Sharif Digital Repository

We study colorings of quasiordered sets (qosets) with a least element 0. To any qoset Q with 0 we assign a graph (called a zerodivisor graph) whose vertices are labelled by the elements of Q with two vertices x; y adjacent if the only elements lying below x and y are those lying below 0. We prove that for such graphs, the chromatic number and the clique number coincide  

Asdefined by Nicholson [W. K. Nicholson. (1977). Lifting idempotents and exchange rings. Trans. Amer. Math. soc. 229: 269-278] an element of a ring is clean if it is the sum of a unit and an idempotent, and a subset of is clean if every element of is clean. In this thesis some like-cleanness properties are introduced and relations between them are investigated.Many equivalent conditions for cleanness of reduced, semiprimitive Gelfand and functional rings are obtained,and finally these results are applied in the rings of continuous functions  

Grobner bases were introduced by Bruno Buchberger in 1965. The terminology acknowledges the influence of Wolfgang Grobner on Buchberger’s work. He introduced a specific generator for ideals in the ring of polynomials over a field and then gave an algorithm for computing of that generator. It leads to solutions to a large number of algorithmic problems that are related to polynomials in several variables. Most notably, algorithms that involve Grobner basis computations allow exact conclusions on the solutions of systems of nonlinear equations, such as the (geometric) dimension of the solution set,the exact number of solutions in case there are finitely many, and their actual computation with... 

Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebraic properties of the edge ideal from the deck of G, that is, from the collection of subgraphs obtained by removing a vertex from G. These properties include the Krull dimension, the Hilbert function, and all the graded Betti numbers. We also state many further questions that arise from our study  

In this thesis, we study the properties of simplicial complexes Δ with the equality I(m) Δ = Im Δ for a given m _>2. The main results are combinatorial characterizations of such complexes in the two-dimensional case  

We study Beck-like coloring of partially ordered sets (posets) with a least element 0. To any poset P with 0 we assign a graph (called a zero-divisor graph) whose vertices are labelled by the elements of P with two vertices x, y adjacent if 0 is the only element lying below x and y. We prove that for such graphs, the chromatic number and the clique number coincide.Also, we give a condition under which posets are not finitely colorable  

Let be a field and R= [x0, . . . , xn−1] be the polynomial ring in n variables over the field . Let G be a finite simple graph with the vertex set V(G) ={0, . . . , n − 1} and the edge set E(G). One can associate a square-free quadratic monomial ideal I(G) = (xixj | {i, j} ∈ E(G)) of R to the graph G. The ideal I(G) is called the edge ideal of G in R. The graph G is called Cohen–Macaulay (resp. Buchsbaum, Gorenstein) over if the ring R/I(G) is Cohen–Macaulay (resp. Buchsbaum, Gorenstein).Let n be a positive integer and let Sn be the set of all nonnegative integers less than n which are relatively prime to n. In this thesis, we investigate structural properties of Cayley graphs generated by... 

The Serre's condition in commutative algebra has long been a subject of study, primarily due to its connection with the normality of rings. We say that a ring $R$ satisfies Serre's condition ($S_n$) if for every $\mathfrak{p} \in \mathrm{Spec}(R)$, we have: \[ \mathrm{depth}R_\mathfrak{p} \geq \mathrm{min}\{n, \mathrm{dim} R_\mathfrak{p} \} \] In this thesis, the significance of this property is first examined from the perspective of commutative algebra, and the prerequisites of commutative algebra and homological algebra are provided. Then, this property is studied from the perspective of combinatorial commutative algebra. This latter subject deals with examining the algebraic properties of... 

Let K be a field and S=K[x0,…,xn-1] be the polynomial ring in n variables over the field K. Let G be a finite undirected graph without loops or multiple edges with the vertex set V(G)={0,…,n-1} and the edge set E(G). One can associate a squarefree quadratic monomial ideal I(G)=of S to the graph G. The ideal I (G) is called the edge ideal of G in S. It is an algebraic object whose invariants can be related to the properties of G and vice versa. The graph G is called Cohen–Macaulay over K (Gorenstein over K) if the ring S/I (G) is Cohen–Macaulay (Gorenstein). Let n ≥ 2 be an integer. The Grimaldi graph represented by G(n) is obtained by letting all the elements of... 

In this project, we developed a theory of monomial preorders, which differ from the classical notion of monomial orders in that they allow ties between monomials since for monomial preorders, the leading ideal is diffrent with monomial orders, our result can be used to study problems where monomial orders fail to give a solution. Some of our results are new even in the clssical case of monomial orders  

set V = {1; : : : ; n}. Let K be a field and let S be the polynomial ring K[x1; : : : ; xn].The edge ideal I(G), associated to G, is the ideal of S generated by the set of squarefree monomials xi xj so that i is adjacent to j. The graph G is Cohen–Macaulay over K if S=I(G) is a Cohen–Macaulay ring. In this project we will explain Herzog-Hibi’s classification of all Cohen–Macaulay bipartite graphs  

Mass-action chemical reaction systems are frequently used in computational biology. The corresponding polynomial dynamical systems are often large (consisting of tens or even hundreds of ordinary differential equations) and poorly parameterized (due to noisy measurement data and a small number of data points and repetitions). Therefore, it is often difficult to establish the existence of (positive) steady states or to determine whether more complicated phenomena such as multi-stationarity exist. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. The focus of this work is on systems with this property, and we say... 

Let (P;≼) be a partially ordered set (poset, briefly) with a least element 0. In this thesis, we deal with zero-divisor graphs of posets. We show that if the chromatic number r(P) and the clique number r(P) (x(r(P)) and !(r(P)), respectively) are finite, then x(r(P)) = !(w(P)) = n in which n is the number of minimal prime ideals of P. We also prove that the diameter of such a graph is either 1, 2 or 3 while its girth is either 3, 4 or 1  

Gauss' theory of the arithmetic of quadratic forms appeared in Disquisitiones Arithmeticae (1801) and in particular Gauss presented a composition law for binary quadratic forms and related it to the arithmetic of quadratic extensions of Q. In the series of papers of “Higher Composition Laws” Manjul Bhargava gave a far reaching generalization of Gauss' composition law and extended it to binary cubic, and a number of other cases. He obtained six composition laws one of which the classical one of Gauss. Since the work of Gauss has had deep impact on number theory and in particular on the arithmetic of quadratic fields, one expects that Bhargava’s theory to lead to new insights in... 

In this paper Alexander Berkovich and Hamza Yesilyurt revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x2 + 5y2. Making use of Ramanujan’s 1 1 summation formula, they establish a new Lambert series identity for Σ1 n;m=1 qn2+5m2 . Conjectures of Fermat and Euler are shown to follow easily from this new formula. But they do not stop there. Employing various formulas found in Ramanujan’s notebooks and using a bit of ingenuity, they obtain a collection of new Lambert series for certain infinite products associated with quadratic forms such as x2+6y2, 2x2+3y2, x2+15y2, 3x2+5y2, x2+27y2, x2+5(y2+z2+w2), 5x2+y2+z2+w2. In the process, they... 

LetK be a biquadratic field. M.-N. Gras and F. Tanoe gave a necessary and sufficient condition that K is monogenic by using a diophantine equation of degree 4 [13]. We consider algebraic extension fields of higher degree. Let F be a Galois extension field over the rationals Q whose Galois group is 2-elementary Abelian. then we shall prove that F of degree graeter than 8, is monogenic if and only if F being field of n'th primitive root of unity under a suitable condition for the case of degree 8  

In this thesis, we give necessary and sufficient conditions for a simplicial com- plex with small codimension to satisfy the Serre’s condition (Sr) as well as the CMt property. We also give a connection between being (Sr) and being CMt. Also, we focus on the dimension of dual modules of local cohomology of Stanley– Reisner rings to obtain a new vector which contains important information on the Serre’s condition (Sr) and the CMt property as well as the depth of Stanley–Reisner rings. We prove some results in this regard including lower bounds for the depth of Stanley–Reisner rings. Further, we give a characterization of (d − 1)-dimensional simplicial complexes with codimension two which are... 

High-quality online services demand reliable and fast packet delivery at the network layer. However, clear evidence documents the existence of compromised routers in the ISP and enterprise networks, threatening network availability and reliability. A compromised router can stealthily drop, modify, inject, or delay packets in the forwarding path to launch Denial-of-Service, surveillance, man-in-the-middle attacks, etc. So researches tried to create intrusion detection methods to identify adversarial routers and switches. To this end, data-plane fault localization (FL) aims to identify faulty links and is an effective means of achieving high network availability. FL protocols use...