Interpretability of first-order linear temporal logics in fork algebras

In this paper we prove theorems on the interpretability of the first-order temporal logics LTL and TL into Fork Algebras. This result is part of a research project on the interpretability of logics in Fork Algebras, and has important applications towards the relational specification of properties of systems within the Ar_gentum tool.     

A study on multi-dimensional products of graphs and hybrid logics

In this work, we address some issues related to products of graphs and products of modal logics. Our main contribution is the presentation of a necessary and sufficient condition for a countable and connected graph to be a product, using a property called intransitivity. We then proceed to describe this property in a logical language. First, we show that intransitivity is not modally definable and also that no necessary and sufficient condition for a graph to be a product can be modally definable. Then, we exhibit a formula in a hybrid language that describes intransitivity. With this, we get a logical characterization of products of graphs of arbitrary dimensions. We then use this characterization to obtain two other interesting results. First, we determine that it is possible to test in polynomial time, using a model-checking algorithm, whether a finite connected graph is a product. This test has cubic complexity in the size of the graph and quadratic complexity in its number of dimensions. Finally, we use this characterization of countable connected products to provide sound and complete axiomatic systems for a large class of products of modal logics. This class contains the logics defined by product frames obtained from Kripke frames that satisfy connectivity, transitivity and symmetry plus any additional property that can be defined by a pure hybrid formula. Most sound and complete axiomatic systems presented in the literature are for products of a pair of modal logics, while we are able, using hybrid logics, to provide sound and complete axiomatizations for many products of arbitrary dimensions.     

On the role of logical connectives for primality and functional completeness of algebras of logics

We consider certain finite universal algebras arising from algebraic semantics in implicational logics. They contain a binary operation → and two constants 0 and 1 satisfying the axioms 0→x=x→1=x→x=1 and 1→x=x valid in most implicational logics. We characterize the completeness (also called primality) of such algebras, i.e. the property that every finitary operation on their universe is a term operation of the algebra (in other words, it is a composition of the basic operations of the algebra). Using clone theory and the knowledge of maximal clones we describe completeness (functional completeness) in terms of nonpreservation of three types of specific relations. If → has a simple property and the algebra contains a binary operation ⊙ with a neutral element 1 and a unary operation ¬ satisfying ¬(1)=0 and ¬x=1 otherwise, the algebra is functionally complete.     

A direct derivation of the optimal linear filter using the maximum principle

The purpose of this paper is to present an alternate derivation of optimal linear filters. The basic technique is the use of a matrix version of the maximum principle of Pontryagin coupled with the use of gradient matrices to derive the optimal values of the filter coefficients for minimum variance estimation under the requirement that the estimates be unbiased. The optimal filter which is derived turns out to be identical to the well-known Kalman-Bucy filter.     

效应代数水平和上的序列积

建立了一族可和序列效应代数的水平和上具有序列积的充分必要条件,进而给出了由序列效应代数的水平和构造序列效应代数的一种方法。     

The evaluation of products and series

The evaluation of an infinite product can be based on a direct expansion to give an infinite series whose terms are given by an iterative formula. When the exact value of the product is known the method can be used to give accelerated convergence to a sum. Examples on the evalutation of constants in prime number theory are given.     

A maximum entropy analysis of global monthly series of rainfall from merged satellite data

This Letter presents the results of analysing a global, satellite‐based monthly precipitation database on a 2.5° latitude‐longitude grid for the 22‐year period from 1979 to 2001 using a direct maximum entropy spectral analysis method. The NOAA Climate Prediction Center (CPC) Merged Analysis of Precipitation (CMAP) database is used to calculate the rainfall cycles for ～10 000 time series covering the whole Earth. It is shown that several cycles other than the annual or seasonal cycles affect the rainfall distribution of many areas, in particular western Europe.     

Method of maximum likelihood for stationary time series models

Whittle [1] proposed a method of obtaining the likelihood function for a linear dynamic model (with rational pulse transfer function and excited by Gaussian signal). In this note a simple derivation of his result is given.     

On direct products of automaton decidable theories

This paper is concerned with connections between two different ways to prove decidability results: methods of automata theory and methods related to products of algebraic systems. Some basic automaton-theoretic conditions on relational structures are presented which give preservation theorems for decidability under certain direct products.     

Complexity of homomorphisms to direct products of graphs

For a graph G, OALG asks whether or not an input graph H together with a partial map g:S→G, S⊆V(H), admits a homomorphism f:H→G such that f|_S=g. We show that for connected graphs G₁, G₂, OAL G₁×G₂ is in P if G₁ and G₂ are trees and NP-complete otherwise.     

On edge connectivity of direct products of graphs

Let λ(G) be the edge connectivity of G. The direct product of graphs G and H is the graph with vertex set V(G×H)=V(G)×V(H), where two vertices (u₁,v₁) and (u₂,v₂) are adjacent in G×H if u₁u₂∈E(G) and v₁v₂∈E(H). We prove that λ(G×Kn)=min{n(n−1)λ(G),(n−1)δ(G)} for every nontrivial graph G and n?3. We also prove that for almost every pair of graphs G and H with n vertices and edge probability p, G×H is k-connected, where k=O(²(n/logn)).     

Minimum cycle bases of direct products of complete graphs

This paper presents a construction of a minimum cycle basis for the direct product of two complete graphs on three or more vertices. With the exception of two special cases, such bases consist entirely of triangles.     

Classification of nilpotent lie algebras of dimension nine whose maximum abelian ideal is of dimension seven

The classification of Nilpotent Lie Algebras has been partially achieved up to dimension eight but this specific problem remained unsolved, since it required an enormous amount of calculations. Nowadays, with the aid of modern computers, it can be solved by designing the proper algorithms. The aim of this paper is to provide the results of the classification of Nilpotent Lie Algebras of dimension nine, whose maximum abelian ideal is of dimension seven.