Showing papers in "Journal of Mathematical Sciences in 2003"

On Fréchet Subdifferentials

Abstract: This survey is devoted to some aspects of the theory of Fréchet subdifferentiation. The selection of the material reflects the interests of the author and is far from being complete. The paper contains definitions and statements of some important results in the field with very few proofs. The author hopes that reading the paper will not be difficult even for those mathematicians whose main scientific interests are not in the field of nonsmooth analysis. The variety of different subdifferentials known by now can be divided into two large groups: “simple” subdifferentials and “strict” subdifferentials. A simple subdifferential is defined at a given point and it does not take into account “differential” properties of a function in its neighborhood. Usually, such subdifferentials generalize some classical differentiability notions (Fréchet, Gâteaux, Dini, etc.). They are not widely used directly because of rather poor calculus. Contrary to simple subdifferentials, the definitions of strict subdifferentials incorporate differential properties of a function near a given point. Usually, strict subdifferentials can be represented as (some kinds of) limits of simple ones. This procedure makes them generalizations of the notion of a strict derivative [14], enriches their properties, and allows constructing satisfactory calculus. The examples of limiting subdifferentials are the generalized differential (the limiting Fréchet subdifferential) [49, 53, 63, 66, 67] and the approximate subdifferential (the limiting Dini subdifferential) [38, 39, 41, 43]. The famous generalized gradient of Clarke [15,17] can also be considered as being a strict subdifferential. The Warga’s derivate container [98,101] also belongs to this class. The limiting subdifferentials proved to be very efficient in nonsmooth analysis and optimization (see [17, 38, 41–44, 49, 50, 52, 53, 62, 63, 67, 69, 74, 75, 93, 95, 97–102]), especially in finite dimensions. When applying limiting subdifferentials in infinite-dimensional spaces, one must be careful about nontriviality of limits in the weak∗ topology. Additional regularity conditions are needed (compact epi-Lipschitzness [7], sequential normal compactness, partial sequential normal compactness [74,75,77], etc.). On the other hand, it is possible to formulate the results without taking limits and thus avoid the above-mentioned difficulties. Such statements are formulated (without additional regularity conditions) in terms of simple subdifferentials calculated at some points arbitrarily close to the point under consideration. They are usually referred to as fuzzy results [10, 12, 27–29, 40, 45, 62, 76, 78, 104]. In general, such results are stronger than the corresponding statements in terms of limiting subdifferentials. In this paper, we discuss only fuzzy results. The paper consists of three sections. Section 1 is devoted to definitions and elementary properties of Fréchet subdifferentials, normal cones, and coderivatives. It partly follows the earlier papers [48, 51], some parts of which have never been published. The main fuzzy results (from the author’s standpoint) in terms of Fréchet subdifferentials are presented in Sec. 2. Some of them are formulated by using strict δ-subdifferentials [55, 61]. The extended extremality notions [61] are discussed in Sec. 3. Being weaker than the traditional definitions, they describe some “almost extremal” points for which the known dual necessary conditions in terms of Fréchet subdifferentials become sufficient. Adopting these extended extremality notions leads to a form of duality in nonsmooth nonconvex optimization. Some constants are defined in the paper which simplify the definitions and statements of the results.

Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem

Abstract: We consider the problem of designing a reflector that transforms a spherical wave front with a given intensity into an output front illuminating a prespecified region of the far-sphere with prescribed intensity. In earlier approaches, it was shown that in the geometric optics approximation this problem is reduced to solving a second order nonlinear elliptic partial differential equation of Monge–Ampere type. We show that this problem can be solved as a variational problem within the framework of Monge–Kantorovich mass transfer problem. We develop the techniques used by the authors in their work “Optical Design of Two-Reflector Systems, the Monge–Kantorovich Mass Transfer Problem and Fermat's Principle” [Preprint, 2003], where the design problem for a system with two reflectors was considered. An important consequence of this approach is that the design problem can be solved numerically by tools of linear programming. A known convergent numerical scheme for this problem was based on the construction of very special approximate solutions to the corresponding Monge–Ampere equation. Bibliography: 14 titles.

Notions of relative interior in Banach spaces

Abstract: Extensions to a Banach space of the equivalent notions of relatively absorbing, non-support, and relative interior points of a convex set in $\reals^n$ are presented. The relations between these extensions are studied, and their basic calculus rules are developed. Several explicit examples and counterexamples in general Banach spaces are given; and the tools for development of further examples are explained. Various implications for infinite dimensional optimization are highlighted.

On Nonstationary Stokes Problem and Navier–Stokes Problem in a Half-Space with Initial Data Nondecreasing at Infinity

Abstract: We prove the solvability of the Stokes problem and the local (in time) solvability of the Navier–Stokes problem in the half-space under the condition that the initial velocity is only bounded and continuous. The proof is based on estimates for the entries of the Green matrix for the Stokes problem. Bibliography: 8 titles.

Projective Transformations of Pseudo-Riemannian Manifolds

Abstract: A projective transformation of a pseudo-Riemannian manifoldM is an automorphism of the induced Riemannian connection of the projective structure that takes geodesics in M to geodesics again. For the first time, the problem of determining Riemannian spaces admitting continuous transformation groups preserving geodesics was considered by Sophus Lie for the case of two-dimensional surfaces (see [306]). However, as Fubini wrote in the preface to [262], “the famous mathematician did not succeed in solving this problem” (which Fubini called the “Lie problem”). Having criticized Lie’s method, which, to Fubini’s opinion, could not be used for the general solution of the problem, Fubini developed his own approach based on the infinitesimal calculus, later named the “Lie differentiation.” The subsequent development of the theory of projective transformations on linear connection spaces is connected with the names of E. Cartan, Eisenhart, Thomas, Knebelman, Schouten, Yano, Egorov, Vranceanu, Kobayashi, and others. It is known that in the spaces of constant curvature S, when considered in the small, the complete projective group coincides with the projective group of pseudo-Euclidean space, i.e., with the group of bilinear substitutions, and depends on n(n+ 2) parameters. In the spaces V n of nonconstant curvature, the order of the complete projective group does not exceed n(n−2)−5 (Egorov [91]), and, moreover, in the majority of cases, this group consists of similarity transformations (homotheties) or isometries. In 1903, in “Turin Academy Notes,” Fubini’s paper “Groups of geodesic transformations” was published [262], which as mentioned above, laid the foundations for a systematic definition and study of Riemannian spaces admitting infinitesimal projective transformations. Later on, Solodovnikov [156–158] continued Fubini’s research and completely solved the problem posed; the works of Fubini and Solodovnikov contain a classification of the Riemannian spaces V , n ≥ 3, in terms of (local) groups of projective transformations which are larger than homothety groups. The conclusions of Fubini and Solodovnikov are based on the assumption that the metrics considered are positive definite. Taking a given signature as a condition considerably complicates the problem and requires a basically new approach for its solution. It was proposed in the papers of the author [5– 11], where the problem of defining all pseudo-Riemannian manifolds with Lorentz signature (+ − . . .−) (Lorentz manifolds) of dimension n ≥ 3 admitting nonhomothetic infinitesimal projective and affine transformations was solved, and for each of them, the maximal projective and affine Lie algebras, together with the homothetic and isometric subalgebras, were defined. This paper includes a survey of the results in the theory of projective transformations of pseudoRiemannian manifolds, in particular, the solution of the classical geometrical problem of determining the pseudo-Riemannian metrics with corresponding geodesics (Sec. 5) and the Lie problem (Sec. 6). By using the technique of skew-normal frames developed by the author in [31] (see Sec. 1), all two-dimensional

Proof of the volume conjecture for torus knots

Abstract: The volume conjecture stated recently by H. Murakami and J. Murakami is proved for the case of torus knots. Bibliography: 9 titles.

Class Numbers of Indefinite Binary Quadratic Forms

Abstract: Let \(h(d)\) be the class number of properly equivalent primitive binary quadratic forms \(ax^2 + bxy + cy^2\) of discriminant \(d = b^2 a - 4ac\). The case of indefinite forms \((d < 0)\) is considered. Assuming that the extended Riemann hypothesis for some fields of algebraic numbers holds, the following results are proved. 1. Let \(\alpha (x)\) be an arbitrarily slow monotonically increasing function such that \(\alpha (x) \to \infty\). Then $$\# \left\{ {p \leqslant \left. x \right|{\text{ }}\left( {\frac{{\text{5}}}{p}} \right) = 1,h(5p^2 ) >(\log p)^{\alpha (p)} } \right\} = o(\pi (x)),$$ where \(\pi (x) = \# \{ p \leqslant x\}\). 2. Let F be an arbitrary sufficiently large positive constant. Then for \(x >x_F\), the relation $$\# \left\{ {p \leqslant \left. x \right|{\text{ }}\left( {\frac{{\text{5}}}{p}} \right) = 1,h(5p^2 ) >F} \right\} \asymp \frac{{\pi (x)}}{F}$$ holds. 3. The relation $$\# \left\{ {p \leqslant \left. x \right|{\text{ }}\left( {\frac{{\text{5}}}{p}} \right) = 1,h(5p^2 ) = 2} \right\} \sim \frac{9}{{19}}A\pi (x)$$ holds, where A is Artin's constant. Hence, for the majority of discriminants of the form \(d = 5p^2\), where \({\left( {\frac{{\text{5}}}{p}} \right) = 1}\) , the class numbers are small. This is consistent with the Gauss conjecture concerning the behavior of \(h(d)\) for the majority of discriminants \(d >0\) in the general case. Bibliography: 22 titles.

Gröbner–Shirshov Bases: From their Incipiency to the Present

Abstract: In this paper, the history and the main results of the theory of Grobner–Shirshov bases are given for commutative, noncommutative, Lie, and conformal algebras from the beginning (1962) to the present time. The problem of constructing a base of a free Lie algebra is considered, as well as the problem of studying the structure of free products of Lie algebras, the word problem for Lie algebras, and the problem of embedding an arbitrary Lie algebra into an algebraically closed one. The modern form of the composition-diamond lemma (the CD lemma) is presented. The rewriting systems for groups are considered from the point of view of Grobner–Shirshov bases. The important role of conformal algebras is treated, the statement of the CD lemma for associative conformal algebras is given, and some examples are considered. An analog of the Hilbert basis theorem for commutative conformalalgebras is stated. Bibliography: 173 titles.

Sensitivity Analysis of Generalized Equations

Abstract: In this paper, we study sensitivity analysis of generalized equations (variational inequalities) with nonpolyhedral set constraints. We use an approach of local reduction of the corresponding constraint set to a convex cone. This leads us to the introduction of an additional term representing a curvature of the constraint set in a linearization of the generalized equations. We also discuss concepts of nondegeneracy and strict complementarity of set constrained problems.

Continuous subdifferential approximations and their applications

Abstract: In this paper, we study continuous approximations to the Clarke subdifferential and the Demyanov– Rubinov quasidifferential. Different methods have been proposed and discussed for the construction of the continuous approximations. Numerical methods for minimization of the locally Lipschitzian functions which are based on the continuous approximations are described and their convergence is studied. To test the proposed methods, numerical experiments have been carried out and discussed in the paper. This paper presents some methods for the continuous approximation of the Clarke subdifferential and the Demyanov–Rubinov quasidifferential of a locally Lipschitzian function. On the basis of the continuous approximations, we propose numerical methods of nonsmooth optimization. The paper presents a survey of some results obtained in [2–10]. The notions of the Clarke subdifferential and the Demyanov–Rubinov quasidifferential play a key role in nonsmooth and nonconvex optimization. The Clarke subdifferential is subject to a calculus in the form of inclusions and the calculus cannot be used in general for the estimation of the subgradients. Unlike the Clarke subdifferential, the quasidifferential has a full-scale calculus, which can be used for its calculation, although its entire calculation requires some operations with polytopes in the n-dimensional space. The number of these polytopes and their vertices can be very large and, therefore, the calculation of the quasidifferential becomes very complicated. In smooth optimization, there exist minimization methods, which, instead of using the gradient, use its approximations through finite differences (forward, backward, and central differences). In [37], a very simple convex nondifferentiable function was presented, for which these finite differences may give no information about the subdifferential. It follows that these finite-difference estimates of the gradient cannot be used for the approximation of the subgradient of the nonsmooth functions. In the past decades, a few methods for the numerical computation of the subgradients were proposed and studied. In [55], Shor proposed special finite differences, which allow one to approximate one subgradient of convex functions. In [56, 57], Studniarski modified Shor’s algorithm and extended it to the class of subregular functions. In [11], Borwein proposed a method for the calculation of one subgradient of the convex functions. Finally, in [46], Nesterov introduced the notion of the lexicographically smooth functions and proposed a method for the calculation of the subgradients. In this paper, we introduce the notion of a discrete gradient as a certain version of finite-difference subgradient estimates. The discrete gradient is defined with respect to a given direction and thus allows one to approximate a directional derivative of a given function. This approximation allows us to propose an effective algorithm for the calculation of a descent direction of a function at a given point. In general, the set of discrete gradients approximates the entire subdifferential, which can present only theoretical interest. For the computation of the descent direction, we need to calculate only a few discrete gradients at a given point, and the proposed algorithm for such a computation is definitive. The lack of continuity of the subdifferential and quasidifferential mappings creates difficulties in the study of methods for the minimization of locally Lipschitz functions. In [59], it was noted that the lack of this property was responsible for the failure of nonsmooth steepest descent algorithms. On

Properties of Spectral Expansions Corresponding to Non-Self-Adjoint Differential Operators

Abstract: This paper is a survey of results in the spectral theory of differential operators generated by ordinary differential expressions and also by partial differential expressions of elliptic type. Our main focus is on the non-self-adjoint case. In contrast to the theory of self-adjoint differential operators, in which a firm foundation of functional analysis methods was laid due to the efforts of many mathematicians, in many respects, a universal conception of approaches to studying the problems arisen was created, and, finally much experience was accumulated in scientific publications for more than a century, the spectral theory of non-self-adjoint operators contains at present fairly many open problems. This does not mean at all that little has been done in this field: a list of all publications devoted to this theme, if it has ever been composed, would look at least like that in the theory of self-adjoint problems. All this is explained by the fact that often to study a new class of non-self-adjoint problems, we need to elaborate new methods using a “fine adjustment” of the functional analysis technique. Of course, the present survey does not claim to be an exhaustive presentation of scientific results and methods of the theory of non-self-adjoint differential operators. Here, we pay considerable attention to the studies carried out at the Chair of General Mathematics of the Department of Computational Mathematics and Cybernetics of the M.V. Lomonosov Moscow State University over a period of more than 30 years. They mainly concern one aspect or another of convergence of spectral expansions related to non-self-adjoint differential operators. The methodology elaborated there turns out to be a fairly effective tool for solving many new problems in this field. The authors try to make the reader familiar with the main results obtained up to now and give an idea of the methods elaborated for their proof. This specific character of the survey explains a certain “narrow specialization” and subjectiveness of the list of literature cited.

Projectively Dual Varieties

Abstract: This text is a draft of the review paper on projectively dual varieties. Topics include dual varieties, Pyasetskii pairing, discriminant complexes, resultants and schemes of zeros, secant and tangential varieties, Ein theorems, applications of projective differential geometry and Mori theory to dual varieties, degree and multiplicities of discriminants, self-dual varieties, etc.

The Rational Krylov Algorithm for Nonlinear Matrix Eigenvalue Problems

Abstract: It is shown how the rational Krylov algorithm can be applied to a matrix eigenvalue problem that is nonlinear in the eigenvalue parameter. Bibliography: 6 titles.

Uniform Estimates of Remainders in Asymptotic Expansions of Solutions to the Problem on Eigenoscillations of a Piezoelectric Plate

Abstract: A method for obtaining estimates of asymptotic remainders is presented. The constants in estimates are independent of the number of the eigenvalue, as well as of the small parameter h, the thickness of the plate. Owing to an information about connections between frequencies of eigenoscillations of the three-dimensional plates and its two-dimensional model obtained under various restrictions to h, it is possible to divide the asymptotics in collective and individual ones. Only in the case of the individual asymptotics, i.e., under rigid restrictions on h, it is possible to construct asymptotic expansions for the corresponding eigenvectors. We consider arbitrarily anizotropic composed cylindrical plates in whcih piezoeffects can dominate along longitudinal directions, as well as along transverse directions. The connectedness of elastic and electric fields Implies the appearance of a nontrivial dissipative components of the operator of the problem under consideration, but its spectrum remains real and positive. Bibliography: 43 titles.

Abstract Stochastic Equations II. Solutions in Spaces of Abstract Stochastic Distributions

Abstract: STOCHASTIC EQUATIONS II. SOLUTIONS IN SPACES OF ABSTRACT STOCHASTIC DISTRIBUTIONS I. V. Melnikova, A. I. Filinkov, and M. A. Alshansky UDC 519.219 Introduction In Part II, we study stochastic evolution equations with additive noise using semigroup methods in the framework of white noise calculus. White noise analysis has been developed during the past three decades by many authors (see, e.g., [1–4] and the references therein). In the present survey, we use the approach of [4], where the theory of R -valued stochastic distributions is presented, and of [5], where the theory of Hilbert space valued stochastic distributions is developed. Our aim is to apply the theory of semigroups of linear operators to stochastic (ordinary and partial) differential equations in cases more general than that described in Part I. Consider, for example, the stochastic heat equation dX(t, x) dt = ∆xX(t, x) +W(t, x), t ∈ [0, T ], x ∈ O = { x ∈ R ; 0 1 in the space S(H)−0 of stochastic distributions with values in H The central point of Part II is the construction of the spaces of H-valued test functions S(H)ρ, ρ ∈ [0, 1], and stochastic distributions S(H)−ρ: S(H)1 ⊂ S(H)ρ ⊂ S(H)0 ⊂ L2(S ;H) ⊂ S(H)−0 ⊂ S(H)−ρ ⊂ S(H)−1 for a separable Hilbert space H. Then we develop the calculus in these spaces and consider basic examples of H-valued stochastic processes: the H-valued weak Wiener process {W (t)} and the H-valued singular white noise process {W(t)}. Generally, both of them take values in S(H)−0 for any fixed t. One can expect that the Q-Wiener processes discussed in Part I belong to L2(S ′;H). It is crucial for our main results on evolution equations that the white noise be (infinitely) differentiable with respect to t (t plays the role of a parameter) in S(H)−1, and in this space, we can give a description of the convergence similar to the convergence in S(R )−1. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 96, Funktsional’nyi Analiz, 2001. 362

On the Sharp Constant in the Small Ball Asymptotics of Some Gaussian Processes under L2-Norm

Abstract: We find the sharp constant in the small L2-deviation asymptotics for a wide class of Gaussian processes including the m-times integrated Wiener process and the m-times integrated Ornstein–Uhlenbeck process. Extremal properties of usual and Euler integration are proved. Bibliography: 19 titles.

New Representations of the Taylor–Stratonovich Expansion

Abstract: The problem of the Taylor–Stratonovich expansion of the Ito random processes in a neighborhood of a point is considered. The usual form of the Taylor–Stratonovich expansion is transformed to a new representation, which includes the minimal quantity of different types of multiple Stratonovich stochastic integrals. Therefore, these representations are more convenient for constructing algorithms of numerical solution of stochastic differential Ito equations. Bibliography: 14 titles.

Some Estimates near the Boundary for Solutions to the Nonstationary Linearized Navier–Stokes Equations

Abstract: In the present paper, new local estimates near the boundary are established for solutions to the nonstationary linearized Navier–Stokes equations are established. Bibliography: 8 titles.

New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium

Abstract: The purpose of the paper is to elaborate qualitative methods for studying the dynamics of rigid bodies interacting with a resisting medium under quasistationarity conditions. This material refers to the qualitative theory of ordinary differential equations, as well as to the dynamics of rigid bodies. In this paper, we use the properties of motion of a body in a medium under conditions of a jet flow past this body. We study the plane and three-dimensional model problems of motion of a body in a resisting medium. New families of phase portraits of variable dissipation systems on twoand three-dimensional manifolds are obtained, and their absolute or relative roughness (i.e., structural stability) is demonstrated. The Jacobi integrable cases of equations of motion of rigid bodies are found, including those in the problem of motion of a spherical pendulum placed in the incoming flow of a medium.

Algorithms for Sat and Upper Bounds on Their Complexity

Abstract: We survey recent algorithms for the propositional satisfiability problem In particular, we consider algorithms having the best current worst-case upper bounds on their complexity We also discuss some related issues: a derandomization of the algorithm of Paturi, Pudlak, Saks, and Zane, the Valiant–Vazirani lemma, and random walk algorithms with the “back button” Bibliography: 47 titles

Estimates of Deviations from Exact Solutions of Elliptic Variational Inequalities

Abstract: In this paper, we present a method of deriving majorants of the difference between exact solutions of elliptic type variational inequalities and functions lying in the admissible functional class of the problem under consideration We analyze three classical problems associated with stationary variational inequalities: the problem with two obstacles, the elastoplastic torsion problem and the problem with friction type boundary conditions The majorants are obtained by a modification of the duality technique earlier used by the author for variational problems with uniformly convex functionals These majorants naturally reflects properties of exact solutions and possess necessary continuity conditions Bibliography: 15 titles

Pontryagin Duality in the Theory of Topological Vector Spaces and in Topological Algebra

Abstract: The theory of topological vector spaces (TVS), being a foundation of modern functional analysis, is now considered as a completely mature, or, to be more specific, dead mathematical discipline. This pessimistic view is based on a picture, where, on the one hand, a well-known system of facts is stated, facts that have been considered classical since the times of Mackey and Grothendieck, and, on the other hand, an opinion exists explicitly or implicitly, that any deviation from this system inevitably tends to a disappointment. “Do not seek any other patterns or connections but those that we have described here, because any of your assumptions will find a counterexample in Nature”; this phrase should have been cited as an afterward to the three textbooks on topological vector spaces translated into Russian [12, 41, 43], which represent the face of this science from the sixties until now. Indeed, after the famous Enflo counterexample [22], the development of the theory of topological vector spaces (which was planned originally as an extension of a clear and simple discipline, linear algebra) was transformed into a sequence of reports on the oddities one can encounter in modern mathematics. The result of this process is a general skepticism about and a loss of interest in this field. The number of papers on the theory of TVS is declining, while the attitude of other mathematicians now consists of polite perplexity or lenient irony. And add to that the fact that the vast majority of specialists in functional analysis do not concern themselves with topological vector spaces but consciously concentrate on Banach theory. (According to Compumath, for example, in 1996 only 7 papers were devoted to locally convex and topological vector spaces, while 612 were devoted to Banach spaces. In the theory of topological algebras, the ratio is 6:90.) The dominant position of Banach theory in functional analysis justifies a natural question: what are the advantages of the Banach case over the more general topological situation? At the conceptual level, the difference becomes most apparent in the theory of topological algebras. Let us cast a critical look. The first impression we will get when comparing Banach and non-Banach theories of topological algebras is as follows. While the concept of Banach algebra emerges naturally from intuitive expectations, faciliated by an acquaintance with general algebra and the theory of Banach spaces (and leads to the profound theory of Banach algebras), the situation with general topological algebras looks completely different. We find here unexpectedly that the very attempts to define a topological algebra (and topological module) lead to results that contradict intuition. Indeed, intuitively it is clear that a “good” definition of topological algebra (and topological module) must fulfill at least the following minimal list of requirements:

Convex hulls of integral points.

Abstract: The convex hull of all integral points contained in a compact polyhedron C is obviously a compact polyhedron. If C is not compact, then the convex hull K of its integral points need not be a closed set. However, under some natural assumptions, K is a closed set and a generalized polyhedron. Bibliography: 11 titles.

On the Structure of a k-Connected Graph

Abstract: For a k-connected graph G, we introduce the notion of a block and construct a block tree. This construction generalizes, for \(k \geqslant 1\), the known constructions for blocks of a connected graph. We apply the introduced notions to describe the set of vertices of a k-connected graph G such that the graph remains k-connected after deleting these vertices. We discuss some problems related to simultaneous deleting of vertices of a k-connected graph without loss of k-connectivity. Bibliography: 5 titles.

Two-Variable Identities in Groups and Lie Algebras

Abstract: We study two-variable Engel-like relations and identities characterizing finite-dimensional solvable Lie algebras and, conjecturally, finite solvable groups and introduce some invariants of finite groups associated with such relations. Bibliography: 29 titles.

Bounds for the distance between nearby jordan and kronecker structures in a closure hierarchy

Abstract: Computing the fine-canonical-structure elements of matrices and matrix pencils are ill-posed problems. Therefore, besides knowing the canonical structure of a matrix or a matrix pencil, it is equally important to know what are the nearby canonical structures that explain the behavior under small perturbations. Qualitative strata information is provided by our StratiGraph tool. Here, we present lower and upper bounds for the distance between Jordan and Kronecker structures in a closure hierarchy of an orbit or bundle stratification. This quantitative information is of importance in applications, e.g., distance to more degenerate systems (uncontrollability). Our upper bounds are based on staircase regularizing perturbations. The lower bounds are of Eckart―Young type and are derived from a matrix representation of the tangent space of the orbit of a matrix or a matrix pencil. Computational results illustrate the use of the bounds. Bibliography: 42 titles.

ON p-ADIC INTEGRATION IN SPACES OF MODULAR FORMS AND ITS APPLICATIONS

Abstract: The purpose of this course is to give an introduction to the theory of p-adic integration with values in spaces of modular forms (elliptic modular forms, Siegel modular forms, . . .). We show that very general p-adic families of modular forms can be constructed as moments of certain p-adic measures on a profinite group Y = lim ←− Yi with values in a formal q-expansion ring like Zp[[q ]] where B is an additive semi-group, and q = {q |ξ ∈ B} the corresponding formally written multiplicative semi-group (for example B = Bn = {ξ = ξ ∈ Mn(Q)|ξ ≥ 0, ξ half-integral} is the semi-group, important for the theory of Siegel modular forms). We discuss some applications of this theory to the construction of certain new p-adic families of modular forms (families of Klingen-Eisenstein series, families of theta-series with spherical polynomials. . .). Main sources of this theory are: • Serre’s theory of p-adic forms as certain formal q-expansions (J.-P. Serre, Formes modulaires et fonctions zeta p-adiques, LNM 350 (1973) 191-268) [Se73]. • Hida’s theory of p-adic modular forms and p-adic Hecke algebras (H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge University Press, 1993 [Hi93]). • Construction of p-adic Siegel-Eisenstein series by the author, see [PaSE]. As an application, we describe a solution of a problem of Coleman-Mazur in [PaTV], using the RankinSelberg method and the p-adic integration in a Banach algebra A. An introductory cours given on November 29 in POSTECH (Pohang, Korea) 0 Introduction Let p be a prime number (we often assume p≥ 5). There are two different ways of introducing p-adic modular forms: the first approach uses formal q-expansions with coefficients in a p-adic ring [Se73], and the second approach is the p-adic interpolation of Galois representations attached to classical automorphic forms. The first approach was extensively developped by Katz [Ka78] for the group G = GL2 over a totally real number field, in order to construct p-adic L-functions for CM-fields using p-adic Hilbert-Eisenstein series. In general, in this q-expansion method a typical p-adic family φ of modular (automorphic) forms is an element of the Serre ring: φ ∈ Λ[[q]] where Λ = Zp[[T ]] is the Iwasawa algebra. In the second approach one considers Λ-adic Galois representations of type ρ : Gal(Q/Q) → GLm(Λ) (“Big Galois representations”, see [Hi86], [Til-U]). These two theories are essentially equivalent if we start from holomorphic automorphic forms on the group G = GL2 over a totally real field, but in other cases there is no direct link between φ and ρ. On the other hand there exist interesting examples of p-adic L-functions Lφ,p and Lρ,p attached to φ and to ρ. In general Lφ,p and Lρ,p should belong to the quotient field L = QuotΛ or to its finite extensions. If ρ interpolates a p-adic family of motives then there are conjectural general definitions of Lρ,p (see [Co-PeRi], [Colm98], [PaAdm]). It would be very interesting to formulate a general Langlands-type conjecture relating Λ-adic automorphic forms and Λ-adic Galois representations. As an application, we describe a solution of a problem of Coleman-Mazur, using the Rankin-Selberg method and the theory of p-adic integration with values in a p-adic algebra A. This problem was stated in "The Eigencurve" (1998), R.Coleman and B.Mazur stated the following as follows: Given a prime p and Coleman’s family {fk′} of cusp eigenforms of a fixed positive slope σ = ordp(αp(k )) > 0, to construct a two variable p-adic L-function interpolating on k the Amice-Velu p-adic L-functions Lp(fk′ ). Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. Such measures come from Eisenstein distributions with values in certain Banach A-modules M = M(N ;A) of families of overconvergent forms over A.

Boundary Estimates for Solutions of the Parabolic Free Boundary Problem

Abstract: Let u and Ω solve the problem $$H(u) = X\Omega ,{\text{ }}u = |Du| = 0{\text{ }}in{\text{ }}Q_1^ + \backslash \Omega ,{\text{ }}u = 0{\text{ }}on{\text{ }}\Pi \cap Q_1 ,$$ where Ω is an open set in $$\begin{gathered} \mathbb{R}_ + ^{n + 1} = \{ (x,t):x \in \mathbb{R}^n ,t \in \mathbb{R}^1 ,x_1 >0\} ,n \geqslant 2,H = \Delta - \partial _t \hfill \\ \hfill \\ \end{gathered} $$ is the heat operator, $$X\Omega $$ denotes the characteristic function of Ω, $$Q_1 $$ is the unit cylinder in ℝn+1, $$Q_1^ + = Q_1 \cap \mathbb{R}_ + ^{n + 1} ,\Pi = \{ (x,t):x1 = 0\} $$ , and the first equation is satisfied in the sense of distributions. We obtain the optimal regularity of the function u, i.e., we show that $$ \in C_x^{1,1} \cap C_t^{0,1} $$ . Bibliography: 6 titles.