The closures of test configurations and algebraic singularity types

Color Jones polynomial is one of the most important quantum invariants in knot theory. Finding the geometric information from the color Jones polynomial is an interesting topic. In this paper, we study the general expansion of color Jones polynomial which includes the volume conjecture expansion and the Melvin-Morton-Rozansky (MMR) expansion as two special cases. Following the recent works on SL(2,C) Chern-Simons theory, we present an algorithm to calculate the higher order terms in general asymptotic expansion of color Jones polynomial from the view of A-polynomial and noncommutative A-polynomial. Moreover, we conjecture that the MMR expansion corresponding to the abelian branch of A-polynomial. Lastly, we give some examples to illustrate how to calculate the higher order terms. These results support our conjecture.

A general method is presented for obtaining successive terms in short wavelength asymptotic expansions of the diffracted field produced by plane acoustic and electromagnetic waves incident on an arbitrary smooth convex surface. By introducing the geodesic coordinate system on arbitrary surfaces of non-constant curvature, both scalar and vector integral equations governing the surface fields are solved directly. The expressions for leading and second-order terms in the asymptotic expansion of the diffracted fields are obtained explicitly and the differences between acoustic and electromagnetic creeping waves are shown.

On the diminishing returns of higher-order terms in asymptotic expansions of bias

Starting with the data of a curve of singularity types, we use the Legendre transform to construct weak geodesic rays in the space of locally bounded metrics on an ample line bundle L over a compact manifold. Using this we associate weak geodesics to suitable filtrations of the algebra of sections of L. In particular this works for the natural filtration coming from an algebraic test configuration, and we show how this recovers the weak geodesic ray of Phong-Sturm.

Modelling physical problems in mathematical form yields the governing equations that may be linear or nonlinear for known and unknown boundaries. The exact solution for those equations may or may not be obtained easily. Hence we seek an analytical approximation solution in terms of asymptotic expansion. In this study, we focus on a singular perturbation in second order ordinary differential equations. Solutions to several perturbed ordinary differential equations are obtained in terms of asymptotic expansion. The aim of this work is to find an approximate analytical solution using the classical method of matched asymptotic expansion (MMAE). The Mathematica computer algebra system is used to perform the algebraic computations. The details procedures will be discussed and the underlying concepts and principles of the MMAE will be clarified. Perturbation problem for linear equation that occurs at one boundary and two boundary layers are discussed. Approximate analytical solution obtained for both cases are illustrated by graph using selected parameter by showing the outer, inner and composite solution separately. Then, the composite solution will be compare to the exact solution to show their accuracy by graph. By comparison, MMAE is found to be one of the best methods to solve singular perturbation problems in second order ordinary differential equation since the results obtained are very close to the exact solution.

First approximation thin shell and higher order thick shell correction theories are derived for nonhomogeneous anisotropic cylindrical elastic shells by use of the method of asymptotic expansion in terms of a small parameter along with Reissner's variational principle. The advantages of employing the asymptotic method, in addition to its systematic nature, are that no a priori kinematic or static assumptions need be made or both, that trasverse stresses develop naturally (even in the thin shell analysis) and that thick shell theories follow automatically. Use of the combined method generates various theories based upon different combinations of axial and circumferential length scales introduced in the nondimensionalization of the coordinates. The first approximation theories derived herein for anisotropic materials have their counterparts for isotropic materials. They are: (1)Simplified Donnell theory; (2)simple bending theory; (3)semimembrane theory; and (4)complete Donnell theory.

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form [Formula: see text] should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion.

An expansion theorem is derived for Rossby normal modes in a closed rectangular basin and the set of Rossby normal modes is proved to be complete. This theorem provides a general linear solution to the initial value problem as well as to the response problem. In particular, the Green's function is obtained for the instantaneous localized torque anywhere in the basin. Weakly nonlinear versions are solved also by the combination of the general linear solution with the asymptotic expansion in terms of small amplitude. Further, an application is suggested to the spectral method of numerical simulation based on Rossby normal modes relevant to the more nonlinear evolution equation on aβ-plane, instead ofsin functions or Chebyshev polynomials, which have been employed conventionally for this purpose.

The application of asymptotic shell theories is demonstrated for the problem of the clamped-free layered, cylindrical shell under internal pressure. In particular, the results of a uniformly valid are compared with those of a specialized refined theory. It is shown that in order to obtain the individual stress gradients for any manner of layering away from the edge, a uniformly valid theory incorporating the effects of all terms included in the specialized first approximation theories is needed. This theory accounts for the presence of all curvatures, even in axisymmetric problems, as likely they should be for anisotropic materials. In contrast, in the specialized theory, some of the curvatures are zero either as a result of the formulation or because of the axisymmetric behavior assumption. The uniformly vaild theory, however, becomes less accurate as the shell thickness increases. One must then rely, as is shown, on a thickness correction theory.

Let (X1,...,X,...,Xn) be independently and identically distributed observations from an exponential family pθ equipped with a smooth prior density w on a real d-dimensional parameter θ. We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, θ0, admits an asymptotic expansion in terms of the Fisher information I(θ0), the prior w, and their first two derivatives. The leading term of the expansion is of the form nd/2c1(d, θ0) and the second order term is of the form n4/2-1c2(d, θ0>, w), with an error term that is o(nd/2-1). We identify the functions c1 and c2 explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at θ0. As a consequence we can give a confidence band for the expected posterior, and we suggest a frequentist refinement for Bayesian testing.

Our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h, valid for h sufficiently small. Such expansions are obtained from the so-called asymptotic expansion inequalities valid in $\mathbf{R}^N$ for $N\geq2$, studies by Schatz in [Math. Comp., 67 (1998), pp. 877–899] and [SIAM J. Numer. Anal., 38 (2000), pp. 1269–1293].

Peristaltic motion of a power law fluid in a two-dimensional channel is studied. Assuming that the wavelength of the peristaltic wave is large in comparison to the mean half-width of the channel, a solution for the stream function is obtained as an asymptotic expansion in terms of slope parameter. Expressions for axial pressure gradient and shear stress are derived. The effect of flow behaviour indexn on the streamline pattern and shear stress is studied and the phenomenon of trapping is discussed.

Vector fields that are discontinuous on codimension-one surfaces are known as Filippov systems and can have attracting periodic orbits involving segments that are contained on a discontinuity surface of the vector field. In this paper we consider the addition of small noise to a general Filippov system and study the resulting stochastic dynamics near such a periodic orbit. Since a straight-forward asymptotic expansion in terms of the noise amplitude is not possible due to the presence of discontinuity surfaces, in order to quantitatively determine the basic statistical properties of the dynamics, we treat different parts of the periodic orbit separately. Dynamics distant from discontinuity surfaces is analyzed by the use of a series expansion of the transitional probability density function. Stochastically perturbed sliding motion is analyzed through stochastic averaging methods. The influence of noise on points at which the periodic orbit escapes a discontinuity surface is determined by zooming into the transition point. We combine the results to quantitatively determine the effect of noise on the oscillation time for a three-dimensional canonical model of relay control. For some parameter values of this model, small noise induces a significantly large reduction in the average oscillation time. By interpreting our results geometrically, we are able to identify four features of the relay control system that contribute to this phenomenon.

Summary We analyze transport properties of ∞uid/solid and solid/solid composites containing flnite arrays of closely spaced rigid inclusions when a host medium is either an elastic matrix or an incompressible ∞uid. The appropriate choice of the number of inclusions and the symmetry of a periodicity cell allows us to introduce simple, yet physically relevant models so that efiective characteristics of homogenized media can be investigated analytically. For various applied loads and shapes of (polydisperse) inclusions we demonstrate the spatial non-uniformity of geometric conflgurations corresponding to either lowest dissipation rate (for ∞uid/solid composites) or to minimal stifiness (for solid/solid composites). In order to flnd the optimal conflgurations, we use a unifled framework based on asymptotic expansions in terms of inter-inclusion distances. Furthermore, we compare efiective transport properties of composite materials containing inclusions with either ∞at or curved boundaries.

An asymptotic method for the Vlasov equation developed in a previous paper is reformulated in the Eulerian representation. Then the theory will be compared with that of Al'tshul' and Karpman; differences between these theories will emphasized.