We prove a robust extension of the quantum adiabatic theorem. The theorem applies to systems that have resonances instead of bound states, and to systems for which just an approximation to a bound state is known. To demonstrate the theorem’s usefulness in a concrete situation, we apply it to shape resonances. Partially Supported by National Science Foundation Grant DMS–0907165. Partially Suppor...

The paper presents an improved collage theorem, valid for a class of signal mappings called Afine Blockwise Aueraging (ABA) Mappings. The .4BA structure is exploited to form a bound depending on norms of collage error signals at several resolutions. Compared to previously published collage theorems, the new theorem provides a much tighter bound on the maximum distance between the original signa...

The purpose of this paper is to obtain rigorous numerical enclosures for solutions Lane–Emden’s equation −Δu=|u|p−1u with homogeneous Dirichlet boundary conditions. We prove the existence a nondegenerate solution u nearby numerically computed approximation û together an explicit error bound, i.e., bound difference between and û. In particular, we focus on sub-square case in which 1<p<2 so that ...

This note aims to give an elementary proof for Toponogov’s theorem in Alexandrov geometry with lower curvature bound. The idea of the comes from fact that, Riemannian geometry, sectional can be embodied geodesic variations.

We propose a PAC-Bayes theorem for the sample-compression setting where each classifier is described by a compression subset of the training data and a message string of additional information. This setting, which is the appropriate one to describe many learning algorithms, strictly generalizes the usual data-independent setting where classifiers are represented only by data-independent message...

Kolmogorov Theorem on the persistence of invariant tori of real analytic Hamiltonian systems is revisited. In this paper we are mainly concerned with the lower bound on the constant of the Diophantine condition required by the theorem. From the existing proofs in the literature, this lower bound turns to be ofO(ε1/4), where ε is the size of the perturbation. In this paper, by means of careful (...

The well-known Bondy’s Theorem [1] guarantees (in terms of vertex degrees) a sufficiently “large” cycle in a block. We show that adding a condition on connectivity of these blocks yields an improvement of the lower bound in Bondy’s Theorem.

We will present many strong partial results towards a classification of exceptional planar/PN monomial functions on finite fields. The techniques we use are the Weil bound, Bezout’s theorem, and Bertini’s theorem.

From the Paley–Wiener 1/4-theorem, the finite energy signal f(t) can be reconstructed from its irregularly sampled values f(k+ k) if f(t) is band-limited and supk j kj < 1=4. We consider the signals in wavelet subspaces and wish to recover the signals from its irregular samples by using scaling functions. Then the way to estimate the upper bound of sup k j kj such that the irregularly sampled s...

We present a short and simple proof of the well-known Cauchy interlace theorem. We use the theorem to improve some lower bound estimates for the spectral radius of a real symmetric matrix.