35 where the expectation value is deened by hA(x)i = R dx e ?x 2 p 2 cosh m (p 2x)A(x) R dx e ?x 2 p 2 cosh m (p 2x) : (62) Inserting in (56) this value of we have searched among the 5 values of g the one which gives the maximum free energy when the stability is marginal (i.e. when (56) is saturated). We have found that the eigenvalue g 5 is the one which gives the maximal free energy. This lea...

By “number theory” I mean algebraic number theory, and by Part 1, roughly the years 1900–1940. The treatment is very sketchy; I hope at some later date to cover the rest of the century and to fill in some of the gaps and correct any errors in this hasty account. At the end of the 19th century, Kurt Hensel introduced the idea of the completion kv of a field k with respect to a discrete valuation...

Nanomaterials (materials at the nanoscale, 10 −9 meters) are extensively used in several industry sectors due to the improved properties they empower commercial products with. There is a pressing need to produce these materials more sustainably. This paper proposes a Multiple Criteria Decision Aiding (MCDA) approach to assess the implementation of green chemistry principles as applied to the pr...

The information age has facilitated knowledge sharing among different parts of the world. Research ideas and methodologies are shared through conferences and publications by MIS researchers in Europe and in North America. However, there has been little movement toward a more uniform research paradigm on the two continents. One reason may be the difference in research tradition, which is strongl...

for 1 ≤ i ≤ m. Suppose that P ⊂ R is a regular prime (R/P is a regular local ring) and 0 6= f ∈ P . The regular local ring R1 = R[ P f ]m where m is a maximal ideal of R[ P f ] containing mR is called a monoidal transform of R. Suppose that V is a valuation ring of the quotient field of S which dominates S (and thus dominates R). Then given a regular prime P of R (or of S) there exists a unique...

We described here all the maximal-sized subsets of the n-cube B n with respect to given diameter and showed the connection between this description and the discrete isoperimetric problem. As a consequence for any number m, 1 m 2 n , the m-element subset of B n with minimal possible diameter was found. Except of it we present here as a second consequence the simple proof of the Theorem of Katona...

Let p be a prime number and let K be a finite extension of Qp. Let R be the valuation ring of K, P the maximal ideal of R, and K̄ = R/P the residue field of K. Let q denote the cardinality of K̄, so K̄ ≃ Fq. For z in K, let ord z denote the valuation of z, and set |z| = q . Let f be a non constant element of K[x1, . . . , xm]. The p-adic Igusa local zeta function Z(s) associated to f (relative to ...

We consider S 2 bundles P and P ′ of totally null planes of maximal dimension over a 4-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product PP ′ of P and P ′ is found to be appropriate for the encoding of both the selfdual and the Einstein-Weyl equations for the 4-metric. This encoding is realized in terms of the properties of certain well defined geometrical obje...

We study the relationship between fields of transseries and residue fields of convex subrings of non-standard extensions of the real numbers. This was motivated by a question of Todorov and Vernaeve, answered in this paper. In this note we answer a question by Todorov and Vernaeve (see, e.g., [35]) concerning the relationship between the field of logarithmic-exponential series from [14] and the...

Let F denote the field of germs at +∞ of Ran,exp-definable unary functions. Starting from its characterization in terms of closure conditions as given by van den Dries, Macintyre and Marker ([3], [4]), we give a similar description of its subring consisting of the germs of polynomial growth. More precisely, denoting this ring by Fpoly and its unique maximal ideal by mpoly, our description picks...