Generic and Cogeneric Monomial Ideals

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions encoded by simplicial complexes. There are numerous equivalent ways to say that a monomial ideal is generic or cogeneric. For a generic monomial ideal, the associated primes satisfy a saturated chain condition, and the Cohen–Macaulay property implies shellability for both the Scarf complex and the Stanley–Reisner complex. Reverse lexicographic initial ideals of generic lattice ideals are generic. Cohen–Macaulayness for cogeneric ideals is characterized combinatorially; in the cogeneric case, the Cohen–Macaulay type is greater than or equal to the number of irreducible components. Methods of proof include Alexander duality and Stanley’s theory of local h -vectors.     

Decomposing Monomial Representations of Solvable Groups

We present an efficient algorithm that decomposes a monomial representation of a solvable groupG into its irreducible components. In contradistinction to other approaches, we also compute the decomposition matrixA in the form of a product of highly structured, sparse matrices. This factorization provides a fast algorithm for the multiplication with A. In the special case of a regular representation, we hence obtain a fast Fourier transform forG . Our algorithm is based on a constructive representation theory that we develop. The term “constructive" signifies that concrete matrix representations are considered and manipulated, rather than equivalence classes of representations as it is done in approaches that are based on characters. Thus, we present well-known theorems in a constructively refined form and derive new results on decomposition matrices of representations. Our decomposition algorithm has been implemented in the GAP share package AREP. One application of the algorithm is the automatic generation of fast algorithms for discrete linear signal transforms.     

Permanental Ideals

We show that the (2  ×  2)-subpermanents of a generic matrix generate an ideal whose height, unmixedness, primary decomposition, the number and structure of the minimal components, resolutions, radical, integral closure and Gröbner bases all depend on the characteristic of the underlying subfield: if the characteristic of the subfield is two, this ideal is the determinantal ideal for which all of these properties are already well known. We show that as long as the characteristic of the subfield is not two, the results are in marked contrast with those for the determinantal ideals.     

Sheaf Cohomology is #P-hard

We prove that computing the dimensions of the cohomology groups of a coherent sheaf on projective space, specified by the kernel, image, or cokernel of a graded matrix, is hard for Valiant's complexity class #P. The same is true of the Euler characteristic, defined as the alternating sum of dimensions.     

Cellular Binomial Ideals. Primary Decomposition of Binomial Ideals

Eisenbud and Sturmfels’ theoretical study assures that it is possible to find a primary decomposition of binomial ideals into binomial ideals over an algebraically closed field. In this paper we complete the algorithms in Eisenbud and Sturmfels (1996, Duke Math. J., 84, 1–45) by filling in the steps for which the authors say they have not been very precise.     

A Control Theory for Boolean Monomial Dynamical Systems

Recently criteria for determining when a certain type of nonlinear discrete dynamical system is a fixed point system have been developed. This theory can be used to determine if certain events modeled by those systems reach a steady state. In this work we formalize the idea of a “stabilizable” discrete dynamical system. We present necessary and sufficient conditions for a Boolean monomial dynamical control system to be stabilizable in terms of properties of the dependency graph associated with the system. We use the equivalence of periodicity of the dependency graph and loop numbers to develop a new O(n ²logn) algorithm for determining the loop numbers of the strongly connected components of the dependency graph, and hence a new O(n ²logn) algorithm for determining when a Boolean monomial dynamical system is a fixed point system. Finally, we show how this result can be used to determine if a Boolean monomial dynamical control system is stabilizable in time O(n ²logn).     

Computing Toric Ideals

Toric ideals are binomial ideals which represent the algebraic relations of sets of power products. They appear in many problems arising from different branches of mathematics. In this paper, we develop new theories which allow us to devise a parallel algorithm and an efficient elimination algorithm. In many respects they improve existing algorithms for the computation of toric ideals.     

Ideals in MV-pairs

An MV-pair is a BG-pair (B, G) (where B is a Boolean algebra and G is a subgroup of the automorphism group of B) satisfying certain conditions. Recently, it was proved by Jenča that, given an MV-pair (B, G), the quotient B/~ _G , where ~ _G is an equivalence relation naturally associated with G, is an MV-algebra, and conversely, to every MV-algebra there corresponds an MV-pair. In this paper, we study relations between congruences of B and congruences of B/~ _G induced by a G-invariant ideal I of B. In addition we bring some relations between ideals in MV-algebras and in the corresponding R-generated Boolean algebras.     

Ideals of an EMV-semiring

ABSTRACT

In this paper, congruences, ideals, and prime ideals of an EMV-semiring and of its associated EMV-algebra are studied. Then EMV-semirings are characterized and it is proved that each EMV-semiring can be embedded into a direct product of a family of MV-semirings as an EMV-semiring. Moreover, another representation of EMV-semirings are presented by EMV-semirings of continuous sections in a sheaf of commutative semirings whose stalks are localizations of EMV-semirings over prime ideals. Also, using the categorical equivalence between EMV-semirings and EMV-algebras, a representation of EMV-algebras are obtained.     

Computing Ideals of Points

We address the problem of computing ideals of polynomials which vanish at a finite set of points. In particular we develop a modular Buchberger–Möller algorithm, best suited for the computation over Q, and study its complexity; then we describe a variant for the computation of ideals of projective points, which uses a direct approach and a new stopping criterion. The described algorithms are implemented in CoCoA, and we report some experimental timings.     

Monomial geometric programming with fuzzy relation inequality constraints with max-product composition

Monomials function has always been considered as a significant and most extensively used function in real living. Resource allocation, structure optimization and technology management can often apply these functions. In optimization problems the objective functions can be considered by monomials. In this paper, we present monomials geometric programming with fuzzy relation inequalities constraint with max-product composition. Simplification operations have been given to accelerate the resolution of the problem by removing the components having no effect on the solution process. Also, an algorithm and a few practical examples are presented to abbreviate and illustrate the steps of the problem resolution.     

Observability Conditions by Polynomial Ideals

In this paper, we give global and local observability conditions of polynomial systems. Each observability is defined in terms of distinguishability of a pair of initial states. The set of pairs of initial states are represented as an affine variety. By using this affine variety, a global observability condition is derived. Then, a local observability condition is presented based on the minimal decomposition of the affine variety and dimension theory.     

Calculating Group Cohomology: Tests for Completion

We want to calculate generators and relations for the mod- p cohomology rings of finite groups using computer technology. For this purpose we develop interactive tests to check whether a specific calculation is complete, in the case that the group is a p -group. The method involves checking some conjectures about the nature of the cohomology rings.     

Monomial reachability and zero controllability of discrete-time positive switched systems

In this paper, monomial reachability and zero controllability properties of discrete-time positive switched systems are investigated. Necessary and sufficient conditions for these properties to hold, together with some interesting examples and some testing algorithms, are provided.     

Ideals and congruences of basic algebras

MV-algebras as well as orthomodular lattices can be seen as a particular case of so-called “basic algebras” which are an alter ego of bounded lattices whose sections are equipped with fixed antitone involutions. The class of basic algebras is an ideal variety. In the paper, we give an internal characterization of congruence kernels (ideals) and find a finite basis of ideal terms, with focus on monotone and effect basic algebras. We also axiomatize basic algebras that are subdirect products of linearly ordered ones.     

Primary Decomposition of Lattice Basis Ideals

We study the primary decomposition of lattice basis ideals. These ideals are binomial ideals with generators given by the elements of a basis of a saturated integer lattice. We show that the minimal primes of such an ideal are completely determined by the sign pattern of the basis elements, while the embedded primes are not. As a special case we examine the ideal generated by the 2  ×  2 adjacent minors of a generic m × n matrix. In particular, we determine all minimal primes in the 3  × n case. We also present faster ways of computing a generating set for the associated toric ideal from a lattice basis ideal.     

Local networking at the Royal Bank of Scotland

Data communications facilities are vital if essential banking operations are to be computerized; branches and head offices that are dependent upon terminals and corporate data to fulfil their functions must have access to such facilities. Back-up facilities are also essential if normal bank functions are not to be disrupted should a fault develop. Investigations that led to the installation of Hyperchannel at the Royal Bank of Scotland are described and the problems to which Hyperchannel provided the solutions are examined. The implementation of the system and plans for future development are discussed.     

Monomial test: Testing the flexural behavior of the degenerated shell element

Shear locking of the degenerated shell element is a result of the development of spurious shear strain by the element, when subjected to high order Kirchhoff displacement fields. The shear locking phenomenon is analyzed in this paper using an analytical test. The effects of the integration scheme, order of the element, order of the modeled Kirchhoff field and the element distortion are analyzed explicitly and quantitatively for the four-node, eight-node and nine-node degenerated shell elements.     

Representations of Regular Ideals in Finite Automata

Regular ideals of a free monoid are shown to be characterized by weak (noninitial) representations in finite automata. The class of kernel ideals is also shown to be invariant under the action of several automata functors and to be equal to the class of zero ideals.