Wyniki wyszukiwania - Biblioteka Nauki

1

Symmetric integrals and trigonometric series

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Cross G. E. , Thomson B. S.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS 1. Introduction............................................................................................5 2. Some preliminary definitions..................................................................6 3. Mařík's symmetric difference..................................................................9 4. Basic definitions...................................................................................11 5. Properties of the second symmetric variation for real functions...........15 6. Measure properties..............................................................................19 7. The integral.........................................................................................23 8. Additivity..............................................................................................26 9. Relations to the James P²-integral.......................................................27 10. Relations to the Burkill SCP-integral...................................................29 11. Mařík's integration by parts formula....................................................36 12. Burkill's integration by parts formula...................................................39 13. An application to trigonometric series.................................................43 14. Some further applications...................................................................47 References...............................................................................................48

2

Second duals of measure algebras

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Dales H. , Lau A. , Strauss D.

Instytut Matematyczny Polskiej Akademii Nauk

EN

Let G be a locally compact group. We shall study the Banach algebras which are the group algebra L¹(G) and the measure algebra M(G) on G, concentrating on their second dual algebras. As a preliminary we shall study the second dual C₀(Ω)'' of the C*-algebra C₀(Ω) for a locally compact space Ω, recognizing this space as C(Ω̃), where Ω̃ is the hyper-Stonean envelope of Ω. We shall study the C*-algebra $B^{b}(Ω)$ of bounded Borel functions on Ω, and we shall determine the exact cardinality of a variety of subsets of Ω̃ that are associated with $B^{b}(Ω)$. We shall identify the second duals of the measure algebra (M(G),∗) and the group algebra (L¹(G),∗) as the Banach algebras (M(G̃),□ ) and (M(Φ),□ ), respectively, where □ denotes the first Arens product and G̃ and Φ are certain compact spaces, and we shall then describe many of the properties of these two algebras. In particular, we shall show that the hyper-Stonean envelope G̃ determines the locally compact group G. We shall also show that (G̃,□ ) is a semigroup if and only if G is discrete, and we shall discuss in considerable detail the product of point masses in M(G̃). Some important special cases will be considered. We shall show that the spectrum of the C*-algebra $L^{∞}(G)$ is determining for the left topological centre of L¹(G)'', and we shall discuss the topological centre of the algebra (M(G)'',□ ).

3

A new theory of Operational Calculus

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Warmus M.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS I. Introduction..................................................................................................................5 II. Ring ℛ. Subring ℱ.....................................................................................................7 III. Class C. Distributions. Congruent functions......................................................10 IV. Distribution ring........................................................................................................20 V. Quotient functions and quotient distributions......................................................29 VI. Sequences of quotient distributions. Distributional convergence..................32 VII. Examples of applications......................................................................................37 VIII. Quotient field K....................................................................................................... 41 IX. A theory corresponding to Laplace Transformation..........................................43 References.....................................................................................................................46

4

An interpolatory estimate for the UMD-valued directional Haar projection

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Lechner R.

Instytut Matematyczny Polskiej Akademii Nauk

EN

We prove an interpolatory estimate linking the directional Haar projection $P^{(ε)}$ to the Riesz transform in the context of Bochner-Lebesgue spaces $L^{p}(ℝⁿ;X)$, 1 < p < ∞, provided X is a UMD-space. If $ε_{i₀} = 1$, the result is the inequality $||P^{(ε)}u||_{L^{p}(ℝⁿ;X)} ≤ C||u||_{L^{p}(ℝⁿ;X)}^{1/𝓣} ||R_{i₀}u||_{L^{p}(ℝⁿ;X)}^{1 - 1/𝓣}$, (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type 𝓣 of $L^{p}(ℝⁿ;X)$. In order to obtain the interpolatory result (1) we analyze stripe operators $S_{λ}$, λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection. The main result on stripe operators is the estimate $||S_{λ}u||_{L^{p}(ℝⁿ;X)} ≤ C·2^{-λ/𝓒}||u||_{L^{p}(ℝⁿ;X)}$, (2) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher cotype 𝓒 of $L^{p}(ℝⁿ;X)$. The proof of (2) relies on a uniform bound for the shift operators Tₘ, $0 ≤ m < 2^{λ}$, acting on the image of $S_{λ}$. Mainly based upon inequality (1), we prove a vector-valued result on sequential weak lower semicontinuity of integrals of the form u ↦ ∫ f(u)dx, where f: Xⁿ → ℝ⁺ is separately convex satisfying $f(x) ≤ C (1 + ||x||_{Xⁿ})^{p}$.

5

Calculus on Lie algebroids, Lie groupoids and Poisson manifolds

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Marle C.-M.

Instytut Matematyczny Polskiej Akademii Nauk

EN

We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its exterior powers, one can define an operator $d_{E}$ similar to the exterior derivative. We present the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for the most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds, whose links with Lie algebroids are very close.

6

Strong shape theory

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Dydak J. , Segal J.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS 1. Introduction..................................................................................................................................... 5 2. Terminology and notation.................................................................................................................... 6 3. Proper maps on contractible telescopes.......................................................................................... 8 4. The strong shape category.................................................................................................................. 13 5. Semi-equivalences............................................................................................................................... 19 6. Geometric characterization of maps inducing strong shape equivalences............................... 21 7. Some classes of maps which induce strong shape equivalence............................................... 27 8. Concluding remarks............................................................................................................................. 35 References.................................................................................................................................................. 38

7

The theory of Marcinkiewicz-Orlicz spaces

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Albrycht J.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS INTRODUCTION.................................................................................................................................................................................................... 3 CHAPTER I. Preliminary concepts..................................................................................................................................................................... 6 CHAPTER II. Classes $\bar{ℳ}$, $\bar\mathscr{N}$, $\bar\mathfrak{M}$, $\bar\mathfrak{N}$ and their properties...................... 12 CHAPTER III. The Marcinkiewicz-Orlicz spaces............................................................................................................................................... 44 CHAPTER IV. Generalized almost-periodic functions..................................................................................................................................... 48 REFERENCES......................................................................................................................................................................................................... 64

8

Free boundary problems for nonstationary Navier-Stokes equations

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Zadrzyńska E.

Instytut Matematyczny Polskiej Akademii Nauk

EN

The results concerning free boundary problems for both incompressible and compressible Navier-Stokes equations are reviewed in this paper.

9

Algebraic Legendrian varieties

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Buczyński J.

Instytut Matematyczny Polskiej Akademii Nauk

EN

Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity (see [Arn74], [Sła91] and references therein). However, complex Legendrian subvarieties are much more rigid and have more exceptional properties. The most remarkable case is the Legendrian subvarieties of projective space; prior to the author's research only few smooth examples of these were known (see [Bry82], [LM07]). Strong restrictions on the topology of such varieties have been found and studied by Landsberg and Manivel ([LM07]). This dissertation reviews the subject of Legendrian varieties and extends some of recent results. The first series of results is related to the automorphism group of any Legendrian subvariety in any projective contact manifold. The connected component of this group (under suitable minor assumptions) is completely determined by the sections of the distinguished line bundle on the contact manifold vanishing on the Legendrian variety. Moreover, its action preserves the contact structure. The relation between the Lie algebra tangent to automorphisms and the sections is given by an explicit formula (see also [LeB95], [Bea07]). This summarises and extends some earlier results of the author. The second series of results is devoted to finding new examples of smooth Legendrian subvarieties of projective space. The examples found by other researchers were some homogeneous spaces, many examples of curves and a family of surfaces birational to some K3 surfaces. Further the author found a couple of other examples including a smooth toric surface and a smooth quasihomogeneous Fano 8-fold. Finally, the author proved that both of these are special cases of a very general construction: a general hyperplane section of a smooth Legendrian variety, after a suitable projection, is a smooth Legendrian variety of smaller dimension. We review all of those examples and also add infinitely many new examples in every dimension, with various Picard rank, canonical degree, Kodaira dimension and other invariants. The original motivation for studying complex Legendrian varieties comes from a 50 years old problem of giving compact examples of quaternion-Kähler manifolds (see [Ber55], [LS94], [LeB95] and references therein). Also Legendrian varieties are related to some algebraic structures (see [Muk98], [LM01], [LM02]). A new potential application to classification of smooth varieties with smooth dual arises in this dissertation.

10

On fans

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Charatonik J. J.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS § 1. Introduction....................................................................................................................................................................... 5 § 2. Preliminaries.................................................................................................................................................................... 6 § 3. Smoothness..................................................................................................................................................................... 7 § 4. Uniform arcwise connectedness................................................................................................................................. 12 § 5. Countable smooth fans.................................................................................................................................................. 14 § 6. Applications. Families of n incomparable countable geometrical fans for each n = 2,3,.................................. 21 § 7. Folding fans...................................................................................................................................................................... 24 § 8. Contractibility.................................................................................................................................................................... 30 § 9. Confluent mappings....................................................................................................................................................... 32 § 10. The summarized table................................................................................................................................................. 35 References............................................................................................................................................................................... 37

11

Global stochastic approximation

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Zieliński R.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS 1. Intuitive background. Statement of the problem...................................................................... 5 2. General structure of global stochastic approximation processes............................................... 7 3. The fundamental theorem on convergence in distribution............................................................ 10 4. Absolute continuity of the limit distribution 4.1. Introductory remarks............................................................................................................. 13 4.2. General case......................................................................................................................... 13 4.3. Uniform experimental design............................................................................................. 14 4.4. Improvement by a randomization....................................................................................... 16 4.5. Problem of optimal experimental design......................................................................... 19 5. Almost sure convergence to global maximum................................................................................ 21 6. A Monte Carlo method.......................................................................................................................... 24 References................................................................................................................................................. 26

12

Direct summands of systems of continuous linear transformations

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Fixman U. , Zorzitto F. A.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction............................................................................................................ 5 1. The category of $C^N$-systems........................................................................... 8 2. The problem of split monomorphisms................................................................ 10 3. Internal hom and tensor product........................................................................... 13 4. Characterizations of split monomorphisms....................................................... 19 5. Reduction to algebraic systems............................................................................ 24 6. Reduction to finite-dimensional indecomposable sources............................ 27 7. The broken chain condition for $C^2$-systems................................................. 32 8. An example for computation of Nli((X, Y), (V, W))................................................ 36 References.................................................................................................................... 41

13

Boolean powers as algebras of continuous functions

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Banaschewski B. , Nelson E.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction................................................. 5 1. Fundamentals................................................ 7 2. Functorial aspects......................................... 11 3. Congruences.................................................. 18 4. Exponent laws................................................ 28 5. Finite A............................................................. 34 6. Boolean ultrapowers..................................... 37 7. Elementary properties.................................. 45 Concluding remarks.......................................... 49 References.......................................................... 50

14

Some logarithmic function spaces, entropy numbers, applications to spectral theory

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Haroske D.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction.........................................................................................5 1. Non-limiting embeddings - a short review........................................8 2. The spaces Lₚ(log L)ₐ on ℝⁿ.........................................................12 2.1. The spaces Lₚ(log L)ₐ(Ω) and $L_{p,q}(log L)ₐ(Ω)$................12 2.2. The spaces $Lₚ(log L)_{-a}(ℝⁿ)$, a > 0...................................20 2.3. The spaces Lₚ(log L)ₐ(ℝⁿ), a > 0.............................................27 2.4. Hölder inequalities...................................................................32 2.5. Examples.................................................................................35 3. Entropy numbers, limiting embeddings.........................................37 4. Applications..................................................................................47 4.1. Eigenvalue distribution............................................................47 4.2. Negative spectrum...................................................................53 5. Homogeneous spaces..................................................................55 References.......................................................................................58

15

Fredholm properties of elliptic operators on ℝⁿ

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Elton D.

Instytut Matematyczny Polskiej Akademii Nauk

EN

We study the Fredholm properties of a general class of elliptic differential operators on ℝⁿ. These results are expressed in terms of a class of weighted function spaces, which can be locally modeled on a wide variety of standard function spaces, and a related spectral pencil problem on the sphere, which is defined in terms of the asymptotic behaviour of the coefficients of the original operator.

16

On the characteristic initial value problem for nonlinear symmetric hyperbolic systems, including Einstein equations

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Cabet A. , Chruściel P. , Wafo R.

Instytut Matematyczny Polskiej Akademii Nauk

EN

We consider a characteristic initial value problem for a class of symmetric hyperbolic systems with initial data given on two smooth null intersecting characteristic surfaces. We prove existence of solutions on a future neighborhood of the initial surfaces. The result is applied to general semilinear wave equations, as well as the Einstein equations with or without sources, and conformal variations thereof.

17

Finite representability and super-ideals of operators

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Heinrich S.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction............................................................................... 5 Notation............................................................................................. 6 1. Finite representability of operators.......................................... 7 2. Super-ideals of operators......................................................... 14 3. Connections with Banach space theory................................. 20 4. Procedures on super-ideals..................................................... 24 5. Examples and applications...................................................... 28 References....................................................................................... 36

18

Convexités dans les espaces vectoriels topologiques généraux

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Turpin P.

Instytut Matematyczny Polskiej Akademii Nauk

FR

TABLE DES MATIÈRES Introduction ...................................................................................................................................................................... 5 Chapitre 0. PRÉLIMINAIRES......................................................................................................................................... 14 0.1. Espaces vectoriels à convergence......................................................................................................... 14 0.2. Fonctions d'Orlicz................................................................................................................................................ 24 0.3. Espaces d'Orlicz................................................................................................................................................. 29 Chapitre I. LIMITES INDUCTIVES DÉNOMBRABLES D'ESPACES TOPOLOGIQUES ÉQUILIBRÉS............. 37 1.1. Généralisation de théorèmes classiques..................................................................................................... 37 1.2. Exemples............................................................................................................................................................. 44 Chapitre II. LE GALBE D'UN OPÉRATEUR, D'UN ESPACE................................................................................... 50 2.1. Bases de filtre galbant un opérateur.............................................................................................................. 51 2.2. Définition du galbe d'un opérateur.................................................................................................................. 53 2.3. Cas particuliers.................................................................................................................................................. 56 2.4. Galbe des structures initiales et finales........................................................................................................ 61 2.5. Galbe des limites inductives vectorielles topologiques dénombrables................................................. 66 Chapitre III. ESPACES D'ORLICZ................................................................................................................................ 71 3.1. Galbe des inclusions $L^ψ_Ω$⊂$L^φ_Ω$................................................................................................... 73 3.2. Galbe des intersections d'espaces d’Orlicz................................................................................................. 81 3.3. Le galbe d'une réunion dénombrable d'espaces d'Orlicz.......................................................................... 87 3.4. Opérateurs entre espaces d’Orlicz de fonctions.......................................................................................... 93 3.5. Compacité d'opérateurs à valeurs dans des espaces d'Orlicz de suites............................................... 96 Chapitre IV. GÉNÉRALISATIONS D'UN THÉORÈME DE MAZUR ET ORLICZ.................................................... 101 Chapitre V. ÉTUDE DES GALBES EN TANT QU'SPACES DE SUITES................................................................ 114 5.1. Deux systèmes fondamentaux d'ensembles de suites équisommables.............................................. 115 5.2. Galbes et galbes stricts.................................................................................................................................... 119 5.3. Ensembles de suites sous-jacents aux galbes et galbes stricts............................................................ 123 5.4. Adhérence dans un galbe de l'ensemble $l^0$ des suites à support fini.............................................. 125 5.5. Galbe engendré par une base de filtre.......................................................................................................... 128 5.6. Calcul du galbe et du galbe strict engendrés par certains points de $l^1$............................................ 133 5.7. Construction d'un espace métrisable ayant un galbe donné.................................................................... 141 Chapitre VI. ESPACES ET OPÉRATEURS GALBES PAR $l^0_t$, $l^0_g$......................................................... 147 6.1. Espaces vectoriels topologiques galbés par $l^0_t$................................................................................. 148 6.2. Espaces vectoriels bornologiques galbés par $l^0_t$............................................................................... 152 6.3. Théorèmes de type Banach-Steinhaus et graphe fermé........................................................................... 156

19

Generalized analytic functions with applications to singular ordinary and partial differential equations

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Ziemian B.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction.............................................................................................................................................5 I. Preliminaries.........................................................................................................................................7 1. A review of classical results in the theory of Laplace integra............................................................7 2. Boundary values of holomorphic functions......................................................................................10 2.1. Distributions as boundary values of holomorphic functions.........................................................10 2.2. Hyperfunctions in one variable....................................................................................................12 3. Mellin analytic functionals, Mellin hyperfunctions and Mellin distributions.........................................14 4. Laplace distributions.........................................................................................................................18 4.1. Convolution of Laplace distributions.............................................................................................21 5. Ecalle distributions.............................................................................................................................23 5.1. Alien derivatives of Ecalle distributions.........................................................................................24 6. Paley-Wiener type theorem for Mellin analytic functionals.................................................................25 6.1. Phragmén-Lindelöf type theorems................................................................................................29 7. The cut-off functions and their Mellin transforms...............................................................................30 8. Modified Cauchy transformation in dimension 1.................................................................................31 II. The theory of generalized analytic functions..........................................................................................33 9. Definition of a generalized analytic function........................................................................................34 10. The Mellin transform of a generalized analytic function.....................................................................35 11. Characterization of GAFs in terms of Mellin transforms.....................................................................37 12. The Borel and Taylor transformations in the class of GAFs..............................................................40 13. Operations on generalized analytic functions...................................................................................40 14. Resurgent functions.........................................................................................................................44 14.1. Alien derivatives of resurgent functions......................................................................................46 14.2. Taylor-Fourier representation of resurgent functions..................................................................47 III. Applications to singular linear differential equations..............................................................................48 15. Special functions as generalized analytic functions...........................................................................48 16. Fuchsian type ODEs with generalized analytic coefficients................................................................52 17. Fuchsian type PDEs with "constant" coefficients................................................................................58 18. GAFs in several variables..................................................................................................................73 19. Fuchsian type PDEs with generalized analytic coefficients................................................................78 Appendices.................................................................................................................................................84 I. The symbol of a distribution in the sense of A. Weinstein. Conormal distributions...................................84 II. Nonlinear singular differential equations.................................................................................................88 1. The case of ordinary differential equations..........................................................................................88 2. The case of partial differential equations.............................................................................................93 References...................................................................................................................................................94 Symbol index.................................................................................................................................................97 Subject index................................................................................................................................................99

20

H-closed and extremally disconnected Hausdorff spaces

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Mioduszewski J. , Rudolf L.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction.......................................................................................................................................................... 5 I. PRELIMINARIES.............................................................................................................................................. 7 § 1. The closures of open subsets in r. o.-equivalent topologies............................................................. 7 § 2. The r. o.-maximal topologies.................................................................................................................... 9 § 3. The H-closed maximal spaces................................................................................................................ 10 § 4. R. o.-equivalence of extensions............................................................................................................... 10 § 5. 0-continuous maps.................................................................................................................................... 11 § 6. The Henriksen-Jerison and skeletal maps........................................................................................... 13 II. H-CLOSED EXTENSIONS OF HAUSDORFF SPACES.................................................................................... 14 § 1. The set of 77-closed extensions of given Hausdorff space............................................................... 14 5 2. Proper maps................................................................................................................................................ 16 § 3. Decompositions of proper maps............................................................................................................ 18 § 4. An application to IT-closed extensions................................................................................................... 19 § 5. The case of compact-like spaces........................................................................................................... 22 § 6. The case of minimal Hausdorff spaces................................................................................................. 25 III. EXTREMALLY DISCONNECTED RESOLUTIONS OF HAUSDORFF SPACES................................. 26 § 1. The set of irreducible maps onto a given Hausdorff space X............................................................ 26 § 2. R. o.-minimal irreducible maps............................................................................................................... 30 § 3. Extremally disconnected resolutions...................................................................................................... 31 IV. COMMUTATION OF H-CLOSED EXTENSIONS AND E. D. RESOLUTIONS...................................... 35 § 1. Commutativity in a pullback diagram...................................................................................................... 35 § 2. Commutativity in a pushout diagram ..................................................................................................... 37 V. PROJECTIVE AND INJECTIVE HAUSDORFF SPACES......................................................................... 39 § 1. H-closed projective spaces. A definition and motivations.................................................................. 41 § 2. The case of compact-like spaces........................................................................................................... 42 § 3. Projectiveness for arbitrary H-closed spaces....................................................................................... 44 § 4. Projectiveness for arbitrary Hausdorff spaces...................................................................................... 45 § 5. Injective extremally disconnected spaces............................................................................................. 46 § 6. Injective Hausdorff spaces....................................................................................................................... 48 Bibliography......................................................................................................................................................... 51