A countable discrete group G is called W?-superrigid (resp. C?-superrigid) if it completely recognizable from its von Neumann algebra L(G) reduced C?-algebra Cr?(G)). Developing new technical aspects in Popa's deformation/rigidity theory we introduce several classes of groups which appear as direct products, semidirect products with non-amenable core and iterations amalgamated free HNN-extensio...

There is no known, tractable, characterization of 3D rigidity of sets of points constrained by pairwise distances or 3D distance constraint graphs. We give a combinatorial approximate characterization of such graphs which we call module-rigidity, which can be determined by a polynomial time algorithm. We show that this property is natural and robust in a formal sense. Rigidity implies module-ri...

This is a short survey paper based on the topics of the author’s research along with his collaborators. Main topics presented in the paper are on the regularity and rigidity theorems for the solutions of the elliptic differential equations. In particular, the author poses several open problems in these topics for further study. The paper contains six sections. Regularity theorems for the ellipt...

We study the curvature condition which uniquely characterizes the hemisphere. In particular, we prove the Min–Oo conjecture for hypersurfaces in Euclidean space and hyperbolic space.

We say a subset Σ ⊆ FN of the free group of rank N is spectrally rigid if whenever T1, T2 ∈ cvN are R-trees in (unprojectivized) outer space for which ‖σ‖T1 = ‖σ‖T2 for every σ ∈ Σ, then T1 = T2 in cvN . The general theory of (non-abelian) actions of groups on R-trees establishes that T ∈ cvN is uniquely determined by its translation length function ‖ · ‖T : FN → R, and consequently that FN its...

A bar framework G(p) in r-dimensional Euclidean space is a graph G = (V,E) on the vertices 1, 2, . . . , n, where each vertex i is located at point p in R. Given a framework G(p) in R, a problem of great interest is that of determining whether or not there exists another framework G(q), not obtained from G(p) by a rigid motion, such that ||q−q || = ||p−p || for all (i, j) ∈ E. This problem is k...

Registration is a technique nowadays commonly used in medical imaging. A drawback of most of the current registration schemes is that all tissue is being considered as non-rigid [1]. Therefore, rigid objects in an image, such as bony structures or surgical instruments, may be transformed non-rigidly. In this paper, we integrate the concept of local rigidity to the FLexible Image Registration To...

UNLABELLED The present study examined changes in the rigidity of interpersonal patterns and symptoms in adolescents (ages 15-18) in a year-long psychodynamic psychotherapy. Seventy-two adolescents (30 in treatment and 42 in a non-treatment "community group") underwent Relationship Anecdote Paradigm (RAP) interviews according to the Core Conflictual Relationship Theme method (CCRT; Luborsky & Cr...

We prove Cheng’s eigenvalue comparison theorems [4] for geodesic balls within the cut locus under weaker geometric hypothesis, Theorems (1.1, 3.1, 3.2) and we also show that there are certain geometric rigidity in case of equality of the eigenvalues. This rigidity becomes isometric rigidity under upper sectional curvature bounds or lower Ricci curvature bounds. We construct examples of smooth m...

We generalize Cauchy’s celebrated theorem on the global rigidity of convex polyhedra in Euclidean 3-space E to the context of circle polyhedra in the 2-sphere S. We prove that any two convex and bounded non-unitary c-polyhedra with Möbiuscongruent faces that are consistently oriented are Möbius-congruent. Our result implies the global rigidity of convex inversive distance circle packings as wel...