Linear Differential Operators for Polynomial Equations

Given a squarefree polynomial P ∈ k₀[ x,y ], k₀a number field, we construct a linear differential operator that allows one to calculate the genus of the complex curve defined by P =  0 (when P is absolutely irreducible), the absolute factorization of P over the algebraic closure of k₀, and calculate information concerning the Galois group of P over ___ k₀(x) as well as overk₀ (x).     

Solutions to the functional equation I(x, y) = I(x, I(x, y)) for three types of fuzzy implications derived from uninorms

This paper investigates an iterative Boolean-like law with fuzzy implications derived from uninorms. More precisely, we characterize the solutions to the functional equation I(x, y) = I(x, I(x, y)) that involve RU-, (U, N)- and QLU-implications generated by the most usual classes of uninorms.     

A study of nonlinear dispersive equations with solitary-wave solutions having compact support

With the use of Adomian decomposition method, the prototypical, genuinely nonlinear K(m,n) equation, u_t+(u^m)_x+(uⁿ)_xxx=0, which exhibits compactons — solitons with finite wavelength — is solved exactly. Two numerical illustrations, K(2,2) and K(3,3), are investigated to illustrate the pertinent features of the proposed scheme. The technique is presented in a general way so that it can be used in nonlinear dispersive equations.     

On identifying codes that are robust against edge changes

Assume that G = (V, E) is an undirected graph, and C ⊆ V. For every v ∈ V, denote I_r(G; v) = {u ∈ C: d(u,v) ≤ r}, where d(u,v) denotes the number of edges on any shortest path from u to v in G. If all the sets I_r(G; v) for v ∈ V are pairwise different, and none of them is the empty set, the code C is called r-identifying. The motivation for identifying codes comes, for instance, from finding faulty processors in multiprocessor systems or from location detection in emergency sensor networks. The underlying architecture is modelled by a graph. We study various types of identifying codes that are robust against six natural changes in the graph; known or unknown edge deletions, additions or both. Our focus is on the radius r = 1. We show that in the infinite square grid the optimal density of a 1-identifying code that is robust against one unknown edge deletion is 1/2 and the optimal density of a 1-identifying code that is robust against one unknown edge addition equals 3/4 in the infinite hexagonal mesh. Moreover, although it is shown that all six problems are in general different, we prove that in the binary hypercube there are cases where five of the six problems coincide.     

Substitutions into propositional tautologies

We prove that there is a polynomial time substitution (y₁,…,yn):=g(x₁,…,xk) with k?n such that whenever the substitution instance A(g(x₁,…,xk)) of a 3DNF formula A(y₁,…,yn) has a short resolution proof it follows that A(y₁,…,yn) is a tautology. The qualification “short” depends on the parameters k and n.     

Modeling of the energies and splitting of the Qx and Qy bands in positional isomers of zinc pyridinoporphyrazines by TDDFT approach: Can TDDFT help distinguishing the structural isomers?

Electronic structures, energies and splitting of the Q_x and Q_y bands for positional isomers of zinc mono-, di-, tri-, and tetra pyridinoporphyrazines as well as parent zinc phthalocyanine were investigated using density functional theory (DFT) and time-dependent (TD) DFT approaches. The influence of the Hartree–Fock exchange on excited state energies and Q_x and Q_y bands splitting were studied using GGA BP86 and hybrid B3LYP and PBE1PBE exchange-correlation functionals. Solvent effects were estimated using the polarized continuum model (PCM) approach and cyclohexane, toluene, or DMSO as solvents. It was found that general trends in the Q_x and Q_y band energies and splitting correlate very well with the available experimental data on pyridinoporphyrazines and follow the trends in HOMO–LUMO and HOMO–LUMO + 1 energy gaps as well as LUMO–LUMO + 1 splitting. TDDFT trends allow estimation of the Q_x and Q_y band energies and splitting in unknown tripyridinoporphyrazines and in individual positional isomers of tetrapyridinoporphyrazines.     

Computing Local Artin Maps,and Solvability of Norm Equations

Let L = K(α) be an Abelian extension of degree n of a number field K, given by the minimal polynomial of α over K. We describe an algorithm for computing the local Artin map associated with the extension L / K at a finite or infinite prime v of K. We apply this algorithm to decide if a nonzero a ∈ K is a norm from L, assuming that L / K is cyclic.     

Gröbner Bases Applied to Finitely Generated Field Extensions

Using a constructive field-ideal correspondence it is shown how to compute the transcendence degree and a (separating) transcendence basis of finitely generated field extensionsk (x) / k(g), resp. how to determine the (separable) degree if k(x) / k(g) is algebraic. Moreover, this correspondence is used to derive a method for computing minimal polynomials and deciding field membership. Finally, a connection between certain intermediate fields of k(x) / k(g) and a minimal primary decomposition of a suitable ideal is described. For Galois extensions the field-ideal correspondence can also be used to determine the elements of the Galois group.     

Taylor approximation for hybrid systems

We propose a new approximation technique for Hybrid Automata. Given any Hybrid Automaton H, we call Approx(H, k) the Polynomial Hybrid Automaton obtained by approximating each formula ϕ in H with the formulae ϕ_k obtained by replacing the functions in ϕ with their Taylor polynomial of degree k. We prove that Approx(H, k) is an over-approximation of H. We study the conditions ensuring that, given any ϵ > 0, some k₀ exists such that, for all k > k₀, the “distance” between any vector satisfying ϕ_k and at least one vector satisfying ϕ is less than ϵ. We study also conditions ensuring that, given any ϵ > 0, some k₀ exists such that, for all k > k₀, the “distance” between any configuration reached by Approx(H, k) in n steps and at least one configuration reached by H in n steps is less than ϵ.     

Mason: morphological simplification

The traditional rounding and filleting morphological filters are biased. Hence, as r grows, the rounding R_r (S) of S shrinks and the filleting F_r (S) grows. A shape S is r-regular when R_r (S) = F_r (S) = S. The combinations F_r (R_r (S)) and R_r (F_r (S)) produce nearly r-regular shapes, but retain a bias: F_r (R_r (S)) is usually smaller than S and R_r (F_r (S)) is larger. To overcome this bias, we propose a new filter, called Mason. The r-mortar M_r (S) of S is F_r (S)–R_r (S), and the stability of a point P with respect to S is the smallest value of r for which P belongs to M_r (S). Stability provides important information about the shape’s imbedding that cannot be obtained through traditional topological or differential analysis tools. F_r (R_r (S)) and R_r (F_r (S)) only affect space in M_r (S). For each maximally connected component of M_r (S), Mason performs either F_r (R_r (S)) or R_r (F_r (S)), choosing the combination that alters the smallest portion of that component. Hence, Mason acts symmetrically on the shape and on its complement. Its output is guaranteed to have a smaller symmetric difference with the original shape than that of either combination F_r (R_r (S)) or R_r (F_r (S)). Many previously proposed shape simplification algorithms were focused on reducing the combinatorial storage or processing costs of a shape at the expense of the smoothness and regularity or altered the shape in regular portions that did not exhibit any high frequency complexity. Mason is the first shape simplification operator that is independent of the particular representation and offers the advantage of preserving portions of the boundary of S that are regular at the desired scale.     

An Efficient Algorithm for the k-Pairwise Disjoint Paths Problem in Hypercubes

A graph G(V, E) (|V|⩾2k) satisfies property A_k if, given k pairs of distinct nodes (s₁, t₁), …, (s_k, t_k) of V(G), there are k mutually node-disjoint paths, one connecting s_i and t_i for each i, 1⩽i⩽k. A necessary condition for any graph to satisfy A_k is that it is (2k−1)-connected. Hypercubes are important interconnection topologies for parallel computation and communication networks. It has been known that hypercubes of dimension n (which are n-connected) satisfy A_⌈n/2⌉. In this paper we give an algorithm which, given k=⌈n/2⌉ pairs of distinct nodes (s₁, t₁), …, (s_k, t_k) in the n-dimensional hypercube, finds the k disjoint paths of length at most n+⌈log n⌉+1 in O(n² log* n) time.     

An Algorithm for the Quillen–Suslin Theorem for Quotients of Polynomial Rings by Monomial Ideals

This paper presents an algorithm for the Quillen–Suslin Theorem for quotients of polynomial rings by monomial ideals, that is, quotients of the form A = k [ x₀, . . . ,x_n ] / I, with I a monomial ideal and k a field. Vorst proved that finitely generated projective modules over such algebras are free. Given a finitely generated module P, described by generators and relations, the algorithm tests whether P is projective, in which case it computes a free basis forP .     

Cylindrical decomposition for systems transcendental in the first variable

We present a generalization of the Cylindrical Algebraic Decomposition (CAD) algorithm to systems of equations and inequalities in functions of the form p(x,f₁(x),…,f_m(x),y₁,…,y_n), where p∈Q[x,t₁,…,t_m,y₁,…,y_n] and f₁(x),…,f_m(x) are real univariate functions such that there exists a real root isolation algorithm for functions from the algebra Q[x,f₁(x),…,f_m(x)]. In particular, the algorithm applies when f₁(x),…,f_m(x) are real exp-log functions or tame elementary functions.     

Scalable 2D Convex Hull and Triangulation Algorithms for Coarse Grained Multicomputers

In this paper we describe scalable parallel algorithms for building the convex hull and a triangulation ofncoplanar points. These algorithms are designed for thecoarse grained multicomputermodel:pprocessors withO(n/p)⪢O(1) local memory each, connected to some arbitrary interconnection network. They scale over a large range of values ofnandp, assuming only thatn⩾p^1+ε(ε>0) and require timeO((T_sequential/p)+T_s(n, p)), whereT_s(n, p) refers to the time of a global sort ofndata on approcessor machine. Furthermore, they involve only a constant number of global communication rounds. Since computing either 2D convex hull or triangulation requires timeT_sequential=Θ(n log n) these algorithms either run in optimal time,Θ((n log n)/p), or in sort time,T_s(n, p), for the interconnection network in question. These results become optimal whenT_sequential/pdominatesT_s(n, p) or for interconnection networks like the mesh for which optimal sorting algorithms exist.     

On the Practical Solution of Genus Zero Diophantine Equations

Let f(X, Y) be an absolutely irreducible polynomial with integer coefficients such that the curve defined by the equation f(X, Y)  =  0 is of genus 0 having at least three infinite valuations. This paper describes a practical general method for the explicit determination of all integer solutions of the diophantine equation f(X, Y)  =  0. Some elaborated examples are given.     

Gröbner Bases over Galois Rings with an Application to Decoding Alternant Codes

We develop a theory of Gröbner bases over Galois rings, following the usual formulation for Gröbner bases over finite fields. Our treatment includes a division algorithm, a characterization of Gröbner bases, and an extension of Buchberger’s algorithm. One application is towards the problem of decoding alternant codes over Galois rings. To this end we consider the module M =  {(a, b) :aS ≡ b  mod x^r} of all solutions to the so-called key equation for alternant codes, where S is a syndrome polynomial. In decoding, a particular solution (Σ, Ω)  ∈ M is sought satisfying certain conditions, and such a solution can be found in a Gröbner basis of M. Applying techniques introduced in the first part of this paper, we give an algorithm which returns the required solution.     

On reliability of the folded hypercubes

In this paper, we explore the 2-extra connectivity and 2-extra-edge-connectivity of the folded hypercube FQ_n. We show that κ₂(FQ_n) = 3n − 2 for n ⩾ 8; and λ₂(FQ_n) = 3n − 1 for n ⩾ 5. That is, for n ⩾ 8 (resp. n ⩾ 5), at least 3n − 2 vertices (resp. 3n − 1 edges) of FQ_n are removed to get a disconnected graph that contains no isolated vertices (resp. edges). When the folded hypercube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.     

Solving Genus Zero Diophantine Equations with at Most Two Infinite Valuations

Let f(X, Y) be an absolutely irreducible polynomial with integer coefficients such that the curve defined by the equation f(X, Y)  =  0 is of genus 0 having at most two infinite valuations. This paper describes a practical general method for the explicit determination of all integer solutions of the diophantine equation f(X, Y)  =  0. Several elaborated examples are given. Furthermore, a necessary and sufficient condition for a curve of genus 0 to have infinitely many integer points is obtained.     

Conditional diagnosability of hypercubes under the comparison diagnosis model

Processor fault diagnosis plays an important role in multiprocessor systems for reliable computing, and the diagnosability of many well-known networks has been explored. Lai et al. proposed a novel measure of diagnosability, called conditional diagnosability, by adding an additional condition that any faulty set cannot contain all the neighbors of any vertex in a system. We make a contribution to the evaluation of diagnosability for hypercube networks under the comparison model and prove that the conditional diagnosability of n-dimensional Hypercube Q_n is 3(n ? 2) + 1 for n ? 5. The conditional diagnosability of Q_n is about three times larger than the classical diagnosability of Q_n.