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 مدل شكارچي-شكار با دفاع القايي و بيماري در شكار كه از ديدگاه تكامل زيستي و اكو-اپيدميولوژي ساخته شده است، مورد مطالعه قرار مي‌گيرد. هدف اصلي اين پايان نامه بررسي تأثير بيماري بر پايداري جمعيت در مدل شكارچي-شكار با دفاع القايي است. ابتدا، مثبت بودن و كرانداري يكنواخت جواب‌هاي مدل اشاره شده اثبات مي‌شود. سپس، وجود و پايداري نقاط تعادل، بخصوص نقاط تعادل همزيستي شكار و شكارچي مورد مطالعه قرار مي‌گيرد. در اين مدل حداكثر نه نقطه تعادل وجود دارد. از يك تبديل پارامتري پيچيده براي بررسي ويژگي‌هاي نقاط تعادل همزيستي مدل استفاده مي‌كنيم. همچنين شرايط كافي براي وجود انشعاب هاپف بدست مي‌آيد. براي تحليل انشعاب هاپف و تعيين جهت سيكل‌هاي حدي در مدل شكارچي-شكار با دفاع القايي و بيماري، ضريب اول لياپانوف در مقادير بحراني پارامتر انشعاب محاسبه مي‌شود. در خاتمه شبيه‌سازي‌هاي عددي براي تكميل نتايج تحليلي انجام مي‌شود  

We introduce the notion of coretractable modules. A module $M$ is said to be coretractable if for every nonzero factor module $M/N$ (where $N \\leq M$), there exists a nonzero $R$-homomorphism $f: M/N \\to M$. We prove that all right (left) modules over a ring are coretractable if and only if the ring is Morita equivalent to a finite product of local right and left perfect rings.  

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}  

   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.

   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  

  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.

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.

  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 $k=1,2,\\ldots \\lceil\\frac{n}{2}\\rceil-1$‎.

   Let T be a triangular algebra over a commutative ring R, _ be an automorphism of T and Z_(T ) be the _-center of T . Suppose that q : T _ T ??! T is an R-bilinear mapping and that Tq : T ??! T is a trace of q. Our aim is to describe the form of Tq satisfying the commuting condition [Tq; x]_ = 0 (resp. the centralizing condition [Tq; x]_ 2 Z_(T ) for all x 2 T . More precisely, we will consider the question of when Tq satisfying the previouse condition has the so-called proper form. We provide sufficient conditions for each centralizing trace of aribitrary mappings on a triangular algebra to be proper and apply this result to describe the centralizing traces of bilinear mappings on certain 0 (resp. the centralizing condition [Tq; x]_ 2 Z_(T ) for all x 2 T . More precisely, we will consider the question of when Tq satisfying the previouse condition has the so-called proper form. We provide sufficient conditions for each centralizing trace of aribitrary mappings on a triangular algebra to be proper and apply this result to describe the centralizing traces of bilinear mappings on certain classical traingular algebras.

A group of stiff problems, their eigenvalues are separated in to two clusters, one contaning the ”stiff” or fast components and one contaning the ”nonstiff” or slow. By using special exponential fitting techniques we develop a h?s. We obtain the size of their stability regions as a function of the order and the fitting condition. We also obtain condition that the coefficients of these methods must satisfy to have a given stiff order for the Prothero-Robinson test equation. Finally, we construct an embedded pair of stability fitted methods of orders 2 and 1 and show its performance by means of several numerical experiments. Key words: Stiff problems; Explicit Runge-Kutta methods; Exponential fitting; Gap in the eigenvalue spectrum.       A group of stiff problems, their eigenvalues are separated in to two clusters, one contaning the ”stiff” or fast components and one contaning the ”nonstiff” or slow. By using special exponential fitting techniques we develop a h?s. We obtain the size of their stability regions as a function of the order and the fitting condition. We also obtain condition that the coefficients of these methods must satisfy to have a given stiff order for the Prothero-Robinson test equation. Finally, we construct an embedded pair of stability fitted methods of orders 2 and 1 and show its performance by means of several numerical experiments. Key words: Stiff problems; Explicit Runge-Kutta methods; Exponential fitting; Gap in the eigenvalue spectrum.    A group of stiff problems, their eigenvalues are separated in to two clusters, one contaning the ”stiff” or fast components and one contaning the ”nonstiff” or slow. By using special exponential fitting techniques we develop a h?s. We obtain the size of their stability regions as a function of the order and the fitting condition. We also obtain condition that the coefficients of these methods must satisfy to have a given stiff order for the Prothero-Robinson test equation. Finally, we construct an embedded pair of stability fitted methods of orders 2 and 1 and show its performance by means of several numerical experiments. Key words: Stiff problems; Explicit Runge-Kutta methods; Exponential fitting; Gap in the eigenvalue spectrum.   

Encompassing many standard notions such as Dedekind finite and reversible rings we introduce and study a new property for subsets of a ring. We give many examples and characterize some rings such as 2-primal rings with the aid of this notion.In ‎addition‎, We study some properties relatedto zero divisors and reversibility in noncommutative rings .  

self-dual and cyclic codes over noncommutative rings

   ‎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.‎

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

  ‎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 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.

  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

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

در اين پايان ­نامه، كاربرد مدل­ هايبيز ناپارامتري در وظايف پردازش زبان طبيعي مورد مطالعه قرار داده شده­ اند. ابتدا روش ­هايبيز ناپارامتري براساس رايج ­ترين توزيع پيشين يعني فرايند ديريكله مورد مطالعهقرار گرفته­ اند. سپس نمايش­ هاي متفاوت از فرايند ديريكله مانند طرح كيسه پوليا،فرايند رستوران چيني و ساختار استيك بريكينگ معرفي شده ­اند. در ادامه به معرفي دو فرايند توليد شده توسط فرايند­هاي ديريكلهيعني فرايند­هايديريكله­ سلسله مراتبي و فرايند­هاي پيتمن يور پرداخته شده است. در پايان 4 راه­ حلپيشنهادي بيز ناپارامتري در وظايف پردازش زبان طبيعي از جمله تقسيم­ بندي كلمه،استخراج عبارت و صف­ بندي، تجزيه­ مستقل از متن و مدل­سازي زبان ارائه شده ­اند.

  ‎‎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.

  An element a of a ring R is nil-clean, if a = e + b, where e2 = e ? R and b is a nilpotentelement, andthering R iscallednil-cleanifeachofitselementsisnil-clean. In [22], it was proved that, for a commutative ring R and an abelian group G, the group ring R[G] is nil-clean, i? R is nil-clean and G is a 2-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally ?nite 2-group over a nil-clean ring is nil-clean, and that the hypercenter of the group G must be a 2-group if a group ring of G is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, i? the ring is a nil-clean ring and the group is a 2-group.Keywords:Nil-clean ring, nil-clean group ring

  Optimal designs have an important role in designing the experiments and help the experimenter to do the experiment in a shorter time and with lower costs. In order to find the optimal designs, one has to consider the optimality criteria which is usually a function of Fisher Information Matrix. In the present thesis, the optimal designs for regression models with interval-valued data are obtained. These data are in fact, observations that are not accurately measurable and are reported as intervals. In this thesis, linear regression models fit these data and an optimal design for them is obtained.

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.  

  ‎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.

  In this theisis, we describe an interactive procedural algorithm for convex multi- objective programming based upon the Tchebyche? method, Wierzbicki’s reference point approach, and the procedure of Michalowski and Szapiro. At each iteration, the decision maker (DM) has the option of expressing his or her objective-function aspirations in the form of a reference criterion vector. Also, the DM has the option of expressing minimally acceptable values for each of the objectives in the form of a reservationvector. Baseduponthisinformation, acertainregionisde?nedforexam- ination. In addition, a special set of weights is constructed. Then with the weights, the algorithm of this paper is able to generate a group of e?cient solutions that provides for an overall view of the current iteration’s certain region. By modi?cation of the reference and reservation vectors, one can ‘‘steer” the algorithm at each itera- tion. From a theoretical point of view, we prove that none of the e?cient solutions obtained using this scheme impair any reservation value for convex problems. The behavior of the algorithm is illustrated by means of graphical representations and an illustrative numerical example. we carry out an extension of the MICA method (modi?ed interactive chebyshev algorithm) for non-convex multiobjective programming. This method is based on the Tchebychev method and in the reference point approach. At each iteration, the decision maker (DM) can provide aspiration levels (desirable values for the objec- tive functions) and also, if the DM wishes, reservation levels (level under which the objective function is not considered acceptable). On the basis of this preferential in- formation, a region of the nondominated objective set is de?ned. In the convex case, considering the aspiration vector as a reference point in an achievement scalarizing function and taking a set of weight vectors, the e?cient solutions generated satisfy the reservation levels. In this work, we analyze the non-convex case. The main re- sult of MICA is veri?ed and demonstrated for the non-convex bi-objective case. The MICA method is not veri?ed in general for multiobjective problems with three or more objective functions, which is demonstrated with a counterexample. we concentrate on reference point based methods in multiobjective programming todemonstrate, asmaincontribution, thatthesolutiontoamultiobjectiveoptimiza- tion problem stays unchanged if the reference point is changed to any point on a set de?nedbymeansoftheoriginalreferencepoint,thenondominatedobjectivesolution and some parameters of the ASF. Concretely, this new set of “equivalent reference points” is the convex linear combination of two straight lines, one containing the original reference point and the other a nondominated objective solution, where the slope of both straight lines is given by the inverses of the weights of the ASF. An illustrative example is used to show the results obtained and an empirical model (application with real data) allows us to highlight possible implications.

‎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 $\\overline{G}$.

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

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.

  This‎ thesis is devoted to the study ‎of‎ properties and ‎‎‎irre‎d‎ucible‎ representations of ‎Le‎ibniz ‎algebras‎.‎ We use the liezation method to translate some fandamental theorems of Lie ‎‎algebras‎‎ ‎such‎ ‎as‎‎ the Engels theorem, the Levi decomposition, ... to Liebniz ‎algebras.‎ Also, we prove that any irreducible representation of ‎a‎ Leibniz algebra can be obtained from an irreducible representation of the ‎semi-‎simple Lie algebra from the Levi decomposition. As an application, we determine the irreducible representations of sl‎2.‎‎‎

  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 .

  در سالهاي اخير متيو فضاهاي متريك جزيي وشبه متريك و شبه متريك وزن دار را معرفي كرد . ايشان به ارتباط بين آنها پرداخت وبيان كرد كه تحت چه شرايطي ميتوان فضاي متريك جزيي وشبه متريك را يه يك متر تبديل كرد. يكي از اهداف پايان نامه اين است كه با استفاده از يك نوع خاص از انقباض ضعيف معرفي شده توسط چريچ وبا كمك گرفتن از توابع كنترل ناپيوسته قضييه نقطه ثابت باناخ را به فضاهاي متريك جزيي گسترش  مي دهيم.

‎In this thesis‎, ‎we compare the abelian subalgebras and ideals of maximal‎‎dimension in some 0px; MARGIN: 0px; -qt-block-indent: 0; -qt-user-state: 0">‎abelian subalgebras of solvable Lie algebras and study the 0px; MARGIN: 0px; -qt-block-indent: 0; -qt-user-state: 0">‎an abelian subalgebra of codimension 2‎. ‎Also‎, ‎we prove that nilpotent Lie algebras with‎‎an abelian subalgebra of codimension 1,2 or 3 contain an abelian ideal with the same dimension‎.‎Furthermore‎, ‎we investigate‎ ‎the structure of Lie algebras with a core-free‎ ‎maximal subalgebra and use them to introduce the conceptes of crown and pre-crown for chief factors‎.

  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

  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.