Theory Of Valuations Research Articles

Resolution of curve and surface singularities in characteristic zero

Preface. Note to the Reader. Terminology. I: Valuation Theory. 1. Marot Rings. 2. Manis Valuation Rings. 3. Valuation Rings and Valuations. 4. The Approximate Theorem for Independent Valuations. 5. Extensions of Valuations. 6. Extending Valuations to Algebraic Overfields. 7. Extensions of Discrete Valuations. 8. Ramification Theory of Valuations. 9. Extending Valuations to Non-Algebraic Overfields. 10. Valuations of Algebraic Function Fields. 11. Valuations Dominating a Local Domain. II: One-Dimensional Semilocal Cohen-Macaulay Rings. 1. Transversal Elements. 2. Integral Closure of One-Dimensional Semilocal Cohen-Macaulay Rings. 3. One-Dimensional Analytically Unramified and Analytically Irreducible CM-Rings. 4. Blowing up Ideals. 5. Infinitely Near Rings. III: Differential Modules and Ramification. 1. Introduction. 2. Norms and Traces. 3. Formally Unramified and Ramified Extensions. 4. Unramified Extensions and Discriminants. 5. Ramification for Quasilocal Rings. 6. Integral Closure and Completion. IV: Formal and Convergent Power Series Rings. 1. Formal Power Series Rings. 2. Convergent Power Series Rings. 3. Weierstrass Preparation Theorem. 4. The Category of Formal and Analytic Algebras. 5. Extensions of Formal and Analytic Algebras. V: Quasiordinary Singularities. 1. Fractionary Power Series. 2. The Jung-Abhyankar Theorem: Formal Case. 3. The Jung-Abhyankar Theorem: Analytic Case. 4. Quasiordinary Power Series.5. A Generalized Newton Algorithm. 6. Strictly Generated Semigroups. VI: The Singularity Zq = XYp. 1. Hirzebruch-Jung Singularities. 2. Semigroups and Semigroup Rings. 3. Continued Factions. 4. Two-Dimensional Cones. 5. Resolution of Singularities. VII: Two-Dimensional Regular Local Rings. 1. Ideal Transform. 2. Quadratic Transforms and Ideal Transforms. 3. Complete Ideals. 4. Factorization of Complete Ideals. 5. The Predecessors of a Simple Ideal. 6. Uniformization. 7. Resolution of Surface Singularities II: Blowing up and Normalizing. Appendices. A: Results from Classical Algebraic Geometry. 1. Generalities. 2. Affine and Finite Morphisms. 3. Products. 4. Proper Morphisms. 5. Algebraic Cones and Projective Varieties. 6. Regular and Singular points. 7. Normalization of a Variety. 8. Desingularization of a Variety. 9. Dimension of Fibres. 10. Quasifinite Morphisms and Ramification. 11. Divisors. 12. Some Results on Projections. 13. Blowing up. 14. Blowing up: the Local Rings. B: Miscellaneous Results. 1. Ordered Abelian Groups. 2. Localization. 3. Integral Extensions. 4. Some Results on Graded Rings and Modules. 5. Properties of the Rees Ring. 6. Integral Closure of Ideals. 7. Decomposition Group and Inertia Group. 8. Decomposable Rings. 9.

Continuous Rotation Invariant Valuations on Convex Sets

The notion of valuation on convex sets can be considered as a generalization of the notion of measure, which is deflned only on the class of convex compact sets. It is well-known that there are important and interesting examples of valuations on convex sets, which are not measures in the usual sense as, for example, the mixed volumes. Basic deflnitions and some classical examples are discussed in Section 2 of this paper. For more detailed information we refer to the surveys [Mc-Sch] and [Mc3]. Throughout this paper all the valuations are assumed to be continuous with respect to the Hausdorfi metric. Note that the theory of valuations which are invariant or covariant with respect to translations belongs to the classical part of convex geometry. There exists an explicit description of translation invariant continuous valuations on 1 and 2 due to Hadwiger [H1] (the case of 2 is nontrivial). Continuous rigid motion invariant valuations on d are completely classifled by the remarkable Hadwiger theorem as linear combinations of the quermassintegrals (cf. [H2] or for a simpler proof [K]).

On Valuations, the Characteristic Polynomial, and Complex Subspace Arrangements

We present a new combinatorial method to determine the characteristic polynomial of any subspace arrangement that is defined over an infinite field, generalizing the work of Blass and Sagan. Our methods stem from the theory of valuations and Groemer's integral theorem. As a corollary of our main theorem, we obtain a result of Zaslavsky about the number of chambers of a real hyperplane arrangement. The examples we consider include a family of complex subspace arrangements, which we call the divisor Dowling arrangement, whose intersection lattice generalizes that of the Dowling lattice. We also determine the characteristic polynomial of interpolations between subarrangements of the divisor Dowling arrangement, generalizing the work of Józefiak and Sagan.

Kinematic formulas for finite lattices

In analogy to valuation characterizations and kinematic formulas of convex geometry, we develop a combinatorial theory of invariant valuations and kinematic formulas for finite lattices. Combinatorial kinematic formulas are shown to have application to some probabilistic questions, leading in turn to polynomial identities for Möbius functions and Whitney numbers.

Convergence structures induced by scales

By a (fine) scale on a lattice L, we mean a (strictly) isotone real-valued function μ on L. The coarsest topology on L such that μ composed with the unary lattice operations becomes continuous is denoted by τ μ . The following convergence structures on L are compared with each other: 1. (i) order convergence, 2. (ii) convergence in the order topology, 3. (iii) τ μ -convergence, 4. (iv) ρ μ -convergence, where ρ μ ( x, y) = μ( x ∨ y) − μ( x ∧ y). We show that for any scale μ on an arbitrary complete lattice, order convergence agrees with ρ μ -convergence and with τ μ -convergence iff μ is a fine continuous scale such that join and meet operations of arbitrary arity are continuous with respect to ρ μ -convergence. Furthermore, for any fine continuous scale μ on a bi-algebraic lattice, order convergence agrees with τ μ -convergence. From these and related results, we derive various applications to the theory of measures and valuations on orthomodular lattices. For example, if μ is a fine scale on a complete orthomodular lattice then order convergence agrees with τ μ -convergence iff μ is continuous and L is algebraic (or atomic and meet-continuous).

Valuations and higher level orders in commutative rings

Valuation theory is one of the main tools for studying higher level orders and the reduced theory of forms over fields, see, for example [BR]. In [MW], the theory of higher level orders and reduced forms was generalized to rings with many units and many of the results for fields carried over to this setting. While it seems desirable to extend these results further, the techniques used for rings with many units will not work for general commutative rings. At the same time, there is a general theory of valuations in commutative rings (see [LM], [M], and [G]), which in [Ma] was used to study orders and the reduced theory of quadratic forms over general commutative rings. Thus it seems natural to ask if the connections between valuations and higher level orders in fields exist in commutative rings. In this paper we use valuation theory to study the space of orders and the reduced Witt ring relative to a higher level preorder in a commutative ring. As in [Ma], we first localize our ring at a multiplicative set, without changing the space of orders, in order to make the valuation theory work better. This is a standard idea from real algebraic geometry. Remarkably, many of the notions, methods, and results for fields carry over to this new setting. We define compatiblity between valuations and orders and preorders, and the ring A(T ) associated to a preorder T , which turns out to be Prufer ring as in the field case. We define the relation of dependency on the set of valuations associated to a preorder and we use this to prove a decomposition theorem for the space of orders. We can then apply this to show that, under a certain finiteness condition, the space of orders is equivalent to the space of orders of a preordered field.

Orderings and real places on commutative rings

Over fields, there is a close relationship between orderings and real places which has been exploited in quadratic form theory [3,5,6, 131 and in the study of the real holomorphy rings and sums of 2n-powers [1,2, 21,221. Also, orderings on an arbitrary commutative ring [4, 141 have proved useful in real algebraic geometry and in quadratic form theory over rings more general than fields. Of course, real places and valuations on fields have played an important role in this theory too. At the same time, there is a general theory of valuations on commutative rings which has been in existence for some time [9, 10,15,23,24]. In this paper, k-places, k a formally real field (in particular, R-places) are considered on an arbitrary commutative ring R. To obtain the good separation properties we want, the places we consider are functions

An elementary based sufficient condition for sums of 2mth powers of polynomials over non-archimedean real closed fields

Using some model-theoretic arguments applied to the theory of real valuations, we obtain a sufficient condition based on elementary properties of ƒ( x), ƒ( x)∈K[x 1,…,x n] , K any real closed field, for ƒ to be sum of 2 mth powers of rational functions in K( x ).

Constructive theory of valuations

Let k be an algebraically closed field of characteristic p ≥ 0, v a discrete rank-one valuation of .We proved, by expliciting the valuation, the existence of discrete rank-one valuations with preassigned singular ideal.Let L be a linear form of Zn.Then the residue field of a discrete valuation v associated with Lisa purely transcendental extension of k, of trascendence degree n-1.In the case n=2, we describe an algorithm that construct the residue field of any discrete rank-one valuation of k2πk, whose center in R2is the maximal ideal, proving that the residue field is a purely transcendental extension of k, of trascendence degree 1. Throughout this paper, k is a fixed algebraically closed field of characteristic p ≥ 0.Let be a collection of indeterminates, and write .

Structural synthesis of multivariable controllers

In this paper the problem of achieving a desired transfer function matrix H d( s) between external inputs and controlled outputs in a linear multivariable system by connecting proper, stabilizing controllers between measured outputs and control inputs is solved in both transfer function and state space settings. The class H d of achievable transfer functions is directly and constructively characterized via the theory of transfer function valuations. For each H d( s) ϵH d, the class of synthesizing controllers is determined. Similar valuation conditions are given for the asymptotic tracking and disturbance rejection problem in which H d( s) is only partially specified. These results extend and complement earlier results. The state space geometric solution of the problem of achieving a desired H d( s) is obtained by formulating it as an equivalent output feedback disturbance rejection problem. A constructive solvability condition in terms of a pair of measurement and control invariant subspaces is given. This requires a nontrivial generalization of the notion of ( C, A, B) pairs. An H d( s) is shown to be admissible if and only if it induces an appropriate pair of invariant subspaces. The signal flow structure and certain factorizability conditions for a robust synthesis of the output feedback disturbance rejection problem are also given which extend earlier results on robust state feedback disturbance rejection. It is shown that every compensator corresponds to a state feedback control law implemented by an unknown input observer. The results of this paper are expected to be useful in the development of parameter optimization or other computer aided design algorithms where response specifications are to be traded against other design criteria such as sensitivity or stability margins since they explicitly characterize the algebraic variety H d in which the response transfer function may lie.

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.

Structural Synthesis of Multivariable Controllers

In this paper the problem of achieving a desired transfer function matrix Hd(s) between external inputs and controlled outputs in a linear multivariable system by connecting proper, stabilizing controllers between measured outputs and control inputs is solved in both transfer function and state space settings. The class Hd of achievable transfer functions is directly and constructively characterized via the theory of transfer function valuations. For each Hd(s) ε Hd, the class of synthesizing controllers is determined. Similar valuation conditions are given for the asymptotic tracking and disturbance rejection problem in which Hd(s) in only partially specified, The problem of achieving a desired Hd(s) is solved by formulating its as an equivalent output feedback disturbance rejection problem. The signal flow structure and certain factorizability conditions for a robust synthesis of the output feedback disturbance rejection problem are also given which extend earlier results on robut state feedback disturbance rejection.

Tests of Market Efficiency of the Chicago Board Options Exchange

Although options have been traded for many centuries,' they are getting increasing attention both in the capital market as an investment tool and in the development of the theory of capital-assets valuations.2 In this work, I investigate the pricing mechanism of call options written on stocks and test the efficiency of a market in which they are traded, the newly created market, the Chicago Board Options Exchange (CBOE). The efficiency of the CBOE is to be tested against a specific model for option pricing developed by Black and Scholes.3 This model was selected because of its attractiveness on theoretical grounds4 and its consistency with the well-known capital-asset-pricing model (CAPM) of Sharpe and Lintner.5 An American call option is a right to buy a specific quantity of a given security at a fixed, predetermined price (the striking price) during a

Ramification theoretic methods in algebraic geometry

Preface Introduction I. General Ramification Theory 1. Elementary lemmas from ideal theory 2. Primary decomposition in non-noetherian rings 3. Integral dependence 4. Linear algebra 5. Discriminant of an algebra 6. The discriminant ideal 7. Galois theory of local rings II. Valuation Theory 8. Ordered abelian groups 9. Valuations 10. Specialization and composition of valuations 11. Ramification theory of valuations 12. Valuations of algebraic function fields III. Noetherian Local Rings 13. The dimension of a noetherian local ring 14. Quadratic transforms IV. Two-Dimensional Local Domains 15. Limits of quadratic sequences 16. Uniformization in a two-dimensional regular local domain 17. Uniformization for a two-dimensional algebraic function field V. Varieties and Transformations 18. Projective varieties 19. Resolution of singularities of algebraic surfaces Bibliography

Book Review: The theory of valuations

Book Review: The theory of valuations

Life office valuations with reference to a reversionary bonus system

The object of this paper is to attempt to evolve a theoretically correct method of valuing life policies in an office having a ‘level’ Compound Reversionary Bonus system. This method can then be used to examine the existing methods of valuation in detail, and also to trace the connexion between the existing methods. The theoretical method of valuation can be approached by a mathematical analysis on strictly theoretical lines. A Compound Reversionary Bonus method has been used in this paper, but similar principles can be applied in the case of a simple reversionary bonus method.It is emphasized at this early stage that I am only concerned with the theory of valuations, and the practical points, such as legal requirements, presentation of valuation results and simplicity of valuation machinery, are not considered.