The Hahn–Banach Theorem for Partially Ordered Totally Convex, Positively Convex and Superconvex Modules

The Hahn–Banach Theorem for partially ordered totally convex modules [3] and a necessary and sufficient condition for the existence of an extension of a morphism from a submodule C ₀ of a partially ordered totally convex module C (with the ordered unit ball of the reals as codomain) to C, are proved. Moreover, the categories of partially ordered positively convex and superconvex modules are introduced and for both categories the Hahn–Banach Theorem is proved.     

Economic Equilibrium: Optimality and Price Decentralization

Mathematical economics has a long history and covers many interdisciplinary areas between mathematics and economics. At its center lies the theory of market equilibrium. The purpose of this expository article is to introduce mathematicians to price decentralization in general equilibrium theory. In particular, it concentrates on the role of positivity in the theory of convex economic analysis and the role of normal cones in the theory of non-convex economies.     

On the Embedding of Ordered Semigroups into Ordered Group

It was shown in [7] that any right reversible, cancellative ordered semigroup can be embedded into an ordered group and as a consequence, it was shown that a commutative ordered semigroup can be embedded into an ordered group if and only if it is cancellative. In this paper we introduce the concept of L-maher and R-maher semigroups and use a technique similar to that used in [7] to show that any left reversible cancellative ordered L or R-maher semigroup can be embedded into an ordered group.     

Gauss-Manin connections for arrangements, III Formal connections

We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a complex rank one local system. We define formal Gauss-Manin connection matrices in the Aomoto complex and prove that, for all arrangements and all local systems, these formal connection matrices specialize to Gauss-Manin connection matrices.

     

A Note on Geometric Structures of Linear Ordered Orthogonal Arrays and (T,M,S)-Nets of Low Strength

The equivalent geometrical configurations of linear ordered orthogonal arrays are determined when their strengths are 3 and 4. Existence of such geometrical configurations is investigated. They are also useful in the study of (T, M, S)-nets.     

Crystal structure and magnetic ordering of RNi Si compounds

The element distributions and the magnetic ordering behaviour of compounds RNi₁₀Si₂ (R = Tb, Dy, Ho, Er, Tm) have been studied by neutron powder diffraction down to temperatures of 1.6 K. The compounds crystallize in an ordered variant of the ThMn₁₂ structure type in the tetragonal space group P4/nmm. An ordered 1:1 distribution of Ni and Si on sites 4d and 4e, respectively, corresponds to a modulation vector [0, 0, 1] with respect to the space group I4/mmm of the ThMn₁₂ structure. TbNi₁₀Si₂ orders antiferromagnetically below T _N = 4.5 K with a magnetic propagation vector of [0, 0, 1/2]. The magnetic Tb moments, 8.97(2) /Tb atom at 1.6 K, are aligned along the c-axis. The Ni sites in TbNi₁₀Si₂ do not carry any ordered magnetic moments. The compounds with R = Dy, Ho, Er, and Tm are paramagnetic down to 1.6 K and 3.0 K, respectively. Received 10 July 2002 / Received in final form 12 September 2002 Published online 29 October 2002     

Interpolation of positive operators

We study problems of interpolation of positive linear operators in couples of ordered Banach spaces. From this viewpoint, we study couples of noncommutative spaces L ₁, L _∞ associated with weights and traces on von Neumann algebras.     

A common generalization for MV-algebras and Łukasiewicz–Moisil algebras

We introduce the notion of n-nuanced MV-algebra by performing a Łukasiewicz–Moisil nuancing construction on top of MV-algebras. These structures extend both MV-algebras and Łukasiewicz–Moisil algebras, thus unifying two important types of structures in the algebra of logic. On a logical level, n-nuanced MV-algebras amalgamate two distinct approaches to many valuedness: that of the infinitely valued Łukasiewicz logic, more related in spirit to the fuzzy approach, and that of Moisil n-nuanced logic, which is more concerned with nuances of truth rather than truth degree. We study n-nuanced MV-algebras mainly from the algebraic and categorical points of view, and also consider some basic model-theoretic aspects. The relationship with a suitable notion of n-nuanced ordered group via an extension of the Γ construction is also analyzed.     

Analysis of atomic arrangement in magnetic Fe–Pt nanoparticles

The order parameter S of Fe–Pt nanoparticles is estimated from X-ray diffraction (XRD) patterns. The total intensity of a diffraction peak is obtained by Rietveld analysis as well as simply integrating the intensity. The Rietveld analysis is found to provide a plausible value of S even for a sample showing an XRD pattern with broad and overlapped peaks. Another order parameter Q, which is obtained from Mössbauer spectra, is introduced, and it is confirmed that Q is equivalent to the probability of Fe atoms being in the L1₀-type atomic arrangement. The coercivity of Fe–Pt nanoparticles is directly proportional to Q, while it vanishes at S=0.4, indicating that the magnetic property of Fe–Pt nanoparticles has a closer relationship to Q than S.     

On then-cutset property

For a partially ordered setP and an elementx ofP, a subsetS ofP is called a cutset forx inP if every element ofS is noncomparable tox and every maximal chain ofP meets {x}∪S. We letc(P) denote the smallest integerk such that every elementx ofP has a cutsetS with ‖S‖?k: Ifc(P)?n we say thatP has then-cutset property. Our results bear on the following question: givenP, what is the smallestn such thatP can be embedded in a partially ordered set having then-cutset property? As usual, 2 ⁿ denotes the Boolean lattice of all subsets of ann-element set, andB _n denotes the set of atoms and co-atoms of 2 ⁿ . We establish the following results: (i) a characterization, by means of forbidden configurations, of whichP can be embedded in a partially ordered set having the 1-cutset property; (ii) ifP contains a copy of 2 ⁿ , thenc(P)?2^[n/2]?1; (iii) for everyn>3 there is a partially ordered setP containing 2 ⁿ such thatc(P)<c(2 ⁿ ); (iv) for every positive integern there is a positive integerN such that, ifB _m is contained in a partially ordered set having then-cutset property, thenm?N.