اين پايان نامه رويكردي مبتني بر روش هم‌محلي با استفاده از بي-اسپلاين‌هاي درجه پنجم براي حل عددي معادلات كورت‌وگ-دو فريس مرتبه بالا (هفتم) پيشنهاد مي‌شود. گسسته‌سازي درجهت زمان از طريق يك روش تفاضل متناهي انجام شده و براي تقريب در جهت مكان از تابع بي-اسپلاين درجه پنجم، و مشتقات آن استفاده شده است. اين تحقيق در نظر دارد مسائل مقدار مرزي مرتبه هفتم را به دستگاه مرتبه پايين\\/تر (چهارم) تبديل نمايد وسپس با استفاده از تابع بي-اسپلاين درجه پنجم جواب تقريبي بدست آورد، همچنين در اين پژوهش ابتدا طريقه بدست آوردن توابع بي-اسپلاين‌ها را بررسي مي‌شود و سپس چگونگي محاسبه مشتقات اين توابع مورد مطالعه قرار مي گيرد. در نتيجه، معادله به يك دستگاه جبري تبديل مي‌شود كه با استفاده از خواص ماتريس‌ها حل مي‌گردد. پايداري روش پيشنهادي نيز مورد بحث قرار گرفته و مشخص شده است كه به‌طور نامشروط پايدار است. مثال‌هاي عددي حل شده تا دقت و پايداري اين تكنيك را نشان مي‌دهند. خطا با استفاده از نرم‌هاي بي‌نهايت و نرم دو محاسبه شده است. در مقايسه با نتايج پيشين، مشاهدات نشان مي‌دهد كه اين روش نسبت به بسياري از روش‌هاي ديگر موجود در منابع، برتري دارد.   

The geometric-quadratic index (GQ) is a recently introduced degree-based topological descriptor, and Kumar et al. observed that it is potentially a very good molecular descriptor. In this thesis, we characterize the extremal (chemical) graphs and trees with respect to (GQ) for a given order and size. Then, the n-vertex trees, unicyclic and bicyclic graphs with the first, second, third, fourth, fifth, and sixth maximum values of this geometric-quadratic index are determined.  

  For a graph G, the first Zagreb index is defined as the sum of the squares of the vertex degrees, while the second Zagreb index is the sum of the products of the degrees of adjacent vertices. The aim of this paper is to completely characterize n-vertex trees with given k ? 1 vertices that have a fixed maximum degree ? ? 3 with respect to the maximal and minimal Zagreb indices. Furthermore, our results provide detailed insights into the structure of extremal trees and are equally applicable to the class of chemical trees.

 در اين پايان نامه يك تقريب جديد براي معادله انتگرو-ديفرانسيل كسري از هردو نوع ولترا و فردهولم در حالت خطي و غيرخطي ايجاد خواهد شد. عليرغم گام هاي مهمي كه در دستيابي به راه حل هاي عددي كارا و نسبتا دقيق در حل معادلات FIDE ها انجام شده است، همچنان شكاف آشكاري براي توسعه يك روش عددي همه كاره و دقيق كه قادر به حل مسائل متنوع FIDE هاي خطي و غيرخطي با عملگر هاي انتگرال باشد، وجود دارد. براي پر كردن اين شكاف، در اين پژوهش از تكنيك هم مكاني بي اسپلاين مكعبي   به عنوان يك رويكرد قوي و سازگار براي حل طيف گسترده اي از معادلات انتگرو-ديفرانسيل كسري در دو نوع خطي و غير خطي با تركيب كردن عملگر هاي انتگرال ولترا و فردهولم پيشنهاد مي شود. اين روش با به كار گيري ويژگي انعطاف پذيري و كارايي محاسباتي خطوط بي اسپلاين مكعبي، با يك تكنيك يكپارچه راه هاي عددي دقيق تري را ارائه مي كند. از لحاظ   حل پذيري (وجود جواب دستگاه بدست آمده ازگسسته سازي مسئله)، تجزيه و تحليل همگرايي و پايداري   مسئله انجام شده، تاييد بيشتري از دقت و قابليت اطمينان روش هم محلي بي اسپلاين مكعبي را بازگو مي كند كه مي تواند بسيار به حل مسائل FIDE ها با پيچيدگي بيشتر كمك كند. به منظور نشان دادن دقت و كارايي روش   پيشنهادي چند مثال عددي آورده شده و با روش­هاي ديگران كه اين مساله را حل نموده اند و در منابع ذكر شده مقايسه شده است.كلمات كل?د?: معادل? انتگرو-د?فرانس?ل ولترا? كسر?،   معادل? انتگرو-د?فرانس?ل فردهولم كسر?، معادل? انتگرو-د?فرانس?ل كسر?،  حساب كسر?،   بي-اسپلاين مكعب?.

   در اين پژوهش، يك روش نوآورانه براي نهان گزاري تصاوير ديجيتال ارائه شده است كه تركيبي از تبديل كسينوسي گسسته (DCT)، تبديل موجك گسسته سه‌سطحي (3L-DWT) و تجزيه مقدار تكين (SVD) است. اين روش با هدف افزايش امنيت، غيرقابل تشخيص بودن و مقاومت طراحي شده و قابليت استخراج واترمارك بدون نياز به تصوير اصلي (واترمارك‌گذاري كور) را فراهم مي‌كند.مراحل اصلي روش پيشنهادي شامل پيش‌پردازش تصوير واترمارك با استفاده از نقشه آرنولد، اعمال تبديل‌هاي DCT و DWT، و تجزيه SVD است. واترمارك در ضرايب فركانس پايين حوزه تبديل تصوير ميزبان جايگذاري مي‌شود تا مقاومت بيشتري در برابر حملات مختلف داشته باشد.نتايج آزمايش‌ها نشان مي‌دهد كه روش پيشنهادي در برابر حملات مختلف مانند فيلترها، نويز، حملات هندسي و حذف رديف/ستون مقاومت بالايي دارد و عملكرد بهتري نسبت به روش‌هاي موجود از خود نشان مي‌دهد. اين روش همچنين امنيت بالايي را با استفاده از نقشه آرنولد تضمين مي‌كند.روش پيشنهادي غيرقابل تشخيص بودن بهتري را تضمين مي‌كند كه مقدار آن 57.6303 dB است و مقاومت بهبود يافته‌اي در برابر حملات فيلتر، نويز نمك و فلفل (  ) و چرخش نسبت به روش‌هاي پيشرفته موجود ارائه مي‌دهد. براي فيلتر ميانه با اندازه‌هاي پنجره مختلف، مقدار WNC اين روش برابر با 1 است كه بيشتر از روش‌هاي موجود است.اين تحقيق ضمن ارائه يك روش بهبود يافته براي واترمارك‌گذاري تصاوير ديجيتال، پتانسيل كاربرد در حوزه‌هاي مختلفي مانند حقوق ديجيتال، پزشكي و امنيت نظامي را دارد.

We study a multi-group SAIRS-type epidemic model with vaccination. the role of asymptomatic and symptomatic infectious individuals is explicitly considered in the transmission pattern of the disease among the groups in which the population is divided. This is a natural extension of the homogeneous mixing SAIRS model with vaccination studied in Ottaviano et. al(2021) to a network of communities. We provide a global stability analysis for the model. We determine the value of the basic reproduction number R0 and prove that the disease-free equilibrium is globally asymptotically stable if R0<1. In the case of the SAIRS model without vaccination, we prove the global asymptotic stability of the disease-free equilibrium also when R0=1. Moreover, if R0>1, the disease-free equilibrium is unstable and a unique equilibrium exists. First, we investigate the local asymptotic stability of the endemic equilibrium and subsequently its global stability, for two variations of the original model. Last, we provide numerical simulations to compare the epidemic spreading on different networks topologies. 

Higher frame rates are very useful for improving medical diagnosis in fast-moving parts of the heart, especiallyin the valves. To this end, we propose a non-polynomial interpolation method for increasing the frame rate in echocardiography. Besides describing the proposed method, we present two additional contributions: (1) we obtain a closed-form solution, which is continuous and infinitely differentiable; (2) we provide an error analysis of the method. The resulting error bound indicates that the interpolation method is reliable. Finally, to show the efficiency of our proposal in temporal super-resolution, i.e., the increase in frame rate, we apply it to three types of datasets, including a 1D signal, a simulated dataset, and B-mode echocardiography images. Our experimental results show that the Mean Squared Error of the proposed method is reduced from 0.6 to 0.3, while having the same computational complexity compared to cubic B-spline. The quantitative results also indicate that, even with lower selection rates, we can reach a high performance reconstruction while the image quality is not degraded significantly.

     در اين پايان‌ نامه روشي قوي براي افزودن واتر مارك به تصاوير ارائه مي شود كه اصلي‌ترين پايه‌هاي آن شامل تبديل موجك ، تجزيه و تحليل حالت تجربي دو بعدي ، تبديل كسينوسي گسسته ، بهينه‌سازي انبوه ذرات و تجزيه و تحليل مقدار تكين است. در طول فرايند تعبيه،   سطح 2 براي تجزيه تصوير پوششي به زيرباندها استفاده مي‌شود. همچنين،  براي تجزيه تصاوير و علامت‌گذاري استفاده مي‌شود. علاوه بر اين، تجزيه و تحليل   بر روي باند انتخاب شده از  اجرا مي‌شود. درفاز بهينه‌سازي، براي جستجوهاي پيچيده و چند بعدي استفاده مي‌شود. عوامل تعبيه و مقياس‌بندي با كمك يك كليد امنيتي تعبيه مي‌شوند. تصوير واتر مارك از طريق فرايند استخراج به‌دست مي آيد. نتايج آزمايشي نشان مي‌دهند كه تكنيك پيشنهادي نسبت به چندين حمله هندسي(اعمال نويز) و غير هندسي قوي است.

   Today, the emergence of generative models based on machine learning has provided a significant reduction in the bit rate of speech codecs. However, in real conditions and in the presence of destructive factors such as noise and distortion, the above processes face serious problems, which is caused by the sensitivity of the maximum likelihood criterion to outliers, as well as the inefficiency of modeling the sum of independent signals with Autoregressive model is used. In this thesis, a method based on predicting variance regularization is introduced to reduce sensitivity to outliers and thus increase system performance. In addition, it is shown that noise reduction to remove unwanted signals can increase the performance significantly. Also, extensive objective evaluations will be presented, which show that the proposed system based on the generative model provides a new coding performance mode for real-time speech signals at 3 Kbit/s.

This thesis investigate a numerical of the convection-diffusion type’s fourth-order singularly perturbed linear and nonlinear boundary value problems. First, the considered linear fourth-order differential equation is converted into a strongly/weakly coupled singularly perturbed system (depending on the coe?cient of the ?rst-order derivative) of two ordinary differential equations with Dirichlet boundary conditions to solve the problem numerically. One of the equations is free from the perturbation parameter in the system. To obtain the solution for this system, we propose a numerical method of quadratic B-splines on an exponentially graded mesh. Convergence analysis shows that the proposed numerical scheme is second-order uniformly convergent in the discrete maximum norm. The nonlinear differential equation is linearized using the quasilinearization technique, and then the proposed approach is applied to the linearized problem. The theoretical outcomes are validated by executing the proposed method on three test problems.   

  In this thesis, we examine theorems about bifurcation from one-dimensional kernels and generalizations of the Crandell-Rabinowitz theorem. Next, using a bifurcation theorem from one dimensional kernels, the bifurcation in Activator-inhibitor systems are discussed. Also, the effect of nonlocal diffusion on bifurcations and the formation of spatially heterogeneous patterns in the case when the rate of dispersion of the inhibitor is small enough, is study. Unstable steady state solutions and existence of Turing instability for the mentioned nonlocal systems is investigated.

In this study, an efficient numerical scheme based on shifted Chebyshev polynomials is established to obtain numerical solutions of the Bagley–Torvik equation. We first derive the shifted Chebyshev operational matrix of fractional derivative. Then, by the use of these operational matrices, we reduce the corresponding fractional order differential equation to a system of algebraic equations, which can be solved numerically by Newton’s method. Furthermore, the maximum absolute error is obtained through error analysis. Finally, numerical examples are presented to validate our theoretical analysis.Keywords: Bagley–Torvikequation; Chebyshev polynomials; Collocation method; Liouville–Caputo derivative

  In recent years, finding the closest target for the under-evaluation decision making units (DMU) has attracted the attention of researchers significantly, and numerous articles have been published in this field. In some of these articles, the related efficiency measure does not satisfy the strong monotonicity property. Since this property plays a very important role in comparing and ranking units, it is very desirable to present methods that, while finding the closest efficient model, their efficiency measure is strongly monotone. Mainly, researches done in this field are divided into the following two general categories: A) The methods that obtain all of the full dimensional efficient facets or their extended versions and then obtain the distance of the DMU under evaluation to these facets. B) Methods that, instead of obtaining full dimensional efficient facets, using some mixed integer linear programming models, implicitly calculate the distance of the under evaluation DMU to the strongly efficient frontier. In both cases, based on the obtained distance, a well-defined strong efficiency measure is introduced. This thesis investigates these methods in detail using some real numerical results.

  ‎Let $V$ be an $n$-dimensional vector space over the finite field consisting of $q$ elements‎. ‎The Grassmann graph of $V$‎, ‎denoted by $\\Gamma_k(V)$‎, ‎is a simple graph whose vertex set is the set of all $k$-dimensional su  aces of $V$‎, ‎with <k<n‎ -‎1$ and two distinct vertices are adjacent if their intersection is $k-1$-dimentional su  ace of $V$‎. ‎Denote by $\\Gamma(n‎, ‎k)_q$ the restriction of $\\Gamma_k(V)$ to the set of all non-degenerate linear $[n‎, ‎k]_q$-codes‎. ‎In this thesis‎, ‎we show that if $n$ is sufficiently large then there exists pairs of codes whose distances in the graphs $\\Gamma_k(V)$ and $\\Gamma(n‎, ‎k)_q$ are distinct‎. ‎Also‎, ‎one 0px; margin-right: 0px; text-indent: 0px;">‎Among other results‎, ‎it is shown that the induced subgraph of $\\Gamma_k(V)$ on projective $[n‎, ‎k]_q$-codes is connected and its diameter is equal to the diameter of the Grassmann graph and the distance between any two vertices coincides with the distance between these vertices in the Grassmann graph‎. ‎Then we study the graphs of simplex codes‎. ‎Finally‎, ‎we prove that binary simplex codes of dimension $3$ are precisely maximal singular su  aces of a non-degenerate quadratic form‎.

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

In this thesis, we study the theory of the (a,k)-regularized resolvent families on a Banachspace. Thesefamiliesincludewell-knowclasses,suchasC0-semigroups,cosine and resolvent families of bounded linear operators. Inparticular,weprovide new insights on the structural properties of the theories of C0-semigroups, strongly continuous cosine families and ?-resolvent families. Key words: one parameter families of bounded operator, C0-semigroups, Cosine families, (?,?)resolvent families, (a,k)-regularized resolvent family, one-zero law.

   Let nq(k‎, ‎d) be the minimum length n for which an [n‎, ‎k‎, ‎d] q-code exists‎. In coding theory‎, ‎there is a natural lower bound on nq(k‎, ‎d) ‎, ‎the Griesmer bound‎: nq(k‎, ‎d) ? gq(k‎, ‎d) =‎‎i=0k-1[dqi] ‎.‎ ‎In this thesis‎,   ‎a lot of new [n‎, ‎4‎, ‎d] 9-codes whose lengths are close to the Griesmer‎ bound were be constructed‎. ‎Also‎, ‎we prove the nonexistence of some linear codes attaining the Griesmer bound‎ using some geometric techniques through projective geometries to determine the‎ exact value of n9(4‎, ‎d) or to improve the known bound on n9(4‎, ‎d) for given values‎ of d‎. ‎Finally‎, ‎the updated table for n9

the main disadavantages of the newton method are that the computation of the hessian matrix is a difficult 

   ‎A ‎‎$‎‎k‎$‎‎-rose graph is a graph consisting of ‎‎$‎‎k‎$‎ cycles that all meet in one vertex. ‎In this thesis, it‎ is ‎shown‎ that except for two specific examples, these rose graphs are determined by the Laplacian spectrum. ‎Then‎ ‎it‎ is proved that if two rose graphs have a so-called universal Laplacian matrix with the same spectrum, then they must be isomorphic. In memory of Horst Sachs (1927–2016), ‎‎we‎ show the specific case of the latter result for the adjacency matrix by using ‎Sachs' ‎theorem and a new result on the number of matchings in the disjoint union of paths.‎‎ ‎T‎hen a new method to determine the degree sequence of cospectral mates of a graph ‎is‎ introduced. ‎Among‎ other results, we prove that ‎all‎‎ $‎2‎$‎-rose ‎graphs,‎ with one exception, are determined by their signless Laplacian ‎spectrum‎.‎

  ‎In addition to their applications in data communication and storage‎, ‎linear codes also have nice applications in combinatorics and cryptography‎. ‎As a special class of linear codes‎, ‎minimal linear codes have important applications in secret sharing and secure two-party computation‎. ‎Constructing minimal linear codes with new and desirable parameters has been an interesting research topic in coding theory and cryptography‎. ‎Ashikhmin and Barg showed that $\\frac{w_{\\min}}{w_{\\max}}>\\frac{q-1}{q}$ is a sufficient condition for a linear code over the finite field$F_q$ to be minimal‎, ‎where $q$ is a prime power‎, ‎$w_{\\min}$ and $w_{max}$ denote the minimum and maximum nonzero weights in the code‎, ‎respectively‎. ‎The first objective of this thesis is to present a sufficient and necessary condition for linear codes over finite fields to be minimal‎. ‎This condition enables us to‎ ‎obtain several infinite families of minimal linear codes with $\\frac{w_{\\min}}{w_{\\max}}\\leq \\frac{q-1}{q}$‎. ‎The second objective of this thesis is to construct infinite families of binary and ternary minimal linear codes‎, ‎violating Ashikhmin and Barg's condition‎. ‎The weight distributions of all these minimal linear codes are also determined‎.  

Let G be a simple graph with Laplacian spectrum  ????_1???_2???????_(n-1)???_n=0. The Laplacian spread of  G  is defined as S_L (G)=?_1-?_(n-1). In this thesis،the newlower bounds on the Laplacian spread of graphs in terms of invariantparameters of graphs such as bandwidth، independence number and vertex connectivity، are obtained. Then we present a new sharp upper bound for  SL(G)and use it to prove the conjecture that ”S_L (G)?n-1” for  t-quasi-regulargraphs when t??(n-3+2/n)  Among other results، it is shown that this conjecture is true for some special graphs such as triangle-free graphs. Finally، we give several sharp lower bounds for  SL(G) as well.  

  Determining the suitable premium for an insurer is one of the most important categories in the insurance industry. Inmostbonus-malussystems premiumbasedonthenumberofclaims Theclaimamountsarenot taken into accont. In this case, policyholders who had accidents with small or large claims are penalized unfairlyinthesameway. Ateventhepolicyholdersmayleavetheinsurancecompanytogetridoftheirbad history claims. In this thesis, in addition to the number of claims, the amount of claim is also considered. Also, in the malus zone, relative premiums softened by introducing and applying deductible  

  ‎‎Let $G$ be an $n$-vertex graph with $m$ edges, maximum degree $\\Delta$‎, ‎average degree $\\overline{d}=\\frac{2m}{n}$‎ ‎and‎‎ ‎clique number $\\omega$ having‎‎Laplacian spectrum $\\mu_1 \\geq \\mu_2 \\geq\\cdots\\geq ‎\\mu_{n-‎1} \\geq \\mu_n = 0$‎. ‎Let $S_k(G)=\\sum_{i=1}^k\\mu_i$ for every \\leq k \\leq n$‎. ‎Also‎, ‎assume ‎that $\\sigma$‎‎ ‎is ‎the number of Laplacian eigenvalues greater than or equal to average degree‎ ‎$\\overline{d}$‎. ‎In this thesis‎, ‎a lower bound for $S_{\\omega‎ -‎1}$ and an upper bound for $S_\\sigma(G)$ in terms of $m,\\Delta,\\omega$ are obtained‎.‎As an application‎, ‎we obtain the stronger bounds for the Laplacian energy $LE =\\sum_{i=1}^n|\\mu_i-\\frac{2m}{n}|$‎, ‎which improve some well known earlier bounds‎.‎Among other results‎, ‎all connected threshold graphs are characterized‎. ‎A Nordhaus-Gaddum-type result for $\\sigma$ is proved‎, ‎too. ‎Finally‎, ‎some relations between $\\sigma$ with other graph invariants are presented‎.

In this thesis, two spectral upper bounds for the k-independencenumber of a graph which is the maximum size of a set of verticesat pairwise distance greater than k, are obtained. Also, we constructgraphs that attain equality for our first bound and show that our secondbound compares favorably to previous bounds on the k-independencenumber. Among other results, some lower bounds for the chromaticand fractional chromatic numbers of a graph in terms of its interiaare presented. Extremal graphs for this bound are investigated, too.Moreover, it is proved that this bounds are not lower bounds for thevector chromatic number of graph. Finally, some Nordhause-Gaddumtype results are proved.  

  ‎In this thesis‎, ‎using variational methods‎, ‎the existence of weak solutions for two margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎For this purpose‎, ‎using certain conditions‎, ‎it is proved that the energy functional satisfies the Palais-Smale condition‎. ‎Then‎, ‎using the Ambrosetti-Rabinowitz condition and the symmetric Mountain Pass Theorem‎, ‎the existance of two nontrivial solutions for the considered equation is proved‎. ‎Also‎, ‎using Lions theorem‎, ‎the concentration of solutions‎, ‎is investigated‎. ‎As an application of the obtained results‎, ‎the existence and concentration of solutions to an elliptic equation with convex-concave nonlinearity is established‎. ‎Finally‎, ‎using the symmetric Mountain Pass Theorem‎, ‎the existence of infinitely many weak solutions to a system of Schrodinger–Kirchhoff equations of fourth-order is investigated

در اين پايان نامه كرانداري وفشردگي عملگرهاي انتگرال ريمان-ليوويل روي فضاهاي موري را بررسي مي كنيم0مشخصه سازي جواب براي معادله انتگرال آبل بدست اورده وباكمك نتايج قضيه نقطه ثابت وجوديكتايي جواب براي مسئله كوشي ثابت خواهد شد  

‎Let $G$ be an $n$-vertex graph with $m$ edges and with Laplacian spectrum $\\mu_1 \\geq \\mu_2 \\geq‎‎\\cdots\\geq \\mu_{n?1} \\geq \\mu_n = 0$‎. ‎The Laplacian energy is defined as $LE =\\sum_{i=1}^n|\\mu_i-\\frac{2m}{n}|$‎.‎In this thesis‎, ‎all graphs with at most four distinct laplacian eigenvalues are studied‎. ‎Also‎, ‎we use these graphs to obtain some upper and lower bounds for the Laplacian energy of an arbitrary graphs‎.‎Among other results‎, ‎we characterize all such graphs which are bipartite or‎‎have exactly one multiple Laplacian eigenvalue‎.‎Let $\\sigma$ be the‎‎largest positive integer such that $\\mu_\\sigma \\geq \\frac{2m}{n}$‎. ‎The graphs satisfying $\\sigma = n‎ -‎1$ are characterized‎.‎Using this‎, ‎we obtain lower bounds for $LE$ in terms of $n‎, ‎m$‎, ‎and the first Zagreb index‎. ‎In‎‎addition‎, ‎some upper bounds for $LE$ in terms of graph invariants such as $n‎, ‎m$‎,‎maximum degree‎, ‎vertex cover number‎, ‎and spanning tree packing number are presented‎.‎Finally‎, ‎we obtain a relation between Laplacian‎‎energy and Laplacian-energy-like invariant of graphs.

‎Let $G =(V(G)‎, ‎E(G))$ be a simple connected graph with diameter $q$ and $k$ be a positive integer $k$ with \\leq k\\leq q$‎. ‎A radio $k$-coloring of‎ ‎$G$ is a mapping $L‎ : ‎V(G) \\rightarrow \\{0‎, ‎1‎, ‎2,\\ldots\\}$ such that $|L (u)-L (v)| > k+1-d(u‎, ‎v)$‎‎for each pair of distinct vertices $u‎, ‎v \\in V(G)$‎, ‎where $d(u‎, ‎v)$‎‎denotes the distance between $u$ and $v$‎. ‎The span $rc_k(L )$ of $L$ is defined as $\\max_{u\\in V(G)} L (u)$; the‎‎radio $k$-chromatic number $rc_k(G)$ of $G$ is $\\min{rc_k(L )}$ over all radio $k$-colorings $L$ of $G$‎. ‎In‎‎this thesis‎, ‎we give some lower and upper bounds of $rc_k(G)$‎, ‎and discuss the sharpness‎‎of these bounds‎. ‎In some cases the necessary and sufficient‎‎conditions for equality of theses bound are given‎, ‎too‎. ‎As an application‎, ‎we obtain lower bounds of the radio‎‎$k$-chromatic number for the cycles‎, ‎grids‎, ‎cubes‎, ‎cartesian products of cycles with either paths or complete graphs‎. %Moreover, ‎we showthat the lower bound of $rc_k(G)$‎, ‎when $G$ is a cube is an improvement of the existing one‎.‎An integer $h$‎, ‎$GFN2252_LABSTRACT_XMLENCODE# < h < rc_{k} (G)$‎, ‎is a hole in a $rc_k$-coloring on $G$‎‎if $h$ is not assigned by it‎. ‎In this paper‎, ‎we construct a larger graph from a graph of a‎‎certain 0px; TEXT-INDENT: 0px; -qt-block-indent: 0; -qt-user-state: 0">‎holes in any $rc_k$-coloring of a graph‎. ‎Exploiting the same property‎, ‎we introduce a‎‎new graph parameter‎, ‎referred as $(k-1)$-hole index of $G$ and denoted by $\\rho_k (G)$‎. ‎We‎‎also explore several properties of $\\rho_k (G)$ including its upper bound and relation with‎‎the path covering number of the complement $\\overline{G}$.

در اين پايان نامه نخست مفهوم انتگرال كسري ريمانليويل وانتگرال كسري اردلي كوبر بيان مي گردد در ادامه وجود جواب هاب معادلات ديفرانسيل كسري با استفاده از قضاياي نقطه ثابت را بررسي مي كنيم

بررسي كنترل در فضاهاي هيلبرت

  The (sign-less) laplacian spectrum of graphs is extensively studied, however, the value of the least sign-less laplacian eigenvalue of graphs and its lower and upper bounds is less considered. The main aim of this thesis is to investigate and optimize the least sign-less Laplacian eigenvalue of some classes of graphs. A connected graph is called unicyclic if it has the same number of vertices and edges. Here, a graph with the maximum least signless Laplacian eigenvalue among all connected unicyclic graphs with fixed order is determined. Also, some relations between the least sign-less Laplacian eigenvalue of a connected n-vertex graph G and its independence number (covering number) are studied. Amongotherresults, wedeterminethegraphswhichhavethe minimum least sign-less Laplacian eigenvalue among all nonbipartite graphs with given either independence number at least n?1 2 or covering number at most n+1 2 .

    Egienvalue problem is one of the most problems in applied mathematics. Amongall of egienvalues the smallest and largest egienvalues have some special importance.Researchers proposed, many numercal methods to solve this problem. In this thesisthe problem of the largest egienvalue of a symetric matrix, convert to a unconstrainedoptimization. Now, we can get a new algorithm by appling an efficient algorithmto solve the generated unconstrained problem. In this thesis, using of a BarzilaiBorwein-like method is proposed. The numerical experimets show the new methodis useful and efficient.

  DNA cryptography is a new branch of cryptography that utilizes DNA as an in- formational and computational carrier with the aid of molecular techniques. Most of the modern encryption algorithms have been broken fully or partially. The world of information security looks for new directions to protect the data and their transmi- tion. The DNA computing in the ?elds of cryptography has been identi?ed as a new hope to create some unbreakable algorithms. In this thesis two digital signatures are discussed and analaysed, the ?rst method uses DNA coding and XOR operation with a symmetric key and the second uses DNA coding, Polymerase Chain Reaction and RSA encryption.

  Let G be a simple graph with the vertex set V(G), r be a non-negative integer and . If the induced subgrsph on S is r-regular, then some upper and lower bounds for the sum of the squares of the entries of the principal eigenvector corresponding to S are presented. Moreover a spectral characterization of families of split graphs, involving their spectral radius and the entries of the principal eigenvector corresponding to the vertices of the maximum independent set is given. An edge-coloring of a graph G with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of G are distinct and the sum of the colors of the edges of G is minimum. The edge-chromatic sum af a graph G is the sum of the colors of edges in a sum edge-coloring of G. In this thesis, We give a polynomial time -approximation algorithm for the edge-chromatic sum problem on r-regular graphs for . Among the other results, the N-completeness of the edge-chromatic sum problem is studied for bipartite graphs and regular graphs. Finally, some upper bounds for the edge-chromatic sum of some split graphs are given.     

  let‎   ‎ ‎be‎ a flat local ring homomorphism, ‎and‎   ‎two‎ finitely generated -‎‎‎mod‎ule‎‎‎.‎ we investigate the interplay between properties of -module‎‎ and the ascent of module structures along local ring homomorphism. ‎W‎e ‎show‎ if ‎satisfies‎   ‎‎(e.g.‎ if ‎ ‎is‎ finitely generated over )‎‎‎ for‎‎‎‎ ,‎‎ then ‎for‎ all ‎ ‎and‎ ‎ ‎has‎ an S -module‎‎ structure that is compatible with its R -module‎‎ structure ‎via‎

Boundary value problems have important applications in various branches of pure and applied sciences, including astrophysics, structural engineering, optimization, and economics.‎‎In some particular situations itis possible to find a general solution of the equation, but in general it is not possible. In most cases, only approximate solutions can be expected. Accordingly, a large number of methods for the numerical solution of BVPs have been proposed in literature.‎  In this thesis, firstly, initial value problems and boundary value problems and also some methods to expriements numerical solutions of the problem   is studied.  In the sequel, by using bernoulli polynomials and imposing reproducing kernel and least square methids a novel numerical method is proposed for solving boundary value problems.Finally, the numerical expriements show that the new methods is efficient.

  ‎In‎ ‎this‎‎ ‎thesis‎,‎‎ ‎first‎‎ ‎we‎‎ ‎investigate‎,‎ ‎the‎‎ ‎existense‎‎ ‎and‎‎ ‎multiplicity‎‎ ‎o‎‎f‎ ‎positive‎‎ ‎solutions‎‎ ‎to‎‎ ‎an‎ ‎N-‎Laplacian‎‎ ‎equation‎‎ ‎in‎‎‎ ‎$ ‎\\mathbb{R}^{N‎‎}‎‎‎ $‎ ‎with‎‎ ‎singular‎‎ ‎and‎‎ ‎exponential‎‎ ‎nonlinearity‎.‎ ‎To‎‎ ‎do‎‎ ‎this‎,‎ ‎we‎‎ ‎use‎‎ ‎the‎‎ ‎Nehari‎‎ ‎manifold‎ ‎method‎.‎ ‎First‎‎, ‎we‎‎ ‎prove‎‎ ‎that‎‎ ‎‎‎local‎‎‎ ‎minima‎‎ ‎of‎‎ ‎the‎‎‎ ‎energy‎ ‎functional‎‎ ‎i‎n‎‎ ‎the‎‎ ‎Nehari‎‎ ‎manifold‎‎ ‎are‎ ‎critical‎‎ ‎‎points‎‎.‎Then‎ ‎we‎‎ ‎divide‎‎ ‎the‎‎ ‎Nehari‎‎ ‎manifold‎‎ ‎in‎‎to‎‎ ‎three‎‎ ‎sets‎‎ ‎corresponding‎‎ ‎to‎‎ ‎local‎‎ ‎maxima‎‎, ‎local‎‎ ‎minima‎‎ ‎and‎‎ ‎points‎‎ ‎of‎‎ ‎inflection‎‎ ‎of‎‎ ‎fibering‎‎ ‎‎maps,‎‎‎ ‎and‎‎ ‎we‎‎ ‎‎find‎‎‎ ‎local‎‎ ‎minima‎‎ ‎of‎‎ ‎‎ ‎the‎‎‎ ‎energy‎ ‎functional‎ ‎in‎‎ ‎these‎‎‎‎ ‎sets‎.‎ ‎In‎‎ ‎the‎‎ ‎next‎‎ ‎part‎‎‎ ‎of‎‎ ‎this‎‎ ‎thesis‎,‎ ‎we‎‎ ‎study‎‎ ‎the‎‎ ‎existence‎‎ ‎of‎‎ ‎a‎ ‎nontrivial‎‎ ‎solution‎‎ ‎for‎‎ ‎a system‎‎ ‎of‎‎ ‎p‎‎-‎Laplacian‎‎‎ ‎equation‎‎ ‎in‎‎ ‎a‎ ‎bounded‎‎ ‎domain‎‎ ‎and‎‎ ‎under‎‎ ‎the‎‎ ‎Dirichlet‎‎‎ ‎bounda‎ry‎‎ ‎condition‎.‎ ‎For‎‎ ‎this‎‎ ‎problem‎‎‎ ‎we‎‎ ‎prove‎‎ ‎that‎‎‎‎ ‎the‎‎‎ ‎energy‎ ‎functional‎ ‎has‎‎ ‎the‎‎ ‎geometry‎‎ ‎of‎‎ ‎mountain‎ ‎pass‎‎. ‎Then‎‎‎‎ ‎using‎‎ ‎the‎‎ ‎S‎addle‎ ‎P‎oint‎‎ ‎Theorem‎‎‎‎ ‎of‎‎ ‎Rabinowitz‎‎‎‎ ‎and‎‎ ‎a‎ ‎generalization‎‎ ‎of‎‎ ‎the‎‎ ‎Landesman‎-‎Lazer‎‎ ‎condition‎,‎ ‎the‎‎ ‎existence‎‎ ‎of‎‎ ‎a‎‎ ‎nontrivial‎‎ ‎solution‎‎ ‎is‎‎ ‎proved‎.‎