Showing papers on "Section (fiber bundle) published in 1996"

Theorems on Regularity and Singularity of Energy Minimizing Maps

Abstract: 1 Analytic Preliminaries.- 1.1 Holder Continuity.- 1.2 Smoothing.- 1.3 Functions with L2 Gradient.- 1.4 Harmonic Functions.- 1.5 Weakly Harmonic Functions.- 1.6 Harmonic Approximation Lemma.- 1.7 Elliptic regularity.- 1.8 A Technical Regularity Lemma.- 2 Regularity Theory for Harmonic Maps.- 2.1 Definition of Energy Minimizing Maps.- 2.2 The Variational Equations.- 2.3 The ?-Regularity Theorem.- 2.4 The Monotonicity Formula.- 2.5 The Density Function.- 2.6 A Lemma of Luckhaus.- 2.7 Corollaries of Luckhaus' Lemma.- 2.8 Proof of the Reverse Poincare Inequality.- 2.9 The Compactness Theorem.- 2.10 Corollaries of the ?-Regularity Theorem.- 2.11 Remark on Upper Semicontinuity of the Density ?u(y).- 2.12 Appendix to Chapter 2.- 2.12.1 Absolute Continuity Properties of Functions in W1,2.- 2.12.2 Proof of Luckhaus' Lemma (Lemma 1 of Section 2.6).- 2.12.3 Nearest point projection.- 2.12.4 Proof of the ?-regularity theorem in case n = 2.- 3 Approximation Properties of the Singular Set.- 3.1 Definition of Tangent Map.- 3.2 Properties of Tangent Maps.- 3.3 Properties of Homogeneous Degree Zero Minimizers.- 3.4 Further Properties of sing u.- 3.5 Definition of Top-dimensional Part of the Singular Set.- 3.6 Homogeneous Degree Zero ? with dim S(?) = n - 3.- 3.7 The Geometric Picture Near Points of sing*u.- 3.8 Consequences of Uniqueness of Tangent Maps.- 3.9 Approximation properties of subsets of ?n.- 3.10 Uniqueness of Tangent maps with isolated singularities.- 3.11 Functionals on vector bundles.- 3.12 The Liapunov-Schmidt Reduction.- 3.13 The ?ojasiewicz Inequality for ?.- 3.14 ?ojasiewicz for the Energy functional on Sn-1.- 3.15 Proof of Theorem 1 of Section 3.10.- 3.16 Appendix to Chapter 3.- 3.16.1 The Liapunov-Schmidt Reduction in a Finite Dimensional Setting.- 4 Rectifiability of the Singular Set.- 4.1 Statement of Main Theorems.- 4.2 A general rectifiability lemma.- 4.3 Gap Measures on Subsets of ?n.- 4.4 Energy Estimates.- 4.5 L2 estimates.- 4.6 The deviation function ?.- 4.7 Proof of Theorems 1, 2 of Section 4.1.- 4.8 The case when ? has arbitrary Riemannian metric.

Existence of multi-bumb solutions for nonlinear schrödinger equations via variational method

Abstract: In this paper, we consider the general semilinear equation (1.6) {epsilon}{sup 2}{Delta}{sub u}-V(x)u+f(u)=O, x {epsilon} R{sup N}. We assume that V(x) satisfies (V1) V(x) is locally Hoelder continuous in R{sup N} and V(x){ge}V{sub O}>O; (V2) there exist k disjoint bounded regions {Omega}{sub 1}, {Omega}{sub 2}...,{Omega}{sub k} such that (1.7) M{sub i}:=inf/{partial_derivative}{Omega}{sub i}/V(x) > {omega}{sub i}: = inf/{Omega}{sub i}V(x), i=1,2,...,k. We also assume that f(u) satisfies (f1) f(u) {epsilon} C{sup 1}(R), f(u){equivalent_to}O for u {le} O and f(u){equivalent_to} O for u > O; (f2) f(u)/u is nondecreasing in u; (f3) O {le} f{sub u}(u) {le} a{sub 1} + a{sub 2}u{sup p-1} for some positive constants a{sub 1}, a{sub 2} and 1 < p < N+2/N-2 (we use the convention now and later that N+2/N-2 should be replaced by {infinity} when N = 1,2); (f4) there exists {beta} {epsilon} (O,1/2) such that F(u) {le} {beta}uf(u), u{ge}O where F(u) = {integral}{sub O}{sup u} f(t) dt. This paper is organized as follows. In Section 2, we scale the equation (1.6) properly and present some results for generalized Palais Smale sequences of a family of modified functionals. In Section 3, we show the existence of critical points of the modified functionals at certain energy levelmore » and in certain neighborhood. Finally, we show the concentration property of these critical points and therefore obtain the solution for equation (1.6). This also finishes the proof of the main theorem. I have learned from the referee that he has just received a closely related paper by del Pino and Felmer, but which is based on different methods. 27 refs.« less

Electronic structure of rare-earth pnictides.

Abstract: The results of first-principles calculations of the electronic band structures, equilibrium lattice constants, cohesive energies, bulk moduli, and magnetic moments are presented for the rare-earth pnictides with the rocksalt structure and chemical formula R-V, where R=Gd, Er, and the group-V elements N, P, and As The linear-muffin-tin-orbital method was used in the atomic sphere approximation The 4f states were treated as localized corelike states with fixed spin occupancies Justifications for this procedure are presented The systems were studied with the 4f spins on all rare-earth ions aligned (ferromagentic phase) and with the spins randomly oriented (paramagentic phase) Within the local spin-density approximation, all systems studied were found to be semimetallic with a hole section of the Fermi surface near \ensuremath{\Gamma} and electron section near X The nitrides, however, have a nearly zero band-gap overlap We estimated quasiparticle self-energy corrections using an approach previously used for semiconductors With these corrections, GdN is found to be a semiconductor in the paramagnetic phase and a semimetal in the ferromagnetic phase ErN, on the other hand, is found to be a semiconductor in both phases All systems correspond to a trivalent state of the rare-earth element and are characterized by ionic bonding The results for the lattice constants and the qualitative conclusion about the semimetallic nature are in agreement with experimental data and with the previous calculations for Gd-pnictides For ErAs, the calculated magnetic exchange splittings, electron and hole concentrations, Fermi-surface cross-sectional areas, and cyclotron masses are in satisfactory agreement with the available Shubnikov--de Haas data on ${\mathrm{Er}}_{\mathit{x}}$${\mathrm{Sc}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As when account is taken of the differences due to the presence of Sc and of the self-energy corrections to the local-density approximation \textcopyright{} 1996 The American Physical Society

Lie algebroids and mechanics

Abstract: We give a formulation of certain types of mechanical systems using the structure of groupoid of the tangent and cotangent bundles to the configuration manifold $M$; the set of units is the zero section identified with the manifold $M$. We study the Legendre transformation on Lie algebroids.

Harmonic analysis of fractal measures

Abstract: We consider affine systems inR n constructed from a given integral invertible and expansive matrixR, and a finite setB of translates,σ bx:=R–1x+b; the corresponding measure μ onR n is a probability measure fixed by the self-similarity $$\mu = \left| B \right|^{ - 1} \sum olimits_{b \in B} {\mu o\sigma _b^{ - 1} } $$ . There are twoa priori candidates for an associated orthogonal harmonic analysis: (i) the existence of some subset Λ inR n such that the exponentials {eiλ·x}Λ form anorthogonal basis forL 2(μ); and (ii) the existence of a certaindual pair of representations of theC *-algebraO N wheren is the cardinality of the setB. (For eachN, theC *-algebraO N is known to be simple; it is also called the Cuntz algebra.) We show that, in the “typical” fractal case, the naive version (i) must be rejected; typically the orthogonal exponentials inL 2(μ) fail to span a dense subspace. Instead we show that theC *-algebraic version of an orthogonal harmonic analysis, namely (ii), is a natural substitute. It turns out that this version is still based on exponentialse iλ·x, but in a more indirect way. (See details in Section 5 below.) Our main result concerns the intrinsic geometric features of affine systems, based onR andB, such that μ has theC *-algebra property (ii). Specifically, we show that μ has an orthogonal harmonic analysis (in the sense (ii)) if the system (R, B) satisfies some specific symmetry conditions (which are geometric in nature). Our conditions for (ii) are stated in terms of two pieces of data: (a) aunitary generalized Hadamard matrix, and (b) a certainsystem of lattices which must exist and, at the same time, be compatible with the Hadamard matrix. A partial converse to this result is also given. Several examples are calculated, and a new maximality condition for exponentials is identified.

The canonical class and the $C^\infty$ properties of Kähler surfaces

Abstract: We give a self contained proof that for Kahler surfaces with non- negative Kodaira dimension, the canonical class of the minimal model and the ( 1)-curves are oriented dieomorphism invariants up to sign. This includes the case pg = 0. It implies that the Kodaira dimension is determined by the underlying dierentiable manifold. We then reprove that the multiplicities of the elliptic bration are determined by the underlying oriented manifold, and that the plurigenera of a surface are oriented dieomorphism invariants. We also compute the Seiberg Witten invariants of all Kahler surfaces of non- negative Kodaira dimension. The proof uses a set up of Seiberg Witten theory that replaces generic metrics by the construction of a localised Euler class of an innite dimensional bundle with a Fredholm section. This makes the techniques of excess intersection available in gauge theory.

Polynomial Methods for Control Systems Design

Abstract: Preface ix.- 1 A Tutorial on H2 Control Theory: The Continuous Time Case.- 1.1 Introduction.- 1.2 LQG control theory.- 1.2.1 Problem formulation.- 1.2.2 Finite horizon solution.- 1.2.3 Infinite horizon solution.- 1.3 H2 control theory.- 1.3.1 Preliminaries.- 1.3.2 State space solution.- 1.3.3 Wiener-Hopf solution.- 1.3.4 Diophantine equations solution.- 1.4 Comparison and examples.- 1.4.1 The LQG as an H2 problem.- 1.4.2 Internal stability.- 1.4.3 Solvability assumptions.- 1.4.4 Non-proper plants.- 1.4.5 Design examples.- 1.5 References.- 2 Frequency Domain Solution of the Standard H? Problem.- 2.1 Introduction.- 2.1.1 Introduction.- 2.1.2 Problem formulation.- 2.1.3 Polynomial matrix fraction representations.- 2.1.4 Outline.- 2.2 Well-posedness and closed-loop stability.- 2.2.1 Introduction.- 2.2.2 Well-posedness.- 2.2.3 Closed-loop stability.- 2.2.4 Redefinition of the standard problem.- 2.3 Lower bound.- 2.3.1 Introduction.- 2.3.2 Lower bound.- 2.3.3 Examples.- 2.3.4 Polynomial formulas.- 2.4 Sublevel solutions.- 2.4.1 Introduction.- 2.4.2 The basic inequality.- 2.4.3 Spectral factorization.- 2.4.4 All sublevel solutions.- 2.4.5 Polynomial formulas.- 2.5 Canonical spectral factorizations.- 2.5.1 Definition.- 2.5.2 Polynomial formulation of the rational factorization.- 2.5.3 Zeros on the imaginary axis.- 2.6 Stability.- 2.6.1 Introduction.- 2.6.2 All stabilizing sublevel compensators.- 2.6.3 Search procedure - Type A and Type B optimal solutions.- 2.7 Factorization algorithm.- 2.7.1 Introduction.- 2.7.2 State space algorithm.- 2.7.3 Noncanonical factorizations.- 2.8 Optimal solutions.- 2.8.1 Introduction.- 2.8.2 All optimal compensators.- 2.8.3 Examples.- 2.9 Conclusions.- 2.10 Appendix: Proofs for section 2.3.- 2.11 Appendix: Proofs for section 2.4.- 2.12 Appendix: Proof of theorem 2.7.- 2.13 Appendix: Proof of the equalizing property.- 2.14 References.- 3 LQG Multivariable Regulation and Tracking Problems for General System Configurations.- 3.1 Introduction.- 3.2 Regulation problem.- 3.2.1 Problem solution.- 3.2.2 Connection with the Wiener-Hopf solution.- 3.2.3 Innovations representations.- 3.2.4 Relationships with other polynomial solutions.- 3.3 Tracking, servo and accessible disturbance problems.- 3.3.1 Problem formulation.- 3.4 Conclusions.- 3.5 Appendix.- 3.6 References.- 4 A Game Theory Polynomial Solution to the H? Control Problem.- 4.1 Abstract.- 4.2 Introduction.- 4.3 Problem definition.- 4.4 The game problem.- 4.4.1 Main result.- 4.4.2 Summary of the simplified solution procedure.- 4.4.3 Comments.- 4.5 Relations to the J-factorization H? problem.- 4.5.1 Introduction.- 4.5.2 The J-factorization solution.- 4.5.3 Connection with the game solution.- 4.6 Relations to the minimum entropy control problem.- 4.7 A design example: mixed sensitivity.- 4.7.1 Mixed sensitivity problem formulation.- 4.7.2 Numerical example.- 4.8 Conclusions.- 4.9 Appendix.- 4.10 References.- 4.11 Acknowledgements.- 5 H2 Design of Nominal and Robust Discrete Time Filters.- 5.1 Abstract.- 5.2 Introduction.- 5.2.1 Digital communications: a challenging application area...- 5.2.2 Remarks on the notation.- 5.3 Wiener filter design based on polynomial equations.- 5.3.1 A general H2 filtering problem.- 5.3.2 A structured problem formulation.- 5.3.3 Multisignal deconvolution.- 5.3.4 Decision feedback equalizers.- 5.4 Design of robust filters in input-output form.- 5.4.1 Approaches to robust H2 estimation.- 5.4.2 The averaged H2 estimation problem.- 5.4.3 Parameterization of the extended design model.- 5.4.4 Obtaining error models.- 5.4.5 Covariance matrices for the stochastic coefficients.- 5.4.6 Design of the cautious Wiener filter.- 5.5 Robust H2 filter design.- 5.5.1 Series expansion.- 5.5.2 The robust linear state estimator.- 5.6 Parameter tracking.- 5.7 Acknowledgement.- 5.8 References.- 6 Polynomial Solution of H2 and H? Optimal Control Problems with Application to Coordinate Measuring Machines.- 6.1 Abstract.- 6.2 Introduction.- 6.3 H.2 control design.- 6.3.1 System model.- 6.3.2 Assumptions.- 6.3.3 The H2 cost function.- 6.3.4 Dynamic weightings.- 6.3.5 The H2 controller.- 6.3.6 Properties of the controller.- 6.3.7 Design procedure.- 6.4 H? Robust control problem.- 6.4.1 Generalised H2 and H? controllers.- 6.5 System and disturbance modelling.- 6.5.1 System modelling.- 6.5.2 Disturbance modelling.- 6.5.3 Overall system model.- 6.6 Simulation and experimental studies.- 6.6.1 System definition.- 6.6.2 Simulation studies.- 6.6.3 Experimental studies.- 6.6.4 H? control.- 6.7 Conclusions.- 6.8 Acknowledgements.- 6.9 References.- 6.10 Appendix: two-DOF H2 optimal control problem.

Interplay of the spin-density-wave state and magnetic field in the organic conductor α-(BEDT-TTF ) 2 KHg(SCN ) 4

Abstract: Magnetic phase diagram of a quasi-two-dimensional organic conductor \ensuremath{\alpha}-(BEDT-TTF${)}_{2}$KHg(SCN${)}_{4}$ is revisited from a viewpoint of magnetic torque measurements in high fields up to 30 T. A phase boundary that is interpreted as a metal-spin density wave (SDW) phase transition is found by using torque measurements. It is shown that this phase boundary is clearly distinguished from so-called kink transition of the magnetoresistance. We demonstrate that the transition temperature defined by the midpoint of the broad phase transition is almost independent on magnetic field up to 23 T. Onset temperature of the transition shifts from about 8 K at H=0 T to higher temperatures with increasing of a magnetic field, and tends to be saturated. The onset line of this transition follows well the theoretical expectation that SDW has to be stabilized by a magnetic field. This allows us to estimate such important band parameters of the quasi-one-dimensional section of the Fermi surface as an effective mass, ${\mathit{m}}^{1\mathrm{D}}$\ensuremath{\simeq}(0.5\ifmmode\pm\else\textpm\fi{}0.1)${\mathit{m}}_{0}$, and an upper limit of an imperfect nesting bandwidth ${\mathit{t}}_{\mathit{c}}$\ensuremath{'}\ensuremath{\simeq}(10 \ifmmode\pm\else\textpm\fi{}1) K. The other phase boundaries determined by the position of the kink and hysteresis properties of the magnetoresistance are interpreted as subphases inside the SDW phase. Inside the SDW phase, we find an additional phase boundary at the temperature-independent field of 23 T, which corresponds to the appearance of de Haas--van Alphen oscillations on a magnetic torque curve. At the 23 T boundary, both the effective mass, m*, and the Dingle temperature, ${\mathit{T}}_{\mathit{D}}$, change their values from m*=(1.67 \ifmmode\pm\else\textpm\fi{} 0.05) ${\mathit{m}}_{0}$ and ${\mathit{T}}_{\mathit{D}}$=3.7--4.0 K in low magnetic field region to (1.95 \ifmmode\pm\else\textpm\fi{} 0.05) ${\mathit{m}}_{0}$ and 2.5--2.8 K in high field region. The latter phenomenon is discussed in terms of a reconstruction of the Fermi surface due to the SDW formation. Hysteresis of the magnetoresistance observed in one of the subphases inside the SDW phase is studied in detail by measuring both the temperature and the magnetic field dependences. \textcopyright{} 1996 The American Physical Society.

Phase-integral method : allowing nearlying transition points

Abstract: 1 Phase-Integral Approximation of Arbitrary Order Generated from an Unspecified Base Function.- 1.1 Introduction.- 1.2 The So-Called WKB Approximation, Its Deficiencies in Higher Order, and Early Attempts to Remedy These Deficiencies.- 1.2.1 Derivation of the WKB Approximation.- 1.2.2 Deficiencies of the WKB Approximation in Higher Order.- 1.2.3 Phase-Integral Approximation of Arbitrary Order, Freed from the First Deficiency.- 1.3 Phase-Integral Approximation of Arbitrary Order, Generated from an Unspecified Base Function.- 1.3.1 Direct Procedure.- 1.3.2 Transformation Procedure.- 1.4 Advantage of Phase-Integral Approximation Versus WKB Approximation in Higher Order.- 1.5 Relations Between Solutions of the Schroedinger Equation and the q-Equation.- 1.5.1 Solutions of the Schroedinger Equation and Solutions of the q-Equation Expressed in Terms of Each Other.- 1.5.2 Ermakov-Lewis Invariant.- 1.6 Phase-Integral Method.- Appendix: Phase-Amplitude Relation.- References.- 2 Technique of the Comparison Equation Adapted to the Phase-Integral Method.- 2.1 Background.- 2.2 Comparison Equation Technique.- 2.2.1 Differential Equation for ?0.- 2.2.2 Determination of the Coefficients An,0 and Bq.- 2.2.3 Differential Equation for ?2N When N > 0.- 2.2.4 Regularity Properties of I2N and ?2N When N > 0.- 2.2.5 Determination of the Coefficients An,2N When N > 0.- 2.2.6 Expressions for ?2 and ?4.- 2.2.7 Behavior of ?2N(z) in the Neighborhood of a First-or Second-Order Pole of Q2(z) When N > 0.- 2.3 Derivation of the Arbitrary-Order Phase-Integral Approximation from the Comparison Equation Solution.- 2.4 Summary of the Procedure and the Results.- References.- Adjoined Papers.- 3 Problem Involving One Transition Zero.- 3.1 Introduction.- 3.2 Comparison Equation Solution.- 3.3 Phase-Integral Approximation Obtained from the Comparison Equation Solution.- References.- 4 Relations Between Different Nonoscillating Solutions of the q-Equation Close to a Transition Zero.- 4.1 Introduction.- 4.2 Comparison Equation Solutions.- 4.3 Comparison Equation Expressions for Nonoscillating Solutions of the q-Equation.- 4.3.1 The Case When Re ? Increases as z Moves Away from t in the Neighborhood of the Anti-Stokes Line A.- 4.3.2 The Case When Re ? Decreases as z Moves Away from t in the Neighborhood of the Anti-Stokes Line A.- 4.3.3 Summary of the Results for the Two Cases in Sections 4.3.1 and 4.3.2.- 4.3.4 Application Illustrating the Consistency of the Formulas Obtained.- 4.4 Simple First-Order Formulas.- 4.5 Relations Between the a-Coefficients Associated with Different q-Functions, in Terms of Which a Given Solution ?(z) is Expressed.- 4.6 Condition for Determination of Regge Pole Positions.- References.- 5 Cluster of Two Simple Transitions Zeros.- 5.1 Introduction.- 5.2 Wave Equation and Phase-Integral Approximation.- 5.3 Comparison Equation.- 5.4 Comparison Equation Solution.- 5.4.1 Determination of ?0(z) and $${\overline K _0}$$.- 5.4.2 Determination of ?2? and $${\overline K _{2\beta }}$$ for ? > 0.- 5.5 Phase-Integral Solution Obtained from the Comparison Equation Solution.- 5.6 Stokes Constants.- 5.7 Application to Complex Potential Barrier.- 5.8 Application to Regge Pole Theory.- Appendix: Phase-Integral Solution Obtained from the Comparison Equation Solution by Straightforward Calculation.- References.- 6 Phase-Integral Formulas for the Regular Wave Function When There Are Turning Points Close to a Pole of the Potential.- 6.1 Introduction.- 6.2 Definitions and Preparatory Calculations.- 6.2.1 Determination of ?0 and A1,0.- 6.2.2 Determination of ?2? and A1,2? for ? > 0.- 6.3 Comparison Equation Corresponding to Scattering States.- 6.3.1 Comparison Equation Solution.- 6.3.2 Phase-Integral Approximation Obtained from the Comparison Equation Solution.- 6.3.3 Behavior of the Wave Function Close to the Origin.- 6.3.4 Summary of Formulas in Section 6.3.- 6.4 Comparison Equation Corresponding to Bound States.- 6.4.1 Quantization Condition.- 6.4.2 Normalized Wave Function.- Appendix: Calculation of q(z) and ?(2n+1).- References.- 7 Normalized Wave Function of the Radial Schroedinger Equation Close to the Origin.- 7.1 Introduction.- 7.2 ?0 > 0.- 7.3 ?0 = 0, ?0 ? 0.- 7.4 Summary of the Results Obtained in the Present Chapter and Discussion of Results Obtained by Previous Authors.- References.- 8 Phase-Amplitude Method Combined with Comparison Equation Technique Applied to an Important Special Problem.- 8.1 Introduction.- 8.2 Quantization Condition.- 8.3 Solution of the Difficulty at the Origin by Means of Comparison Equation Solutions Expressed in Terms of Coulomb Wave Functions.- 8.4 Application to a Two-Dimensional Anharmonic Oscillator.- References.- 9 Improved Phase-Integral Treatment of the Combined Linear and Coulomb Potential.- 9.1 Introduction.- 9.2 Energy Levels.- 9.3 Expectation Values.- Appendix: Expressions for Phase-Integral Quantities in Terms of Complete Elliptic Integrals.- References.- 10 High-Energy Scattering from a Yukawa Potential.- 10.1 Introduction.- 10.2 Phase Shifts.- 10.3 Probability Density at the Origin.- Appendix: Numerical Solution of the Schroedinger Equation.- References.- 11 Probabilities for Transitions Between Bound States in a Yukawa Potential, Calculated with Comparison Equation Technique.- 11.1 Introduction.- 11.2 Phase-Integral Formulas.- 11.3 Comparison Equation Formulas.- References.- Author Index.

Generalized Fourier-Mukai transforms

Abstract: The paper sets out a generalized framework for Fourier-Mukai transforms and illustrates their use via vector bundle transforms. A Fourier-Mukai transform is, roughly, an isomorphism of derived categories of (sheaves) on smooth varieties X and Y. We show that these can only exist if the first Chern class of the varieties vanishes and, in the case of vector bundle transforms, will exist if and only if there is a bi-universal bundle on XxY which is "strongly simple" in a suitable sense. Some applications are given to abelian varieties extending the work of Mukai.

Plane curves with a big fundamental group of the complement

Abstract: Let $C \s \pr^2$ be an irreducible plane curve whose dual $C^* \s \pr^{2*}$ is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincar\'e group $\pi_1(\pr^2 \se C)$ contains a free group with two generators. If the geometric genus $g$ of $C$ is at least 2, then a subgroup of $G$ can be mapped epimorphically onto the fundamental group of the normalization of $C$, and the result follows. To handle the cases $g=0,1$, we construct universal families of immersed plane curves and their Picard bundles. This allows us to reduce the consideration to the case of Pl\"ucker curves. Such a curve $C$ can be regarded as a plane section of the corresponding discriminant hypersurface (cf. [Zar, DoLib]). Applying Zariski--Lefschetz type arguments we deduce the result from `the bigness' of the $d$-th braid group $B_{d,g}$ of the Riemann surface of $C$.

On the isomorphisms and automorphism groups of circulants

Abstract: Denote by C n (S) the circulant graph (or digraph). Let M be a minimal generating element subset of Z n , the cyclic group of integers modulo n, and $$\tilde M = \left\{ {\left. {m, - m} \right|m \in M} \right\}$$ In this paper, we discuss the problems about the automorphism group and isomorphisms of C n (S). When $$M \subseteq S \subseteq \tilde M$$ , we determine the automorphism group of C n (S) and prove that for any T ⊆ Z n , C n (S) ? C n (T) if and only if T = ?S, where ? is an integer relatively prime to n. The automorphism groups and isomorphisms of some other types of circulant graphs (or digraphs) are also considered. In the last section of this paper, we give a relation between the isomorphisms and the automorphism groups of circulants.

Summability of the Solutions of Nonlinear Elliptic Equations With Right Hand Side Measures

Abstract: , only the function \m!summability of u" is increasing .Other results concerning existence, uniqueness, regularity and generalizations can befound in the papers quoted in the references.2. Regularity resultsIn this section we present an improvement of the summability of the solution uof theDirichlet problem (1.6).Theorem 2.1. The solution uof (1.6) given by Theorem 1.1 is such that, for every >

Holomorphic bundles on O(-k) are algebraic

Abstract: We show that holomorphic bundles on O(-k) for k > 0 are algebraic. We also show holomorphic bundles on O(-1) are trivial outside the zero section.

Topology of energy surfaces and existence of transversal Poincare sections

Abstract: Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not

Deformation quantization of compact Kähler manifolds via Berezin-Toeplitz operators

Abstract: This talk reports on results on the deformation quantization (star products) and on approximative operator representations for quantizable compact Kahler manifolds obtained via Berezin-Toeplitz operators. After choosing a holomorphic quantum line bundle the Berezin-Toeplitz operator associated to a differentiable function on the manifold is the operator defined by multiplying global holomorphic sections of the line bundle with this function and projecting the differentiable section back to the subspace of holomorphic sections. The results were obtained in (respectively based on) joint work with M. Bordemann and E. Meinrenken.

Topology of energy surfaces and existence of transversal Poincar\'e sections

Abstract: Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not exist a transverse and complete Poincar\'e section. We show that there are topological obstacles for its existence such that only in the cases of $S^1\times S^2$ and $T^3$ such a Poincar\'e section can exist.

Massively parallel simulations of motions in a Gaussian velocity field

Abstract: The purpose of the present note is to describe the details of a set of simulation tools which we developed in order to study the statistical properties of the solutions of the equation: $$ d{{X}_{t}} = \vec{v}(t,{{X}_{t}})dt $$ when \( \{ \vec{v}(t,x);t0,x \in {{\mathbb{R}}^{2}}\} \) is a stationary and homogeneous Gaussian field with a spectrum of Kolmogorov type. The study is motivated by problems of transport of passive tracer particles at the surface of a two dimensional medium. We are mostly concerned with mathematical modeling of problems from oceanography and we think of the surface of the ocean as a physical medium to which our modeling efforts could apply. For this reason we shall sometimes use the terminology drifters for the passive tracers. The programs have been written for the MASPAR II. We describe the different forms of the simulations and we give numerical results which illustrate the transport properties of such a random medium. These results lead to the formulation of several precise mathematical conjectures which we discuss in the last section.

A remark on the jet bundles over the projective line

Abstract: This is a footnote of a recent interesting work of Cohen, Manin and Zagier, where they, among other things, produce a natural isomorphism between the sheaf of (n-1)-th order jets of the n-th tensor power of the tangent bundle of a Riemann surface equipped with a projective structure and the sheaf of differential operators of order n (on the trivial bundle) with vanishing 0-th order part. We give a different proof of this result without using the coordinates, and following the idea of this proof we prove: Take a line bundle L with $L^2 = T$ on a Riemann surface equipped with a projective structure. Then the jet bundle $J^n(L^n)$ has a natural flat connection with $J^n(L^n) = S^n(J^1(L))$. For any $m >n$ the obvious surjection $J^m(L^n) \rightarrow J^n(L^n)$ has a canonical splitting. In particular, taking $m = n+1$, one gets a natural differential operator of order $n+1$ from $L^n$ to $L^{-n-2}$.

On 2-distributions in 8-dimensional vector bundles over 8-complexes

Abstract: It is shown that Z2-index of a 2-distribution in an 8-dimensional spin vector bundle over an 8-complex is independent of the 2-distribution. Necessary and sufficient conditions for the existence of 2-distributions in such vector bundles are given in terms of characteristic classes and a certain secondary cohomology operation. In some cases this operation is computed.

Remarks on the complementability of spaces of Bochner integrable functions in spaces of vector measures

Abstract: In the paper [5] L. Drewnowski and the author proved that if $X$ is a Banach space containing a copy of $c_0$ then $L_1({\mu },X)$ is {\it not} complemented in $cabv({\mu },X)$ and conjectured that the same result is true if $X$ is any Banach space without the Radon-Nikodym property. Recently, F. Freniche and L. Rodriguez-Piazza ([7]) disproved this conjecture, by showing that if $\mu$ is a finite measure and $X$ is a Banach lattice not containing copies of $c_0$, then $L_1({\mu },X)$ is complemented in $cabv({\mu },X)$. Here, we show that the complementability of $L_1({\mu },X)$ in $cabv({\mu },X)$ together with that one of $X$ in the bidual $X^{\ast\ast}$ is equivalent to the complementability of $L_1({\mu },X)$ in its bidual, so obtaining that for certain families of Banach spaces not containing $c_0$ complementability occurs (Section 2), thanks to the existence of general results stating that a space in one of those families is complemented in the bidual. We shall also prove that certain quotient spaces inherit that property (Section 3).

Objects with dense diagonals

Abstract: The purpose of this note is to show that, in a finitely complete category X with a proper ( ɛ, M)-factorization system for morphisms and a closure operator c w.r.t. the class \(M\, \subseteq \,mono\,(X)\) in the sense of [DG], the full subcategory Δ( c) of those objects X ∈ X for which the diagonal δx: $${\delta _{\text{X}}}{\text{ : X }} \to {\text{ X}}$$ is c-dense, satisfies all the stability properties that one expects a category of “connected” objects to have. In fact, subject to suitable conditions on the given data, we show that ∇(c) is closed in X under ɛ -images, c-dense extensions, direct products, and under chained sinks. The first three closure properties appear essentially in [DT], Section 7.8, but not the crucial fourth property, which exhibits ∇(c) as a component subcategory in the sense of [Ti]; see also [T] and [C].

Origination of Self-Complementary Planar Structures and Discovery of Their Constant-Impedance Property

Abstract: According to the result obtained in Chapter 3, section 31, the input impedances, Z1and Z2, for mutually complementary planar structures are given by relationship (33), that is \(Z_{1}Z_{2}= (Z_{0}/2)^{2}\) where Z0 is the intrinsic impedance of the medium, which is approximately equal to 120 π[Ω] in free space This relationship was derived by the author in 1948 [11–13] As mentioned in section 31, however, the same expression had already been obtained by several other investigators, after making various assumptions, as the relationship between the input impedances for a slot antenna and its complementary wire antenna [31–34] However, expression (41) in the present theory is derived without any assumptions being made, and it is always exact for any shape of mutually complementary structure Therefore, expression (41) is an innovative and generalized relationship for a pair of arbitrarily shaped complementary planar structures Nevertheless, the originality or novelty of the author’s theory was not appreciated when it first appeared

Theta line bundles and the determinant of the Hodge bundle

Abstract: In this note we examine the question of expressing the determinant of the push forward of a symmetric line bundle on an abelian fibration in terms of the pull back of the relative dualizing sheaf via the zero section.

Great sphere foliations and manifolds with curvature bounded above

Abstract: The survey is devoted to Toponogov's conjecture, that {\it if a complete simply connected Riemannian manifold with sectional curvature $\le 4$ and injectivity radius $\ge \pi/2$ has extremal diameter $\pi/2$, then it is isometric to CROSS}. In Section 1 the relations of problem with geodesic foliations of a round sphere are considered, but the proof of conjecture on this way is not complete. In Section 2 the proof based on recent results and methods for topology and volume of Blaschke manifolds is given.

Back reaction on the topological degrees of freedom in (2+1)-dimensional spacetime.

Abstract: We investigate the back-reaction effect of the quantum field on the topological degrees of freedom in a (2+1)-dimensional toroidal universe, scrM\ensuremath{\simeq}${\mathit{T}}^{2}$\ifmmode\times\else\texttimes\fi{}R. Constructing a homogeneous model of the toroidal universe, we examine explicitly the back-reaction effect of the Casimir energy of a massless, conformally coupled scalar field, with a conformal vacuum. The back reaction causes an instability of the universe: The torus becomes thinner and thinner as it evolves, while its total two-volume (area) becomes smaller and smaller. The back reaction caused by the Casimir energy can be compared with the influence of the negative cosmological constant: Both of them make the system unstable and the torus becomes thinner and thinner in shape. On the other hand, the Casimir energy is a complicated function of the Teichm\"uller parameters (${\mathrm{\ensuremath{\tau}}}^{1}$,${\mathrm{\ensuremath{\tau}}}^{2}$) causing highly nontrivial dynamical evolutions, while the cosmological constant is simply a constant. Since the spatial section is a two-torus, we shall write down the partition function of this system, fixing the path-integral measure for gravity modes, with the help of the techniques developed in string theories. We show explicitly that the partition function expressed in terms of the canonical variables corresponding to the (redundantly large) original phase space is reduced to the partition function defined in terms of the physical-phase-space variables with a standard Liouville measure. This result is compatible with the general theory of the path integral for the first-class constrained systems. \textcopyright{} 1996 The American Physical Society.