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A new mechanism for symmetry breaking is proposed which naturally avoids the constraints following from the usual theorems of symmetry breaking. In the context of super-symmetry, for example, the breaking may be consistent with a vanishing…
A new method of metric space investigation, based on classification of its finite subspaces, is suggested. It admits to derive information on metric space properties which is encoded in metric. The method describes geometry in terms of only…
Recent links between Finsler Geometry and the geometry of spacetimes are briefly revisited, and prospective ideas and results are explained. Special attention is paid to geometric problems with a direct motivation in Relativity and other…
In this paper, we study the distance problem in the setting of finite p-adic rings. In odd dimensions, our results are essentially sharp. In even dimensions, we clarify the conjecture and provide examples to support it. Surprisingly,…
Thanks to its versatility, its simplicity, and its fast convergence, ADMM is among the most widely used approaches for solving a convex problem in distributed form. However, making it running efficiently is an art that requires a fine…
Even though it has been almost a century since quantum mechanics planted roots, the field has its share of unresolved problems. It could be the result of a wrong mathematical structure providing inadequate understanding of the quantum…
Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces,…
The historical developments of conformal transformations and symmetries are sketched: Their origin from stereographic projections of the globe, their blossoming in two dimensions within the field of analytic complex functions, the generic…
Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a very difficult problem with many applications. While it is hopeless to expect much in general, we know a surprising amount about these…
In various areas of modern physics and in particular in quantum gravity or foundational space-time physics it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be…
Large-scale eigenvalue problems arise in various fields of science and engineering and demand computationally efficient solutions. In this study, we investigate the subspace approximation for parametric linear eigenvalue problems, aiming to…
The purpose of this note is to present several aspects of concentration phenomena in high dimensional geometry. At the heart of the study is a geometric analysis point of view coming from the theory of high dimensional convex bodies. The…
We consider Calderon's inverse problem with partial data in dimensions $n \geq 3$. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility…
We initiate the study of the norm-squared of the momentum map as a rigorous tool in infinite dimensions. In particular, we calculate the Hessian at a critical point, show that it is positive semi-definite along the complexified orbit, and…
This is a survey of our research on geometric structures of projective embeddings and includes some topics of our talks in several symposia during 1990-99. We clarify our main problem, which is to construct a kind of geometric composition…
This paper focuses on investigating generalized relative interior notions for sets in locally convex topological vector spaces with particular attentions to graphs of set-valued mappings and epigraphs of extended-real-valued functions. We…
Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group…
The paper is devoted to developing subdifferential theory for set-valued mappings taking values in ordered infinite-dimensional spaces. This study is motivated by applications to problems of vector and set optimization with various…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
An enquiry into the physical nature of time and space and into the ontology of quantum mechanics from a metageometric perspective, resulting from the belief that geometric thought and language are powerless to farther understanding of these…