An Interesting Class of Operators with unusual Schatten-von Neumann behavior

We consider the class of integral operators Q_\f on $L^2(\R_+)$ of the form (Q_\f f)(x)=\int_0^\be\f (\max\{x,y\})f(y)dy. We discuss necessary and sufficient conditions on $\phi$ to insure that $Q_{\phi}$ is bounded, compact, or in the Schatten-von Neumann class \bS_p, $1<p<\infty$. We also give necessary and sufficient conditions for $Q_{\phi}$ to be a finite rank operator. However, there is a kind of cut-off at $p=1$, and for membership in \bS_{p}, $0<p\leq1$, the situation is more complicated. Although we give various necessary conditions and sufficient conditions relating to Q_{\phi}\in\bS_{p} in that range, we do not have necessary and sufficient conditions. In the most important case $p=1$, we have a necessary condition and a sufficient condition, using $L^1$ and $L^2$ modulus of continuity, respectively, with a rather small gap in between. A second cut-off occurs at $p=1/2$: if \f is sufficiently smooth and decays reasonably fast, then \qf belongs to the weak Schatten-von Neumann class \wS{1/2}, but never to \bS_{1/2} unless \f=0. We also obtain results for related families of operators acting on $L^2(\R)$ and $\ell^2(\Z)$. We further study operations acting on bounded linear operators on $L^{2}(\R^{+})$ related to the class of operators Q_\f. In particular we study Schur multipliers given by functions of the form $\phi(\max\{x,y\})$ and we study properties of the averaging projection (Hilbert-Schmidt projection) onto the operators of the form Q_\f.Comment: 87 page

Seeded Graph Matching via Large Neighborhood Statistics

We study a well known noisy model of the graph isomorphism problem. In this model, the goal is to perfectly recover the vertex correspondence between two edge-correlated Erd\H{o}s-R\'{e}nyi random graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. For seeded problems, our result provides a significant improvement over previously known results. We show that it is possible to achieve the information-theoretic limit of graph sparsity in time polynomial in the number of vertices $n$. Moreover, we show the number of seeds needed for exact recovery in polynomial-time can be as low as $n^{3\epsilon}$ in the sparse graph regime (with the average degree smaller than $n^{\epsilon}$) and $\Omega(\log n)$ in the dense graph regime. Our results also shed light on the unseeded problem. In particular, we give sub-exponential time algorithms for sparse models and an $n^{O(\log n)}$ algorithm for dense models for some parameters, including some that are not covered by recent results of Barak et al

Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs

General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph H, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and Schur triples in random subsets of integers. In the second part of the paper we return to the subgraph counts in random graphs and provide upper tail estimates in the rooted case.Comment: 15 page

Monotone graph limits and quasimonotone graphs

The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object which may be represented by a symmetric function $W$ on $[0,1]$, i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a `quasi-monotonicity' property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and $L^1$ norms of kernels of the form $W_1-W_2$ with $W_1$ and $W_2$ monotone that may be of interest in its own right; no such inequality holds for general kernels.Comment: 38 page

Random trees with superexponential branching weights

We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors $w_n$ associated to the vertices of the tree and depending only on their individual degrees $n$. We focus on the case when $w_n$ grows faster than exponentially with $n$. In this case the measures on trees of finite size $N$ converge weakly as $N$ tends to infinity to a measure which is concentrated on a single tree with one vertex of infinite degree. For explicit weight factors of the form $w_n=((n-1)!)^\alpha$ with $\alpha >0$ we obtain more refined results about the approach to the infinite volume limit.Comment: 19 page

An Asymptotically-Optimal Sampling-Based Algorithm for Bi-directional Motion Planning

Bi-directional search is a widely used strategy to increase the success and convergence rates of sampling-based motion planning algorithms. Yet, few results are available that merge both bi-directional search and asymptotic optimality into existing optimal planners, such as PRM*, RRT*, and FMT*. The objective of this paper is to fill this gap. Specifically, this paper presents a bi-directional, sampling-based, asymptotically-optimal algorithm named Bi-directional FMT* (BFMT*) that extends the Fast Marching Tree (FMT*) algorithm to bi-directional search while preserving its key properties, chiefly lazy search and asymptotic optimality through convergence in probability. BFMT* performs a two-source, lazy dynamic programming recursion over a set of randomly-drawn samples, correspondingly generating two search trees: one in cost-to-come space from the initial configuration and another in cost-to-go space from the goal configuration. Numerical experiments illustrate the advantages of BFMT* over its unidirectional counterpart, as well as a number of other state-of-the-art planners.Comment: Accepted to the 2015 IEEE Intelligent Robotics and Systems Conference in Hamburg, Germany. This submission represents the long version of the conference manuscript, with additional proof details (Section IV) regarding the asymptotic optimality of the BFMT* algorith

CaCu2(SeO3)2Cl2: spin-1/2 Heisenberg chain compound with complex frustrated interchain couplings

We report the crystal structure, magnetization measurements, and band-structure calculations for the spin-1/2 quantum magnet CaCu2(SeO3)2Cl2. The magnetic behavior of this compound is well reproduced by a uniform spin-1/2 chain model with the nearest-neighbor exchange of about 133 K. Due to the peculiar crystal structure, spin chains run in the direction almost perpendicular to the structural chains. We find an exotic regime of frustrated interchain couplings owing to two inequivalent exchanges of 10 K each. Peculiar superexchange paths grant an opportunity to investigate bond-randomness effects under partial Cl-Br substitution.Comment: Extended version: 9 pages, 7 figures, 4 table

Vibrational spectral studies and crystal analysis of coordination complexes

There are many physical or physico-chemical methods available for studying the various properties of chemical compounds. These methods can be conveniently classified into classes which are determined by the purpose of the investigation. Those used to elucidate molecular structure fall into two classes. The first, which includes x-ray, neutron, and electron diffraction yields detailed information about the whole structure of the molecule. These techniques introduce a metrical element into the understanding of the complex, revealing the lengths of the chemical bonds, the angles between them, and other structural details such as the various forms of isomerism in the topological and conformational arrangements of polydentate ligands about the metal centre. The second gives fragmentary information concerning individual bonds of a particular group of atoms in the molecule. This class includes optical rotatory dispersion, circular dichrosism, the measurement of electric and magnetic moments, and various kinds of spectroscopy of the region ranging from microwave to ultraviolet. The last two groups should yield information mostly in the area of the electronic structure of the molecule. The last group covers vibrational spectroscopy which originates in the vibrations of the nuclei constituting a molecule. Vibrational spectra are observed both as infrared and Raman spectra, and the frequencies of the vibrational transitions are determined by the masses of the constituent atoms, the molecular geometry and the interatomic forces. The intensities of infrared and Raman spectra are related to the changes in dipole moment and polarisability, respectively. Attempts have been made to analyse the vibrational spectra quantitatively. Though there has been some success, work on many aspects (such as intensity in relation to bonding) is sparse. Nevertheless, the understanding of the molecular and the electronic structure of some molecules can be much extended by employing a combination of methods. This thesis consists of two main sections. The first describes the vibrational spectral studies of diarsine complexes. Most of the spectral work is centred around the low infrared frequency region. Raman spectra of two complexes are reported and characterised by their metal-ligand sensitive absorptions. The frequencies of these assignments are used to calculate the frequencies of the infrared active metal-ligand vibrations. The frequencies assigned to the metal-ligand vibration of diarsine complexes are analysed in terms of the change in electronic configuration of the transition element. It is hoped that the inferences obtained from the vibrational may elucidate the bonding in these complexes. These results are checked against those derived from the data given in x-ray and electron spin resonance (e.s.r.) spectral studies. The second main section gives a description of the crystal and the molecular structure of cobalt triethylenetetramine glycinato dichloride, -2'-(RS)-(Co(trien)(gly)) Cl2.H2O. It belongs to a series of cobalt(III) polyamino compounds containing asymmetric nitrogen centres. These polyamino complexes can display various forms of isomerism in their topological and confrontational arrangements. The reasons for the study of this particular complex are many, and are given in the Introduction to Crystal Studies. However, the primary reason is that precise molecular geometry is of great importance as a check on the predicted molecular geometry derived from energy minimisation techniques

Moderate deviations via cumulants

The purpose of the present paper is to establish moderate deviation principles for a rather general class of random variables fulfilling certain bounds of the cumulants. We apply a celebrated lemma of the theory of large deviations probabilities due to Rudzkis, Saulis and Statulevicius. The examples of random objects we treat include dependency graphs, subgraph-counting statistics in Erd\H{o}s-R\'enyi random graphs and $U$-statistics. Moreover, we prove moderate deviation principles for certain statistics appearing in random matrix theory, namely characteristic polynomials of random unitary matrices as well as the number of particles in a growing box of random determinantal point processes like the number of eigenvalues in the GUE or the number of points in Airy, Bessel, and $\sin$ random point fields.Comment: 24 page

Phase relationships between two or more interacting processes from one-dimensional time series. I. Basic theory.

A general approach is developed for the detection of phase relationships between two or more different oscillatory processes interacting within a single system, using one-dimensional time series only. It is based on the introduction of angles and radii of return times maps, and on studying the dynamics of the angles. An explicit unique relationship is derived between angles and the conventional phase difference introduced earlier for bivariate data. It is valid under conditions of weak forcing. This correspondence is confirmed numerically for a nonstationary process in a forced Van der Pol system. A model describing the angles’ behavior for a dynamical system under weak quasiperiodic forcing with an arbitrary number of independent frequencies is derived