Showing papers on "Section (fiber bundle) published in 2012"

Test configurations and Okounkov bodies

Abstract: We associate to a test configuration for a polarized variety a filtration of the section ring of the line bundle. Using the recent work of Boucksom and Chen we get a concave function on the Okounkov body whose law with respect to Lebesgue measure determines the asymptotic distribution of the weights of the test configuration. We show that this is a generalization of a well-known result in toric geometry. As an application, we prove that the pushforward of the Lebesgue measure on the Okounkov body is equal to a Duistermaat-Heckman measure of a certain deformation of the manifold. Via the Duisteraat-Heckman formula, we get as a corollary that in the special case of an effective C-x-action on the manifold lifting to the line bundle, the pushforward of the Lebesgue measure on the Okounkov body is piecewise polynomial.

Quantization of conic Lagrangian submanifolds of cotangent bundles

Abstract: Let $M$ be a manifold and $\Lambda$ a compact exact connected Lagrangian submanifold of $T^*M$. We can associate with $\Lambda$ a conic Lagrangian submanifold $\Lambda'$ of $T^*(M\times R)$. We prove that there exists a canonical sheaf $F$ on $M\times R$ whose microsupport is $\Lambda'$ outside the zero section. We deduce the already known results that the Maslov class of $\Lambda$ is $0$ and that the projection from $\Lambda$ to $M$ induces isomorphisms between the homotopy groups.

Optical Frequency Comb Generation Using Dual-Mode Injection-Locking of Quantum-Dash Mode-Locked Lasers: Properties and Applications

Abstract: In this paper, we describe generation and application of wide narrow linewidth optical frequency combs using dual-mode injection-locking of InP quantum-dash mode-locked lasers. First, the dependence of the RF locking-range on the device's absorber voltage is experimentally investigated. Under optimized absorber voltage, a continuous wide RF locking-range of ${\approx}{\rm 400}~{\rm MHz}$ is achievable for lasers with 21 GHz repetition rate. The total RF locking-range of ${\approx}{\rm 440}~{\rm MHz}$ is possible considering locking-range for positive and negative absorber voltages. This wide tuning ${>}2{\%}$ of the repetition rate, a record for a monolithic mode-locked laser, is reported from a two-section device without any additional passive section or extended-cavity for repetition rate tuning. It is shown that the effective RF locking-range in dual-mode injection corresponds to the optical locking-range and repetition rate tuning under CW injection, which is wider when the free-running mode-locking operation is “less stable.” The widest comb consists of 35 narrow lines within 10 dB of the peak, spanning ${\approx}{\rm 0.7}~{\rm THz}$ and generating 3.7 ps pulses. Second, we show the first demonstration of multi pump phase-synchronization of two 10 Gb/s DPSK channels in a phase-sensitive amplifier using dual-mode injection-locking technique. The phase-sensitive amplifier based on the “black box” scheme shows more than 7 dB phase-sensitive gain and error free performance for both input channels with 1 dB penalty.

Polar manifolds and actions

Abstract: A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. We show how to construct a manifold admitting a polar group action by prescribing its isotropy groups along a fundamental domain in the section. This generalizes a classical construction for cohomogeneity-one manifolds.We give many examples showing the richness of this class of group actions and relate the topology of the section to the topology of the manifold.

Vector Bundles on Degenerations of Elliptic Curves and Yang-Baxter Equations

Abstract: In this paper the authors introduce the notion of a geometric associative $r$-matrix attached to a genus one fibration with a section and irreducible fibres. It allows them to study degenerations of solutions of the classical Yang-Baxter equation using the approach of Polishchuk. They also calculate certain solutions of the classical, quantum and associative Yang-Baxter equations obtained from moduli spaces of (semi-)stable vector bundles on Weierstrass cubic curves.

The quantum Lefschetz hyperplane principle can fail for positive orbifold hypersurfaces

Abstract: We show that the Quantum Lefschetz Hyperplane Principle can fail for certain orbifold hypersurfaces and complete intersections. It can fail even for orbifold hypersurfaces defined by a section of an ample line bundle.

Large Deviation Principles for Random Walk Trajectories. I

Abstract: This paper deals with a random walk $S_n:=\xi_1+\cdots+\xi_n$, $n=0,1,\ldots,$ in the $d$-dimensional Euclidean space ${\mathbb R}^d$, where $S_0=0$ and $\xi_k$ are independent identically distributed random vectors satisfying Cramer's moment conditions. For random polygons with nodes at the points $(\frac{k}{n},\frac{1}{x}S_k)$, $k=0,1,\ldots,n,$ we obtain the logarithmic asymptotics of the large deviation probabilities in different trajectory spaces when $x\sim \alpha_0 n$, $\alpha_0>0$, as $n\to\infty.$ The results include the so-called local and extended large deviation principles (l.d.p.'s) (see ite15) that hold in those cases where the “usual” l.d.p. does not apply. The paper consists of three parts. Part I has two sections. Section 1 presents the key concepts and some facts concerning the l.d.p. in arbitrary metric spaces. In section 2 we formulate the “strong” versions of the “usual” l.d.p. in the large deviation zones that were obtained earlier in [A. A. Borovkov, Theory Probab. Appl., 12 (1967)...

Rational Points and Arithmetic of Fundamental Groups: Evidence for the Section Conjecture

Abstract: Part I Foundations of Sections.- 1 Continuous Non-abelian H1 with Profinite Coefficients.-2 The Fundamental Groupoid.- 3 Basic Geometric Operations in Terms of Sections.- 4 The Space of Sections as a Topological Space.- 5 Evaluation of Units.- 6 Cycle Classes in Anabelian Geometry.- 7 Injectivity in the Section Conjecture.- Part II Basic Arithmetic of Sections.- 7 Injectivity in the Section Conjecture.- 8 Reduction of Sections.- 9 The Space of Sections in the Arithmetic Case and the Section Conjecture in Covers.- Part III On the Passage from Local to Global.- 10 Local Obstructions at a p-adic Place.- 11 Brauer-Manin and Descent Obstructions.- 12 Fragments of Non-abelian Tate-Poitou Duality.- Part IV Analogues of the Section Conjecture.- 13 On the Section Conjecture for Torsors.- 14 Nilpotent Sections.- 15 Sections over Finite Fields.- 16 On the Section Conjecture over Local Fields.- 17 Fields of Cohomological Dimension 1.- 18 Cuspidal Sections and Birational Analogues.

The Quantum Lefschetz Hyperplane Principle Can Fail for Positive Orbifold Hypersurfaces

Abstract: We show that the Quantum Lefschetz Hyperplane Principle can fail for certain orbifold hypersurfaces and complete intersections. It can fail even for orbifold hypersurfaces defined by a section of an ample line bundle.

F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds

Abstract: The Mordell-Weil group of an elliptically fibered Calabi-Yau threefold X contains information about the abelian sector of the six-dimensional theory obtained by compactifying F-theory on X. After examining features of the abelian anomaly coefficient matrix and U(1) charge quantization conditions of general F-theory vacua, we study Calabi-Yau threefolds with Mordell-Weil rank-one as a first step towards understanding the features of the Mordell-Weil group of threefolds in more detail. In particular, we generate an interesting class of F-theory models with U(1) gauge symmetry that have matter with both charges 1 and 2. The anomaly equations --- which relate the Neron-Tate height of a section to intersection numbers between the section and fibral rational curves of the manifold --- serve as an important tool in our analysis.

Cubic fourfolds containing a plane and a quintic del Pezzo surface

Abstract: We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C_8 and C_14 has five irreducible components. In the component corresponding to the existence of a tangent conic to the sextic degeneration curve of the quadric bundle, we prove that the general member is both pfaffian and has nontrivial Brauer class. Such cubic fourfolds also provide twisted derived equivalences between K3 surfaces of degree 2 and 14, hence further corroboration of Kuznetsov's derived categorical conjecture on the rationality of cubic fourfolds.

Supercongruences for a truncated hypergeometric series

Abstract: The purpose of this note is to obtain some congruences modulo a power of a prime $p$ involving the truncated hypergeometric series $$\sum_{k=1}^{p-1} {(x)_k(1-x)_k\over (1)_k^2}\cdot{1\over k^a}$$ for $a=1$ and $a=2$. In the last section, the special case $x=1/2$ is considered.

Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes

Abstract: In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes Nα(t), Nβ(t), t>0, we have that \(N_{\alpha}(N_{\beta}(t)) \stackrel{\mathrm{d}}{=} \sum_{j=1}^{N_{\beta}(t)} X_{j}\), where the Xjs are Poisson random variables. We present a series of similar cases, where the outer process is Poisson with different inner processes. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns compositions of the form \(N_{\alpha}(\tau_{k}^{ u})\), ν∈(0,1], where \(\tau_{k}^{ u}\) is the inverse of the fractional Poisson process, and we show how these compositions can be represented as random sums. Furthermore we study compositions of the form Θ(N(t)), t>0, which can be represented as random products. The last section is devoted to studying continued fractions of Cauchy random variables with a Poisson number of levels. We evaluate the exact distribution and derive the scale parameter in terms of ratios of Fibonacci numbers.

Fast finite-energy planes in symplectizations and applications

Abstract: We define the notion of fast finite-energy planes in the symplectization of a closed 3-dimensional energy level $M$ of contact type. We use them to construct special open book decompositions of $M$ when the contact structure is tight and induced by a (non-degenerate) dynamically convex contact form. The obtained open books have disk-like pages that are global surfaces of section for the Hamiltonian dynamics. Let $S \subset \R^4$ be the boundary of a smooth, strictly convex, non-degenerate and bounded domain. We show that a necessary and sufficient condition for a closed Hamiltonian orbit $P\subset S$ to be the boundary of a disk-like global surface of section for the Hamiltonian dynamics is that $P$ is unknotted and has self-linking number -1.

Operator and commutator moduli of continuity for normal operators

Abstract: We study in this paper properties of functions of perturbed normal operators and develop earlier results obtained in \cite{APPS2}. We study operator Lipschitz and commutator Lipschitz functions on closed subsets of the plane. For such functions we introduce the notions of the operator modulus of continuity and of various commutator moduli of continuity. Our estimates lead to estimates of the norms of quasicommutators $f(N_1)R-Rf(N_2)$ in terms of $\|N_1R- RN_2\|$, where $N_1$ and $N_2$ are normal operator and $R$ is a bounded linear operator. In particular, we show that if $0<\a<1$ and $f$ is a Holder function of order $\a$, then for normal operators $N_1$ and $N_2$, $$ \|f(N_1)R-Rf(N_2)\|\le\const(1-\a)^{-2}\|f\|_{Ł_\a}\|N_1R-RN_2\|^\a\|R\|^{1-\a}. $$ In the last section we obtain lower estimates for constants in operator Holder estimates.

Ricci curvature and $L^p$-convergence

Abstract: A main goal in this talk is to introduce the notion of the L-convergence of tensor fields with respect to the Gromov-Hausdorff topology given in [1]. The precise statement is as follows: Let n be a positive integer, K a real number and let (M∞,m∞, υ) be the GromovHausdorff limit metric measure space of a sequence of renormalized pointed complete ndimensional Riemannian manifolds {(Mi,mi, vol)}i∈N with RicMi ≥ K(n− 1) and M∞ = {m∞}, where vol := vol/volB1(mi). Cheeger-Colding showed that the cotangent bundle π 1 : T M∞ → M∞ of M∞ exists in some sense. It is a fundamental property of the cotangent bundle that every Lipschitz function f on a Borel subset A of M∞ has the canonical section df(x) ∈ T ∗ xM∞ for a.e. x ∈ A. We also define the tangent bundle π 0 : TM∞ → M∞ of M∞ by the dual vector bundle of T M∞ and denote the dual section of df by ∇f : A → TM∞. For r, s ∈ Z≥0, let π s : T r sM∞ := ⊗r i=1 TM∞ ⊗ ⊗r+s i=r+1 T M∞ → M∞. For A ⊂ M∞, we put T r sA := (π r s) −1(A). We will denote by ⟨·, ·⟩ the canonical metric on T r sM∞ and by L(T r sA) the space of L -sections of T r sA over A. Let r, s ∈ Z≥0, R > 0, 1 < p < ∞ and Ti ∈ L(T r sBR(mi)) for every i ≤ ∞ with supi≤∞ ||Ti||Lp < ∞, where BR(mi) := {xi ∈ Mi;xi,mi < R} and xi,mi is the distance between xi and mi. We now consider the following question: Question: Can we define the following?

Krasnoselskii-type fixed-point theorems for weakly sequentially continuous mappings

Abstract: In this article, we establish some fixed-point results of Krasnoselskii type for the sum of two weakly sequentially continuous mappings that extend previous ones. In the last section, we apply such results to study the existence of solutions to a nonlinear integral equation modelled in a Banach space.

Kähler–Einstein metrics on strictly pseudoconvex domains

Abstract: Extending the results of Cheng and Yau it is shown that a strictly pseudoconvex domain \({M\subset X}\) in a complex manifold carries a complete Kahler–Einstein metric if and only if its canonical bundle is positive, i.e. admits an Hermitian connection with positive curvature. We consider the restricted case in which the CR structure on \({\partial M}\) is normal. In this case M must be a domain in a resolution of the Sasaki cone over \({\partial M}\) . We give a condition on a normal CR manifold which it cannot satisfy if it is a CR infinity of a Kahler–Einstein manifold. We are able to mostly determine those normal CR three-manifolds which can be CR infinities. We give many examples of Kahler–Einstein strictly pseudoconvex manifolds on bundles and resolutions. In particular, the tubular neighborhood of the zero section of every negative holomorphic vector bundle on a compact complex manifold whose total space satisfies c1 < 0 admits a complete Kahler–Einstein metric.

Extensions of Serrin’s Uniqueness and Regularity Conditions for the Navier–Stokes Equations

Abstract: Consider a smooth bounded domain \({\Omega \subseteq {\mathbb{R}}^3}\) , a time interval [0, T), 0 < T ≤ ∞, and a weak solution u of the Navier–Stokes system. Our aim is to develop several new sufficient conditions on u yielding uniqueness and/or regularity. Based on semigroup properties of the Stokes operator we obtain that the local left-hand Serrin condition for each \({t\in (0,T)}\) is sufficient for the regularity of u. Somehow optimal conditions are obtained in terms of Besov spaces. In particular we obtain such properties under the limiting Serrin condition \({u \in L_{\rm loc}^\infty([0,T);L^3(\Omega))}\). The complete regularity under this condition has been shown recently for bounded domains using some additional assumptions in particular on the pressure. Our result avoids such assumptions but yields global uniqueness and the right-hand regularity at each time when \({u \in L_{\rm loc}^\infty([0,T);L^3(\Omega))}\) or when \({u(t)\in L^3(\Omega)}\) pointwise and u satisfies the energy equality. In the last section we obtain uniqueness and right-hand regularity for completely general domains.

Large deviations of empirical measures of zeros on riemann surfaces

Abstract: This is a continuation of a project on large deviations for the empirical measures of zeros of random holomorphic sections of random line bundles over a Riemann surface X. In a previous article with O. Zeitouni (arXiv:0904.4271), we proved an LDP for random polynomials in the genus zero case. In higher genus, there is a Picard variety of line bundles and so the line bundle L is a random variable as well as the section s. The space of pairs (L, s) is known as the "vortex moduli space". The zeros of (L, s) fill out the configuration space $X^{(N)}$ of $N$ points of $X$. The LDP shows that the configurations concentrate at one equilibrium measure exponentially fast. The new features of the proof involve Abel-Jacobi theory, the prime form and bosonization.

The special linear version of the projective bundle theorem

Abstract: A special linear Grassmann variety SGr(k,n) is the complement to the zero section of the determinant of the tautological vector bundle over Gr(k,n). For a representable ring cohomology theory A(-) with a special linear orientation and invertible stable Hopf map \eta, including Witt groups and MSL[\eta^{-1}], we have A(SGr(2,2n+1))=A(pt)[e]/(e^{2n}), and A(SGr(2,2n)) is a truncated polynomial algebra in two variables over A(pt). A splitting principle for such theories is established. We use the computations for the special linear Grassmann varieties to calculate A(BSL_n) in terms of the homogeneous power series in certain characteristic classes of the tautological bundle.

Parameter Identification of Hysteretic Models using Partial Curve Mapping

Abstract: This paper describes a new method for curve matching essential to the solution of inverse problems represented by material parameter identification. Hysteretic response curves are specifically addressed as a general class. The method is based on Partial Curve Mapping (PCM) of the experimental and computed curves. This methodology involves a curve matching metric which is computed using the volume between the test curve and the computed curve section. A number of examples are presented to demonstrate the capability.

Electronic structure and fermiology of superconducting LaNiGa 2

Abstract: We report electronic structure calculations for the layered centrosymmetric superconductor LaNiGa${}_{2}$, which has been identified as having a possible triplet state based on evidence for time reversal symmetry breaking. The Fermi surface has several large sheets and is only moderately anisotropic, so that the material is best described as a three-dimensional metal. These include sections that are open in the in-plane direction as well as a section that approaches the zone center. The density of states is high and primarily derived from Ga $p$ states, which hybridize with Ni $d$ states. Comparing with experimental specific heat data, we infer a superconducting $\ensuremath{\lambda}\ensuremath{\le}0.55$, which implies that this is a weak to intermediate coupling material. However, the Ni occurs in a nominal ${d}^{10}$ configuration in this material, which places the compound far from magnetism. Implications of these results for superconductivity are discussed.

Generalized Harish-Chandra Modules

Abstract: This course is an introduction to algebraic methods in the infinite-dimensional representation theory of semisimple Lie algebras over the complex numbers. In the first section we present basic definitions and theorems concerning Harish-Chandra modules, Fernando–Kac subalgebras associated to \(\mathfrak{g}\)-modules, generalized Harish-Chandra modules, and the special case of weight modules. Work of Kostant allows us to demonstrate that not all simple \(\mathfrak{g}\)-modules are generalized Harish-Chandra modules. In the second section we discuss the Zuckerman derived functors and several of their important properties. We tailor this section to the theory of algebraic constructions of generalized Harish-Chandra modules. In the third section we summarize the main results in our joint work with Ivan Penkov on the classification of generalized Harish-Chandra modules having a “generic” minimal \(\mathfrak{k}\)-type. This classification makes extensive use of the Zuckerman derived functors in the context of pairs \((\mathfrak{g},\mathfrak{k})\) where \(\mathfrak{g}\) is a semisimple Lie algebra and \(\mathfrak{k}\) is a subalgebra of \(\mathfrak{g}\) which is reductive in \(\mathfrak{g}\). We also utilize the theory of the cohomology of the nilpotent radical of a parabolic subalgebra with coefficients in an infinite-dimensional \((\mathfrak{g},\mathfrak{k})\)-module. The crucial point of this section is that we do not assume that \(\mathfrak{k}\) is a symmetric subalgebra of \(\mathfrak{g}\).

GEOMETRY OF ${\mathcal{P}} R$-WARPED PRODUCTS IN PARA-KÄHLER MANIFOLDS

Abstract: In this paper, we initiate the study of ${\mathcal{P}} R$-warped products in para-K a hler manifolds and prove some fundamental results on such submanifolds. In particular, we establish a general optimal inequality for ${\mathcal{P}}R$-warped products in para-K a hler manifolds involving only the warping function and the second fundamental form. Moreover, we completely classify ${\mathcal{P}} R$-warped products in the flat para-K a hler manifold with least codimension which satisfy the equality case of the inequality. Our results provide an answer to the Open Problem (3) proposed in [19, Section 5].

Beam dynamics design of a new radio frequency quadrupole for beam-current upgrade of the Japan Proton Accelerator Research Complex linac

Abstract: We have performed the beam-dynamics design of an ${\mathrm{H}}^{\ensuremath{-}}$ radio frequency quadrupole (RFQ) for the beam-current upgrade of the Japan Proton Accelerator Research Complex linac (RFQ III). LINACSrfqDES was set up to support the conventional design method, i.e., design with CURLI-type shaper followed by constant bore radius and with constant intervane voltage, supplemented by keeping the equipartitioned condition in the gentle buncher section. For the particle simulation, LINACSrfqSIM was used. The obtained transmission, transverse and longitudinal emittance are 98.5%, $0.21\ensuremath{\pi}\text{ }\text{ }\mathrm{mm}\text{ }\mathrm{mrad}$, and $0.11\ensuremath{\pi}\text{ }\text{ }\mathrm{MeV}\text{ }\mathrm{deg}$ for the input beam current of 60 mA and normalized rms emittance of $0.20\ensuremath{\pi}\text{ }\text{ }\mathrm{mm}\text{ }\mathrm{mrad}$. This design satisfies the requirements of RFQ III.

Symmetric differentials and the fundamental group

Abstract: Esnault asked whether every smooth complex projective variety with infinite fundamental group has a nonzero symmetric differential (a section of a symmetric power of the cotangent bundle). In a sense, this would mean that every variety with infinite fundamental group has some nonpositive curvature. We show that the answer to Esnault's question is positive when the fundamental group has a finite-dimensional representation over some field with infinite image. This applies to all known varieties with infinite fundamental group. Along the way, we produce many symmetric differentials on the base of a variation of Hodge structures. One interest of these results is that symmetric differentials give information in the direction of Kobayashi hyperbolicity. For example, they limit how many rational curves the variety can contain.

Some Common Fixed Point Theorems in Menger Space

Abstract: This paper consists four sections. First section is central to the text. In second section, we generalize the results of Kohli and Vashistha [1] for pairs of mappings using weakly compatible maps. Third section deals the results for pair of weakly compatible maps along with property (E.A.) using different types of control functions, which generalize the results of Kohli and Vashistha [1] and Kubiaczyk and Sharma [2]. Fourth section is concerned with results for occasionally weakly compatible maps and generalizes, extends and unifies several well known comparable results in literature.

Presheaves of symmetric tensor categories and nets of C*-algebras

Abstract: Motivated by algebraic quantum field theory, we study presheaves of symmetric tensor categories defined over the base of a space, intended as a spacetime. Any section of a presheaf (that is, any "superselection sector", in the applications that we have in mind) defines a holonomy representation whose triviality is measured by Cheeger-Chern-Simons characteristic classes, and a non-abelian unitary cocycle defining a Lie group gerbe. We show that, given an embedding in a presheaf of full subcategories of the one of Hilbert spaces, the section category of a presheaf is a Tannaka-type dual of a locally constant group bundle (the "gauge group"), which may not exist and in general is not unique. This leads to the notion of gerbe of C*-algebras, defined on the given base.