We study learnability in the online learning model. We define several complexity measures which capture the difficulty of learning in a sequential manner. Among these measures are analogues of Rademacher complexity, covering numbers and fat shattering dimension from statistical learning theory. Relationship among these complexity measures, their connection to online learning, and tools for boun...

Based on the recently developed Finite Integration Method (FIM) for solving one-dimensional ordinary and partial differential equations, this paper extends the technique to higher dimensional partial differential equations. The main idea is to extend the first order finite integration matrices constructed by using either Ordinary Linear Approach (OLA) (uniform distribution of nodes) or Radial B...

The purpose of this paper is to prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally polarized abelian surface defined over a number field. Most of our argument works in arbitrary dimension and the restriction on the dimension ≤ 2 is used only at the last st...

The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VCdimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e(logn) . The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, e √ ...

A strong case exists for the introduction of burst non-uniform sampling (NUS) in the direct dimension of NMR spectroscopy experiments. The resulting gaps in the NMR free induction decay can reduce the power demands of long experiments (by switching off broadband decoupling for example) and/or be used to introduce additional pulses (to refocus homonuclear coupling, for example). The final EXtend...

In this paper properties and construction of designs under a centered version of the L2-discrepancy are analyzed. The theoretic expectation and variance of this discrepancy are derived for random designs and Latin hypercube designs. The expectation and variance of Latin hypercube designs are significantly lower than that of random designs. While in dimension one the unique uniform design is als...

An expression for the transmitted fraction of the uniform and parallel radiation incident upon the coupler between two rectangular light pipes of different physical dimensions is derived. It is a function of the pipe dimension ratio and the angle of the incident radiation. It is shown that for radiation nearly parallel to the pipe axis, it is possible to design a coupler for any pipe dimension ...

We compute the minimal angle spread with respect to uniform distribution in probability simplex. The resulting optimization problem is analytically solved. formula provided shows that approaches zero as dimension tends infinity. also discuss an application cognitive science.

We analyze the problem of boundary observability of the finite-difference space semi-discretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the mesh size) boundary observability for the solutions obtained by the two-grid preconditioner introduced by Glowinski [6]. Our method uses previously known uniform observability inequalities for low frequency solutio...

A bstract. The three level operator finite difference schemes on nonuniform on time grids in llilbert spaces of finite dimension are consid­ ered. A priori estimates for uniform stability on the initial conditions are received under natural assumptions on operators and non uniform time grids. The results obtained here are applied to study stability o f the three levels weighted schemes of secon...