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
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Problems involving cracks are of particular importance in structural mechanics, and gave rise to many interesting mathematical techniques to treat them. The difficulties stem from the singularities of domains, which yield lower regularity of solutions. Of particular interest are techniques which allow us to identify cracks and defects from the mechanical properties. Long before advent of mathematical modeling in structural mechanics, defects were identified by the fact that they changed the sound of a piece of material when struck. These techniques have been refined over the years. This volume gives a compilation of recent mathematical methods used in the solution of problems involving cracks, in particular problems of shape optimization. It is based on a collection of recent papers in this area and reflects the work of many authors, namely Gilles Frémiot (Nancy), Werner Horn (Northridge), Jiří Jarušek (Prague), Alexander Khludnev (Novosibirsk), Antoine Laurain (Graz), Murali Rao (Gainesville), Jan Sokołowski (Nancy) and Carol Ann Shubin (Northridge). We review the techniques which can be used for numerical analysis and shape optimization of problems with cracks and of the associated variational inequalities. The mathematical results include sensitivity analysis of variational inequalities, based on the concept of conical differential introduced by Mignot. We complete results on conical differentiability obtained for obstacle problems, by results derived for cracks with non-penetration condition and parabolic variational inequalities. Numerical methods for some problems are given as an illustration. From the point of view of applied mathematics numerical analysis is a necessary ingredient of applicability of the models proposed. We also extend the result on conical differentiability to the case of some evolution variational inequalities. The same mathematical model can be represented in different ways, like primal, dual or mixed formulations for an elliptic problem. We use such possibilities for models with cracks. For the shape sensitivity analysis, in Chapters 1 to 3 we give a thorough introduction to the use of first and second order shape derivatives and their application to problems involving cracks. In Chapter 1, for the convenience of the reader, we provide classical results on shape sensitivity analysis in smooth domains. In Chapter 2, the results on the first order Eulerian semi-derivative in domains with cracks are presented. Of particular interest is the so-called structure theorem for the shape derivative. In Chapter 3, the results on the Fréchet derivative in domains with cracks are presented as well, for first and second order derivatives, using a technique different from that in Chapter 2. In Chapter 4, we extend those ideas to Banach spaces, and give some applications of this extended theory. The polyhedricity of convex sets is considered in the spirit of [71], [87], in the most general setting. These abstract results can be applied to sensitivity analysis of crack problems with non-linear boundary conditions. The results obtained use non-linear potential theory and are interesting on their own. In Chapter 5, several techniques for the study of cracked domains with non-penetration conditions on the crack faces in elastic bodies are presented. The classical crack theory in elasticity is characterized by linear boundary conditions which do not correspond to the physical reality since the crack faces can penetrate each other in this model. In this chapter, non-penetration conditions on the crack faces are considered, which leads to a non-linear problem. The model is presented and the shape sensitivity analysis is performed. Chapter 6 is devoted to the newly developed smooth domain method for cracks. In that chapter the problem on a domain with a crack is transformed into a new problem on a smooth domain. This approach is useful for numerical methods. In \cite{belh} this formulation is used combined with mixed finite elements, and some error estimates are derived for the finite element approximation of variational inequalities with non-linear condition on the crack faces. We give applications of this method to some classical problems. Finally, in Chapter 7 we study integro-differential equations arising from bridged crack models. This is a classical technique, but we introduce a few modern approaches to it for completeness sake.
Contents Introduction.................................................................................................................................................. 3 § 1. System $\mathscr{S}$ of a propositional calculus...................................................................... 4 § 2. System $\mathscr{S}*$..................................................................................................................... 5 § 3. $\mathscr{S}*$-algebras.................................................................................................................. 9 § 4. The algebra of set designations of $\mathscr{S}*$.................................................................... 11 § 5. Models of the system $\mathscr{S}*$............................................................................................ 13 § G. Completeness theorem................................................................................................................... 17 § 7. Formalized theory of fields of sets.................................................................................................. 20 § 8. Classical elementary theory of Boolean algebras....................................................................... 23 § 9. Elementary theories of $\mathscr{S}$-algebras based on $\mathscr{S}$-logic.................. 26 References................................................................................................................................................. 29
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New sufficient conditions for the existence of an invariant measure for nonexpansive Markov operators defined on Polish spaces are presented. These criteria are applied to iterated function systems, stochastically perturbed dynamical systems and Poisson stochastic differential equations. We also estimate the Ledrappier version of capacity for invariant measures.
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An h-space is a compact set with respect to a quasi-metric and endowed with a Borel measure such that the measure of a ball of radius r is equivalent to h(r), for some function h. Applying an approach introduced by Triebel in [28] we define Besov spaces of generalised smoothness on h-spaces. We describe the techniques and tools used in this construction, namely snowflaked transforms and charts. This approach relies on using what is known for function spaces on some fractal sets, which are themselves defined as traces of convenient function spaces on ℝⁿ. It has turned out to be important to obtain new properties and characterisations for the elements of these spaces, for example, to guarantee the independence of the charts used. So we also present results for Besov spaces of generalised smoothness on ℝⁿ and some special fractal sets, namely characterisations by differences and a homogeneity property (on ℝⁿ) and non-smooth atomic decompositions.
CONTENTS Part I 1. Introduction...................................................................................................................................................... 5 2. Preliminaries.................................................................................................................................................. 6 2.1. Notation........................................................................................................................................................ 6 2.2. Local preliminaries.................................................................................................................................... 6 3. The space $\mathscr{A}$............................................................................................................................. 9 3.1. Generalities................................................................................................................................................. 9 3.2. Linear transformations of $\mathscr{A}$............................................................................................... 10 3.3. Global measure.......................................................................................................................................... 11 4. Lattices and convex bodies.......................................................................................................................... 11 4.1. Lattices......................................................................................................................................................... 11 4.2. Convex bodies............................................................................................................................................. 13 5. An analogue of Minkowski's convex body theorem................................................................................. 15 5.1. Convex body theorem................................................................................................................................ 15 5.2. Applications of theorem 2......................................................................................................................... 16 6. Successive minima....................................................................................................................................... 18 6.1. Preliminaries............................................................................................................................................... 18 6.2. The product of successive minima; an upper bound.......................................................................... 19 6.3. The product of successive minima; a lower bound............................................................................ 22 6.4. Applications to algebraic number theory................................................................................................ 24 7. T-adeles........................................................................................................................................................... 32 7.1. The general theory for T-adeles............................................................................................................... 32 7.2. Two special cases..................................................................................................................................... 35 Part II 1. Introduction ..................................................................................................................................................... 37 2. Topology in $\mathscr{G}$................................................................................................................... 37 2.1. Two topologies on $\mathscr{G}$........................................................................................................... 37 2.2. Comparison of the two topologies.......................................................................................................... 39 3. Compactness for lattices............................................................................................................................. 41 3.1. Two topologies on the lattice space....................................................................................................... 41 3.2. An important lemma................................................................................................................................... 43 3.3. An analogue of Mahler’s compactness theorem................................................................................. 44 4. The Chabauty topology................................................................................................................................. 45 5. T-adeles ......................................................................................................................................................... 47 References.......................................................................................................................................................... 49
CONTENTS Chapter 0...............................................................................................................................................................................5 0.1. Introduction..................................................................................................................................................................5 0.2. Preliminary results.......................................................................................................................................................9 Chapter I..............................................................................................................................................................................16 I.1. Best approximation in finite-dimensional subspaces of ℒ(B,D)....................................................................................16 I.2. Kolmogorov's type criteria for spaces of compact operators; general case.................................................................26 I.3. Criteria for the space $K(C_K(T))$.............................................................................................................................30 I.4. The case of sequence spaces....................................................................................................................................38 Chapter II.............................................................................................................................................................................43 II.1. Extensions of linear operators from hyperplanes of $l^{(n)}_∞$.................................................................................43 II.2. Minimal projections onto hyperplanes of $l^{(n)}_1$...................................................................................................52 II.3. Strongly unique minimal projections onto hyperplanes of $l^{(n)}_∞$ and $l^{(n)}_1$...............................................59 II.4. Minimal projections onto subspaces of $l^{(n)}_∞$ of codimension two......................................................................71 II.5. Uniqueness of minimal projections onto subspace of $l^{(n)}_∞$ of codimension two................................................75 II.6. Strong unicity criterion in some space of operators....................................................................................................79 Chapter III.............................................................................................................................................................................83 III.1. Extensions of linear operators from finite-dimensional subspaces I...........................................................................83 III.2. Extensions of linear operators from finite-dimensional subspaces II..........................................................................90 III.3. Algorithms for seeking the constant $W_m$..............................................................................................................97 References..........................................................................................................................................................................99 Index..................................................................................................................................................................................102 Index of symbols................................................................................................................................................................102
CONTENTS Introduction............................................................................................................................................................................... 5 Chapter 0. PRELIMINARIES 0.1. (Preliminary remarks and notation)............................................................................................................................. 9 0.2. (Notation — continuation).............................................................................................................................................. 10 0.3. (Notation and some definitions).................................................................................................................................. 10 0.4. (Statement of problems; definition of solutions of differential functional equations)......................................... 12 0.5. (Equivalence of problems: differential and integral; definition of solutions of integral equations).................. 14 Chapter I. EXISTENCE AND UNIQUENESS OF SOLUTIONS AND THE CONVERGENCE OF SUCCESSIVE APPROXIMATIONS IN COMPACT SETS 1.1. Notation and definitions................................................................................................................................................. 17 1.2. Uniqueness...................................................................................................................................................................... 18 1.3. Existence and successive approximations................................................................................................................ 19 1.4. Existence without uniqueness...................................................................................................................................... 22 1.5. Some generalizations of the results from 1.2-1.4..................................................................................................... 23 1.6. Some supplementary remarks..................................................................................................................................... 24 Chapter II. LOCAL AND GLOBAL EXISTENCE AND UNIQUENESS 2.1. Notation and definitions................................................................................................................................................. 27 2.2. Union of solutions........................................................................................................................................................... 28 2.3. Global uniqueness.......................................................................................................................................................... 29 2.4. Definition of the condition (W) and some remarks................................................................................................... 31 2.6. Local existence of solutions.......................................................................................................................................... 32 2.6. Lemmas............................................................................................................................................................................ 33 2.7. Limits of solutions on the boundary............................................................................................................................. 36 2.8. Prolongations................................................................................................................................................................... 38 2.9. Global existence under the assumptions on uniqueness...................................................................................... 39 2.10. Global existence without uniqueness....................................................................................................................... 41 2.11. Global existence without uniqueness by the method of A. Bielecki, T. Dłotko and M. Kuczma...................... 43 2.12. Existence of solutions under the assumptions (Y) and (Ỹ)................................................................................... 46 2.13. Local convergence of successive approximations under the assumptions (V)............................................... 47 2.14. Remarks on some generalizations........................................................................................................................... 49 Chapter III. CONTINUOUS DEPENDENCE OF SOLUTIONS ON GIVEN FUNCTIONS 3.1. Continuous dependence on $λ, ψ, {φ^a}$............................................ 51 3.2. Continuous dependence on ƒ....................................................... 52
CONTENTS Introduction......................................................................................................................................... 5 Chapter I Elementary topological characterizations of fundamental dimension........................... 6 1. Characterizations of fundamental dimension..................................................................... 6 2. The fundamental dimension of components of compacta.............................................. 9 3. The fundamental dimension of the union of two compacta............................................. 10 Chapter II Cohomology groups over local systems and generalized local systems................... 13 1. Local systems of groups......................................................................................................... 13 2. Cohomology with coefficients in local systems.................................................................. 16 3. The Künneth formula 4. Generalized local systems..................................................................................................... 20 Chapter III Homological characterizations of fundamental dimension........................................... 22 1. Deformability of maps and the number................................................................................ 23 2. Obstructions to deformability.................................................................................................. 24 3. Coefficients of cyclicity and ℱ-continua................................................................................. 25 4. Continua with fundamental dimension ≥ 3........................................................................ 28 5. Two algebraic lemmas............................................................................................................ 29 6. Continua with fundamental dimension equal to 1............................................................. 31 7. Continua with fundamental dimension equal to 2............................................................. 33 8. The main results....................................................................................................................... 34 Chapter IV Applications of the homological characterizations of fundamental dimension to the study of some special problems................................................................................................. 37 1. The fundamental dimension of the Cartesian product of a closed manifold and a continuum........................................................................................................................................ 37 2. The fundamental dimension of the Cartesian product of a curve and a continuum... 38 3. An example of a finite-dimensional continuum with an infinite family of shape factors and the fundamental dimension of the Cartesian product of polyhedra........................... 42 4. The fundamental dimension of the union of two compacta and of the quotient space............................................................................................................................................................ 43 5. The fundamental dimension of the suspension of a compactum.................................. 44 6. The fundamental dimension of the Cartesian product of approximative 1-connected compacta............................................................................................................................. 46 7. The fundamental dimension of a subset of manifold....................................................... 48 Final remarks and problems................................................................................................................... 50 References.................................................................................................................................................. 52 Index of symbols........................................................................................................................................ 54
Contents Introduction.................................................................................................................................................. 5 I. Preliminaries........................................................................................................................................... 6 II. Main theorem.......................................................................................................................................... 8 III. The categories $k_1, ..., k_{35}$ are rich......................................................................................... 11 IV. Forms $a_1, ..., a_{10}$ and the cone............................................................................................. 27 V. Classification of thin categories......................................................................................................... 40 VI. Operations $o_1, o_2$ and reducing of a category by these operations.................................. 48 Appendix...................................................................................................................................................... 52 References................................................................................................................................................. 53
This work is devoted to the solvability of semilinear equations (*) Lx + f(x) = y, x ∈ D(L) ⊂ E, y ∈ F, where E, F are real Banach spaces and L: D(L) → F is a linear operator with dimKerL = codimR(L) = ∞. We introduce the notion of a generalized A-proper mapping f(x) associated with the operator L and show that some classes of monotone-type mappings (i.e. $(M_L)$, $(M_L)_+$, $(S_L)$ or $(S_L)_+$) are nontrivial examples of A-proper mappings. Using the topological transversality, we develop the continuation method for L-condensing A-proper mappings and obtain solvability results for the equation (*). The abstract results for A-proper mappings are applied to the problem of time-periodic solutions of semilinear wave equations. We introduce a generalized coincidence degree called the Browder-Petryshyn-Mawhin coincidence degree.
FR
TABLE DES MATIÈRES 1. Introduction...............................................................................................................5 2. Notation ....................................................................................................................7 3. Factorisation fredholmienne et les applications A-propres........................................8 4. Exemples des applications A-propres - applications des types monotones.............10 5. Propriétés des applications A-propres....................................................................28 6. Applications L-condensantes..................................................................................32 7. Applications aux problèmes de coïncidence............................................................45 8. Théorie du degré de coïncidence...........................................................................56 9. Application au système d'équations d'ondes semilinéaires.....................................61 Références.................................................................................................................65
CONTENTS Introduction..............................................................................................5 I. Basic definitions and notation..............................................................6 M-bases and finite-dimensional decompositions....................................6 Some geometric properties of Banach spaces.......................................9 II. Constructions of equivalent norms.....................................................12 III. (p,q)-estimates in interpolation spaces..............................................21 IV. Geometric properties of operators....................................................26 Nearly uniformly convex operators........................................................27 Nearly uniformly smooth operators.........................................................30 V. Factoring operators through nearly uniformly convex spaces...........35 Factorizations and geometric properties of operators..........................35 The case of spaces with finite-dimensional decompositions.................40 References.............................................................................................45
CONTENTS 1. Introduction .......................................................................................................................................................... 5 2. Notation and preliminary remarks............................................................................................................................ 7 3. A geometric approach to the calculus of variations............................................................................................... 9 4. Multisymplectic manifolds and a multiphase structure of a classical field theory........................................... 19 5. A multiphase structure of General Relativity............................................................................................................ 22 6. The Cauchy problem and ADMW coordinates in General Relativity................................................................... 26 7. A symplectic structure in the set of solutions of field equations.......................................................................... 29 8. A symplectic structure in the set of Einstein metrics.............................................................................................. 36 9. The gauge distribution and the action of the diffeomorphism group.................................................................. 39 10. Degrees of freedom and a superphase space -for General Relativity............................................................. 46 11. A pseudo-differential structure in the space ℋ. A Lie algebra of functionals on ℋ....................................... 48 12. A variational principle for General Relativity............................................................................................................ 55 13. The Hamilton-Jacobi equation in lagrangian field theories................................................................................ 57 14. The Hamilton-Jacobi equation in General Relativity............................................................................................. 62 15. Proofs............................................................................................................................................................................. 66 Appendix. Proof of the ellipticity of the operator AA*..................................................................................................... 79 References.......................................................................................................................................................................... 82
CONTENTS Introduction.......................................................................................................................................................... 5 Chapter I. Some preliminary lemmas............................................................................................................ 8 Chapter II. Weighted $H^p$ spaces of analytic functions.......................................................................... 13 1. Behaviour at the boundary....................................................................................................................... 13 2. Maximal function characterization........................................................................................................... 15 3. Atomic decomposition.............................................................................................................................. 20 4. Dual spaces............................................................................................................................................... 27 Chapter III. $H^p$ spaces associated with the space of homogeneous type (R, w(x)dx).................... 31 1. The space $\mathfrak{H}^1(w(x)dx)$...................................................................................................... 31 2. The spaces $\mathfrak{H}^p(w(x)dx)$ for p < 1.................................................................................... 33 Chapter IV. Applications and examples.......................................................................................................... 40 1. A weighted Hilbert transform.................................................................................................................... 40 2. Equivalence between the space of radial functions in $H^1(R^n)$ and the space of even functions in $\mathfrak{H}^1(|r|^{n-1}dr)$..................................................................................... 40 3. Integral operators in the line obtained by restricting to radial functions some systems of Riesz transforms in higher dimensions................................................................................................. 44 4. The kernel $z^{-2}$...................................................................................................................................... 54 References............................................................................................................................................................. 58
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Consider a Hidden Markov Model (HMM) such that both the state space and the observation space are complete, separable, metric spaces and for which both the transition probability function (tr.pr.f.) determining the hidden Markov chain of the HMM and the tr.pr.f. determining the observation sequence of the HMM have densities. Such HMMs are called fully dominated. In this paper we consider a subclass of fully dominated HMMs which we call regular. A fully dominated, regular HMM induces a tr.pr.f. on the set of probability density functions on the state space which we call the filter kernel induced by the HMM and which can be interpreted as the Markov kernel associated to the sequence of conditional state distributions. We show that if the underlying hidden Markov chain of the fully dominated, regular HMM is strongly ergodic and a certain coupling condition is fulfilled, then, in the limit, the distribution of the conditional distribution becomes independent of the initial distribution of the hidden Markov chain and, if also the hidden Markov chain is uniformly ergodic, then the distributions tend towards a limit distribution. In the last part of the paper, we present some more explicit conditions, implying that the coupling condition mentioned above is satisfied.
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comité de rédaction: Czesław Bessaga, Stanisław mazur, Władysław Orlicz, Aleksander Pełczyński, Stefan Rolewicz, Wiesław Żelazko Front page of Volume II, p.1-1 Tables des Matières, p.1-4 Préface, p.5-5 Publications de Stefan Banach, p.7-11 S. Banach: Front page and preface to "THÉORIE DES OPÉRATIONS LINÉAIRES" (MONOGRAFIE MATEMATYCZNE V.1), p.13-18 S. Banach: THÉORIE DES OPÉRATIONS LINÉAIRES (MONOGRAFIE MATEMATYCZNE T.1), p.19-222 C. Bessaga, A. Pełczyński: SOME ASPECTS OF THE PRESENT THEORY OF BANACH SPACES, p.223-304 S. Banach: SUR LES OPÉRATIONS DANS LES ENSEMBLES ABSTRAITS ET LEUR APPLICATION AUX ÉQUATIONS INTÉGRALES, p.305-348 S. Banach: SUR LE PROLONGEMENT DE CERTAINES FONCTIONNELLES, p.349-350 S. Banach: SUR LA CONVERGENCE PRESQUE PARTOUT DE FONCTIONNELLES LINÉAIRES, p.355-364 S. Banach, H. Steinhaus: SUR LE PRINCIPE DE LA CONDENSATION DE SINGULARITÉS, p.365-374 S. Banach: SUR LES FONCTIONNELLES LINÉAIRES, p.375-380 S. Banach: SUR LES FONCTIONNELLES LINÉAIRES II, p.381-395 S. Banach, S. Saks: SUR LA CONVERGENCE FORTE DANS LE CHAMP LP, p.396-401 S. Banach: ÜBER METRISCHE GRUPPEN, p.402-411 S. Banach, S. Mazur: EINE BEMERKUNG ÜBER DIE KONVERGENZMENGEN VON FOLGEN LINEARER OPERATIONEN, p.412-415 S. Banach, C. Kuratowski: SUR LA STRUCTURE DES ENSEMBLES LINÉAIRES, p.416-419 S. Banach, S. Mazur: ZUR THEORIE DER LINEAREN DIMENSION, p.420-430 S. Banach, S. Mazur: SUR LA DIMENSION LINÉAIRE DES ESPACES FONCTIONNELS, p.431-433 S. Banach: DIE THEORIE DER OPERATIONEN UND IHRE BEDEUTUNG FÜR DIE ANALYSIS, p.434-441 S. Banach: ÜBER HOMOGENE POLYNOME IN (L2), p.442-449 S. Banach: ÜBER DAS „LOI SUPRÊME" VON J. HOENE-WROŃSKI, p.450-457 S. Banach: SUR LA DIVERGENCE DES INTERPOLATIONS, p.458-464 S. Banach: REMARQUES SUR LES GROUPES ET LES CORPS MÉTRIQUES (RÉDIGÉ D'APRÈS UNE NOTICE POSTHUME PAR S. HARTMAN), p.465-468 Tables des Matières du Volume I, p.469-470
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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.
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