Preparation Theorem Research Articles

Low-frequency dispersion characteristics of a multilayered coaxial cable

This paper provides an exact asymptotic analysis regarding the low-frequency dispersion characteristics of a multilayered coaxial cable. A layer-recursive description of the dispersion function is derived that is well suited for asymptotic analysis. The recursion is based on two well-behaved (meromorphic) subdeterminants defined by a perfectly electrically conducting (PEC) and a perfectly magnetically conducting termination, respectively. For an open waveguide structure, the dispersion function is a combination of two such functions, and there is only one branch point that is related to the exterior domain. It is shown that if there is one isolating layer and a PEC outer shield, then the classical Weierstrass preparation theorem can be used to prove that the low-frequency behavior of the propagation constant is governed by the square root of the complex frequency, and an exact analytical expression for the dominating term of the asymptotic expansion is derived. It is furthermore shown that the same asymptotic expansion is valid to its lowest order even if the outer shield has finite conductivity and there is an infinite exterior region with finite nonzero conductivity. As a practical application of the theory, a high-voltage direct current (HVDC) power cable is analyzed and a numerical solution to the dispersion relation is validated by comparisons with the asymptotic analysis. The comparison reveals that the low-frequency dispersion characteristics of the power cable is very complicated and a first-order asymptotic approximation is valid only at extremely low frequencies (below 1 Hz). It is noted that the only way to come to this conclusion is to actually perform the asymptotic analysis. Hence, for practical modeling purposes, such as with fault localization, an accurate numerical solution to the dispersion relation is necessary and the asymptotic analysis is useful as a validation tool.

Lebesgue classes and preparation of real constructible functions

We call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. For any q>0 and constructible functions f and μ on E×Rn, we prove a theorem describing the structure of the set{(x,p)∈E×(0,∞]:f(x,⋅)∈Lp(|μ|xq)}, where |μ|xq is the positive measure on Rn whose Radon–Nikodym derivative with respect to the Lebesgue measure is |μ(x,⋅)|q:y↦|μ(x,y)|q. We also prove a closely related preparation theorem for f and μ. These results relate analysis (the study of Lp-spaces) with geometry (the study of zero loci).

Integration of semialgebraic functions and integrated Nash functions

When integrating semialgebraic functions one has to leave the semialgebraic setting. For example, one gets the global logarithm and iterated antiderivatives of algebraic power series such as the arctangent. We show that it is enough to enlarge the semialgebraic functions by these functions to completely describe parameterized integrals of semialgebraic functions. To realize this we close the rings of algebraic power series in arbitrary dimension under taking antiderivatives. We analyze these rings profoundly. In particular we show that the Weierstrass division theorem and the Weierstrass preparation theorem hold. This allows us to apply model theoretic results to obtain an explicit description of parameterized integrals of semialgebraic functions. Finally, we investigate the structure generated by the integrated algebraic power series.

The asymptotic analysis of multiple imaginary characteristic roots for LTI delayed systems based on Puiseux–Newton diagram

The paper proposes a novel procedure for the asymptotic expansions of root loci around multiple imaginary roots of an exponential polynomial, which is necessary for the stability analysis of the LTI systems with commensurate delays. With the LTI delay systems given as exponential polynomials (also called quasi-polynomial), we seek to characterise the asymptotic behaviours of the characteristic roots of such systems in an algebraic way and determine whether the imaginary roots cross from one half-plane into another or only touch the imaginary axis. According to the Weierstrass preparation theorem, the quasi-polynomial equation is equivalent to an algebraic equation in the neighbourhood of a singular point. Furthermore, our result gives an explicit expression of the coefficients of the algebraic equation in infinite power series of delay parameter, and the determinations of such power series coefficients refer to the computation of residues of memorphic functions. Subsequently, the classic Puiseux–Newton diagram algorithm can be used to calculate the algebraic expansions of the reduced equation directly. Thus, the asymptotic behaviours of root loci around singular points of the quasi-polynomial equation are obtained. Some illustrative simulations are given to check the validity of the proposed method on asymptotic analysis with a powerful software.

Weierstrass preparation and algebraic invariants

We prove a form of the Weierstrass Preparation Theorem for normal algebraic curves over complete discrete valuation rings. While the more traditional algebraic form of Weierstrass Preparation applies just to the projective line over a base, our version allows more general curves. This result is then used to obtain applications concerning the values of u-invariants, and on the period–index problem for division algebras, over fraction fields of complete two-dimensional rings. Our approach uses patching methods and matrix factorization results that can be viewed as analogs of Cartan’s Lemma.

On the structure of p-adic subanalytic functions and sets

We prove a p-adic version of the Lion–Rolin preparation theorem. As a consequence, we obtain a cell decomposition theorem, which can be viewed as an extension of Denef’s cell decomposition theorem to the p-adic analytic case. Using these theorems, we give shorter and more explicit proofs of some of the results by Denef and van den Dries on p-adic subanalytic sets.

Elementary analytic methods in higher ramification theory

This paper offers proofs of a number of standard results in the higher ramification theory of discretely valued fields, using as tools only the Weierstrass Preparation Theorem and the theory of the Newton polygon and copolygon. The propositions proved are the functoriality of the Hasse–Herbrand transition function, Herbrandʼs Theorem, Senʼs Theorem on the rapidity of approach to the identity of successive p -power iterates of an invertible series in characteristic p , and the Hasse–Arf Theorem.

An inductive analytic criterion for flatness

We present a constructive criterion for flatness of a morphism of analytic spaces X -> Y or, more generally, for flatness over Y of a coherent sheaf of modules on X. The criterion is a combination of a simple linear-algebra condition in codimension zero and a condition in codimension one which can be used together with the Weierstrass preparation theorem to inductively reduce the fibre dimension of the morphism.

Bornes effectives des fonctions d'approximation des solutions formelles d'équations binomiales

The aim of this paper is to give an effective version of the Strong Artin Approximation Theorem for binomial equations. First we give an effective version of the Greenberg Approximation Theorem for polynomial equations, then using the Weierstrass Preparation Theorem, we apply this effective result to binomial equations. We prove that the Artin function of a system of binomial equations is bounded by a doubly exponential function in general and that it is bounded by an affine function if the order of the approximated solutions is bounded.

Weierstrass preparation theorem for noncommutative rings

A power series over a complete local ring can be canonically decomposed into the product of an invertible power series and a unital polynomial whose degree coincides with the number of the first invertible coefficient. This statement is known as the Weierstrass preparation theorem. It follows from a more general statement known as the Weierstrass division theorem. The present article contains a detailed proof of generalizations of the Weierstrass preparation theorem and the Weierstrass division theorem for the so-called rings of skew power series. Such rings arise in number theory, and first in studies of formal groups over local fields. Bibliography: 3 titles.

Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings

This paper investigates the geometry of the expansion RQ of the real field R by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusinski–Lion–Rolin). To this end, we study non-standard models R of the universal diagram T of RQ in the language L augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation theorem, upon which the majority of fundamental results in analytic geometry rely, but which is unavailable in the general quasianalytic geometry. The basic tools applied here are transformation to normal crossings and decomposition into special cubes. The latter method, developed in our article [32], combines modifications by blowing up with a suitable partitioning. Via an analysis of L-terms and infinitesimals, we prove the valuation property for functions given by L-terms, and next the exchange property for substructures of a given model R. Our proofs are based on the concepts of analytically independent as well as active and non-active infinitesimals, introduced in this article. Further, quantifier elimination for T is established through model-theoretic compactness. The universal theory T is thus complete and o-minimal, andRQ is its prime model. Under the circumstances, every definable function is given piecewise by L-terms, and therefore the previous results concerning L-terms generalize immediately to definable functions. In this fashion, we obtain the valuation property and preparation theorem for quasi-subanalytic functions. Finally, a quasi-subanalytic version of Puiseux’s theorem with parameter is demonstrated. AMS Classification: Primary 32S45, 14P15, 32B20; Secondary 03C10, 26E10, 03C64.

La somme des faux degrés—un mystère en théorie des invariants

We continue the study of the polynomiality of the semicentre Sz ( p ) of the enveloping algebra of a parabolic subalgebra p of a semisimple Lie algebra g , motivated by its truth when g is of type A or C [F. Fauquant-Millet, A. Joseph, Semi-centre de l'algèbre enveloppante d'une sous-algèbre parabolique d'une algèbre de Lie semi-simple, Ann. Sci. École Norm. Sup. (4) 38 (2) (2005) 155–191] and when p = b , a Borel subalgebra [A. Joseph, A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. Algebra 48 (1977) 241–289] and p = g (Chevalley). We construct a linear map of Sz ( b ) into Sz ( g ) and show it to be an isomorphism just in types A and C. We link this to the difficulty of proving the polynomiality of Sz ( p ) outside types A and C. It leads to “false degrees” defined by underlying combinatorial structure. These are the true degrees when the bounds in [F. Fauquant-Millet, A. Joseph, Semi-centre de l'algèbre enveloppante d'une sous-algèbre parabolique d'une algèbre de Lie semi-simple, Ann. Sci. École Norm. Sup. (4) 38 (2) (2005) 155–191] coincide and polynomiality ensues. We show that these false degrees always sum to c ( p ) : = 1 2 ( dim p + index ( p ) ) which can fail for the true degrees when they are defined. Finally we prove the Tauvel–Yu conjecture on the index of a parabolic.

Enveloping algebras of Slodowy slices and the Joseph ideal

Let G be a simple algebraic group over an algebraically closed field \mathbb k of characteristic 0 , and \mathfrak g=\text{Lie}\,G . Let (e,h,f) be an \mathfrak{sl}_2 -triple in \mathfrak g with e being a long root vector in \mathfrak g . Let (\,\cdot\,,\,\cdot\,) be the G -invariant bilinear form on \mathfrak g with (e,f)=1 and let \chi\in\mathfrak g^* be such that \chi(x)=(e,x) for all x\in\mathfrak g . Let {\mathcal S} be the Slodowy slice at e through the adjoint orbit of e and let H be the enveloping algebra of {\mathcal S} ; see [31]. In this note we give an explicit presentation of H by generators and relations. As a consequence we deduce that H contains an ideal of codimension 1 which is unique if \mathfrak g is not of type \mathrm A . Applying Skryabin's equivalence of categories we then construct an explicit Whittaker model for the Joseph ideal of U(\mathfrak g) . Inspired by Joseph's Preparation Theorem we prove that there exists a homeomorphism between the primitive spectrum of H and the spectrum of all primitive ideals of infinite codimension in U(\mathfrak g) which respects Goldie rank and Gelfand–Kirillov dimension. We study highest weight modules for the algebra H and apply earlier results of Miličić–Soergel and Backelin to express the composition multiplicities of the Verma modules for H in terms of some inverse parabolic Kazhdan–Lusztig polynomials. Our results confirm in the minimal nilpotent case the de Vos–van Driel conjecture on composition multiplicities of Verma modules for finite {\mathcal W} -algebras. We also obtain some general results on the enveloping algebras of Slodowy slices and determine the associated varieties of related primitive ideals of U(\mathfrak g) . A sequel to this paper will treat modular aspects of this theory.

Some results of the local theory of smooth functions

We present results of the investigation of the local behavior of smooth functions in neighborhoods of their regular and critical points and prove theorems on the mean values of the functions considered similar to the Lagrange finite-increments theorem. We also study the symmetry of the derivative of an analytic function in the neighborhood of its multiple zero, prove new statements of the Weierstrass preparation theorem related to the critical point of a smooth function with finite smoothness, determine a nongradient vector field of a function in the neighborhood of its critical point, and consider one critical case of stability of an equilibrium position of a nonlinear system.

A proof of the valuation property and preparation theorem

The purpose of this article is to present a short model-theoretic proof of the valuation property for a polynomially bounded o-minimal theory $T$. The valuation property was conjectured by van den Dries, and proved for the polynomially bounded case by van

A preparation theorem for Weierstrass systems

It is shown that Lion and Rolin’s preparation theorem for globally subanalytic functions holds for the collection of definable functions in any expansion of the real ordered field by a Weierstrass system.

A preparation theorem for codimension-one foliations

After gluing foliated complex manifolds, we derive a preparation-like theorem for singularities of codimension-one foliations and planar vector fields (in the real or complex setting). Without computation, we retrieve and improve results of Levinson-Moser for functions, Dufour-Zhitomirskii for nondegenerate codimension-one foliations (proving in turn the analyticity), Str?oyzyna- y Zo
ladek for non degenerate planar vector fields and Bruno-?Ecalle for saddle-node foliations in the plane.

Ruled surfaces and generic coverings

In this paper we study normal singularities of complex analytic surfaces. In particular, since a normal surface has only isolated singularities (see, e.g., [10]), we restrict our attention to germs (S, s) where S is a connected normal surface, s ∈ S and S s is non-singular. We begin by setting some standard notation. A normal generic covering (S,π) (ngc in the sequel) is a finite holomorphic map π :S → C2 from a connected normal surface S to the complex plane C2, which is an analytic covering branched over a curve B ⊂C2, such that the fiber over a smooth point of B is supported on degπ − 1 distinct points. A ngc is called smooth if S is non-singular. Two ngcs (S1,π1), (S2,π2) are called (analytically) equivalent if there exists an isomorphism φ :S1 → S2 such that π1 = π2 ◦ φ. In the sequel, we will consider equivalent ngcs to be the same covering. The main interest in ngcs comes from the well-known fact that, by Weierstrass preparation theorem, given an analytic surface S ⊂ C, a generic projection S π −→ C2 is (at least locally, in order to insure degπ < ∞) a ngc branched over a curve (see [4]). Classically, one would like to reconstruct every ngc starting from downstairs data (i.e. in C2, like the branch curve B). Over a non-singular point of B , π is locally (in S) equivalent to the map of the complex plane to itself which takes (x, y) to (x, y) with a = 1,2. The main point is then to study germs of ngcs where the branch curve is a singular germ of a plane curve. We can then restrict to the case in which the branch locus B has only one singular point, which we may assume to be the origin O . For a fixed curve B there are three natural problems related to ngcs: the existence problem (there exists a ngc branched over B?), the smoothness problem (there exists a smooth ngc branched over B?), and the uniqueness problem (under which hypothesis is the covering unique? This is related to a conjecture of Chisini, see [8]).

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.

EXPANSIONS OF ALGEBRAICALLY CLOSED FIELDS II: FUNCTIONS OF SEVERAL VARIABLES

Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every definable function which is meromorphic on Knis necessarily a rational function. We finally discuss definable analogues of complex analytic manifolds, with possible connections to the model theoretic work on compact complex manifolds, and present two examples of "nonstandard manifolds" in our setting.