The rigidity conjecture

A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. In this paper we state a conjecture which describes how topological classes are organized into rigidity classes.     

Some properties of rigidity mappings

The bounds of minimal rank of differentials of rigidity mappings are obtained. They depend on the structural scheme and on the positions of fastened points. Two hypotheses are introduced. One on presence in the set of minimal rank of the rigidity mapping of a construction with a zero-length lever. Another—on the unboundedness of the sets of constant rank of the rigidity mapping in the case of their positive dimension. In some cases these hypotheses are proved.     

具有非零最小弯曲刚度梁在多载荷情况作用下的等强度设计

用最小余能原理对静不定梁进行等强度设计的解析方法被推广到有指定的非零最小弯曲刚度约束和多载荷工作情况的情形.并提出了一个求解等强度设计的数值方法,因而对有任意截面形式的梁,在几个任意分布的载荷情况作用下,且考虑最小弯曲刚度约束这一最一般情形得到了统一的等强度设计解法.     

Completeness of the Floquet solutions to the problem of a solid oscillating in a fluid

We prove the completeness of the Floquet solutions to the parabolic equation describing small oscillations of a fluid-solid system. The symmetry axis of the solid is fixed inside a container of an arbitrary shape which is filled with an incompressible viscous fluid. The solid oscillates torsionally under the action of an elastic force with time periodic rigidity.     

Stress distribution in the vicinity of a circular hole in an orthotropic spherical shell with finite shear rigidity

The authors consider the problem of stress concentration in the vicinity of a circular hole in an orthotropic spherical shell with finite shear rigidity. The case in which the hole edges are reinforced by a thin elastic ring and some associated special cases are investigated.     

Deformations of hyperbolic convex polyhedra and cone-3-manifolds

The Stoker problem, first formulated in Stoker (Commun. Pure Appl. Math. 21:119–168, 1968), consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means of the double construction, this problem is intimately related to rigidity issues for 3-dimensional cone-manifolds. In Mazzeo and Montcouquiol (J. Differ. Geom. 87(3):525–576, 2011), two such rigidity results were proven, implying that the infinitesimal version of the Stoker conjecture is true in the hyperbolic and Euclidean cases. In this second article, we show that local rigidity holds and prove that the space of convex hyperbolic polyhedra with given combinatorial type is locally parametrized by the set of dihedral angles, together with a similar statement for hyperbolic cone-3-manifolds.     

On the dynamics of generic non-Abelian free actions

We investigate some global generic properties of the dynamics associated to non-Abelian free actions in certain special cases. The main properties considered in this paper are related to the existence of dense orbits, to ergodicity and to topological rigidity. We first deal with them in the case of conservative homeomorphisms of a manifold and C ¹-diffeomorphisms of a surface. Groups of analytic diffeomorphisms of a manifold which, in addition, contain a Morse-Smale element and possess a generating set close to the identity are considered as well. From our discussion we also derive the existence of a rigidity phenomenon for groups of skew-products which is opposed to the phenomenon present in Furstenbergs celebrated example of a minimal diffeomorphism that is not ergodic (cf. [Ma]).     

On the spectrum of a multipoint boundary value problem for a fourth-order equation

We study the spectral properties of a multipoint boundary value problem for a fourth-order equation that describes small deformations of a chain of rigidly connected rods with elastic supports. We study the dependence of the spectrum of the boundary value problem on the rigidity coefficients of the supports. We show that the spectrum of the boundary value problem splits into two parts, one of which is movable under changes of the rigidity coefficients and the other remains fixed. As the rigidity coefficients grow, the eigenvalues corresponding to the movable part of the spectrum grow as well; moreover, the double degeneration of some eigenvalues is possible.     

Some estimates for the torsional rigidity of composite rods

A well‐known problem in elasticity consists in placing two linearly elastic materials (of different shear moduli) in a given plane domain Ω, so as to maximize the torsional rigidity of the resulting rod; moreover, the proportion of these materials is prescribed. Such a problem may not have a classical solution as the optimal design may contain homogenization regions, where the two materials are mixed in a microscopic scale. Then, the optimal torsional rigidity becomes difficult to compute. In this paper we give some different theoretical upper and lower bounds for the optimal torsional rigidity, and we compare them on some significant domains. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pseudolinear equivalent constant rigidity concept for analyzing welding residual deformation

To ascertain the extent of deformation due to the thermal cycles caused by welding it calls for solving a complex thermal elasto-plastic problem, which is non-linear and involves plastic deformation of the medium at high temperature varying in both time and space. Analytical solutions turned out to be inadequate. At the same time conventional numerical techniques proved to be highly time consuming and thereby prohibitively expensive in real life situations. The concept of pseudolinear equivalent constant rigidity system was developed in this investigation for thermo-mechanical analysis of plates undergoing variation of rigidity due to a continuously changing temperature profile as is encountered in welding situations. The initial non-linear problem with modulus varying with temperature was transformed into a pseudolinear equivalent system of constant rigidity that was solved by applying linear analysis.     

Investigation of the natural vibrations of plates made of composite materials with the help of the finite-element method

The modes and frequencies of small natural vibrations in a cavity of thin plates are investigated on the basis of the method of finite elements in displacements. The effect of the rigidity characteristics of a material on the natural frequencies and modes of vibrations of flat cantilever vanes whose tapered edge is restrained is studied. An example is given of the use of the finite-element method to determine the natural vibrations of an orthotropic plate in the shape of an airplane wing. Quadrangular and modified triangular bending finite elements are used to simulate a continuous system. A mass-element matrix is constructed on the basis of a four-term ("contracted") polynomial determining the deflection that permits a significant reduction of the order of the solving system.Translated from Mekhanika Polimerov, No. 2, pp. 284–288, March–April, 1976.     

On Spaces of Infinitesimal Motions and Three Dimensional Henneberg Extensions

We investigate certain spaces of infinitesimal motions arising naturally in the rigidity theory of bar and joint frameworks. We prove some structure theorems for these spaces and, as a consequence, are able to deduce some special cases of a long standing conjecture of Graver, Tay and Whiteley concerning Henneberg extensions and generically rigid graphs.     

Interpolating between torsional rigidity and principal frequency

A one-parameter family of variational problems is examined that interpolates between torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. The associated partial differential equation is derived, which is shown to have positive solutions in many cases. Results are obtained regarding extremal domains and regarding variations of the domain or the parameter.     

Rigidity degrees of indecomposable modules over representation-finite self-injective algebras

The rigidity degree of a generator-cogenerator determines the dominant dimension of its endomorphism algebra, and is closely related to a recently introduced homological dimension — rigidity dimension. In this paper, we give explicit formulae for the rigidity degrees of all indecomposable modules over representation-finite self-injective algebras by developing combinatorial methods from the Euclidean algorithm. As an application, the rigidity dimensions of some algebras of types A and E are given.     

Dynamical systems on graphs through the signless Laplacian matrix

There is a deep and interesting connection between the topological properties of a graph and the behaviour of the dynamical system defined on it. We analyse various kind of graphs, with different contrasting connectivity or degree characteristics, using the signless Laplacian matrix. We expose the theoretical results about the eigenvalue of the matrix and how they are related to the dynamical system. Then, we perform numerical computations on real-like graphs and observe the resulting system. Comparing the theoretical and numerical results, we found a perfect consistency. Furthermore, we define a metric which takes into account the “rigidity” of the graph and enables us to relate all together the topological properties of the graph, the signless Laplacian matrix and the dynamical system.     

HOMOLOGY RIGIDITY OF GRASSMANNIANS

Applying the theory of Grbner basis to the Schubert presentation for the cohomology of Grassmannians [2], we extend the homology rigidity results known for the classical Grassmanians to the exceptional cases.     

Pseudo‐reflection phenomena for singularities in thin elastic shells

We consider problems of statics of thin elastic shells with hyperbolic middle surface subjected to boundary conditions ensuring the geometric rigidity of the surface. The asymptotic behaviour of the solutions when the relative thickness tends to zero is then given by the membrane approximation. It is a hyperbolic problem propagating singularities along the characteristics. We address here the reflection phenomena when the propagated singularities arrive to a boundary. As the boundary conditions are not the classical ones for a hyperbolic system, there are various cases of reflection. Roughly speaking, singularities provoked elsewhere are not reflected at all at a free boundary, whereas at a fixed (or clamped) boundary the reflected singularity is less singular than the incident one. Reflection of singularities provoked along a non‐characteristic curve C are also considered. Copyright © 2003 John Wiley & Sons, Ltd.     

Design of Microstrip Antennas with Composite Laminates Considering Their Structural Rigidity

Two types of conformal load-bearing antenna structures (CLAS) were designed with microwave composite laminates and Nomex honeycomb cores to secure both the structural rigidity and a good electrical performance. One was a 4 × 8 array for the synthetic-aperture radar (SAR) system and the other was a 5 × 2 array for the wireless local-area network (LAN) system. The design was based on a wide bandwidth, high polarization purity, low losses, and high structural rigidity. The design, fabrication, and structural/electrical performances of the antenna structures were studied. Their flexural behavior was examined by three-point bending, impact, and buckling tests. The electrical measurements were in a good agreement with simulation results. The complex antenna structures obtained have good flexural characteristics. The design of this antenna structure is extended to give a useful guide for sandwich panel manufacturers as well as antenna designers.     

Regularization of optimal design problems for bars and plates,part 1

In this paper, we consider a number of optimal design problems for elastic bars and plates. The material characteristics of rigidity of an elastic nonhomogeneous medium are taken as the control variables. A linear functional of the solutions to the equilibrium boundary-value problem is minimized under additional restrictions upon the control variables.Special variations of the control within a narrow strip provide a necessary condition for a strong local minimum (Weierstrass test). This necessary condition can be used for the detailed analysis of the following problems: bar of extremal torsional rigidity; optimal distribution of isotropic material with variable shear modulus within a plate; and optimal orientation of principal axes of elasticity in an orthotropic plate. For all of these cases, the stationary solutions violate the Weierstrass test and therefore are not optimal. This is because, in the course of the approximation of the optimal solution, the curve dividing zones occupied by materials with different rigidities displays rapid oscillations sweeping over a two-dimensional region. Within this region, the material behaves as a composite medium assembled of materials of the initial class.     

Stable Ground States for the Relativistic Gravitational Vlasov–Poisson System

We consider the three dimensional gravitational Vlasov–Poisson (GVP) system in both classical and relativistic cases. The classical problem is subcritical in the natural energy space and the stability of a large class of ground states has been derived by various authors. The relativistic problem is critical and displays finite time blow up solutions. Using standard concentration compactness techniques, we however show that the breaking of the scaling symmetry allows the existence of stable relativistic ground states. A new feature in our analysis which applies both to the classical and relativistic problem is that the orbital stability of the ground states does not rely as usual on an argument of uniqueness of suitable minimizers—which is mostly unknown—but on strong rigidity properties of the transport flow, and this extends the class of minimizers for which orbital stability is now proved.