Arithmetical Rank Research Articles

Width and Local Homology Dimension for Triangulated Categories

Let T be a compactly generated triangulated category. In this paper, the width and local homology dimension of an object X with respect to a homogeneous ideal a, widthR(a,X) and hd(a,X), respectively, are introduced. The local nature and some basic properties of widthR(a,X) and hd(a,X) are provided. In addition, we give an upper bound and lower bound of widthR(a,X). What is more, we give the relationship between the local homology dimension hd(a,X) and the arithmetic rank of a and dimR.

Arithmetical rank and cohomological dimension of generalized binomial edge ideals

Let [Formula: see text] be a connected and simple graph on the vertex set [Formula: see text]. To the graph [Formula: see text] one can associate the generalized binomial edge ideal [Formula: see text] in the polynomial ring [Formula: see text]. We provide a lower bound for the cohomological dimension of [Formula: see text]. We also study when [Formula: see text] is a cohomologically complete intersection. Finally, we show that the arithmetical rank of [Formula: see text] equals the projective dimension of [Formula: see text] in several cases.

On the dimension of dual modules of local cohomology and the Serre's condition for the unmixed Stanley–Reisner ideals of small height

In this paper, we focus on the dimension of dual modules of local cohomology of Stanley–Reisner rings to obtain a new vector. This vector 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 (Sd−3) but they are not Cohen–Macaulay. By using this characterization, we obtain a condition to equality of projective dimension of the Stanley–Reisner rings and the arithmetical rank of their Stanley–Reisner ideals. Moreover, our characterization allows us to compute the h-vectors and give a negative answer to a known question regarding these vectors.

Utilization of fuzzy critical path method and fuzzy program evaluation and review technique for building a hydroelectric power station

In this paper, fuzzy critical path method and fuzzy program evaluation and review technique are used to calculate the earliest project completion time for constructing a hydroelectric power plant project. Fuzzy trapezoidal numbers are used to estimate the activity time and determine the range of pessimistic to optimistic variation of time. Furthermore, the minimum and maximum times of project completion duration were calculated by using arithmetic operations and ranking of fuzzy trapezoidal numbers. These hybrid methods are able to deal with the limitations associated with classical critical path method and program evaluation and review technique. The fuzzy techniques were applied to network activities in a manner similar to the classical methods for optimizing the project completion duration, thereby minimizing the cost of the project. Analysis was carried out to determine the critical path through the use of fuzzy critical path method. The fuzzy program evaluation and review technique was also used to determine the probability of completing the project at a scheduled time. These two methods were then compared and the most probable scenarios were analyzed. Finally, it was concluded that fuzzy program evaluation and review technique is better than fuzzy critical path method and more efficient in terms of early project completion time.

Reducers and K 0 with support

Let R be a commutative noetherian ring, be a Serre subcategory of the category of finitely generated R-modules and be the category of finitely generated projective R-modules. We define invariants based on the categories of chain complexes from homologically supported in with support between 0 and n () and the K-group with support similar to the stable range in classical K-theory. We introduce a notion called a reducer which we use to express the class of a complex in terms of classes of complexes of smaller amplitude and use these to study and bound the above defined invariants by standard invariants like arithmetic rank, grade and projective dimension. We also give conditions for to be isomorphic to

EVALUATING STRATEGIC EDUCATIONAL PLANNING PRACTICES AT THE MINISTRY OF EDUCATION IN THE SULTANATE OF OMAN

The current study aimed to identify the reality of planning practices for workers in educational planning agencies in the Ministry of Education in the Sultanate of Oman. The problem arose out of the existence of many aspects of deficiency and shortcomings in the exercise of the planning tasks assigned to them in these bodies, in addition to the presence of other problems experienced by these devices, which limit the efficiency and effectiveness of workers to exercise the tasks assigned to them. To achieve the objectives of the study, a questionnaire was formed consisting of (50) items, which were classified according to four axes represented by planning practices for workers in educational planning agencies, namely: tasks of educational planning agencies, administrative organization, planning cadres, educational information. The study sample consisted of (208) individuals with positions: department manager, deputy director, head of department, and a technical member in the planning departments of the Ministry and educational governorates in the Sultanate of Oman, and statistical processing of data was carried out using arithmetic averages, standard deviations and rank. The results of the study showed: The planning practices of workers in educational planning agencies in the four fields of study: tasks of planning organs, administrative organization, planning cadres, educational information were of a moderate degree.

SYMBOLIC ANALYTIC SPREAD: UPPER BOUNDS AND APPLICATIONS

AbstractThe symbolic analytic spread of an ideal $I$ is defined in terms of the rate of growth of the minimal number of generators of its symbolic powers. In this article, we find upper bounds for the symbolic analytic spread under certain conditions in terms of other invariants of $I$. Our methods also work for more general systems of ideals. As applications, we provide bounds for the (local) Kodaira dimension of divisors, the arithmetic rank, and the Frobenius complexity. We also show sufficient conditions for an ideal to be a set-theoretic complete intersection.

P.450 Overvisted... and overmedicated? regular medication regimen in psychiatric frequent attenders

Water pollution is a problem globally, where untreated human and industrial waste is directly discharged into water bodies. Choosing a suitable wastewater treatment technology (WTT) is vital. We develop a novel multi-criteria group decision-making framework to evaluate four WTTs (Anaerobic Anoxic Oxic, Triple Oxidation Ditch, Anaerobic Single Oxidation Ditch, Sequencing Batch Reactor Activated Sludge). It consists of four phases: (1) collecting incomplete linguistic pairwise comparisons; (2) characterizing this matrix by incomplete interval type-2 fuzzy sets; (3) repairing incomplete information to reach predefined consistency/consensus levels; and (4) calculating the arithmetic ranking values. We demonstrate the feasibility and superiority of the model, inviting experts to evaluate the WTTs against eight criteria. Finally, we conduct a series of comparative analyses and a sensitivity analysis. The model gives a thorough measure of WTTs across multiple criteria. The suitable WTT is different for small or medium towns, and big cities in developing or developed countries.

Unmixedness and arithmetic properties of matroidal ideals

Let R=k[x_1,ldots ,x_n] be the polynomial ring in n variables over a field k and let I be a matroidal ideal of degree d. In this paper, we study the unmixedness properties and the arithmetical rank of I. Moreover, we show that ara(I)=n-d+1. This answers the conjecture made by Chiang-Hsieh (Comm Algebra 38:944–952, 2010, Conjecture).

Incomplete interval type-2 fuzzy preference relations based on a multi-criteria group decision-making model for the evaluation of wastewater treatment technologies

Water pollution is a problem globally, where untreated human and industrial waste is directly discharged into water bodies. Choosing a suitable wastewater treatment technology (WTT) is vital. We develop a novel multi-criteria group decision-making framework to evaluate four WTTs (Anaerobic Anoxic Oxic, Triple Oxidation Ditch, Anaerobic Single Oxidation Ditch, Sequencing Batch Reactor Activated Sludge). It consists of four phases: (1) collecting incomplete linguistic pairwise comparisons; (2) characterizing this matrix by incomplete interval type-2 fuzzy sets; (3) repairing incomplete information to reach predefined consistency/consensus levels; and (4) calculating the arithmetic ranking values. We demonstrate the feasibility and superiority of the model, inviting experts to evaluate the WTTs against eight criteria. Finally, we conduct a series of comparative analyses and a sensitivity analysis. The model gives a thorough measure of WTTs across multiple criteria. The suitable WTT is different for small or medium towns, and big cities in developing or developed countries.

On determinantal ideals and algebraic dependence

Let X be a matrix with entries in a polynomial ring over an algebraically closed field K. We prove that, if the entries of X outside some -submatrix are algebraically dependent over K, the arithmetical rank of the ideal of t-minors of X drops at least by one with respect to the generic case; under suitable assumptions, it drops at least by k if X has k zero entries. This upper bound turns out to be sharp if , since it then coincides with the lower bound provided by the local cohomological dimension.

The arithmetical rank of k-complete ideals

We introduce the notions of algebraic and arithmetic derivation. As an application, we use the combinatorial decomposition of an ideal to provide a constructive method to find the algebraic invariants, as the arithmetical rank, for a family of squarefree monomial ideals, the $k$--complete ideals $I_k^n,$ also known as squarefree Veronese ideals of degree $k$.

Reducing liveness to safety in first-order logic

We develop a new technique for verifying temporal properties of infinite-state (distributed) systems. The main idea is to reduce the temporal verification problem to the problem of verifying the safety of infinite-state systems expressed in first-order logic. This allows to leverage existing techniques for safety verification to verify temporal properties of interesting distributed protocols, including some that have not been mechanically verified before. We model infinite-state systems using first-order logic, and use first-order temporal logic (FO-LTL) to specify temporal properties. This general formalism allows to naturally model distributed systems, while supporting both unbounded-parallelism (where the system is allowed to dynamically create processes), and infinite-state per process. The traditional approach for verifying temporal properties of infinite-state systems employs well-founded relations (e.g. using linear arithmetic ranking functions). In contrast, our approach is based the idea of fair cycle detection. In finite-state systems, temporal verification can always be reduced to fair cycle detection (a system contains a fair cycle if it revisits a state after satisfying all fairness constraints). However, with both infinitely many states and infinitely many fairness constraints, a straightforward reduction to fair cycle detection is unsound. To regain soundness, we augment the infinite-state transition system by a dynamically computed finite set, that exploits the locality of transitions. This set lets us define a form of fair cycle detection that is sound in the presence of both infinitely many states, and infinitely many fairness constraints. Our approach allows a new style of temporal verification that does not explicitly involve ranking functions. This fits well with pure first-order verification which does not explicitly reason about numerical values. In particular, it can be used with effectively propositional first-order logic (EPR), in which case checking verification conditions is decidable. We applied our technique to verify temporal properties of several interesting protocols. To the best of our knowledge, we have obtained the first mechanized liveness proof for both TLB Shootdown, and Stoppable Paxos.

Arithmetical rank of binomial ideals

In this paper, we investigate the arithmetical rank of a binomial ideal J. We provide lower bounds for the binomial arithmetical rank and the J-complete arithmetical rank of J. Special attention is paid to the case where J is the binomial edge ideal of a graph. We compute the arithmetical rank of such an ideal in various cases.

Local cohomology of Du Bois singularities and applications to families

In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,\mathfrak{m})$ be a local ring; we prove that if $R_{\text{red}}$ is Du Bois, then $H_{\mathfrak{m}}^{i}(R)\rightarrow H_{\mathfrak{m}}^{i}(R_{\text{red}})$ is surjective for every $i$. We find many applications of this result. For example, we answer a question of Kovács and Schwede [Inversion of adjunction for rational and Du Bois pairs, Algebra Number Theory 10 (2016), 969–1000; MR 3531359] on the Cohen–Macaulay property of Du Bois singularities. We obtain results on the injectivity of $\operatorname{Ext}$ that provide substantial partial answers to questions in Eisenbud et al. [Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput. 29 (2000), 583–600] in characteristic $0$. These results can also be viewed as generalizations of the Kodaira vanishing theorem for Cohen–Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen–Macaulayness of the defining ideal of Du Bois singularities, which are characteristic-$0$ analogs and generalizations of results of Singh–Walther and answer some of their questions in Singh and Walther [On the arithmetic rank of certain Segre products, in Commutative algebra and algebraic geometry, Contemporary Mathematics, vol. 390 (American Mathematical Society, Providence, RI, 2005), 147–155]. We extend results on the relation between Koszul cohomology and local cohomology for $F$-injective and Du Bois singularities first shown in Hochster and Roberts [The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117–172; MR 0417172 (54 #5230)]. We also prove that singularities of dense $F$-injective type deform.

The arithmetical rank of the edge ideals of cactus graphs

ABSTRACTWe prove that, for the edge ideal of a cactus graph, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes. The bound is sharp, but in many cases, it can be improved. Moreover, we show that the edge ideal of a Cohen–Macaulay graph that contains exactly one cycle or is chordal or has no cycles of length 4 and 5 is a set-theoretic complete intersection.

Arithmetical rank of strings and cycles

Let $R$ be a polynomial ring over a field $K$. To a given squarefree monomial ideal $I \subset R$, one can associate a hypergraph $H(I)$. In this article, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $R/I$ when $H(I)$ is a string or a cycle hypergraph.

A note on rational normal scrolls

We give a general upper bound for the arithmetical rank of the ideals generated by the 2-minors of scroll matrices with entries in an arbitrary commutative unit ring.

The arithmetical rank of the edge ideals of graphs with pairwise disjoint cycles

We prove that, for the edge ideal of a graph whose cycles are pairwise vertex-disjoint, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes.

Cohen Macaulayness and Arithmetical Rank of Generalized Theta Graphs

In this paper, we study some algebraic invariants of the edge ideal of generalized theta graphs, such as arithmetical rank, big height and height. We give an upper bound for the difference between the arithmetical rank and big height. Moreover, all Cohen-Macaulay (and unmixed) graphs of this type will be characterized.