A Geometric Theory of Nonlinear Morphoelastic Shells

Many thin three-dimensional elastic bodies can be reduced to elastic shells: two-dimensional elastic bodies whose reference shape is not necessarily flat. More generally, morphoelastic shells are elastic shells that can remodel and grow in time. These idealized objects are suitable models for many physical, engineering, and biological systems. Here, we formulate a general geometric theory of nonlinear morphoelastic shells that describes both the evolution of the body shape, viewed as an orientable surface, as well as its intrinsic material properties such as its reference curvatures. In this geometric theory, bulk growth is modeled using an evolving referential configuration for the shell, the so-called material manifold. Geometric quantities attached to the surface, such as the first and second fundamental forms, are obtained from the metric of the three-dimensional body and its evolution. The governing dynamical equations for the body are obtained from variational consideration by assuming that both fundamental forms on the material manifold are dynamical variables in a Lagrangian field theory. In the case where growth can be modeled by a Rayleigh potential, we also obtain the governing equations for growth in the form of kinetic equations coupling the evolution of the first and the second fundamental forms with the state of stress of the shell. We apply these ideas to obtain stress-free growth fields of a planar sheet, the time evolution of a morphoelastic circular cylindrical shell subject to time-dependent internal pressure, and the residual stress of a morphoelastic planar circular shell.     

Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew’s triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange–Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange–d’Alembert–Pontryagin and Hamilton–d’Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.     

A Geometric Heat-Flow Theory of Lagrangian Coherent Structures

We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection–diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or heat equation. By this coordinate transformation, the reversible effects of advection are separated from the irreversible joint effects of advection and diffusion. In this framework, LCSs express themselves as (boundaries of) metastable sets under the Lagrangian diffusion process. In the case of spatially homogeneous isotropic diffusion, averaging the time-dependent family of Lagrangian diffusion operators yields Froyland’s dynamic Laplacian. In the associated geometric heat equation, the distribution of heat is governed by the dynamically induced intrinsic geometry on the material manifold, to which we refer as the geometry of mixing. We study and visualize this geometry in detail, and discuss connections between geometric features and LCSs viewed as diffusion barriers in two numerical examples. Our approach facilitates the discovery of connections between some prominent methods for coherent structure detection: the dynamic isoperimetry methodology, the variational geometric approaches to elliptic LCSs, a class of graph Laplacian-based methods and the effective diffusivity framework used in physical oceanography.

     

State-Space Regularization: Geometric Theory

Regularization of nonlinear ill-posed inverse problems is analyzed for a class of problems that is characterized by mappings which are the composition of a well-posed nonlinear and an ill-posed linear mapping. Regularization is carried out in the range of the nonlinear mapping. In applications this corresponds to the state-space variable of a partial differential equation or to preconditioning of data. The geometric theory of projection onto quasi-convex sets is used to analyze the stabilizing properties of this regularization technique and to describe its asymptotic behavior as the regularization parameter tends to zero. Accepted 26 April 1996     

Potential Theory of Geometric Stable Processes

In this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate Brownian motions with geometric stable subordinators. More precisely, we establish the asymptotic behaviors of the Green function and the Lévy density of symmetric geometric stable processes. The asymptotics of these functions near zero exhibit features that are very different from the ones for stable processes. The Green function behaves near zero as 1/(|x|^d log ²|x|), while the Lévy density behaves like 1/|x|^d. We also study the asymptotic behaviors of the Green function and Lévy density of subordinate Brownian motions with iterated geometric stable subordinators. As an application, we establish estimates on the capacity of small balls for these processes, as well as mean exit time estimates from small balls and a Harnack inequality for these processes. The research of this author is supported in part by MZT grant 0037118 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167. The research of this author is supported in part by a joint US-Croatia grant INT 0302167. The research of this author is supported in part by MZT grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167.     

Geometric Theory of Reduction of Nonlinear Control Systems

The foundations of a differential geometric theory of nonlinear control systems are described on the basis of categorical concepts (isomorphism, factorization, restrictions) by analogy with classical mathematical theories (of linear spaces, groups, etc.).     

Geometric Models of the Statistical Theory of Fragmentation

We propose an approach to describing a medium fragmentation process based on studying the stochastic geometry of the medium states. This approach allows accounting for the interrelation of the produced fragments relative to their positions and, in particular, allows taking the size of the fragmenting object into account. We use this approach to analyze a one-dimensional model—a stochastic process with discrete time and a phase space consisting of partitions into fragments of the real axis. We derive the driving equation for the partition function with respect to sizes and prove the existence of a limit distribution.     

Wedge Dislocation in the Geometric Theory of Defects

We consider a wedge dislocation in the framework of elasticity theory and the geometric theory of defects. We show that the geometric theory quantitatively reproduces all the results of elasticity theory in the linear approximation. The coincidence is achieved by introducing a postulate that the vielbein satisfying the Einstein equations must also satisfy the gauge condition, which in the linear approximation leads to the elasticity equations for the displacement vector field. The gauge condition depends on the Poisson ratio, which can be experimentally measured. This indicates the existence of a privileged reference frame, which denies the relativity principle.     

Quantum Mechanics as a Complete Physical Theory

We show that the principles of a complete physical theory and the conclusions of the standard quantum mechanics do not irreconcilably contradict each other as is commonly believed. In the algebraic approach, we formulate axioms that allow constructing a renewed mathematical scheme of quantum mechanics. This scheme involves the standard mathematical formalism of quantum mechanics. Simultaneously, it contains a mathematical object that adequately describes a single experiment. We give an example of the application of the proposed scheme.     

Some Classical Problems of Geometric Approximation Theory in Asymmetric Spaces

Mathematical Notes - For unbounded subsets E of the complex plane, we obtain conditions that are necessary or sufficient so that, for any compact set K that does not divide the plane, the simple...     

转动系统的相对论性分析力学理论

本文讨论了转动相对论力学理论，主要是建立转动系统的相对论性分析力学理论·构造转动系统的相对论性广义动能函数Ｔｒ＝∑ｎｉ＝１Ｉ０ｉΓｉ２（１－１－θ·２ｉ／Γｉ２）和广义加速度能量函数Ｓｒ＝１２∑ｎｉ＝１Ｉｉ（θ·ｉ·θ¨ｉ）２Γｉ２－θ·２ｉ＋θ¨２ｉ，给出其Ｈａｍｉｌｔｏｎ原理和三种不同形式的Ｄ′Ａｌｅｍｂｅｒｔ原理；对于完整约束系统，建立了转动系统的相对论性Ｌａｇｒａｎｇｅ方程、Ｎｉｅｌｓｅｎ方程、Ａｐｐｅｌ方程和Ｈａｍｉｌｔｏｎ正则方程；对于非完整约束系统，建立了转动系统的相对论性Ｒｏｕｔｈ方程、Чаплыгин方程、Ｎｉｅｌｓｅｎ方程和Ａｐｐｅｌ方程；并给出转动系统的相对论性Ｎｏｅｔｈｅｒ守恒律     

A Geometric Characterization of Poly-antimatroids

The concept of "antimatroid with repetition" was coined by Bjorner, Lovasz and Shor in 1991 as an extension of the notion of antimatroid in the framework of non-simple languages [Björner A., L. Lovász, and P. R. Shor, Chip-firing games on graphs, European Journal of Combinatorics 12 (1991), 283–291]. There are some equivalent ways to define antimatroids. They may be separated into two categories: antimatroids defined as set systems and antimatroids defined as languages. For poly-antimatroids we use the set system approach. In this research we concentrate on interrelations between geometric, algorithmic, and lattice properties of poly-antimatroids. Much to our surprise it turned out that even the two-dimensional case is not trivial.     

Quantum Mechanics and Representation Theory: The New Synthesis

Quantum mechanics and representation theory, in the sense of unitary representations of groups on Hilbert spaces, were practically born together between 1925–1927, and have continued to enrich each other till the present day. Following a brief historical introduction, we focus on a relatively new aspect of the interaction between quantum mechanics and representation theory, based on the use of K-theory of C ^*-algebras. In particular, the study of the K-theory of the reduced C ^*-algebra of a locally compact group (which for a compact group is just its representation ring) has culminated in two fundamental conjectures, which are closely related to quantum theory and index theory, namely the Baum–Connes conjecture and the Guillemin–Sternberg conjecture. Although these conjectures were both formulated in 1982, and turn out to be closely related, so far there has been no interplay between them whatsoever, either mathematically or sociologically. This is presumably because the Baum–Connes conjecture is nontrivial only for noncompact groups, with current emphasis entirely on discrete groups, whereas the Guillemin–Sternberg conjecture has so far only been stated for compact Lie groups. As an elementary introduction to both conjectures in one go, indicating how the latter can be generalized to the noncompact case, this paper is a modest attempt to change this state of affairs.     

多频常微分方程的几何奇异扰动理论

在该文中,作者把常微分方程的几何奇异扰动理论推广到具多个频率的系统,同时给出了一个例子来说明主要理论．     

Localization in Adiabatic Shear Flow Via Geometric Theory of Singular Perturbations

Journal of Nonlinear Science - We study localization occurring during high-speed shear deformations of metals leading to the formation of shear bands. The localization instability results from the...     

A Geometric Analysis of Phase Retrieval

Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of m measurements, \(y_k = \left| \varvec{a}_k^* \varvec{x} \right| \) for \(k = 1, \ldots , m\), is it possible to recover \(\varvec{x} \in \mathbb C^n\) (i.e., length-n complex vector)? This generalized phase retrieval (GPR) problem is a fundamental task in various disciplines and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretic explanations. In this paper, we take a step toward bridging this gap. We prove that when the measurement vectors \(\varvec{a}_k\)’s are generic (i.i.d. complex Gaussian) and numerous enough (\(m \ge C n \log ^3 n\)), with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) There are no spurious local minimizers, and all global minimizers are equal to the target signal \(\varvec{x}\), up to a global phase, and (2) the objective function has a negative directional curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a second-order trust-region algorithm.