Radial Deformations and Cavitation in Riemannian Manifolds with Applications to Membrane Shells

This study is a geometric version of Ball's work, Philos. Trans. Roy. Soc. London Ser. A 306 (1982), no. 1496, 557-611. Radial deformations in Riemannian manifolds are singular solutions to some nonlinear equations given by constitutive functions and radial curvatures. A geodesic spherical cavity forms at the center of a geodesic ball in tension by means of given surface tractions or displacements. The existence of such solutions depends on the growth properties of the constitutive functions and the radial curvatures. Some close relationships are shown among radial curvature, the constitutive functions, and the behavior of bifurcation of a singular solution from a trivial solution. In the incompressible case the bifurcation depends on the local properties of the radial curvature near the geodesic ball center but the bifurcation in compressible case is determined by the global properties of the radial curvatures. A cavity forms at the center of a membrane shell of isotropic material placed in tension by means of given boundary tractions or displacements when the Riemannian manifold under question is a surface of $\R^3$ with the induced metric. In addition, cavitation at the center of ellipsoids of $\R^n$ is also described if the Riemannian manifold under question is $(\R^n g)$ where $g(x)$ are symmetric, positive matrices for $x\in\R^n.$Comment: 61 page

Space of Infinitesimal Isometries and Bending of Shells

We discuss infinitesimal isometries of the middle surfaces and present some characteristic conditions for a function to be the normal component of an infinitesimal isometry. Our results show that those characteristic conditions depend on the Gaussian curvature of the middle surfaces: Normal components of infinitesimal isometries satisfy an elliptic problem, or a parabolic one, or a hyperbolic one according to the middle surface being elliptic, or parabolic, or hyperbolic, respectively. In those cases, a problem of determining an infinitesimal isometry is changed into that of 1-dimension. Then we apply those results to the energy functionals of bending of shells which has been obtained as two-dimensional problems by the limit theory of Gamma-convergence from the three-dimensional nonlinear elasticity. Therefore the limit theory of Gamma-convergence reduces to be a one-dimensional problem in the those cases.Comment: 51 page

Linear Strain Tensors on Hyperbolic Surfaces and Asymptotic Theories for Thin Shells

We perform a detailed analysis of the solvability of linear strain equations on hyperbolic surfaces. We prove that if the surface is a smooth noncharacteristic region, any first order infinitesimal isometry can be matched to an infinitesimal isometry of an arbitrarily high order. The implications of this result for the elasticity of thin hyperbolic shells are discussed

Boundary controllability for the quasilinear wave equation

We study the boundary exact controllability for the quasilinear wave equation in the higher-dimensional case. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another location under some geometrical condition. The Dirichlet action and the Neumann action are studied, respectively. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the quasilinear wave equation. A criterion of exact controllability is given, which based on the sectional curvature of the Riemann metric. Some examples are presented to verify the global exact controllability.Comment: 47page

Uniform Sobolev Resolvent Estimates for the Laplace-Beltrami Operator on Compact Manifolds

In this paper we continue the study on the resolvent estimates of the Laplace-Beltrami operator $\Delta_g$ on a compact manifolds $M$ with dimension $n\geq3$. On the Sobolev line $1/p-1/q=2/n$ we can prove that the resolvent $(\Delta_g+\zeta)^{-1}$ is uniformly bounded from $L^p$ to $L^q$ when $(p,q)$ are within the admissible range $p\leq2(n+1)/(n+3)$ and $q\geq2(n+1)/(n-1)$ and $\zeta$ is outside a parabola opening to the right and a small disk centered at the origin. This naturally generalizes the previous results in \cite{Kenig} and \cite{bssy} which addressed only the special case when $p=2n/(n+2), q=2n/(n-2)$. Using the shrinking spectral estimates between $L^p$ and $L^q$ we also show that when $(p,q)$ are within the interior of the admissible range, one can obtain a logarithmic improvement over the parabolic region for resolvent estimates on manifolds equipped with Riemannian metric of non-positive sectional curvature, and a power improvement depending on the exponent $(p,q)$ for flat torus. The latter therefore partially improves Shen's work in \cite{Shen} on the $L^p\to L^2$ uniform resolvent estimates on the torus. Similar to the case as proved in \cite{bssy} when $(p,q)=(2n/(n+2),2n/(n-2))$, the parabolic region is also optimal over the round sphere $S^n$ when $(p,q)$ are now in the admissible range. However, we may ask if the admissible range is sharp in the sense that it is the only possible range on the Sobolev line for which a compact manifold can have uniform resolvent estimate for $\zeta$ being ouside a parabola.Comment: A few details revise

Color kinematic symmetric (BCJ) numerators in a light-like gauge

Color-ordered tree level scattering amplitudes in Yang-Mills theories can be written as a sum over terms which display the various propagator poles of Feynman diagrams. The numerators in these expressions which are obtained by straightforward application of Feynman rules are not satisfying any particular relations, typically. However, by reshuffling terms, it is known that one can arrive at a set of numerators which satisfy the same Jacobi identity as the corresponding color factors. By extending previous work by us we show how this can be systematically accomplished within a Lagrangian framework. We construct an effective Lagrangian which yields tree-level color-kinematic symmetric numerators in Yang-Mills theories in a light-like gauge at five-points. The five-point effective Lagrangian is non-local and it is zero by Jacobi identity. The numerators obtained from it respect the original pole structure of the color-ordered amplitude. We discuss how this procedure can be systematically extended to higher order.Comment: 36 page

QCD Corrections to the Charged-Higgs-Boson Decay of a Heavy Top Quark

It is shown that up to an over all scale the lowest-order QCD corrections to $t\to H^+b$ and to $t\to W^+b$ are the same in the heavy top limit. Asymptotically, they are given by $-{4\alpha_s\over 3\pi}[{\pi^2\over 3}-{5\over 4}]$, resulting in a reduction in the decay rate by about $9\%$, rather than $6\%$ reported previously in the literature. This is verified explicitly by an analytic calculation. The application of the equivalence theorem to this process is also discussed.Comment: 12 pages, UPR-0508T, UM-TH-92-1

Constraints and Generalized Gauge Transformations on Tree-Level Gluon and Graviton Amplitudes

Writing the fully color dressed and graviton amplitudes, respectively, as ${\bf A}= =$ and ${\bf A}_{gr}=$, where $|A>$ is a set of Kleiss-Kuijf color-ordered basis, $|N>,$|\tilde N> $and$|C>$are the similarly ordered numerators and color coefficients, we show that the propagator matrix$M$has$(n-3)(n-3)!$independent eigenvectors$|\lambda ^0_j>$with zero eigenvalue, for$n$-particle processes. The resulting equations$ = 0$are relations among the color ordered amplitudes. The freedom to shift$|N> \to |N> +\sum_j f_j|\lambda ^0_j>$and similarly for$|\tilde N>$, where$f_j$are$(n-3)(n-3)!$arbitrary functions, encodes generalized gauge transformations. They yield both BCJ amplitude and KLT relations, when such freedom is accounted for. Furthermore,$f_j$ can be promoted to the role of effective Lagrangian vertices in the field operator space.Comment: 22 pages, JHEP version, Appendix A expanded, one typo fixe

Deep Zero-Shot Learning for Scene Sketch

We introduce a novel problem of scene sketch zero-shot learning (SSZSL), which is a challenging task, since (i) different from photo, the gap between common semantic domain (e.g., word vector) and sketch is too huge to exploit common semantic knowledge as the bridge for knowledge transfer, and (ii) compared with single-object sketch, more expressive feature representation for scene sketch is required to accommodate its high-level of abstraction and complexity. To overcome these challenges, we propose a deep embedding model for scene sketch zero-shot learning. In particular, we propose the augmented semantic vector to conduct domain alignment by fusing multi-modal semantic knowledge (e.g., cartoon image, natural image, text description), and adopt attention-based network for scene sketch feature learning. Moreover, we propose a novel distance metric to improve the similarity measure during testing. Extensive experiments and ablation studies demonstrate the benefit of our sketch-specific design.Comment: 5 pages, 3 figures, IEEE International Conference on Image Processing (ICIP

Evaluation of the CHY Gauge Amplitude

The Cachazo-He-Yuan (CHY) formula for $n$-gluon scattering is known to give the same amplitude as the one obtained from Feynman diagrams, though the former contains neither vertices nor propagators explicitly. The equivalence was shown by indirect means, not by a direct evaluation of the $(n\! - \!3)$-dimensional integral in the CHY formula. The purpose of this paper is to discuss how such a direct evaluation can be carried out. There are two basic difficulties in the calculation: how to handle the large number of terms in the reduced Pfaffian, and how to carry out the integrations in the presence of a $\sigma$-dependence much more complicated than the Parke-Taylor form found in a CHY double-color scalar amplitude. We have solved both of these problems, and have formulated a method that can be applied to any $n$. Many examples are provided to illustrate these calculations.Comment: Version to appear in Physical Review