Application of an idea of Voronoĭ to lattice zeta functions

A major problem in the geometry of numbers is the investigation of the local minima of the Epstein zeta function. In this article refined minimum properties of the Epstein zeta function and more general lattice zeta functions are studied. Using an idea of Voronoĭ, characterizations and sufficient conditions are given for lattices at which the Epstein zeta function is stationary or quadratic minimum. Similar problems of a duality character are investigated for the product of the Epstein zeta function of a lattice and the Epstein zeta function of the polar lattice. Besides Voronoĭ type notions such as versions of perfection and eutaxy, these results involve spherical designs and automorphism groups of lattices. Several results are extended to more general lattice zeta functions, where the Euclidean norm is replaced by a smooth norm.     

Application of an idea of Vorono? to John type problems

Using an idea of Vorono?, many John type and minimum position problems in dimension d can be transformed into more accessible geometric problems on convex subsets of the -dimensional cone of positive definite quadratic forms. In this way, we prove several new John type and minimum position results and give alternative versions and extensions of known results. In particular, we characterize minimum ellipsoidal shells of convex bodies and, in the typical case, show their uniqueness and determine the contact number. These results are formulated also in terms of the circumradius of convex bodies. Next, circumscribed ellipsoids of minimum surface area of a convex body and the corresponding minimum position problem are studied. Then we investigate John type characterizations of minimum positions of a convex body with respect to moments and the product of a moment and the moment of the polar body. The technique used in this context, finally, is applied to obtain corresponding results for the mean width and the surface area.     

Application of the trigonal curve to the Blaszak–Marciniak lattice hierarchy

We develop a method for constructing algebro-geometric solutions of the Blaszak–Marciniak (BM) lattice hierarchy based on the theory of trigonal curves. We first derive the BM lattice hierarchy associated with a discrete (3×3)-matrix spectral problem using Lenard recurrence relations. Using the characteristic polynomial of the Lax matrix for the BM lattice hierarchy, we introduce a trigonal curve with two infinite points, which we use to establish the associated Dubrovin-type equations. We then study the asymptotic properties of the algebraic function carrying the data of the divisor and the Baker–Akhiezer function near the two infinite points on the trigonal curve. We finally obtain algebro-geometric solutions of the entire BM lattice hierarchy in terms of the Riemann theta function.     

Application of Maslov’s formula to finding an asymptotic solution to an elastic deformation problem

We propose a scheme of bifurcation analysis of equilibrium configurations of a weakly inhomogeneous elastic beam on an elastic base under the assumption of two-mode degeneracy; this scheme generalizes the Darinskii-Sapronov scheme developed earlier for the case of a homogeneous beam. The consideration of an inhomogeneous beam requires replacing the condition that the pair of eigenvectors of the operator from the linear part of the equation (at zero) is constant by the condition of the existence of a pair of vectors smoothly depending on the parameters whose linear hull is invariant with respect to . It is shown that such a pair is sufficient for the construction of the principal part of the key function and for analyzing the branching of the equilibrium configurations of the beam. The construction of the required pair of vectors is based on a formula for the orthogonal projection onto the root subspace of (from the theory of perturbations of self-adjoint operators in the sense of Maslov). The effect of the type of inhomogeneity of the beam on the formof its deflection is studied.     

Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itô's Formula

We prove a uniform bound for the density, p _t (x), of the solution at time t(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p _t (x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al. ⁽⁵⁾     

Riemann surfaces with longest systole and an improved Voronoï algorithm

In this paper we introduce a new method in order to find the Riemann surface M of a fixed topological type with the longest systole; it is based on a cell decomposition of the Teichmüller space of M. The method also works in the Euclidean case and is similar to the so-called Voronoï algorithm for positive definite quadratic forms, or equivalently, for lattice sphere packings. In particular, we give a new proof of Rogers' theorem.     

Essential Spectra of a 3 × 3 Operator Matrix and an Application to Three-Group Transport Equations

In this paper we study spectral properties of a 3 × 3 block operator matrix with unbounded entries and with domain consisting of vectors which satisfy certain relations between their components. It is shown that, under certain conditions, this block operator matrix defines a closed operator, and the essential spectra of this operator are determined. These results are applied to a three-group transport equation.     

The Application of Kernel Smoothing to Time Series Data

There are already a lot of models to fit a set of stationary time series, such as AR, MA, and ARMA models. For the non-stationary data, an ARIMA or seasonal ARIMA models can be used to fit the given data. Moreover, there are also many statistical softwares that can be used to build a stationary or non-stationary time series model for a given set of time series data, such as SAS, SPLUS, etc. However, some statistical softwares wouldn＇t work well for small samples with or without missing data, especially for small time series data with seasonal trend. A nonparametric smoothing technique to build a forecasting model for a given small seasonal time series data is carried out in this paper. And then, both the method provided in this paper and that in SAS package are applied to the modeling of international airline passengers data respectively, the comparisons between the two methods are done afterwards. The results of the comparison show us the method provided in this paper has superiority over SAS＇s method.     

A Geometric Uncertainty Principle with an Application to Pleijel’s Estimate

Let \({\Omega \subset \mathbb{R}^2}\) be an open, bounded domain and \({\Omega = \bigcup_{i = 1}^{N} \Omega_{i}}\) be a partition. Denote the Fraenkel asymmetry by \({0 \leq \mathcal{A}(\Omega_i) \leq 2}\) and write $$D(\Omega_i) := \frac{|\Omega_{i}| - {\rm min}_{1 \leq j \leq N}{|\Omega_{j}|}}{|\Omega_{i}|}$$ with \({0 \leq D(\Omega_{i}) \leq 1}\) . For N sufficiently large depending only on \({\Omega}\) , there is an uncertainty principle $$\left(\sum_{i=1}^{N}{\frac{|\Omega_{i}|}{|\Omega|}{\mathcal{A}}(\Omega_i)}\right) + \left(\sum_{i=1}^{N}{\frac{|\Omega_i|}{|\Omega|}D(\Omega_i)}\right) \geq \frac{1}{60000}.$$ The statement remains true in dimensions \({n \geq 3}\) for some constant \({c_{n} > 0}\) . As an application, we give an (unspecified) improvement of Pleijel’s estimate on the number of nodal domains of a Laplacian eigenfunction and an improved inequality for a spectral partition problem.     

Optimal Hölder Continuity for a Class of Degenerate Elliptic Problems with an Application to Subsonic-Sonic Flows

In this paper, we first study a class of elliptic equations with anisotropic boundary degeneracy. Besides establishing the existence, uniqueness and comparison principle, we obtain the optimal Hölder estimates for weak solutions by the estimates in the Campanato space. Based on such Hölder estimates, we then investigate subsonic-sonic flows with singularities at the sonic curves in a symmetric convergent nozzle with straight wall for an approximate model of the potential flow equation. It is proved that the perturbation problem of the symmetric subsonic-sonic flow is solvable and the symmetric subsonic-sonic flow is stable.     

A Free Boundary Problem for an Elliptic–Parabolic System: Application to a Model of Tumor Growth

Abstract

In this article, we study a free boundary problem for a system of two partial differential equations, one parabolic and other elliptic. The system models the growth of a tumor with arbitrary initial shape. We establish the existence and uniqueness of a solution for some time interval. In the special case where we only have the elliptic equation, the problem coincides with the Hele–Shaw problem.     

A Discrete-Time Clark–Ocone Formula and its Application to an Error Analysis

In this paper, we will establish a discrete-time version of Clark(–Ocone–Haussmann) formula, which can be seen as an asymptotic expansion in a weak sense. The formula is applied to the estimation of the error caused by the martingale representation. Throughout, we use another distribution theory with respect to Gaussian rather than Lebesgue measure, which can be seen as a discrete Malliavin calculus.     

A proof of uniqueness of the Gurariĭ space

We present a short and elementary proof of isometric uniqueness of the Gurari? space.     

A Two-Level Procedure for the Global Optimum Design of Composite Modular Structures—Application to the Design of an Aircraft Wing

This work concerns a two-level procedure for the global optimum design of composite modular structures. The case-study considered is the least weight design of a stiffened wing-box for an aircraft structure. The method is based on the use of the polar formalism and on a genetic algorithm. In the first level of the procedure, the optimal structure is designed as it was composed by a single equivalent layer, while a laminate realizing the optimal structure is found in the second level. The method is able to automatically find the optimal number of modules, no simplifying assumptions are used, and it can be easily generalized to other problems. The work is divided into two parts: the theoretical formulation in this first part, the genetic procedure and some numerical examples in the second one.     

A Two-Level Procedure for the Global Optimum Design of Composite Modular Structures—Application to the Design of an Aircraft Wing

This work concerns a two-level procedure for the global optimum design of composite modular structures. The case-study considered is the least weight design of a stiffened wing-box for an aircraft structure. The method is based on the use of the polar formalism and on a genetic algorithm. In the first level of the procedure, the optimal structure is designed as composed by a single equivalent layer, while a laminate realizing the optimal structure is found in the second level. The method is able to automatically find the optimal number of modules; no simplifying assumptions are used and it can be easily generalized to other problems. The work is divided into two parts: the theoretical formulation in the first part, the genetic procedure and some numerical examples in this second one.     

Kinematics of Mechanisms to the Second Order – Application to the Closed Mechanisms

The degree of freedom of a closed mechanism is the dimension of a subset M of R ⁿ , M being the inverse image of the unity by the closure function f : (q ₁, ..., q _n ) f(q ₁, ..., q _n ), where q ₁, ..., q _n are the articular coordinates. We first study the regular points for the mapping f from R ⁿ into the Lie group of displacements and, second, study the singularities of the mapping f. The classical theory of mechanisms considers, often implicitly, that f is a subimmersion. Here, the calculations are made in a larger case, up to second order, and the results are then slightly different. The case of such classical mechanisms as Bennett, Bricard, and Goldberg mechanisms, justify the considerations of this more general framework and the example of a Bricard mechanism is chosen as an application of the method.     

Bott Generator and Its Application to an Almost Complex Structure on S~6

The Bott generator of the homotopy groupπ2k-1U(∞) is used to construct an almost complex structure on S6, which is integrable except a small neighborhood.     

Symmetric Exponential Integrators with an Application to the Cubic Schrödinger Equation

In this article, we derive and study symmetric exponential integrators. Numerical experiments are performed for the cubic Schrödinger equation and comparisons with classical exponential integrators and other geometric methods are also given. Some of the proposed methods preserve the L ²-norm and/or the energy of the system.