Let  R  e a Noetherian local ring, and let  C  e a semidualizing  R-module. In this paper, we present someresults concerning the vanishing of  Ext  and finite injective dimension of  Hom. Additionally, we extendthese results in terms of finite  C-injective dimension of  Hom. We also investigate the consequencesof some of these extensions in the case where  R  is Cohen–Macaulay and  C  is a canonical module forR. Furthermore, we provide positive answers to the Auslander–Reiten conjecture for finitely generatedR-module  M  uch that  IC  -idR(HomR(M, R)) < ?or  M ?AC  (R)  with  IC  -idR(HomR(M, M)) < ?.Moreover, we derive a number of criteria for a semidualizing  R-module  C  to be a canonical module for  Rin terms of the vanishing of Ext and the finite C-injective dimension of Hom.

Let $R$ be a commutative, Noetherian, Local ring and $\\mathfrak{a}$ , $\\mathfrak{b}$ are parameters ideals of $R$ such that $\\mathfrak{a}\\subseteq\\mathfrak{b}.$ thus $\\Hom_R(R/\\mathfrak{a},R/\\mathfrak{b})$ is a free module over $R/\\mathfrak{a}$ of rank one.Now let $M$ be a finited generated $R$-module. in this work, we study the structure of such modules of homomorphisms $\\Hom_R(R/\\mathfrak{a},M/\\mathfrak{b}M)$ that $M$ is not Cohen-Macaulay. Our main Results start with small dimension then we generalize to higher dimensions.\\textbf{Keywords}:\\textit{System of parameters, Depth, Dimension, Cohen-Macaulay, Parameter ideal, Indecomposable module, Torsion submodule, Torsion functor}  

   In this thesise,we stady the commuting graph,the power graph and the enhance power graph of agroup G that are denoted respectively Com(G),Pow(G) .and EPow(G).Furthermor,we introduce a new graph that is called the deep commuting graph of the group G   The vertex set of these graphs are the element of G and two elements of G are joined in the deep commuting graph if the pre -image of these   .elements.commute in each central extension of G      It is proved that deep commuting graph of G is between the commuting graph and the enhance power gra ph.       :Key word   .Deep commuting,Schur multiplier,Central extensio  

   Abstract    In the last two decades, autonomous vehicles have received a widespread use and attention. In this thesis, navigation of such machines is investigated. Path planning is one of the important part of this navigation. Duo to the long-term use of roads and lack of their maintenance, the roads which autonomous vehicles need to pass have tracks. In these cases, while an autonomous vehicle passes through these cracked areas at high speed, it will increase the sense of bump and even deviate from the originally planned route. Further, this may potentially cause vehicle damage. In this thesis, an adjustable speed navigation method, called Bat-Pigeon algorithms is investigated. First a review on heuristic and meta-heuristic algorithms is presented. After that, Bat-algorithm and Pigeon-algorithm for optimization are studied. Next, an image processing technique is introduced and by using of that process a combination of Bat- and Pigeon- algorithms, that is Bat-Pigeon-algorithm, is investigated for navigation and path planning of autonomous vehicles.       Keywords: Heuristic algorithm, Meta-heuristic algorithm, Swarm-intelligence optimization, Bat-algorithm, Pigeon-algorithm, Autonomous vehicle, Bat-Pigeon algorithm, Path planning, Road track detection, Image processing.   

  In this thesis, We study and analyze high-order Runge–Kutta convolution quadrature (RKCQ)methods for obtaining the numerical solution of nonlinear fractional integro-differential equations(FIDEs) with weakly singular kernels. Wefirst study the existence and uniqueness of solutions for the original problem. Then, the convergence and stability results of the RKCQ method are obtained. Finally, some numerical experiments are reported to illustrate the effectiveness of the proposed schemes.

  مدولهاي درون منظم موضوع بسياري از مقالات در طول شصت سال گذشته بوده كه فوچز اين سوال را مطرح كرد كه كدام گروه آبلي درون منظم هستند. گلاز و ويكلس در [19[ و رنگسومي در [30 [به اين سوال براي طبقات بزرگي از گروههاي آبلي پاسخ دادند. اما مسئله همچنان باز است. وير ? در [3? [جز اولين كساني بود كه روي مدول هاي درون منظم روي حلقه هاي دلخواه بررسي كرد، او بيشتر بر مدول تصويري تمركز داشت. لي و همكارانش، بعدها در [23 [تحقيقات كلي تري در مورد مدولهاي درون منظم انجام دادند. حلقه هاي منظم يكه يكطرفه و مدولهاي درون منظم يكه يكطرفه، براي اولين بار توسط ارليچ در [10 ،11 [مورد مطالعه قرار گرفت. لي و ژانگ در [38 [در مورد اين موضوع توضيح دادند. همچنين مدولهاي درون منظم قوي توسط لي و همكارانش در [23 ،38 [مورد بحث قرار گرفتند كه اين مدول ها را “مدولهاي درون منظم آبلي” ناميدند. [همچنين گلاز و ويكلس در [19 ،([نتايجي در مورد ايده آل درون منظم(براي مثال، گروه هاي آبلي ) ثابت كرد. در مطالب پيش رو هدف بررسي سه مورد خواهد بود. ابتدا چندين نتيجه كلي در مورد اشكال مختلف “درون منظم” را ثابت خواهيم كرد، كه بر تئوري توسعه يافته قبلي بسط داده شده است. سپس بسياري از نتايج شناخته شده در مورد “درون منظم” در گروههاي آبلي را به مدولها روي حلقه هاي جابه جايي با طيف نوتري گسترش خواهيم داد. در نهايت تعميم مفيدي از مدولهاي درون منظم (روي حلقه هاي جابهجايي) را تعريف مي كنيم كه آن را مدولهاي درون منظم ضعيف مي ناميم و بسياري از ويژگيهاي اين مدولها را بررسي خواهيم كرد

  ‎In this thesis‎, ‎we study the anticenter(Lie-center) of Leibniz algebras and give several concepte that are related to this notion‎. Also give some bounds on the dimension of hypercenter of a Liebniz algebra.‎‎ Furthermore, we study Lie-central extensions and we obtain a six-term exact sequence of Lie-homology groups associated to a Lie-central extension. This allow us to characterize Lie-stem ‎extensions,‎ stem-covers and Lie-capable Liebniz algebras.

  ‎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.

In this thesis, we have examined the necessary and sufficient conditions for a group ring to be weakly nil-neat.  The first chapter contains the primitive concepts and definitions. In the seconds chapter, we define nil-clean group ring and weakly nil-clean group ring. The main purpose of the third chapter is to examine the weakly nil-clean group ring.

  We study the relationship between depth and regularity of a homogeneous ideal I and those of (I, f ) and I : f , where f is a linear form or a monomial.

In this thesis three numerical methods based on cubic ‎and‎ sextic B-spline for numerical solution nonlinear singular boundary value problems considered. First method based on uniform mesh is of second-order and second method based on non-uniform mesh is of forth-order and third method based on sextic B-spline is of seventh-order. In this thesis the proposed methods not only approximate the solution but also approximate this derivatives and error analysis and convergence of these methods have been analyzed. The end eight linear and nonlinear examples are given to show the applicability and performance methods which show the applicability and performance of the proposed methods.‎  

در اين پايان نامه برخي از روش ها از مرتبه دو را براي چهار نوع اسپلاين انتگرالي درجه دو بررسي شده است.ثابت شده است كه اسپلاين انتگرالي درجه دو داراي همگرايي در تقريب مقدار تابع و تقريب مشتقات مرتبه دوم در نقاط ميان بازه اي يكنواخت هستند.و همچنين بي اسپلاين درجه دوم براي درون يابي يك تابع جلو انتگرال با استفاده از مقادير معلوم انتگرال در زير بازه ها به جاي مقادير تابع در گره ها استفاده مي شود .اين درون يابي اسپلاين انتگرالي درجه دو ناميده مي شود

The main point of this thesis is to prove that therer are several serier. 

   ‎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.‎

  Let G be a finite group and H be a subgroup of G. The Chermak-Delgado measureof H with respect to G is defined as mG(H) =|H| |CG(H)|. The set of all subgroupsof G with maximal Chermak-Delgado measure, denoted CD(G), is a sublattice of thelattice of all subgroups of G. In this thesis, we prove that if H   CD(G) then H issubnormal in G and prove if K is a finite group then CD(G K) = CD(G) CD(K).Also, we describe CD(G?Cp) where G has a non-trivial center and p is an odd primeand determine conditions for a wreath product to be a member of its own Chermak-Delgado lattice. Finally, we characterize the structure of finite groups G whoseChermak– Delgado lattice is the interval[G=Z(G)] = {H   L(G)|Z(G) H   G}.

  A ring R is called left comorphic if, for each a ? R, there exists b ? R such that Ra = l(b)and r(a) = bR. Examples include (von Neumann) regular rings, and Z p n for a prime p and n ? 1.

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.  

  AbstractThe concept of a module M being almost N-injective, where N is some module, was introduced by Baba (1989). For a given module M, the class of modules N, for which M is almost N-injective, is not closed under direct sums. Baba gave a necessary and sufficient condition under which a uniform, finite length module U is almost V-injective, where V is a finite direct sum of uniform, finite length modules, in terms of extending properties of simple submodules of V. Let M be a uniform module and V be a finite direct sum of indecomposable modules. Some conditions under which M is almost V-injective are determined, thereby Baba’s result is generalized. A module M that is almost M-injective is called an almost self-injective module. Commutative indecomposable rings and von Neumann regular rings that are almost self-injective are studied. It is proved that any minimal right ideal of a von Neumann regular, almost right self-injective ring, is injective. This result is used to give an example of a von Neumann regular ring that is not almost right self-injective.Keywords: Almost injective module, Direct sum, Finite length module, Uniform module, Indecomposable module, Self-injective module, Indecomposable ring, Von Neumann regular rin

  This thesis studies vanishing of Ext modules over Cohen–Macaulay local rings. The main result of this thesis implies that the Auslander–Reiten conjecture holds for maximal Cohen–Macaulay modules of rank one over Cohen–Macaulay normal local rings. It also recovers a theorem of Avramov– Buchweitz–S¸ega and Hanes–Huneke, which shows that the Tachikawa conjecture holds for Cohen–Macaulay generically Gorenstein local rings. This conjecture is widely open in general, even for modules overcommutativeNoetherianlocalrings. Oversuchrings,weprovethatalargeclass of ideals satisfy the extension condition proposed in the aforementioned conjecture of Auslander and Reiten. Keywords: Auslander-Reitenconjecture,Cohen-Macaulyring,Normalring,Gornestein ring, Free module, Weakly m-full and Ext module.

   A cubic trigonometric B-spline collocation approach for the numerical solution of fractional sub-diffusion equation is presented in this paper. The approach is based on the usual finite difference scheme to discretize the time derivative while the approximation of the secondorder derivative with respect to space is obtained by the cubic trigonometric B-spline functions with the help of Grünwald–Letnikov discretization of the Riemann–Liouville derivative.

AbstractIn this thesis, we will consider two numerical methods to approximate of solution of theExtended Fisher–Kolmogorov equation. Both methods under study are >methods, At first a nonlinear high-order difference scheme will be described to solve theExtended-Fisher-Kolmogorov equation . Existence and uniqueness conditions of the solutionwill be analyesed, by utilizing the energy method we proved that the convergent order in maximumnorm is two in temporal direction and four in spatial direction. Solving of numericalresults verifed the theoretical results. In addition a second-order three-level linearly implicit finitedifference method will be studied for solving the extended Fisher–Kolmogorov equation inboth 1D and 2D . The existence and uniqueness of the proposed scheme is proved. In additionBy verifying the convergence and stability of the method, proved that method is second-orderconvergent both in time and space variables, and the method is almost unconditionally stable  

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.  

‎A ring with unity is called ni-clean if every element can be expressed as sum of a nilpotent and an idempotent‎. ‎In this thesis‎, ‎we characterize the nil clean matrix rings over fields‎, ‎in fact‎, ‎we prove that for a field $F$ the ring $M_n(F)$ is nil-clean if and only if $F\\cong \\  {Z}_2$‎. ‎As an application‎, ‎we obtain a complete characterization of the finite rank Abelian groups‎ ‎with nil clean endomorphism ring‎. ‎ ‎For a finite commutative ring $R$‎, ‎the nil-clean graph $G_N(R)$ is a simple graph such that the vertex set is the ring $R$ and two ring elements $a$ and $b$ are adjacent if $a+b$ is nil clean in $R$‎. ‎Graph theoretic properties like girth‎, ‎dominating set‎, ‎diameter etc‎. ‎of nil clean graph have been studied.  

A numerical scheme for a fractional sub-diffusion problem using parametric quantic spline

  ‎Let $G$ be a finite group and $\\gamma(G)$ and $l(G)$‎‎denote the number of conjugacy 0px; MARGIN: 0px; -qt-block-indent: 0; -qt-user-state: 0">‎subgroups of $G$ and the number of conjugacy 0px; MARGIN: 0px; -qt-block-indent: 0; -qt-user-state: 0">‎non-normal non-nilpotent subgroups of $G$‎, ‎respectiverly‎. ‎In this thesis‎, ‎we prove that if $G$ is a solvable group‎, ‎then‎ ‎$\\gamma(G) \\geq 2^{|\\pi(G)|-2}$ and if $G$ is a non-solvable‎‎group‎, ‎then $l(G) \\leq |\\pi(G)|$ and $\\gamma(G) \\geq |\\pi(G)|+1$‎, ‎where $|\\pi(G)|$ is the number of prime devisors of $|G|$‎. ‎Also‎, ‎we give a classification of all groups that equality holds in former relation.

‎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.

‎Stochastic differential equations ‎(SDEs) ‎and ‎random‎ differential equations ‎(‎RDE‎s)‎ have been used for many years in a wide range of applications‎. ‎Since these equations dont   have exact ‎solutions‎‎, ‎their numerical solutions have attracted lots of researchers‎. ‎Although classical numerical schemes for ‎ordinary‎ differential ‎equations‎‎   can be often used pathwise for RDEs‎, ‎they rarely attain their traditional order‎.A big class of these equations are Affine ‎random‎ differential equations which have many applications‎. ‎New forms of higher order Taylor-like schemes and also some multi-step ‎methods‎ for RDEs are derived for ‎this‎ kind of equations‎.In the following   ‎numerical schemes based on ‎Itô expansion for ‎RDEs that are driven by an Itô diffusion process‎, ‎i.e‎. ‎the solution of an Itô SDE are considered‎. ‎Under some ‎conditions‎ their convergence will be proposed and some numerical examples are included to confirm ‎the ‎theoretical result‎.

One of the continuity conditions identified by Utumi on self-injective rings isthe $C_3$-condition, where a module $M$ is called a $C_3$-module if whenever $A$ and $B$ are directsummands of $M$ and $A \\cap B = 0$, then $A \\oplus B$ is a summand of $M$. In addition to injectiveand direct-injective modules, the 0px; TEXT-INDENT: 0px; -qt-block-indent: 0">indecomposable and regular modules. Indeed, every commutative ring is a $C_3$-ring. Inthis thesis provide a general and unified treatment of the above mentioned 0px; TEXT-INDENT: 0px; -qt-block-indent: 0">modules in terms of the $C_3$-condition, and establish new characterizations of several wellknown classes of rings.

  ‎Nonlinear ‎monotone ‎system ‎of ‎equations ‎is ‎one ‎of ‎the ‎most ‎important ‎problems ‎in ‎applied ‎mathematics ‎where ‎arise ‎in ‎various ‎applications ‎such ‎as ‎subproblems ‎in ‎the ‎generalized ‎proximal ‎algorithms ‎with ‎Bregman ‎distance. ‎There ‎are ‎many ‎various ‎methods ‎to ‎solve ‎this ‎problem, ‎such ‎as ‎Newtons ‎method,quasi‎-newton method and modified version of them. The important weakness of these methods especially for large scale problems is the need to calculate Jacobian matrix in each iteration and solving the corresponding ‎system ‎of linear equations.‎‎projection based algorithms are one of the efficient methods for solving the monotone nonlinear system of equations. In this thesis, Motivated by some conjugate gradient method and the ‎pro‎jection ‎technique ‎two ‎new ‎families ‎of ‎projection ‎based ‎methods ‎is ‎provided ‎to ‎solve ‎nonlinear ‎monotone ‎system ‎of ‎equations. ‎The ‎obtained ‎numerical ‎results ‎show ‎that ‎the ‎methods ‎are ‎effective ‎and ‎efficient.‎‎‎‎

  Let R be a commutative Noetherian local ring and E the minimal injective cogenerator of the category of R-modules. An R-module M is (Matlis) reflexive if the natural evaluation map M ?? HomR(HomR(M, E), E) is an isomorphism. We prove that if S is a multiplicatively closed subset of R and M is an Rsmodule which is reflexive as an R-module, then M is a reflexive Rs-module. The converse holds when S is the complement of the union of finitely many nonminimal primes of R, but fails in general. Let (R, m) denote a local ring with E = ER(R/m) the injective hull of the residue field. Let p ? SpecR denote a prime ideal with dim R/p = 1, and let ER(R/p) be the injective hull of R/p. As the main result we prove that the Matlis dual HomR(ER(R/p, E) is isomorphic to Rcp, the completion of Rp, if and only if R/p is complete. In the case of R a one dimensional domain there is a complete description of Q ?R Rb in terms of the completion Rb. Keywords: Matlis reflexive, Injective hull, Completion, One dimensional domain, Matlis duality, Minimal injective cogenerator

  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

  In this thesis‎, ‎a counterexample ‎is‎‎ ‎given‎‎ to the persistence and non-increasing depth properties‎. ‎M‎oreover sequentially cohen-macaulay graphs ‎are‎‎ ‎considered‎‎ (by adding whiskers)‎. ‎Also‎ ‎we introduce the polarization of koszul cycles and use it to form a basis for the koszul homology of   is polarization of the monomail ideal $I$ in the polynomail ring S=K[ , … , ).‎ ‎Then we study the depth function of powers of edge ideals of whisker graphs‎. ‎We also express the realation between persistence property and depth stability. In ‎fact,‎ for a connected finite simple graph G‎ we show that dstab (I(G)   < l (I(G)) . ‎For trees we give a stronger bound for dstab(I(G))‎. ‎We also shaw for any two integers 1 ? a < b there exists a tree for which dstab(I(G)) = a and l(I(G))=b.