Related papers: Existence, Regularity, and Properties of Generaliz…

Asymptotic properties of random regular graphs are object of extensive study in mathematics. In this note we argue, based on theory of spin glasses, that in random regular graphs the maximum cut size asymptotically equals the number of…

In this short note, we give an elementary proof of a universal approximation theorem for neural networks with three hidden layers and increasing, continuous, bounded activation function. The result is weaker than the best known results, but…

The aim of this text is to provide an elementary and self-contained exposition of Gromov's argument on topological overlap (the presentation is based on Gromov's work, as well as two follow-up papers of Matousek and Wagner, and of…

We first propose a generalization of the image conjecture [Z3] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent…

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely…

This paper theoretically investigates the following empirical phenomenon: given a high-complexity network with poor generalization bounds, one can distill it into a network with nearly identical predictions but low complexity and vastly…

This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions. (1) Isoparametric functions on Riemannian manifolds, including exotic spheres. The existences and…

Topological mapping of a large physical system on a graph, and its decomposition using universal measures is proposed. We find inherent limits to the potential for optimization of a given system and its approximate representations by…

We give conjectures on the "asymptotic" behaviour of the Hilbert series of (quotients by) generic ideals in the exterior algebra, as the number of variables tend to infinity. Our conjectures are supported by extensive computer calculations.

The minimum number of observations such that the maximum likelihood estimator in a Gaussian graphical model exists with probability one is called the maximum likelihood threshold of the underlying graph G. The natural algebraic relaxation…

We investigate the validity the generalized second law of thermodynamics in a general braneworld model with curvature correction terms on the brane and in the bulk, respectively. Employing the derived entropy expression associated with the…

It is often stated that a phase of standard, decelerated cosmological expansion is characterised by the absence of global event horizons, while a phase of accelerated expansion is associated with the absence of particle horizons. This is…

This paper formalises the concepts of weakly and weakly regularly persistent input trajectory as well as their link to the Observability Grammian and the existence and uniqueness of solutions of Moving Horizon Estimation (MHE) problems.…

If our observable Universe is only a tiny region of a vastly larger and conformally older spacetime, then the usual formulations of the classical flatness and horizon problems of the Hot Big Bang can be reinterpreted as artifacts…

What is going on (as of August 2008) at the interface between theoretical general relativity, string-inspired models, and observational astrophysics? Quite a lot. In this mini-survey I will make a personal choice and focus on four specific…

We investigate the validity of Thorne's hoop conjecture in non-axisymmetric spacetimes by examining the formation of apparent horizons numerically. If spaces have a discrete symmetry about one axis, we can specify the boundary conditions to…

This short communication advances the hypothesis that the observed fractal structure of large-scale distribution of galaxies is due to a geometrical effect, which arises when observational quantities relevant for the characterization of a…

This paper explores recent progress related to constraint maps. Building on the exposition in [14], our goal is to provide a clear and accessible account of some of the more intricate arguments behind the main results in this work. Along…

We introduce classes of graphs with bounded expansion as a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank…

We prove that in a closed Riemannian manifold with dimension between $3$ and $7$, either there are minimal hypersurfaces with arbitrarily large area, or there exist uncountably many stable minimal hypersurfaces. Moreover, the latter case…