On the Equations De.ning Curves in a Polynomial Algebra

Let A be a commutative Noetherian ring of dimension n (n ≥ 3). Let I be a local complete intersection ideal in A[T] of height n. Suppose I/I ² is free A[T]/I-module of rank n and (A[T]/I) is torsion in K ₀(A[T]). It is proved in this article that I is a set theoretic complete intersection ideal in A[T] if one of the following conditions holds: (1) n ≥ 5, odd; (2) n is even, and A contains the field of rational numbers; (3) n = 3, and A contains the field of rational numbers.     

Lattice-Ordered Matrix Algebras Over Real GCD-Domains

Let R ? ? be a GCD-domain. In this article, Weinberg's conjecture on the n × n matrix algebra M _n (R) (n ≥ 2) is proved. Moreover, all the lattice orders (up to isomorphisms) on a full 2 × 2 matrix algebra over R are obtained.     

Existence of an Almost Periodic Solution in a Difference Equation with Infinite Delay

In order to obtain the existence of an almost periodic functional difference equation x(n + 1) = ?(n,x_n ),n ∈ Z ⁺ and where x_n is defined by x_n (s) = x(n + s) for s ∈ Z ^?, on an axiomatic phase space B, we consider a certain stability property, which is referred to as BS-stable under disturbances from Ω(?) with respect to K, this stability implies ρ-stable under disturbances from Ω(?) with respect to compact set K.     

Parameter identification in nonlocal nonlinear evolution equations

We develop an approximation framework for identifying parameters in a general class of nonautonomous, nonlocal and nonlinear evolution equations. After establishing existence and uniqueness of solutions, we present a convergence theory for Galerkin approximations to inverse problems involving these equations. Our approach relies on the theory of maximal monotone operators in Banach spaces. An application to a nonautonomous nonlinear integral equation arising in heat flow is also discussed.     

Microlocal Analysis and Renormalization in Finite Temperature Field Theory

We reassess the problem of renormalization in finite temperature field theory (FTFT). A new point of view elucidates the relation between the ultraviolet divergences for T = 0 and T ¬ = 0 theories and makes clear the reason why the ultraviolet behavior keeps unaffected when we consider the FTFT version associated to a given quantum field theory (QFT). The strength of the derivation one lies on the Hörmander’s criterion for the existence of products of distributions in terms of the wavefront sets of the respective distributions. The approach allows us to regard the FTFT both imaginary and real time formalism at once in a unified way in the contour ordered formalism.     

Distinguishing non-standard natural numbers in a set theory within Łukasiewicz logic

In , a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of ω” is truth value 1 in any model of , and we prove an analogy of Hájek’s theorem with a very simple procedure.      

An example on ordered Banach algebras

Let be a complex unital Banach algebra. We consider the Banach algebra ordered by the algebra cone , and investigate the connection between results on ordered Banach algebras and the right bound of the numerical range of elements in .

     

Set partitions and moments of random variables

It is well known that the sequence of Bell numbers (Bn)_n?0 (Bn being the number of partitions of the set [n]) is the sequence of moments of a mean 1 Poisson random variable τ (a fact expressed in the Dobiński formula), and the shifted sequence (Bn+1)_n?0 is the sequence of moments of 1+τ. In this paper, we generalize these results by showing that both and (where is the number of m-partitions of [n], as they are defined in the paper) are moment sequences of certain random variables. Moreover, such sequences also are sequences of falling factorial moments of related random variables. Similar results are obtained when is replaced by the number of ordered m-partitions of [n]. In all cases, the respective random variables are constructed from sequences of independent standard Poisson processes.     

Shadows of ordered graphs

Isoperimetric inequalities have been studied since antiquity, and in recent decades they have been studied extensively on discrete objects, such as the hypercube. An important special case of this problem involves bounding the size of the shadow of a set system, and the basic question was solved by Kruskal (in 1963) and Katona (in 1968). In this paper we introduce the concept of the shadow ∂G of a collection G of ordered graphs, and prove the following, simple-sounding statement: if n∈N is sufficiently large, |V(G)|=n for each G∈G, and |G|<n, then |∂G|?|G|. As a consequence, we substantially strengthen a result of Balogh, Bollobás and Morris on hereditary properties of ordered graphs: we show that if P is such a property, and |Pk|<k for some sufficiently large k∈N, then |Pn| is decreasing for k?n<∞.