فهرست مطالب

نشریه پژوهشهای ریاضی
سال ششم شماره 4 (پیاپی 15، زمستان 1399)

  • تاریخ انتشار: 1399/12/05
  • تعداد عناوین: 16
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  • حمیده آرام*، رعنا خوئیلر، نسرین ده گردی صفحات 501-508

    فرض کنید  گرافی از مرتبه  و اندازه  باشد. اگر  مقادیر ویژه ماتریس لاپلاسین باشند، آن گاه انرژی لاپلاسین گراف  به صورت معرفی می شود. در این مقاله بررسی انرژی لاپلاسین در گراف ها را ادامه می دهیم و کران های جدیدی برای انرژی لاپلاسین در گراف ها به دست می آوریم.

    کلیدواژگان: ماتریس لاپلاسین، مقادیر ویژه ماتریس لاپلاسین، انرژی لاپلاسین
  • محمود افشاری*، سعید طهماسبی، شبنم شادمان صفحات 509-520

    در روند بررسی و شناخت جوامع آماری، تحلیل داده های به دست آمده از این جوامع، امری مهم و ضروری تلقی می شود. یکی از روش های مناسب در تحلیل داده ها، بررسی ساختاری تابع برازش شده به وسیله این داده ها است. تبدیل موجک، یکی از ابزارهای بسیار قوی در تحلیل چنین توابع است و ساختار  ضرایب موجک  اهمیت خاصی دارد. در این مقاله، ضمن معرفی تبدیل موجک و فرآیند  خود برگشتی میانگین متحرک با حافظه طولانی مدت، ساختار ماتریس کوواریانس موجکی این فرآیند بررسی و سپس  پارامترهای این مدل به روش بیزی وبر پایه موجک ها برآورد می شوند.  در پایان با استفاده از شبیه سازی، عملکرد و کارایی برآورد پیشنهادی، در مقایسه با دو روش برآوردیابی دیگر ارزیابی می شود. نتایج نشان دهنده عملکرد خوب این روش هست.

    کلیدواژگان: تبدیل موجک، ضرایب موجکی، حافظه طولانی مدت، استنباط بیزی
  • احسان انجیدنی* صفحات 521-526

    در این مقاله‏، زیرجمعی بودن توابع روی عملگرها‏ی مثبت را بدون فرض یکنوایی عملگری و تحدب عملگری بررسی می کنیم. گیریم ‎‎$‎A‎$‎‏ و ‎‎$‎B‎$‎‏ عملگرهای مثبت روی یک فضای هیلبرت ‎‎$‎‎mathcal{H}‎$‎‎‏ باشند و ‎‎$‎0leq AB+BA‎$‎‏. فرض کنید برای عملگر‎ ‏‎‎‎$‎$‎‎E=(A+B)^{-frac{1}{2}}left(A^2+B^2right)(A+B)^‎{‎-frac{1}{2}}‎,‎$$‏ بازه‎ باز ‎‎$(‎m_E,M_E)‎$‏،‎ که‎ در آن‏، ‎‎$‎m_‎E‎$‎‏ و ‎‎$‎M_E‎$‎‏ کران های عملگر ‎‎$‎E‎$‎‏ هستند‏،‏ با طیف های مربوط به عملگرهای ‎‎$‎A‎$‎‏ و ‎‎$‎B‎$‎‏ اشتراک نداشته باشد.‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎ ‏ در این صورت‎‎‏‏، برای هر تابع پیوسته ‎‎$‎g:(0,infty) ‎rightarrow‎‎mathbb{R}^+‎$‎‎‏ که برای آن‏، تابع ‎‎$‎f(t)=frac{g(t)}{t}‎$‎‏ محدب و نزولی باشد‏، خواهیم داشت ‎‎$‎‎$‎g(A+B)leq c(m,M,f)(g(A)+g(B)),‎$‎‎$‎‎ ‏که در آن‏، ‎‎$‎m‎$‎‏ و ‎‎$‎M‎$‎‏ کران های عملگر ‎‎$‎A+B‎$‎‏ هستند و ‎‎$‎‎$‎‎c(m,M,f):=max_{mleq tleq M}left{frac{‎frac{f(M)-f(m)}{M-m}t+‎frac{Mf(m)-mf(M)}{M-m}}{f(t)‎}right}‎.‎$‎‎$‎./files/site1/files/64/3Anjidani.pdf

    کلیدواژگان: ترتیب ماتریسی، نامساوی عملگری زیرجمعی، تابع یکنوا، تابع محدب، نامساوی عملگری ینسن
  • اسمعیل بابلیان، فاطمه چیت ساز، علی داوری* صفحات 527-540

    ایده اصلی این مقاله، استفاده از چندجمله ای های چبیشف برای حل معادلات انتگرال-دیفرانسیل تاخیری فردهلم خطی با مراتب بالا است. معمولا حل این معادلات به روش های تحلیلی امکان پذیر نیست یا در صورت امکان بسیار مشکل است. در این روش معادله مورد نظر به وسیله روابط ماتریسی بین چندجمله ای های چبیشف و مشتقات آنها به دستگاه  معادلات خطی تبدیل می شود. ماتریس های عملیاتی عملگرهای تاخیر و مشتق همراه با روش تایو برای محاسبه ضرایب مجهول بسط چبیشف جواب استفاده می شوند. همگرایی روش بررسی شده است. مثال های عددی، اعتبار و کارایی روش ارایه شده را نشان می دهند. هم چنین نتایج حاصل از روش با نتایج موجود مقایسه  شده است.

    کلیدواژگان: معادله دیفرانسیل تاخیری، معادله انتگرال-دیفرانسیل تاخیری فردهلم، روش تائو، ماتریس عملیاتی، چندجمله ای های چبیشف
  • روح الله بخشنده چمازکتی* صفحات 541-548

    در این مقاله به محاسبه تقارن های لی (تقارن های کیلینگ) و تقارن های نوتری متریک هذلولوی می پردازیم. هم چنین یک آنالیز جبر تقارن های لی و نوتری متناظر با این متریک هذلولوی را بیان کرده و قوانین بقا با استفاده از تقارن های نوتری محاسبه شده اند.

    کلیدواژگان: تقارن لی، تقارن نوتری، قانون بقا، متریک ریمانی، محک ناوردایی، فضا زمان
  • سمیرا پورخواجویی، ساره گلی، امیر هاشمی* صفحات 549-562

    موضوع اصلی این مقاله، بررسی پایه مرزی برای یک ایده آل نقاط است. برای این منظور ابتدا الگوریتمی برای محاسبه ایده آل مرتب و پایه مرزی نظیر آن برای یک ایده آل نقاط (که دارای چندگانگی هستند) می پردازیم. ایده آل نقاط از کاربردهای مختلفی در علوم و مهندسی برخوردار است که در این مقاله ما به کاربرد آن در یافتن مدل آماری بهینه اشاره می کنیم. در پایان، پس از بیان مطالب مورد نیاز، با استفاده از روش های ارایه شده در این مقاله، به محاسبه مدل های مختلف برای مثالی بر مبنای داده های واقعی و توضیح کارآیی مدل های ارایه شده می پردازیم.

    کلیدواژگان: پایه مرزی، ایده آل نقاط، الگوریتم بوخبرگر-مولر، طرح آزمایش ها، رگرسیون رده بندی ریاضی (2010): 13P10، 14Q99، .68W30
  • حسن پیریائی، غلامحسین یاری*، رحمان فرنوش صفحات 563-578

    در این مقاله علاوه بر برآوردهای حداکثر درست نمایی و بیز، از روش جدید برآورد ای-بیز برای پارامتر مجهول و توابع قابلیت اعتماد و نرخ خطر توزیع نمایی معکوس تعمیم یافته، استفاده می شود. محاسبات براساس داده های سانسور نوع 2 و تحت تابع زیان درجه دوم خطا انجام می شود. این برآوردها براساس یک توزیع پیشین مزدوج برای پارامتر مجهول به دست می آیند. برای محاسبه این برآوردها، از سه توزیع پیشین متفاوت برای ابرپارامترها به منظور مقایسه نتایج استفاده شده است. رفتار مجانبی برآوردهای ای-بیز و ارتباط بین آنها نیز بحث می شود. در نهایت مقایسه ای بین برآوردهای حداکثر درست نمایی، بیز و ای-بیز با حجم نمونه مختلف و با استفاده از روش شبیه سازی مونت کارلو انجام می شود.

    کلیدواژگان: برآورد ای-بیز، داده های سانسور نوع 2، قابلیت اعتماد، نرخ خطر، شبیه سازی مونت کارلو
  • قربانعلی حقیقت دوست*، جعفر اوج بگ صفحات 579-586

    در این مقاله توپولوژی رویه های هم انرژی غیرتکین برای سیستم هامیلتونی با دو درجه آزادی روی مخروط واقع در یک میدان پتانسیلی توصیف شده است. هم چنین روش یافتن ناوردای توپولوژیکی سیستم های هامیلتونی انتگرال پذیر از حالت فشرده به رویه های دوار نافشرده توسیع داده شده است.

    کلیدواژگان: سیستم های هامیلتونی، رویه های هم انرژی، ناورداهای فومنکو-زی شانگ، میدان های پتانسیلی
  • پریسا رحیم خانی، یدالله اردوخانی* صفحات 587-600

    در این مقاله، روشی عددی برای حل معادلات دیفرانسیل تاخیری  بیان می شود. هدف اصلی، معرفی تابع های تکه ای براساس تابع های تیلور کسری در محاسبات کسری است. هم چنین یک فرمول بندی کلی برای  ماتریس عملیاتی انتگرال کسری این توابع نتیجه گرفته می شود. این ماتریس با روش هم مکانی برای تبدیل حل این مسئله به حل یک دستگاه از معادلات جبری، استفاده می شود. مثال هایی برای نشان دادن کاربرد روش حاضر، آورده می شود.

    کلیدواژگان: معادلات دیفرانسیل تاخیری کسری، روش هم مکانی، تابع تیلور کسری، مشتق کاپوتو، ماتریس عملیاتی
  • عباس زیوری کاظم پور*، اباصلت بداغی صفحات 601-608

    فرض کنید A و B دو جبر باناخ و B یک A-مدول باشد. دراین مقاله تحت شرایط خاص ثابت می کنیم که هرشبه (n+1)-همریختی  ژوردان f یک شبه (n)-همریختی ژوردان است و هر هرشبه (n)-همریختی ژوردان یک (n)-همریختی ژوردان است.

    کلیدواژگان: n-همریختی، n-همریختی ژوردان، شبه n-همریختی ژوردان
  • صدیقه شرفیان، علیرضا سهیلی*، عبدالساده نیسی صفحات 609-620

    قیمت گذاری اختیارات نقش بسیار مهمی در کنترل و مدیریت ریسک دارد. بحث قیمت گذاری نیازمند فرآیند مدل سازی، روش های حل و اجرای مدل با داده های واقعی در یک بازار بررسی شده است. در این مقاله در نظر داریم یک مدل برای دارایی پایه مبتنی بر مدل های تصادفی کسری که نوع خاصی از رفتار تغییرات دارایی های تصادفی است را بیان کنیم. علاوه بر آن یک روش عددی مبتنی بر توابع پایه شعاعی ارایه می دهیم که جواب های دقیق تری نسبت به روش های بررسی شده دیگران دارد. پایداری این روش نیز بررسی می شود. سرانجام مدل حاصل را بر داده های واقعی بازار سکه با استفاده از نرم افزار متلب اجرا می کنیم. امید است با مطالعه این مقاله یک رویکرد جدیدی در قیمت گذاری مشتقات در مطالعات بازارهای آن کشور صورت گیرد.

    کلیدواژگان: مشتق کسری، معادله بلک شولز کسری، روش توابع پایه شعاعی
  • اکبر طیبی*، مراد بهادری، حسن صادقی صفحات 621-630

    در این مقاله با استفاده از مفهوم مترهای متقارن کروی، مترهای λ -هم ارز تصویری را به عنوان تعمیمی طبیعی از مترهای هم ارز تصویری تعریف می کنیم. سپس، مثال های غیربدیهی از مترهای λ-هم ارز تصویری ارایه می کنیم. فرض کنید F و  F  دو متریک λ-هم ارز تصویری روی منیفلد M باشند. ابتدا رابطه بین ژیودزی های F و  F   را به دست می آوریم.  سپس ثابت می کنیم که هر ژیودزی ازF  مضربی از یک ژیودزی F  می شود و برعکس. در انتها ثابت می کنیم که مترهای داگلاس، مترهای ویل و مترهای داگلاس- ویل تعمیم یافته همگی پایاهای λ-هم ارز تصویری هستند.

    کلیدواژگان: پایای تصویری، متر مسطح تصویری، مترهای هم ارز تصویری، متر داگلاس، متر ویل، متر داگلاس- ویل تعمیم یافته
  • فرشته فروزش*، فرهاد سجادیان، مهتا بدرود صفحات 631-644

    در این مقاله، به معرفی توپولوژی معکوس در BL - جبرها می پردازیم و فشردگی، هاسدورف، {0}_T و  {T_{1بودن توپولوژی معکوس روی مجموعه همه فیلترهای اول مینیمال  BL- جبر A بررسی می شود. هم چنین ظریفتر بودن توپولوژی زاریسکی از توپولوژی معکوس روی (Min (A  نشان داده می شود. در این راستا، ملاحظه می شود که تحت شرایطی این دو توپولوژی روی  (Min (A معادل می شوند. در پایان، با معرفی مفهوم کم- توسیع نشان داده می شود نگاشت بین فیلترهای اول مینیمال آنها نسبت به دو توپولوژی معکوس و زیر فضایی زاریسکی، پیوسته است.

    کلیدواژگان: فیلترهای اول مینیمال، توپولوژی زاریسکی، توپولوژی معکوس، کم- توسیع. رده بندی ریاضی (2010): uxx55 و 25G03
  • قدرت الله فصیحی رامندی*، شاهرود اعظمی صفحات 645-660

    در این مقاله ابتدا مفاهیم مقدماتی منیفلد سایا را یاد آوری می کنیم بعد شار ریچی-بورگویگنون که تعمیمی از شار ریچی و شار یامابه است را روی منیفلدهای سایا معرفی می کنیم. سپس  با استفاده از میدان برداری دیتورک معادله شار ریچی-بورگویگنون روی منیفلدهای سایا  را به معادله دیگری تحویل یافته می کنیم که خطی سازی این معادله دیفرانسیل با مشتقات جزیی اکیدا سهموی است و با  قضایای معادلات دیفرانسیل سهموی  با مشتقات جزیی نشان می دهیم که  تحت شرایطی شار ریچی-بورگویگنون روی منیفلدهای سایا با شرط آغازین دارای جواب است و این جواب یکتا است. هم چنین، در نهایت نشان می دهیم که هر جواب از شار ریچی-بورگویگنون روی منیفلدهای سایا بسته (فشرده و بدون مرز) خود متشابه است و سالیتون متناظر با آن سالیتون مانا است.

    کلیدواژگان: شار هندسی، سالیتون، منیفلد سایا، خود-متشابه
  • محسن قاسمی*، رضوان ورمزیار صفحات 661-670

    یک گراف را 1-منظم گوییم هرگاه گروه خودریختی های آن به صورت منظم روی کمان ها عمل کند. در این مقاله گراف های 1-منظم از ظرفیت چهار و مرتبه  که در آن  یک عدد اول است، رده بندی شده است.

    کلیدواژگان: گراف های s - انتقالی، گراف های متقارن، گراف های کیلی.
  • حنیف میرزایی* صفحات 671-686

    ارتعاشات سیستم های مختلف مانند جرم-فنر، نخ کشسان، میله و غیره به صورت مسئله مقدار ویژه ماتریس ژاکوبی مدل بندی می شوند. مسئله تعیین ماتریس ژاکوبی با استفاده از داده های طیفی معلوم، مسئله مقدار ویژه معکوس ماتریس ژاکوبی گفته می شود. در این مقاله، ماتریس ژاکوبی  با استفاده از دو طیف و یک داده اضافی بازسازی می شود. یکی از طیف ها مقادیر ویژه ماتریس  و طیف دیگر مقادیر ویژه زیر ماتریس حاصل از حذف هم زمان دو سطر و ستون ماتریس  است. شرایط لازم و کافی روی داده های طیفی را برای حل پذیری مسئله معکوس ارایه کرده و الگوریتم هایی برای تعیین ماتریس ژاکوبی  ارایه می گردد. سرانجام با ارایه چند مثال عددی ماتریس ژاکوبی و سیستم جرم- فنر متناظر بازسازی می شوند.

    کلیدواژگان: مسئله مقدار ویژه معکوس، ماتریس ژاکوبی، داده های طیفی، سیستم جرم-فنر
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  • Hamideh Aram*, R. Khoeelar, Nasrin Dehgardi Pages 501-508
  • Mahmod Afshari*, Saeed Tahmasbi, Shabnam Shadman Pages 509-520
    Introduction

    The data obtained from observing a phenomenon over time is very common. One of the most popular models in time series and signal processing is the Autoregressive moving average model (ARMA). If the investigated time series has long memory, autoregressive fractional moving average model or in other words (ARFIMA), would be appropriate. The ARFIMA (p, d, q) model was first introduced by [1]. Classic methods for modeling, inference and estimating these processes lead to complex calculations of covariance structure and likelihood functions that make data processing difficult.Wavelet transform is one of the most powerful tools in analyzing such functions and best performs these functions from different time and location  perspectives as well as high and low frequencies. Wavelet transform due to the decreasing correlation property is a very efficient method in analysis and inference for estimating long-term memory processes.The values ​​obtained from wavelet transforms for long-term memory processes, in spite of the complex covariance structure of these processes, the wavelet coefficients are almost uncorrelated and thus much easier to handle [2]. The dense covariance structure of such processes makes it difficult to accurately calculate the maximum likelihood function of data sets [3]. In these cases, the Bayesian method can be easily used to calculate wavelet coefficients. In this paper, while briefly introducing wavelet transform in section two, the ARFIMA model and covariance matrix structure of this model is investigated in section three.  In section four, theBayesian estimation of ARFIMA parameters based on wavelets are calculated. At the end section, ‎we survey the theoretical outcomes with numerical computation by using simulation to described purpose estimation.

    Material and methods

    In this scheme, first we explain wavelet transformation, ARFIMA model and covariance matrix structure of this model. By using wavelet decomposition, Bayesian estimation of ARFIMA model parameters are calculated. The performance of purpose estimation is assessed with simulated data for comparing with respect to another estimators.

    Results and discussion

    We discuss in detail wavelet transformation and autoregressive fractional  moving average model with long memory. The structure of covariance matrices of wavelet coefficients and Bayesian wavelet estimation of parameters are investigated.
    At the end we used simulation study to examine our proposed estimation. Notice that, obtained results confirm that proposed estimation is better than another.

    Conclusion

    The following conclusions were drawn from this research.Wavelet transform due to the decreasing correlation property is very efficient method in estimating long-term memory processes.  The main purpose of this paper is to provide a new wavelet estimation of ARFIMA model parameters via Bayesian method.The main characteristic of this method is that it can be easily used and therefore many calculations are reduced.The proposed method can be applied for estimating of parameters in the simulation.According to the figures, we conclude   that Bayesian wavelet estimation of autoregressive process parameters is appropriate and better than with respect to other estimations.According to the table, by increasing the sample size, standard error of proposed estimator is decreased, so it was shown that the new proposed method is better with respect to others.

    Keywords: Wavelet transformation, Wavelet coefficients, Long term memory, Bayesian inference
  • Ehsan Anjidani* Pages 521-526

    ‎‎‎‎In ‎this ‎paper, ‎we ‎investigate ‎the ‎subadditivity ‎of ‎functions ‎on positive ‎operators ‎without ‎operator ‎monotonicity ‎and ‎operator ‎convexity: Let ‎‎$‎A‎$ ‎and ‎$‎B‎$ ‎be positive operators on a Hilbert space ‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎$‎‎mathcal{H}‎$ ‎satisfying‎ ‎‎$‎0leq AB+BA‎$. Suppose that for the operator ‎‎‎‎$$‎E=(A+B)^{-frac{1}{2}}left(A^2+B^2right)(A+B)^‎{‎-frac{1}{2}}‎,$$‎‎ the open interval ‎$‎(m_E,M_E)‎$ where, ‎$‎m‎_E$ ‎and ‎‎$‎M_E‎$ ‎are ‎bounds ‎of ‎operator ‎‎$‎E‎$‎,‎ ‎does ‎not ‎intersect ‎the ‎spectrums ‎of ‎operators ‎‎$‎A‎$ ‎‎and ‎‎$‎B‎$‎.‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎ Then, for every continuous function ‎‎‎‎$‎g:(0,infty)‎rightarrow‎‎mathbb{R}^+‎$ ‎for ‎which ‎the function‎ ‎‎$‎f(t)=frac{g(t)}{t}‎$ is convex and decreasing, we have ‎‎‎ ‎‎$‎‎$‎g(A+B)leq c(m,M,f)(g(A)+g(B)),‎$‎‎$‎‎‎ where, ‎$‎m‎$ ‎and ‎‎$‎M‎$ ‎are ‎bounds ‎of ‎operator ‎‎$‎A+B‎$ ‎and‎‎‎‎‎ ‎‎$‎‎$‎‎c(m,M,f):=max_{mleq tleq M}left{frac{‎frac{f(M)-f(m)}{M-m}t+‎frac{Mf(m)-mf(M)}{M-m}}{f(t)‎}right}‎.‎$‎‎$‎.

    Keywords: Matrix order, subadditive operator inequality, monotone function, convex function, Jensen’s operator inequality. functions, Linear Algebra Appl. 465 (2015) 401–411
  • Babolian, Fatemeh Chitsaz, Ali Davari* Pages 527-540

    The main aim of this paper is to apply the Chebyshev polynomials for the solution of the linear Fredholm integro-differential-difference equation of high orders. It is usually difficult to analytically solve this equation. Our approach consists of reducing the problem to a set of linear equations by means of the matrix relations between the Chebyshev polynomials and their derivatives. The operational matrices of delay and derivative together with the Tau method are then utilized to evaluate the unknown coefficients of Chebyshev expansion of the solution.  The convergence analysis is studied. Illustrative examples show the validity and applicability of the presented technique. Also, a comparison is made with existing results. 

    Introduction

    The integro-difference equations arise in different applications such as biological, physical and engineering problems. In recent years, there has been a growing interest in the numerical treatment of the integro-differential-difference equations. Since the mentioned equations are usually difficult to solve analytically, numerical methods are required. Several numerical methods were used such as successive approximation method, Adomian decomposition method, the Taylor collocation method, Haar wavelet method, Legendre wavelets method, wavelet-Galerkin method, monotone iterative technique, Walsh series method, etc. In this work, we develop a framework to obtain the numerical solution of the s-order linear Fredholm integro-differential-difference equation with variable coefficients. under the mixed conditions whereand are known continuous functions. Here, the real coefficients and  are given constants. Our approach consists of reducing the problem to a set of linear equations by expanding the solution  in terms of Chebyshev polynomials. The operational matrices of delay and derivative are given. These matrices together with the Tau method utilized to evaluate the unknown coefficients of expansion. The Tau method has been originally proposed by Lanczos for ordinary differential equations and extended by Ortiz. The method consists of expanding the required approximate solution as the elements of a complete set of orthogonal polynomials. Recently there have been several published works in the literature on the applications of the Tau method.

    Conclusion

    This paper deals with the solution of linear Fredholm integro-differential-difference equations of high order with variable coefficients. Our approach was based on the Chebyshev Tau method which reduces a linear Fredholm integro-differential-difference equation into a set of linear algebraic equations. Numerical results show that this approach can solve the problem effectively. The approach, with some modifications, can be employed to solve differential-difference equations and Fredholm integro-differential equations.

    Keywords: Differential-difference equation, Fredholm integro-differential-difference equation, Tau method, Operational matrix, Chebyshev polynomials
  • Rohollah Bakhshandeh Chamazkoti* Pages 541-548
    Introduction

    ‎  Symmetries of an equation are closely related to conservation‎ laws‎. ‎Noetherchr('39')s theorem provides a method for finding conservation laws of differential‎ equations arising from a known Lagrangian and having a known Lie symmetry‎.An algorithmic method to determine conservation laws for systems of‎ Euler-Lagrange equations which their Noether symmetries are known is  Noether‎ theorem‎. ‎This theorem relies on the availability of a Lagrangian and the‎ corresponding Noether symmetries which leave invariant the action integral‎. One can find the geodesic equations  from the variation of the geodesic‎ Lagrangian defined by the given metric and since the Noether symmetries‎ are a subgroup of the Lie group of Lie symmetries of these equations‎, ‎one should expect‎ a relation of theNoether symmetries of this Lagrangian with the projective‎ collineations of the metric or with its degenerates‎‎. ‎Recently Bokhari et al‎. have published many papers about the relation of the Noether symmetries‎ and Lie point symmetrys  and conservation laws of some special spacetimes‎.‎‎Tsamparlis and Paliathanasis have calculated the Lie point symmetries and the Noether symmetries explicitly‎ together with the corresponding linear and quadratic first integrals‎ for the Schwarzschild spacetime and the Friedman Robertson Walker (FRW)‎ spacetime‎. ‎More than they succeeded in establishing a connection between the Lie‎ symmetries of the geodesic equations in a Riemannian‎ space with the collineations of the metric‎. ‎

    Results and discussion

    In this paper, Lie and Noether symmetries and conservation laws for the hyperbolic model metric of SL(2, R) geometry in a Riemannian space are obtained. Then the point symmetries of the one parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian that means Noether symmetries are found and then the conservation laws associated to the system of geodesic equations are calculated via Noether’s theorem.

    Conclusion

    The following conclusions were drawn from this research.The Lie point symmetry algebra of the hyperbolic model metric SL(2, R) geometry ‎has six dimensions which would include the Noether symmetries‎, have an additional basis element  which is dilation in the arclength‎.The Lie point symmetry  commutes with all other symmetries‎. As a more discussion in Lie algebra analysis‎, ‎one may find the Lie algebra of‎ Lie point symmetry corresponding to the metric is not semisimple or not solvable‎, because it has degenerated Killing form‎. There are five conserved flows corresponding to the Noether symmetries of given metric.

    Keywords: Conservation laws‎, ‎Lie point symmetry‎, ‎LagrangianRiemannian‎ space, Noether theorem‎, ‎Hyperbolic model metric
  • Samira Poukhajouei, Sare Goli, Amir Hashemi* Pages 549-562
    Introduction

    Border bases are a generalization of Gröbner bases for zero-dimensional ideals which have attracted the interest of many researchers recently. More precisely, border bases provide a new method to find a structurally stable monomial basis for the residue class ring of a polynomial ideal and this yields a special generating set for the ideal possessing many nice properties.Given a finite set of points, finding the set of all polynomials vanishing on it (so-called either ideal of points or vanishing ideal of the set of points) has numerous applications in several fields in Mathematics and other sciences. In 1982, Buchberger and Möller proposed an algorithm to compute a Gröbner basis for an ideal of points. This algorithm proceeds by performing Gaussian elimination on a generalized Vandermonde matrix. In 2006, Farr and Gao presented an incremental algorithm to compute a Gröbner basis for an ideal of points. The main goal of their paper is to calculate a Gröbner basis for the vanishing ideal of any finite set of points under any monomial ordering, and for points with nontrivial multiplicities they adapt their algorithm to compute the vanishing ideal via Taylor expansions.The method of border bases is a beneficial tool to obtain a set of polynomial models identified by experimental design and regression. The utilization of Gröbner bases theory in experimental design was introduced by Pistone and Wynn. However, using Gröbner bases we cannot find all possible models which form structure of an order ideal for an experiment. For example, if we consider the design {(-1,1),(1,1),(0,0),(1,0),(0,-1)}, the model {} cannot be computed by Gröbner bases method. This fact is expected this method relies on monomial orderings.

    Material and methods

    In this paper, we first present the Buchberger-Möller and Farr-Gao algorithms and then by applying these algorithms, we describe an algorithm which computes a border basis for the ideal of points corresponding to the input set of points with nontrivial multiplicity. In addition, we focus on presenting different models related to an experiment by using the concept of monomial bases for the residue class ring of a polynomial ideal.

    Results and discussion

     As we mentioned earlier, Buchberger-Möller algorithm is an efficient algorithm to compute a Gröbner basis for an ideal of points. We describe a simpler presentation of this algorithm in which we use the function NormalForm which receives as input a linear polynomial p and a Grbner basis G = { , . . . , } of linear polynomials in , . . . ,  and returns f and q=[, . . . ,  ] where f is the remainder of the division of p by G and p=+· · ·++r. Furthermore, we compare the efficiency of this algorithm with the function VanishingIdeal of Maple.  Given a finite set of points, we consider the case in which some points in the set have nontrivial multiplicity. Based on the Farr-Gao algorithm, we prepare an algorithm that computes a border basis for the vanishing ideal of the finite set of points by using Taylor expansions. Suppose that n is the number of factors in an experiment. An experimental design is a finite set  of points. The set of all polynomials vanishing at the design is called a design ideal. Regression analysis is a useful statistical process for the investigation of relationships between a response (or dependent) variable and one or more predictor (or independent) variables. When there is more than one predictor variable in a regression model, the model is a multiple linear regression model which we can call polynomial model. Suppose a random sample of size n is given (then we have exactly n data points are observed from (Y,X). The expession error? is the model for multiple linear regression where chr('39')s are called slopes or regression coefficients. Also, representing the merged effects of the predictor variables on the response variable is called interaction effect. By using multiple linear regression, we can analyze models containing interaction effects. For example, let us consider the following model +error. By substituting  and  , we have a multiple linear regression as follows +error. In addition, multiple R-squared or R2 is a statistical measure that states the square of the relationship between the predicted response value and response value. It should be noted that multiple R-squared is always any value between 0 and 1, where a value closer to 1 informing that a greater proportion of variance is computed for the model. Statistically, a high multiple R-squared shows a well-fitting regression model. Also in multiple regression, tolerance is used as an indicator of multicollinearity. Tolerance may be said to be the opposite of the coefficient of determination and is obtained as . All other things equal, researchers desire higher levels of tolerance, as low levels of tolerance are known to affect adversely the results associated with a multiple regression analysis. The smaller the tolerance of a variable, the more redundant is its contribution to the regression (i.e., it is redundant with the contribution of other independent variables). In the regression equation, if the tolerance of any of the variables is equal to zero (or very close to zero), the regression equation cannot be evaluated (the matrix is said to be ill-conditioned in this case, and it cannot be inverted).

    Conclusion

    The following conclusions were drawn from this research.We present a simpler variant of Buchberger- Möller algorithm (which seems to be easier for the implementation issue) for computing a border basis for an ideal of points.
     We present an algorithm that incrementally computes a border basis for the vanishing ideal of any finite set of points in which some points have multiplicity.We provide good statistical polynomial models which are more suitable for practical applications due to the stability of border bases models compared with Gröbner bases models.

    Keywords: Border basis, Ideal of points, Buchberger-Möller algorithm, Experimental design, Regression
  • Hassan Piriaei, Gholamhossein Yari*, Rahman Farnoosh Pages 563-578

    Introduction  :

     This paper is concerned with using the Maximum Likelihood, Bayes and a new method, E-Bayesian, estimations for computing estimates for the unknown parameter, reliability and hazard rate functions of the Generalized Inverted Exponential distribution. The estimates are derived based on a conjugate prior for the unknown parameter. E-Bayesian estimations are obtained based on three different prior distributions of the hyper parameters. Asymptotic behaviors of E-Bayesian estimations and relations among them have been discussed. The results are computed based on type-II censoring and squared error loss function. Finally, a comparison among the Maximum Likelihood, Bayes and E-Bayesian estimation methods in different sample sizes are made, using the Monte Carlo simulation, which shows that the new method is more efficient than other old methods and is easy to operate.

    Method

    Suppose the Generalized Inverted Exponential distribution and its unknown parameter, reliability and hazard rate functions. Then the estimates of functions of interest are derived based on type II censored samples of this distribution, using the Monte Carlo simulation. 

    Results :

     Results show that the E-Bayesian method is more efficient than other old methods and is easy to operate. Also, the asymptotic behaviors of three E-Bayesian estimations are the same.

    Keywords: E-Bayesian estimation, Type-II censoring, Reliability, Hazard Rate, Monte Carlo simulation
  • Ghorbanali Haghiighatdoost* Pages 579-586

    The theory of topological classification of integrable Hamiltonian systems with two degrees of freedom due to Fomenko and his school‎. ‎On the basis of this theory we give a topological Liouville classification of the integrable Hamiltonian systems with two degrees of freedom‎. ‎Essentially‎, ‎to an integrable system with two degrees of freedom which is restricted to a nonsingular 3-dimensional iso-energy manifold. Fomenkochr('39')s theory ascribes in an effective way a certain discrete invariant which has the structure of a graph with numerical marks‎. ‎This invariant‎, ‎which is called the marked molecule or the Fomenko-Zieschang invariant‎, ‎gives a full description (up to Liouville equivalence) of the Liouville foliation for the system.The topological classification of integrable Hamiltonian systems corresponding to the Liouville equivalence in potential fields on surfaces of revolution for surfaces that is diffeomorphic with 2-dimensional sphere, contains a wide classes of mechanical systems that describes the motion of a particle on a 2-dimensional sphere with revolution metric, which has been studied.In this paper, the topology of non-singular iso-energy surfaces for a Hamiltonian system with two degrees of freedom on a cone located in a potential field is described. Also, the method of finding the topological invariant of integrable Hamiltonian systems is extended from compact case to non-compact rotating surfaces.

    Keywords: Hamiltonian System, Iso-energy Surfaces, Fomenko-Zieschang invariant, Potential field
  • Parisa Rahimkhani, Yadollah Ordokhani* Pages 587-600
    Introduction

    Fractional calculus has been used to model physical and chemical processes that are found to be best described by fractional differential equations. Recently, fractional calculus has attracted much attention since it plays an important role in many fields of science and engineering. Fractional delay differential equations (FDDEs) are a class of  fractional differential equations that  the rate of change of unknown function depends not only on the values of unknown function for the same time value but also on previous time values. The solution of delay differential equations not only requires information of current state, but also requires some information about the previous state. FDDEs have received considerable recent attention and been proven to model many real life problems. For most of fractional order delay differential equations, exact solutions are not known. Therefore different numerical methods have been developed and applied for providing approximate solutions. The objective of this paper is to define the new fractional-order piecewise functions based on the Taylor  polynomials for solving the FDDEs. This method is accurate and easy to implement in solving FDDEs.

    Material and methods

    In this paper, first we construct piecewise functions based on the fractional-order Taylor functions. Then, we calculate the  fractional integral operational matrix for the fractional Taylor piecewise functions. This matrix and collocation method  are utilized to reduce FDDEs to a system of algebraic equations which can be solved via a suitable numerical method.

    Results and discussion

    We apply mentioned paper for solving some test problems to highlight the significant features of our technique. Also, we compare our numerical results with multiquadric approximation scheme. The reported results demonstrate that there is a good agreement between approximate solution and exact solution. We plot the numerical solutions obtained by the presented method for various values of α with the exact solution.  It is obvious from these Figures  that, as α is close to integer value, numerical solutions converge to the exact solution.

    Conclusion

    The following conclusions were drawn from this paper. Fractional-order Taylor piecewise functions  have three degrees of freedom (m, n, α)  but Taylor polynomials have one degree of freedom (m).Instead of converting fractional-order Taylor piecewise functions  into other functions, we have obtained the  fractional-order Taylor piecewise functions operational matrix directly. The main characteristic of this method is that it reduces under study problem to a system of algebraic equations which can be easily solved by an iterative method.

    Keywords: Fractional delay differential equations, Collocation method, Fractional-Taylor function, Caputo derivative, Operational matrix
  • Abbas Zivari Kazempour*, Abasalt Bodaghi Pages 601-608

    Let A and B be Banach algebras and B be a right A-module. In this paper, under special hypotheses we prove that every pseudo (n+1)-Jordan homomorphism f:A----> B is a pseudo n-Jordan homomorphism and every pseudo n-Jordan homomorphism is an n-Jordan homomorphism.

    Keywords: n-homomorphism, n-Jordan homomorphism, Pseudo n-Jordan homomorphism
  • Sedighe Sharifian, Ali R. Siheili*, A. Neisy Pages 609-620
    Introduction

    Fractional Differential Calculus (FDC) began in the 17th century and its initial discussions were related to the works of Leibniz, Lagrange, Abel and others. In recent decades, the fractional differential equations have been considered in different fields such as fluid flow, electromagnetics, engineering, economics and finance. In the early 1970s, Black and Scholes introduced their famous model for pricing option. This model is one of the most popular models in the financial market and plays an important role in determining the price of a high-risk asset in financial modeling field. In this equation, researchers seek to obtain the option value by numerical or analytical methods or to extract new pricing models that reflect the real financial market. The Black-Scholes equation is based on some assumptions which caused constraints in the market. Some advanced models such as jump-diffusion, stochastic interest rate, and stochastic volatility models have been proposed to remove these constraints. Wang and Meng (2010) show that the distribution of stock returns have long-range dependence property which is not consistent with the classical Black -Scholes equation assumptions. This led us to use the ​​fractional modeling in this paper which is derived by applying the fractional specifications of stock market suggesting by Mandelbrot (1963). So, in this paper, we reach to the fractional Black-Scholes model by replacing fractional Brownian motion instead of standard Brownian motion in the classical Black-Scholes equation. The fractional Black-Scholes equation gives better solutions than classical Black-Scholes model to our data which their distribution of stock price depend on long-range. So, in this paper, we solve the fractional Black- Scholes equation and use the combination of radial basis functions and finite difference methods to solve the fractional Black-Scholes equation. This method is flexible because it does not depend on the position of points, and in comparison with other methods, it has a short run time in high dimensions.

    Material and methods

    In this paper, we reach to the fractional Black-Scholes equation by using the fractional underlying asset that follows the fractional Brownian motion. The model represents the price behavior of an European option. It is based on the stochastic behavior of underlying asset which is priced by fractional models. We also apply the radial basis function method to solve this model. In this method, we do not necessarily need to have points with equal distances and the convergence rate can be exponential. Therefore, this method can provide more acceptable solutions than the other numerical methods. It should be noted that the fractional derivative of the pricing function is approximated by the Caputo fractional derivative. The stability of the proposed method has also been studied.

    Results and discussion  :

    In this paper, we provide the numerical results for the fractional Black-Scholes equation on real data of the coin option by radial basis function method. We receive the data of working days from 95/10/01 to 96/01/06 from Tehran Stock Exchange website by Excel software and uploaded them to MATLAB software format. The parameters required by the model are also estimated by using data. We obtain option price for different α by using these parameters and present the results in figures. Figures show that if the price of the underlying asset is lower than the exercise price, the option price decreases by the increasing of the fractional order (α), which means that if the holder chooses not to exercise the contract, he will incur less loss. Increasing α plays an important role in decreasing option price, if the option price decreases, the increasing α will make the purchaser incur less loss. On the other hand, when α increases, if the underlying asset price is above the strike price, the call option price increases, and the holder will make a gain by exercising it. We also predict the price of the coin for the next 5 days by different α. The results are presented in a table and are compared with real price of the market, which show that the method is efficient and the fractional Black-Scholes equation has better performance than the classical Black-Scholes equation.

    Conclusion

    The purpose of this paper is to model the fractional Black -Scholes equation and to solve it by radial basis function method. First we modeled the fractional Black-Scholes equation by using fractional underlying asset and then we solved this equation by radial basis function method. Finally, the efficiency of this method is shown by using the real data of the coin option. The numerical results present that the method is efficient, and the fractional Black-Scholes equation performs better than the classical Black- Scholes equation .The existence of α in the fractional Black -Scholes equation has drawn the prices closer to the real price.

    Keywords: Fractional derivative, Fractional Black-Scholes equation, Radial basis function method
  • Akbar Tayebi*, Morad Bahadori, Hassan Sadeghi Pages 621-630
    Introduction

       In this paper, by using the concept of spherically symmetric Finsler metric, we define the notion of -projectively related metrics as an extension of projectively related metrics. We construct some non-trivial examples of -projectively related metrics. Let F and   be two -projectively related metrics on a manifold M.  We find the relation between the geodesics of F and   and prove that any geodesic of  F is a multiple of a geodesic of   and the other way around. There are several projective invariants of Finsler metrics, namely, Douglas metrics, Weyl metrics and generalized Douglas-Weyl curvature. We prove that the Douglas metrics, Weyl metrics and generalized Douglas-Weyl metrics are -projective invariants.

    Material and methods

    First we obtain the spray coefficients of a spherically symmetric Finsler metric. By considering it, we define -projectively related metrics which is a generalization of projectively related Finsler metrics. Then we find the geodesics of two -projectively related metrics. We obtain the relation between Douglas, Weyl and generalized Douglas-Weyl curvatures  of two -projectively related metrics.

    Results and discussion

    We find the Douglas curvature, Weyl curvature and generalized Douglas-Weyl curvature of two -projectively related Finsler metrics. These calculations tell us that these class of Finsler metrics are -projective invariants.

    Conclusion

    The following conclusions were drawn from this research. We prove that the Douglas curvature, Weyl curvature and generalized Douglas-Weyl curvature are -projective invariants. Let F and   be two -projectively related metrics on a manifold M.  We show that F is a Berwald metric if and only if  is a Berwald metric..

    Keywords: Projective invariant, Projectively flat metric, Projectively related metrics, Douglas metric, Weyl metric, Generalized Douglas-Weyl metric
  • Fereshteh Forouzesh*, Farhad Sajadian, Mahta Bedrood Pages 631-644

    In this paper, we introduce Inverse topology in a BL-algebra A and prove the set of all minimal prime filters of A, namely Min(A) with the Inverse topology is a compact space, Hausdorff, T0  and T1-Space. Then, we show that Zariski topology on Min(A) is finer than the Inverse topology on Min(A). Then, we investigate what conditions may result in the equivalence of these two topologies. Finally, we define min-extension in  BL-algebra  and  show that the mapping on Min(A) with respect to both the Zariski and the Inverse topology is continuous.

    Keywords: minimal prime filter, zariski topology, inverse topology, min-extension
  • Ghodratallah Fasihi Ramandi*, Shahroud Azami Pages 645-660
    Introduction

    After pioneering work of Hamilton in 1982, Ricci flow and other geometric flows became as one of the most interesting topics in both mathematics and physics. In the present paper, firstly, we summarize some introductory concepts about contact manifolds. Then, the notion of Ricci-Bourgoignon flow as a generalization of Ricci and Yamabe flows is introduced. Using De Turck vector field, the equation of Ricci-Bourgoignon flow has been reduced to another equation which its linearization is a strictly parabolic partial differential equation. According to theory of partial differential equation, we have showed that for ρ< and a given initial condition the Ricci-Bourgoignon flow has a unique solution for a short time. Finally, we show that every solution of Ricci-Bourgoignon flow on a closed (compact without boundary) contact manifold is self-similar and the corresponding soliton is steady.

    Material and methods

    In this scheme, first we summarized some basic concepts on contact manifolds. Then, equation of Ricci-Bourgoignon flow on contact manifolds is introduced. Using De Turck vector filed and theory of PDE’s, short time existence and uniqueness solution for such equation is obtained.

    Results and discussion

    We obtained a condition for which Ricci-Bourgoignon flows with initial condition have a unique solution for a short time. Also, our results show that every solution of Ricci-Bourgoignon flow on a closed contact manifold is self-similar and the corresponding soliton is steady.

    Conclusion

    The following conclusions were drawn from this research.Short time existence and uniqueness theorem for Ricci-Bourgoignon flow examined in this paper.Our results showed that solutions of this equation on a closed contact manifold are self-similar and their corresponding solitons are steady.Regardless of the dimension of underlying contact manifold, we showed that for ρ< the Ricci-Bourgoignon flow with given initial condition has a unique solution for a short time.

    Keywords: Geometric flow- Soliton- Contact manifold- self-similar solution.n
  • Mohsen Ghasemi*, Rezvan Varmazyar Pages 661-670

    A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, tetravalent one-regular graphs of order 11p2, where p is a prime, are classied.

    Keywords: s-Transitive graphs, ‎‎Symmetric graphs, Cayley graphs
  • Hanif Mirzaei* Pages 671-686
    Introduction

    Many problems in sciences and engineering can be studied by mathematical models. These models are classified as direct problems and inverse problems. In the structural vibrations, analysis and estimation of the behavior of system e.g. response of the system for an external force and natural frequencies from the known physical parameters is called a direct problem. Determination or estimation of the system physical parameters such as density, mass, stiffness and cross sectional area from the behavior of the system is called an inverse problem. A class of inverse problems which physical parameters determined from the spectral data (eigenvalues, eigenvectors, or both) is called inverse eigenvalue problem. There are many systems such as mass- spring system, vibrating Rods and Beams which are modeled as an eigenvalue problem. Free vibrations of a mass- spring system and discretization of a rod and Sturm-Liouville equations lead to Jacobi matrix eigenvalue problem. Inverse eigenvalue problem for Jacobi matrix is determination of entries using some spectral data. Different algorithms have been presented for constructing a Jacobi matrix. In this paper, we construct a Jacobi matrix and the corresponding mass-spring system using some new spectral data.

    Material and methods

    We try to construct a Jacobi matrix from two spectra and one extra data. For this purpose, using given spectral data, we find the required data of well-known Lancsoz method. Then applying Lancsoz method, we construct a positive definite Jacobi matrix. Finally, according to the relations between Jacobi matrix, mass and stiffness matrices, we obtain corresponding mass-spring system.

    Results and discussion

    Necessary and sufficient conditions on given spectral data for solvability of the inverse eigenvalue problem are presented.We find two algorithms for constructing positive definite Jacobi matrix and the corresponding mass-spring system.We solve some examples using the given algorithms. There is a good agreement between the spectral data of constructed matrix and initial given data.

    Conclusion

    The following results are obtained from this research.We find two algorithms for constructing a Jacobi matrix using two spectra and one extra data.It is observed that, for a set of spectral data, there might be exist more than one solution.It seems that, one may extend the method of this paper for matrix eigenvalue problem which arise in discretization of vibrating rod using finite element method.

    Keywords: Inverse eigenvalue problem, Jacobi matrix, Spectral data, Mass-Spring system