Real Valuations and Orderings on a Module

The purpose of this article is to introduce the notion of real valuations on modules over commutative rings. These real valuations are characterized by their associated valuation triples, and a necessary and sufficient condition for a module to possess a real valuation is established. Moreover, the close interplay between valuations and orderings on a module is investigated by introducing the compatibility of valuations with orderings. Via the compatibility of valuations with orderings, the reality of valuations on a module also is characterized.     

Valuation Extensions of Filtered and Graded Algebras

In this note we relate the valuations of the algebras appearing in the noncommutative geometry of quantized algebras to properties of sublattices in some vector spaces. We consider the case of algebras with PBW-bases and prove that under some mild assumptions, the valuations of the ground field extend to a noncommutative valuation. Later we introduce the notion of F-reductor and graded reductor and reduce the problem of finding an extending noncommutative valuation to finding a reductor in an associated graded ring having a domain for its reduction.     

Global analytic geometry

Motivated by the well-known lack of archimedean information in algebraic geometry, we define, formalizing Ostrowski's classification of seminorms on Z, a new type of valuation of a ring that combines the notion of Krull valuation with that of a multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations artificially introduced in arithmetic geometry by the theory of schemes. This also allows us to define a notion of global analytic space that reconciles Berkovich's notion of analytic space of a (Banach) ring with Huber's notion of non-archimedean analytic spaces. After defining natural generalized valuation spectra and computing the spectrum of Z and Z[X], we define analytic spectra and sheaves of analytic functions on them.     

Theory of valuations on manifolds, II

This article is the second part in the series of articles where we are developing theory of valuations on manifolds. Roughly speaking valuations could be thought as finitely additive measures on a class of nice subsets of a manifold which satisfy some additional assumptions.The goal of this article is to introduce a notion of a smooth valuation on an arbitrary smooth manifold and establish some of the basic properties of it.     

Theory of valuations on manifolds, III. Multiplicative structure in the general case

This is the third part of a series of articles where the theory of valuations on manifolds is constructed. In the second part of this series the notion of a smooth valuation on a manifold was introduced. The goal of this article is to put a canonical multiplicative structure on the space of smooth valuations on general manifolds, thus extending some of the affine constructions from the first author's 2004 paper and, from the first part of this series.

     

Uniform valuations on planar ternary rings

We introduce a new notion of valuations on planar ternary rings(PTRs), which allows us to extend some results concerning the relations between orderings and valuations of fields to PTRs. Our concept generalizes van Maldeghem's notion of PTRs with valuation, coordinatizing the buildings at infinity of discrete triangle buildings.     

Valuations of glued near hexagons

Valuations of near polygons were introduced in [ 12 ] as an important tool for classifying dense near polygons. In the present article, we will introduce the class of the semi‐diagonal valuations. These valuations live in glued near hexagons. A glued near hexagon S can be coordinatized by a pair of admissible triples; such triples consist of a Steiner system , a group G, and a certain nice map . We will give a necessary and sufficient condition for the existence of semi‐diagonal valuations in in terms of these two admissible triples. For two classes of glued near hexagons, we will use this condition to determine all semi‐diagonal valuations. Each semi‐diagonal valuation will also give rise to a hyperplane of the glued near hexagon. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 35–48, 2007     

Valuations with Crofton formula and Finsler geometry

Valuations admitting a smooth Crofton formula are studied using Geometric Measure Theory and Rumin's cohomology of contact manifolds. The main technical result is a current representation of a valuation with a smooth Crofton formula. A geometric interpretation of Alesker's product is given for such valuations. As a first application in Finsler geometry, a short proof of the theorem of Gelfand–Smirnov that Crofton densities are projective is derived. The Holmes–Thompson volumes in a projective Finsler space are studied. It is shown that they induce in a natural way valuations and that the Alesker product of the k-dimensional and the l-dimensional Holmes–Thompson valuation is the (k+l)-dimensional Holmes–Thompson valuation.     

A ruled residue theorem for function fields of conics

The ruled residue theorem characterises residue field extensions for valuations on a rational function field. Under the assumption that the characteristic of the residue field is different from 2 this theorem is extended here to function fields of conics. The main result is that there is at most one extension of a valuation on the base field to the function field of a conic for which the residue field extension is transcendental but not ruled. Furthermore the situation when this valuation is present is characterised.     

Valuations on lattice polytopes

Let L be a lattice (that is, a Z-module of finite rank), and let L=P(L) denote the family of convex polytopes with vertices in L; here, convexity refers to the underlying rational vector space V=Q⊗L. In this paper it is shown that any valuation on L satisfies the inclusion-exclusion principle, in the strong sense that appropriate extension properties of the valuation hold. Indeed, the core result is that the class of a lattice polytope in the abstract group L=P(L) for valuations on L can be identified with its characteristic function in V. In fact, the same arguments are shown to apply to P(M), when M is a module of finite rank over an ordered ring, and more generally to appropriate families of (not necessarily bounded) polyhedra.     

Evaluation codes and plane valuations

We apply tools coming from singularity theory, as Hamburger–Noether expansions, and from valuation theory, as generating sequences, to explicitly describe order functions given by valuations of 2-dimensional function fields. We show that these order functions are simple when their ordered domains are isomorphic to the value semigroup algebra of the corresponding valuation. Otherwise, we provide parametric equations to compute them. In the first case, we construct, for each order function, families of error correcting codes which can be decodified by the Berlekamp–Massey–Sakata algorithm and we give bounds for their minimum distance depending on minimal sets of generators for the above value semigroup.     

Valuation theory: A constructive view

The theory of valuations on fields is developed in the constructive spirit of Errett Bishop. As a consequence of the general theory we are able to construct all nonarchimedean valuations on algebraic number fields and compute their ramification indices and residue class degrees. The notion of a field with a valuation for which the infimum of the values of any polynomial function can be computed plays an important role. Numerous limiting counterexamples are provided.     

Rank One Discrete Valuations of Power Series Fields

In this article we study rank one discrete valuations of the field k((X ₁,…, X _n )) whose center in k[[X ₁,…, X _n ]] is the maximal ideal. In Sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote Section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in Section 5.

The constructions given in these sections are not effective in the general case, because we need either to use Zorn's lemma or to know explicitly a section σ of the natural homomorphism R _v  → Δ _v between the ring and the residue field of the valuation v.

However, as a consequence of this construction, in Section 7, we prove that k((X ₁,…, X _n )) can be embedded into a field L((Y ₁,…, Y _n )), where L is an algebraic extension of k and the “extended valuation” is as close as possible to the usual order function.     

Saturation for valuations on two-dimensional regular local rings

We study valuations centered in a regualar local ring of dimension two. We define the notion of saturation with respect to such a valuation, extending the classical definitions. The invariants associated in a natural way to the valuation are related with the saturated ring and som geometric properties are deduced. Received February 16, 1998; in final form February 9, 1999     

Pseudo-valuations on pre-logics

Based on the Buşneag’s model [1,2,3], the notion of pseudo-valuations (valuations) on a pre-logic is introduced, and a pseudo-metric is induced by a pseudo-valuation on pre-logics. Related properties are investigated, and conditions for a real-valued function on a pre-logic to be a pseudo-valuation are discussed. Using the notion of (pseudo) valuation, we show that the binary operation ∗ in pre-logics is uniformly continuous.     

Real Closed Valuation Rings

The real closed valuation rings, i.e., convex subrings of real closed fields, form a proper subclass of the class of real closed domains. It is shown how one can recognize whether a real closed domain is a valuation ring. This leads to a characterization of the totally ordered domains whose real closure is a valuation ring. Real closures of totally ordered factor rings of coordinate rings of real algebraic varieties are very frequently valuation rings. In particular, the real closure of the coordinate ring of a curve is an SV-ring (i.e., the factor rings modulo prime ideals are valuation rings). Real closed valuation rings play a role in the definition of real closed rings, as well as in the construction of real closures of rings and porings. They can also be used for the study of univariate differentiable semi-algebraic functions. This leads to the notion of differentiablility of semi-algebraic functions along half branches of curves.     

Even valuations on convex bodies

The notion of even valuation is introduced as a natural generalization of volume on compact convex subsets of Euclidean space. A recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on Grassmannians. This connection can be described in part using generating distributions for symmetric compact convex sets. We also explore some consequences of these characterization results in convex and integral geometry.

     

A Valuation Theory for Nonassociative Quaternion Algebras

A nonassociative quaternion algebra over a field F is a 4-dimensional F-algebra A whose nucleus is a separable quadratic extension field of F. We define the notion of a valuation ring for A, and we also define a value function on A with values from a totally ordered group. We determine the structure of the set on which the function assumes non-negative values and we prove that, given a valuation ring of A, there is a value function associated to it if and only if the valuation ring is integral and invariant under proper F-automorphisms of A.     

Fair dynamic valuation of insurance liabilities: Merging actuarial judgement with market- and time-consistency

In this paper, we investigate the fair valuation of insurance liabilities in a dynamic multi-period setting. We define a fair dynamic valuation as a valuation which is actuarial (mark-to-model for claims independent of financial market evolutions), market-consistent (mark-to-market for any hedgeable part of a claim) and time-consistent, extending the work of Dhaene et al. (2017) and Barigou and Dhaene (2019). We provide a complete hedging characterization for fair dynamic valuations. Moreover, we show how to implement fair dynamic valuations through a backward iterations scheme combining risk minimization methods from mathematical finance with standard actuarial techniques based on risk measures.     

Horizon Valuations for Mathematical Programming in Financial Planning

Using a linear programming model for the financial planning of an organization requires the specification of a horizon date and a valuation of the firm at that date. Given perfect information about future opportunities, an exact valuation procedure should lead to the same optimal solution of the model regardless of the choice of horizon date. Even in the absence of perfect information, conventional valuations fall far shorter of this ideal than they need to. It is shown that for a modest increase in the size of the linear programme, better valuations can be achieved and, most importantly, valuations which consider the impact on the value of the firm of post horizon constraints and liabilities as well as post-horizon opportunities.