Theory Of Valuations Research Articles

Valuations on Concave Functions and Log-Concave Functions

Recently, the theory of valuations on function spaces has been rapidly growing. It is more general than the classical theory of valuations on convex bodies. In this paper, all continuous, SL(n) and translation invariant valuations on concave functions and log-concave functions are completely classified, respectively.

Contact measures in isotropic spaces

We revisit the contact measures introduced by Firey, and further developed by Schneider and Teufel, from the perspective of the theory of valuations on manifolds. This reveals a link between the kinematic formulas for area measures studied by Wannerer and the integral geometry of curved isotropic spaces. As an application we find explicitly the kinematic formula for the surface area measure in Hermitian space.

Valuation Leveling on Rational Function Fields

Since a new field of research in the theory of valuations has been opened by the notion of symmetric extensions of a valuation on a field K to K(X1,…, Xn), with respect to the indeterminates X1,…, Xn, it makes sense to look for applications of this theory in dealing with arbitrary valuations on rational function fields. This paper investigates the conditions and methods of transforming asymmetric valuations on rational function fields into symmetric ones - procedure that will be called valuation leveling - the motivation being the opportunity of applying the specific results from the theory of symmetric valuations in the general case.

Reified valuations and adic spectra

We revisit Huber’s theory of continuous valuations, which give rise to the adic spectra used in his theory of adic spaces. We instead consider valuations which have been reified, i.e., whose value groups have been forced to contain the real numbers. This yields reified adic spectra which provide a framework for an analogue of Huber’s theory compatible with Berkovich’s construction of nonarchimedean analytic spaces. As an example, we extend the theory of perfectoid spaces to this setting.

Kinematic formulas for tensor valuations

Abstract We prove new kinematic formulas for tensor valuations and simplify previously known Crofton formulas by using the recently developed algebraic theory of translation invariant valuations. The heart of the paper is the computation of the Alesker–Fourier transform on the large class of spherical valuations, which is achieved by differential-geometric and representation theoretical tools. We also describe in explicit form the product and convolution of tensor valuations.

Real valuations and the limits of multivariate rational functions

The purpose of this paper is to investigate the limits of multivariate rational functions with the aid of the theory of real valuations. The following is one of our main results. For two nonzero polynomials f, g ∈ ℝ[x1,…,xn] and (a1,…,an) ∈ ℝn, the (finite) limit of the rational function [Formula: see text] at (a1,…,an) does not exist if and only if (1) there exists a sequence u1(x),…,un(x) of polynomials over ℝ in one variable x such that ui(0) = ai for i = 1,…,n, g(u1(x),…,un(x)) ≠ 0, but [Formula: see text]; or (2) there exist two sequences u1(x),…,un(x) and w1(x),…,wn(x) of polynomials over ℝ in one variable x such that ui(0) = wi(0) = ai for i = 1,…,n, g(u1(x),…,un(x)) ⋅ g(w1(x),…,wn(x)) ≠ 0, but [Formula: see text].

Fear or Greed? Duty or Solidarity? Motivations and Stages of Moral Reasoning

As institutional economists recognize the limits of the canonical self-interest assumption, the lack of a theory of human valuation specifying the determinants of individuals’ utility judgments renders the prediction of behavior in collective action dilemmas virtually impossible. This problem hinders our ability to devise institutions to help cope with a variety of pressing social dilemmas that persist. We suggest that scholars can overcome this difficulty by integrating models of sociocognitive and moral development into the framework of institutional analysis and that such integration aligns with a new revolutionary shift in the social sciences. This shift is evidenced by the “rehabilitation” of key notions once downplayed due to the dominance of positivism. We present results from hypothesis testing linked to a cognitivist-developmental theory of human valuations. These results demonstrate that cooperative motivations and choices in public goods provision dilemmas are associated with further stages of interior development of the participants. We conclude that institutions addressing choices in morally relevant conflicts of action should be designed to promote swifter movement of individuals along the path of interior growth.

Integral geometry of complex space forms

We show how Alesker's theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this approach yields a new formula expressing the volumes of the tubes about a totally real submanifold in terms of its intrinsic Riemannian structure. We also show by direct calculation that the Lipschitz-Killing valuations stabilize the subspace of invariant angular curvature measures, suggesting the possibility that a similar phenomenon holds for all Riemannian manifolds. We conclude with a number of open questions and conjectures.

Linear extension sums as valuations on cones

The geometric and algebraic theory of valuations on cones is applied to understand identities involving summing certain rational functions over the set of linear extensions of a poset.

Towards a Behavioral Algebraic Theory of Logical Valuations

Logical matrices are widely accepted as the semantic structures that most naturally fit the traditional approach to algebraic logic. The behavioral approach to the algebraization of logics extends the applicability of the traditional methods of algebraic logic to a wider range of logical systems, possibly encompassing many-sorted languages and non-truth-functional phenomena. However, as one needs to work with behavioral congruences, matrix semantics are unsuited to the behavioral setting. In [5], a promising version of algebraic valuation semantics was proposed in order to fill in this gap. Herein, we define the class of valuations that should be canonically associated to a logic, and we show, by means of new meaningful bridge results, how it is related to the behaviorally equivalent algebraic semantics of a behaviorally algebraizable logic.

A fuzzy similarity inference method for fuzzy reasoning

In this paper, Fuzzy Similarity Inference (FSI) is investigated. First, the axiomatic definition of the fuzzy similarity measure is introduced. It is a generalization of the similarity measure. Some formulas are given to compute the fuzzy similarity of two fuzzy sets. Second, a novel fuzzy reasoning, called fuzzy similarity inference, is proposed. Computational formulas for both fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT) are obtained. Based on the theory of partial valuations, FSI method is put under the framework of logic semantically. Furthermore, the generalized problem of FSI is investigated. Computational formulas for α-FSI FMP and α-FSI FMT are obtained. At last, reversibility properties of FSI are proved.

Quaternionic plurisubharmonic functions and their applications to convexity

The goal of this article is to present a survey of the recent theory of plurisubharmonic functions of quaternionic variables, and its applications to theory of valuations on convex sets and HKT-geometry (HyperK\ahler with Torsion). The exposition follows the articles math.CV/0104209, math.CV/0208005, math.MG/0401219 by the author and math.CV/0510140 by M. Verbitsky and the author.

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.

Stable valued fields

We are concerned with a class of valued fields, called stable. We propound an extension of a notion in the monograph by S. Bosch, U. Guntzer, and R. Remmert (Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry, Springer, Berlin (1984)), namely, that of a (ultrametric) norm on groups, rings, algebras, and vector spaces, to the case where the value of the norm is taken from an arbitrary (not necessarily Archimedean) linearly ordered Abelian group (using — as in the general theory of valuations — the version of a logarithmic norm). Our main result extends Proposition 6 in the cited monograph to the general case, thereby making it possible to use the technique of Cartesian spaces to deliver further results on stable valued fields.

Theory of Valuations on Manifolds: A Survey

This is a non-technical survey of a recent theory of valuations on manifolds constructed in [A10], [A11], [AF] and [A12], and actually a guide to this series of articles. We also review some recent related results obtained by a number of people.

Translation invariant valuations in the space of planes

The paper is related to the problem of measure generation in the space IE of planes in IR3 by combinatorial, translation invariant valuations. General results concerning that problem have been derived in 1994 and 1996 (this journal vol. 31, no. 4. and vol. 33, no. 4, respectively). The purpose of the present article is to give a proof of two geometrical identities on which the theorem on valuations in the space IE can be based. The article consists of the motivational first part that contains the basic concepts from the theory of combinatorial valuations and measure generation in IE, and the second that gives a proof of the identities in question.

Plotting missing points and branches of real parametric curves

This paper is devoted to the study (from the theoretic and algorithmic point of view) of the existence of points and branches non-reachable by a parametric representation of a rational algebraic curve (in n-dimensional space) either over the field of complex numbers or over the field of real numbers. In particular, we generalize some of the results on missing points in (J. Symbolic Comput. 33, 863–885, 2002) to the case of space curves. Moreover, we introduce for the first time and we solve the case of missing branches. Another novelty is the emphasis on topological conditions over the curve for the existence of missing points and branches. Finally, we would like to point out that, by developing an “ad hoc” and simplified theory of valuations for the case of parametric curves, we approach in a new and unified way the analysis of the missing points and branches, and the proposal of the algorithmic solution to these problems.

Theory of valuations on manifolds, I. Linear spaces

This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of an affine space. In this article we still work only with linear spaces. We introduce a space of smooth (non-translation invariant) valuations on a linear spaceV. We present three descriptions of this space. We describe the canonical multiplicative structure on this space generalizing the results from [4] obtained for polynomial valuations.

Isometric full embeddings of [formula omitted] into [formula omitted

We show that there is up to isomorphism a unique isometric full embedding of the dual polar space DW ( 2 n − 1 , q ) into the dual polar space DH ( 2 n − 1 , q 2 ) . We use the theory of valuations of near polygons to study the structure of this isometric embedding. We show that for every point x of DH ( 2 n − 1 , q 2 ) at distance δ from DW ( 2 n − 1 , q ) the set of points of DW ( 2 n − 1 , q ) at distance δ from x is a so-called SDPS-set which carries the structure of a dual polar space DW ( 2 δ − 1 , q 2 ) . We show that if n is even, then the set of points at distance at most n 2 − 1 from DW ( 2 n − 1 , q ) is a geometric hyperplane of DH ( 2 n − 1 , q 2 ) and we study some properties of these new hyperplanes.

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.