متغیر مناسب ابتدا معادله را به یک معادله جدید با جواب هموارتر تبدیل می کنیم و سپس روش هم محلی موردنظر را روی آن پیاده سازی می کنیم. آنالیز همگرایی روش را بررسی کرده و نتایج عددی حاصل از اعمال روش پیشنهادی را گزارش می دهیم.
    کلیدواژگان: معادلات دیفرانسیل کسری غیرخطی تک مرتبه ای، روش هم محلی، چندجمله ای های ژاکوبی، آنالیز همگرایی
  • سعید مقصودی* صفحات 221-228
    فرض کنید  یک گروه موضعا فشرده،  یک تابع وزن و   فضای توابع اندازه پذیر روی   باشد که اساسا کراندار و در بینهایت صفر می شوند. در این مقاله توپولوژی موضعا محدب  را روی فضای وزندار  بررسی می کنیم. نشان می دهیم که دوگان   با این توپولوژی برابر فضای باناخ  است. علاوه بر این، برخی ویژگی های فضای    با توپولوژی مذکور را بررسی می کنیم.
    کلیدواژگان: گروه موضعا فشرده، توپولوژی موضعا محدب، فضای لبگ وزندار، دوگان
  • فرشید میرزایی*، افسون حمزه صفحات 229-236
    در این مقاله، ضمن بیان ویژگی هایی از توابع موجک هار، به ارائه روشی برای تقریب جواب انتگرال وینر کسری با پارامتر هرست  با  استفاده از این توابع می پردازیم. هم چنین تجزیه و تحلیل خطای روش مورد نظر ارائه شده است. این روش را روی چند مثال پیاده سازی کرده و نتایج عددی را در قالب جدول مقادیر خطا ارائه می دهیم. دقت مطلوبی از نتایج در مورد تعداد کمی از نقاط در مثال های ارائه شده مشاهده می شود.
    کلیدواژگان: توابع موجک هار، انتگرال های وینر کسری، حرکت براونی
  • مرتضی میرزایی ازندریانی*، مهدی رحیمی صفحات 237-252
    در این مقاله انواع جدیدی از دوگان ها و دوگان های تقریبی در فضاهای هیلبرت را با استفاده از ضرب گرها، عملگرهای وارون پذیر و نشانه ها معرفی می کنیم. تاکنون مقالات متعددی در مورد دوگان های تقریبی و کاربردهای آن ها نوشته شده که در این مقالات دوگان های تقریبی برای دنباله های بسل بررسی شده اند. در این جا دوگان های تقریبی را برای دنباله های دلخواه در یک فضای هیلبرت تعریف کرده، آن ها را با دوگان های تقریبی بسل مورد مقایسه قرارداده و نشان می دهیم با وجود این که این دوگان های تقریبی لزوما تمام خواص دوگان های تقریبی بسل را ندارند اما می توانند در بازسازی سیگنال ها مفید واقع شوند. علاوه بر این، نتایج جدیدی برای دوگان های تقریبی بسل به دست می آوریم.
    کلیدواژگان: فضای هیلبرت، دنباله بسل، قاب، دوگان تقریبی، ضرب گر، بازسازی سیگنال
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  • Elnaz Babaei, Elham Hashemizadeh* Pages 121-136
    Introduction
    A system of integral equations can describe different kind of problems in sciences and engineering. There are many different methods for numerical solution of linear and nonlinear system of integral equations.
    Material and methods
    This paper proposed a numerical method based on modification of Hat functions for solving system of Fredholm-Hammerstein integral equations. The proposed method reduced a system of integral equation to a system of algebraic equations that can be solved easily by known methods.
    Results and discussion
    For showing the accuracy and capability of the proposed method, some numerical examples are proposed that their results compared by results of other methods, and shows the capability and the superiority of this method to other existed methods.
    Also this paper derived the computational cost and the error analysis of the proposed method.
    Conclusion
    The following conclusions were drawn from this research.


    This paper proposed a numerical method based on modification of Hat functions for solving system of Fredholm-Hammerstein integral equations.
    The proposed method reduced a system of integral equation to a system of algebraic equations that can be solved easily by known methods.
    The presented error analysis and solved problems show capability and the superiority of this method to other existed methods../files/site1/files/52/1.pdf
    Keywords: Fredholm-Hammerstein integral equations, Modification of Hat functions, Operational matrix, System of integral equations
  • Michele Benzi, Fatemeh Panjeh Ali Beik*, Sayyed–Hasan Azizi Chaparpordi, Zohreh Rouygar Pages 137-150
    In this paper, we develop some stationary iterative schemes in block forms for solving double saddle point problem. To this end, we first generalize the Jacobi iterative method and study its convergence under certain condition. Moreover, using a relaxation parameter, the weighted version  of the Jacobi method together with its convergence analysis are considered. Furthermore, we extend a method from the class of Gauss-Seidel iterative method and establish its convergence properties under a certain condition. In addition, the block successive overrelaxation (SOR) method is used to construct an iterative scheme to solve the mentioned double saddle point problem and its convergence properties are analyzed. In order to illustrate the efficiency of the proposed methods, we report some numerical experiments  for a class of saddle point problems arising from the modeling of liquid crystal directors using finite elements../files/site1/files/52/2.pdf
    Keywords: Double saddle point problem, Block Gauss-Seidel method, Block SOR method, Convergence, Liquid crystal
  • Masoud Sabzevari* Pages 151-156
    Introduction
    In this paper and by employing some certain techniques and results arisen in the theory of Tanaka, we provide a short proof for the maximum conjecture on the rigidity of Beloshapka's models of the specific CR dimension one and of length . As a consequence, we realize that the Lie group of biholomorphisms (CR automorphisms) associated with each of these models only consists of linear maps.   
    Material and methods
    Two major approaches employed so far to study Beloshapka's maximum conjecture have been 1) the approach of envelope of holomorphy, applied by Ilya Kossovskii to confirm the mentioned conjecture in the specific lengths three and four and 2) Cartan's approach of solving (biholomorphic) equivalence problems, employed by the present author in CR dimension one. Both of these approaches are geometric. In this paper, we introduce and employ the algebraic approach of applying some techniques in Tanaka's theory of transitive prolongations to study the already mentioned conjecture in the specific CR dimension one. The proofs are based upon some results concerning Tanaka prolongation of rank two fundamental algebras of lengths , achieved by Medori and Nacinovich (1997). Here, we prove first an algebraic parallel version of Beloshapka's conjecture and employ the results to solve this geometric open problem in CR dimension one.
    Results and discussion
    As is the main goal of this paper, we confirm the maximum conjecture on the rigidity of Beloshapka's CR models in CR dimension one. As a consequence, we realize that each biholomorphic deformation of these models is linear. It may be worth to notice that the maximum conjecture in CR dimension one has been confirmed before by the present author, using the Cartan approach of solving equivalence problems. But, here we provide a much shorter proof for this result.
    Conclusion
    The following conclusions were drawn from this research.


    We confirm Beloshapka's maximum conjecture in CR dimension one.
    It is introduced a new approach to consider this conjecture.
    Comparing the performance of the Tanaka approach, employed in this paper, with two approaches of "envelope of holomorphy" and "Cartan's theory", this algebraic approach seems the best weapon, introduced so far, to attack the maximum conjecture. ./files/site1/files/52/3.pdf
    Keywords: Totally nondegenerate CR models, Maximum conjecture, Tanaka prolongation
  • Reza Sazeedeh*, Fatemeh Savoji Pages 157-164
    Throughout this paper‎, (R, m) is a‎ commutative Noetherian local ring with the maximal ideal m. ‎The following conjecture proposed by Bass [1]‎, ‎has been‎ proved by Peskin and Szpiro [2] for almost all rings: ‎
    (B) If R admits a finitely generated R-module of‎ finite injective dimension‎, ‎then R is Cohen-Macaulay.
    ‎The problems treated in this paper are closely related to the‎ following generalization of  Bass conjecture which is still wide‎ open:‎
    (GB) If R admits a finitely generated R-module of‎ finite Gorenstein-injective dimension‎, ‎then R is‎ Cohen-Macaulay.
    ‎  Our idea goes back to the first steps of the solution of Bass conjecture given by Levin and Vasconcelos in 1968 [3] when R admits a‎ finitely generated R-module of injective dimension 1‎.
    Levin and Vasconcelos indicate that if‎ is a non-zerodivisor‎, ‎then for‎ every finitely generated R/xR-module M‎, ‎there is‎ ‎. ‎Using this fact‎, ‎they construct a‎ finitely generated R-module of finite injective dimension in‎ the case where R is Cohen-Macaulay (the converse of Conjecture‎ B)‎.
    ‎ In this paper we study the Gorenstein injective dimension of local cohomology‎. ‎We also show that if R is Cohen-Macaulay‎ with minimal multiplicity‎, ‎then every finitely generated module‎ of finite Gorenstein injective dimension has finite injective‎ dimension.‎
    ‎We prove that a Cohen-Macaulay local ring‎ has a finitely generated module of‎ finite Gorenstein injective dimension.‎
    ./files/site1/files/52/4.pdf
    Keywords: Gorenstein injective, Cohen-Macaulay ring, Local cohomology modules
  • Ildar Sadeghi*, Ali Hassanzadeh Pages 165-174
    Introduction
    In the last few years there is a growing interest in the theory of quasi-metric spaces and other related structures such as quasi-normed cones and asymmetric normed linear spaces, because such a theory provides an important tool in the study of several problems in theoretical computer science, approximation theory, applied physics, convex analysis and optimization. Many works on general topology and functional analysis have recently been obtained in order to extend the well-known results of the classical theory of normed linear spaces to the framework of asymmetric normed linear spaces and quasi-normed cones.
    ‎An abstract cone is analogous to a real vector space‎, ‎except that we take  as the set‎ ‎of scalars‎. ‎ In 2004, O‎. ‎Valero introduced the normed cones and proved some closed graph and open mapping results for normed cones. Also Valero defined and studied some properties of quotient normed cones. P. Selinger studied the norm properties of a cone with its order properties and proved Hahn-Banach theorems in these cones under the appropriate conditions. Valero and his colleagues discussed the metrizability of the unit ball of the dual of a normed cone and the isometries of normed cones. Other properties are investigated in a series of papers by Romaguera, Sanchez Perez and Valero.
     The Bishop-Phelps theorem is a fundamental theorem in functional analysis which has many applications in the geometry of Banach spaces and optimization theory. The classical Bishop-Phelps theorem states that “the set of support functionals for a closed bounded convex subset  of a real Banach space X, is norm dense in  and the set of support points of  is dense in the boundary of ".  Indeed, E. Bishop and R. R. Phelps answer a question posed by ‎Victor Klee in 1958. We give an analogue to the normed cones‎, in fact we show that in a continuous normed cone the set of support points of a closed convex set is a dense subset of the boundary under the appropriate hypothesis.
    Conclusion
    In this paper the notion of support points of convex sets in normed cones is introduced and it is shown that in a continuous normed cone, under the appropriate conditions, the set of support points of a bounded Scott-closed convex set is nonempty. We also present a Bishop-Phelps type Theorem for normed cones../files/site1/files/52/5.pdf
    Keywords: Support point, Normed cone, Bishop-Phelps theorem
  • Mohammad Tabatabaie, Bentolhoda Sadathosseini* Pages 175-186
    In this paper we study the concept of Arveson spectrum on locally compact hypergroups and for an important class of compact countable hypergroups .


    In thiis paper we study the concept of Arveson spectrum on locally compact hypergroups and develop its basic properties for an important class of compact countable hypergroups ../files/site1/files/52/6.pdf
    Keywords: hypergroups, Arveson spectrum, w*–algebra
  • Pages 187-196
    Abstract
    In this paper, we consider the problem of means in several multivariate log-normal distributions and propose a useful method called as generalized variable method. Simulation studies show that suggested method has a appropriate size and power regardless sample size. To evaluation this method, we compare this method with traditional MANOVA such that the actual sizes of the two methods are close but the power of test and coverage probability of proposed methods are better than MANOVA in most cases specially when the sample sizes are small. Therefore, we can use this method when the variance-covariance matrices are not equal and there is not a suitable method../files/site1/files/52/7.pdf
    Keywords: Generalized Variable, Multivariate log-normal distribution, Size, power of test, Coverage probability, Cholesky decomposition
  • Reza Orfi* Pages 197-204
    Let G be a finite non-abelian group of order p^4 . In this paper we give a structure theorem for the Sylow p-subgroup, Aut_p(G)  , of the automorphism group of G../files/site1/files/52/8.pdf
    Keywords: Automorphism group, Central Automorphisms, finite p-group
  • Pages 205-220
    Introduction
    The modeling of many real-life physical systems leads to a set of fractional differential equations. Also fractional differential equations appear in various physical processes such as viscoelasticity and viscoplasticity, modeling of polymers and proteins, transmission of ultrasound waves, signal processing, control theory, etc. Most of fractional differential equations especially their nonlinear types do not have exact analytic solution, so numerical methods must be used. Therefore many authors have worked on the numerical solutions of this kind of equations. In recent years, many numerical methods have emerged, such as, the Adomian decomposition method, the Homotopy method, the multistep method, the extrapolation method, the spline collocation method, the product integration method and the predictor-corrector method. But most of the aforementioned methods consider the linear type of equations without a reliable theoretical justification. Then providing an efficient numerical scheme to approximate the solutions of nonlinear fractional differential equations is worthwhile and new in the literature. The main object of this paper is to develop and analyze a high order numerical method based on the collocation method when applies the orthogonal Jacobi polynomials as bases functions for the single term nonlinear fractional differential equations.
    Material and methods
    Due to the well-known existence and uniqueness theorems the solutions of the fractional differential equations typically suffer from singularity at the origin. Consequently direct application of the Jacobi collocation method may lead to very weak numerical results. To fix this difficulty, we introduce a smoothing transformation that removes the singularity of the exact solution and enables us to approximate the solution with a satisfactory accurate result. Convergence analysis of the proposed scheme is also presented which demonstrates that the regularization process improves the smoothness of the input data and thereby increases the order of convergence.
    Results and discussion
    We illustrate some test problems to show the effectiveness of the proposed scheme and to confirm the obtained theoretical predictions. In overall, the reported results justify that the proposed regularization strategy works well and the obtained approximate solutions have a good accuracy. To show the applicability of our approach we solve a practical example which is developed for a micro-electrical system (MEMS) instrument that has been designed primary to measure the viscosity of fluids that are encounter during oil well exploration using the proposed scheme.  Moreover, we make a comparison between our scheme and the operational Tau method to show the efficiency of our technique. The reported results approve the superiority of the proposed approach. Finally, we consider a problem that we do not have access to its exact solution. In this case, we use the “Variational Iteration Method (VIM)” as a qualitatively correct picture of the exact solution (the source solution) to evaluate the precision of the proposed technique. The obtained results approve that our approach produces the approximate solution which is in a good agreement with source ones.
    Conclusion
    The following conclusions were drawn from this research.

    A reliable numerical method based on the Jacobi collocation method to approximate the solutions of a class of nonlinear fractional differential equations was developed.
    To achieve an efficient approximation a regularization strategy was proposed that improves the smoothness of the input data and enables us to obtain an approximate solution with a satisfactory accuracy.
    Convergence analysis of the proposed method was investigated which confirmed the high order of convergence of the proposed method../files/site1/files/52/9.pdf
    Keywords: Single term nonlinear fractional differential equations, Collocation method, Jacobi polynomials, Convergence analysis
  • Saeid Maghsoudi* Pages 221-228
    Introduction
    Let G be a locally compact group with a fixed left Haar measure λ  and   be a weight function on G;  that is a Borel measurable function  with   for all .   We denote by  the set of all measurable  functions  such that ; the group algebra of  G  as defined in [2]. Then   with the convolution product “*” and the norm   defined by   is a Banach algebra known as Beurling algebra. We denote by n(G,) the topology generated by the  norm .    Also, let  denote the space of all measurable functions   with , the Lebesgue space as defined in [2].
    Then   with   the product  defined by , the   norm  defined by  , and the complex conjugation as involution is a commutative algebra. Moreover,  is the dual of . In fact, the mapping   is an isometric isomorphism.
     We denote by the -subalgebra of  consisting of all functions  on G such that for each , there is a compact subset K of G for which
    .  For a study of in the unweighted case see  [3,6].
     We introduce and study a locally convex topology  on  such that  can be identified with the strong dual of . Our work generalizes  some interesting results of  [15] for group algebras to a more general setting of weighted group algebras. We also show that (,)  could be a normable or bornological space only if G is compact. Finally, we prove that  is complemented in   if and only if G is compact. For some similar recent studies see [4,7,8,10,12-14]. One may be interested to see the work [9] for an application of these results.
    Main
    results
    We denote by   the set of increasing sequences of compact subsets of G and by ℛ the set of increasing sequences  of real numbers in  divergent to infinity. For any  and , set and note that  is a convex balanced absorbing set in the space . It is easy to see that the family of all sets  is a base of neighbourhoods of zero for a locally convex topology on  see for example [16]. We denote this topology by .  Here we use some ideas from  [15], where this topology has been introduced and studied for  group algebras.
    Proposition 2.1 Let G be a locally compact group, and  be a weight function on G.   The norm topology n(G,) on  coincides with the topology  if and only if G is compact.
    Proposition 2.2 Let G be a locally compact group, and  be a weight function on G.  Then the dual of (,)  endowed with the strong topology can be identified with endowed with -topology.
    Proposition 2.3 Let G be a locally compact group, and  be a weight function on G.  Then the following assertions are equivalent:a) (,)  is barrelled.
    b) (,)  is bornological.
    c) (,)  is metrizable.
    d) G  is compact.
    Proposition 2.4 Let G be a locally compact group, and  be a weight function on G.  Then  is not complemented in ../files/site1/files/52/10.pdf
    Keywords: Locally compact group, Locally convex topology, Weighted Lebesgue space, Dual
  • Pages 229-236
    Introduction
    The stochastic calculus plays an important role in the study of stochastic integral equations and stochastic differential equations. The fractional Brownian motion has many applications in different branches of sciences such as economics, physics and biology.
    In many situations, the exact solution of these equations are not available or finding their exact solution is a very difficult process. Thus, finding an accurate and efficient numerical method for solving stochastic differential equations, and stochastic integral equations is important. Researchers have applied various numerical methods such as Dirichlet forms, Euler approximation, Skorohod integral, etc. In this paper, we used Haar wavelet functions for solving fractional Wiener integrals. Moreover, the error analysis of the proposed method is investigated. 
    Material and methods
    In this scheme, first we present the properties of the Haar wavelet functions then an efficient method based on these functions is proposed to estimate the solution of fractional Wiener integral with Hurst parameter
    Results and discussion
    We solve two numerical examples by using present method to demonstrate the efficiency and simplicity of the present method. For different values of , mean of error and standard deviation of error are shown in the tables. The obtained results confirm that proposed method enables us to find reasonable approximate solutions.
    Conclusion
    The Haar wavelet is the simplest possible wavelet, so proposed method is easy to implement and it is a powerful mathematical tool to obtain the numerical solution of various kind of problems. ./files/site1/files/52/11.pdf
    Keywords: Haar wavelet functions, Fractional wiener integrals, Brownian motion
  • Morteza Mirzaee Azandaryani*, Mehdi Rahimi Pages 237-252
    Paper pages (237-252)
    Introduction
    ‎Frames for Hilbert spaces were first introduced by Duffin and‎ ‎Schaeffer in 1952 to study some problems in nonharmonic‎ ‎Fourier series‎, ‎reintroduced in 1986 by Daubechies‎, ‎Grossmann and‎ ‎Meyer‎. ‎Various generalizations of frames have been introduced and many applications of them in different branches have been presented‎.
    ‎‎Bessel multipliers in Hilbert spaces were introduced by Peter Balazs‎. ‎As we know in frame theory‎, ‎the composition of the synthesis and analysis operators of a frame is called the frame operator‎. ‎A multiplier for two Bessel sequences is an operator that combines the analysis operator‎, ‎a multiplication pattern with a fixed sequence‎, ‎called the symbol‎, ‎and the synthesis operator‎. ‎Bessel multipliers have useful applications‎, ‎for example they are used for solving approximation problems and they have applications as time-variant filters in acoustical signal processing‎‎. We mention that many generalizations of Bessel multipliers have been introduced, also multipliers have been studied for non-Bessel sequences.
    ‎Approximate duals in frame theory have important applications‎, ‎especially are used for the reconstruction of signals when it is difficult to find alternate duals‎. ‎Approximate duals are useful for wavelets‎, ‎Gabor systems and in sensor modeling‎. ‎Approximate duality of frames in Hilbert spaces was recently investigated by Christensen and Laugesen and some interesting applications of approximate duals were obtained‎. ‎For example‎, ‎it was shown that how approximate duals can be obtained via perturbation theory and some applications of approximate duals to Gabor frames especially Gabor frames generated by the Gaussian were presented‎. Afterwards, many authors studied approximate duals of Bessel sequences and many properties and generalizations of them were presented. In this note, we consider approximate duals for arbitrary sequences.
    Results and discussion
    In this paper, we introduce some new kinds of duals and approximate duals in Hilbert spaces using multipliers, invertible operators and symbols. Many papers about approximate duals and their applications have been written so far which in these papers approximate duals have been considered for Bessel sequences. Here, we introduce approximate duals for arbitrary sequences in a Hilbert space, compare them with Bessel approximate duals and we show that they can be useful for the reconstruction of signals though they do not have all of the properties of Bessel approximate duals. Moreover, we obtain some new results for Bessel approximate duals.
    Conclusion
    The following conclusions were drawn from this research.

    New kinds of duals and approximate duals for arbitrary sequences are introduced using multipliers, invertible operators and symbols.
    Duals and approximate duals of non-Bessel sequences are compared with the Bessel ones and some differences between them are shown by presenting various examples.
    Some properties and applications of duals and approximate duals of non-Bessel sequences are stated.
    Some new results about duals and approximate duals of Bessel sequences are obtained especially some important concepts such as closeness of Bessel sequences, nearly Parseval frames and multipliers with constant symbols are related to approximate duals of frames.  ./files/site1/files/52/12.pdf
    Keywords: Hilbert space, Bessel sequence, Frame, Approximate dual, Multiplier, Reconstruction of signals