Related papers: Variants of VC dimension and their applications to…
We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu (1935) and Bhatia and Davis (2000) concerning measures on the line to several dimensions. This is…
Particle physics models where there are large hidden extra dimensions are currently on the focus of an intense activity. The main reason is that these large extra dimensions may come with a TeV scale for quantum gravity (or string theory)…
Recently, the authors introduced the theory of high-arity PAC learning, which is well-suited for learning graphs, hypergraphs and relational structures. In the same initial work, the authors proved a high-arity analogue of the Fundamental…
We study the evolution of biased domain walls in the early universe. We explicitly discuss the roles played by the surface tension and volume pressure in the evolution of the walls, and quantify their effects by looking at the collapse of…
The possibility of physics in multiple time dimensions is investigated. Drawing on recent work by Walter Craig and myself, I show that, contrary to conventional wisdom, there is a well-posed initial value problem--deterministic, stable…
Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this enterprise is to find the…
We study the generalization capacity of group convolutional neural networks. We identify precise estimates for the VC dimensions of simple sets of group convolutional neural networks. In particular, we find that for infinite groups and…
We study the implications of translation invariance on the tangent dynamics of extended dynamical systems, within a random matrix approximation. In a model system, we show the existence of hydrodynamic modes in the slowly growing part of…
Efficient multiphysics models that can adapt to the varying complexity of physical processes in space and time are desirable for modeling fluid migration in the subsurface. Vertical equilibrium (VE) models are simplified mathematical models…
Motivated by recent developments in Hamiltonian variational principles, Hamiltonian variational integrators, and their applications such as to optimization and control, we present a new Type II variational approach for Hamiltonian systems,…
An overview is given of recent developments in the field of Dirac equations generalized to curved space-times. An illustrative discussion is provided. We conclude with a variation of Dirac's large-number hypothesis which relates a number of…
We introduce the notion of angular values for deterministic linear difference equations and random linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear…
The so-called spectral dimension is a scale-dependent number associated with both geometries and field theories that has recently attracted much attention, driven largely though not exclusively by investigations of causal dynamical…
Consider briefly the equations of fluid dynamics-they describe the enormous wealth of detail in all the interacting physical elements of a fluid flow-whereas in applications we want to deal with a description of just that which is…
This paper presents a series of new results for domain adaptation in the multi-view learning setting. The incorporation of multiple views in the domain adaptation was paid little attention in the previous studies. In this way, we propose an…
This work studies embedding of arbitrary VC classes in well-behaved VC classes, focusing particularly on extremal classes. Our main result expresses an impossibility: such embeddings necessarily require a significant increase in dimension.…
This paper shows that the $\mathrm{VC}_2$-dimension of a subset of $\mathbb{F}_p^n$ known as the 'quadratic Green-Sanders example' is at least 3 and at most 501. The upper bound confirms a conjecture of Terry and Wolf, who introduced this…
The notion of the flow introduced by Kitaev is a manifestly topological formulation of the winding number on a real lattice. First, we show in this paper that the flow is quite useful for practical numerical computations for systems without…
Planck-scale dynamical dimensional reduction is attracting more and more interest in the quantum-gravity literature since it seems to be a model independent effect. However different studies base their results on different concepts of…
We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid…