Overlap properties of geometric expanders

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of the simplices induced by the images of its hyperedges. In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence $\{H_n\}_{n=1}^\infty$ of arbitrarily large $(d+1)$-uniform hypergraphs with bounded degree, for which $\inf_{n\ge 1} c(H_n)>0$. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of $(d+1)$-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant $c=c(d)$. We also show that, for every $d$, the best value of the constant $c=c(d)$ that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete $(d+1)$-uniform hypergraphs with $n$ vertices, as $n\rightarrow\infty$. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any $d$ and any $\epsilon>0$, there exists $K=K(\epsilon,d)\ge d+1$ satisfying the following condition. For any $k\ge K$, for any point $q \in \mathbb{R}^d$ and for any finite Borel measure $\mu$ on $\mathbb{R}^d$ with respect to which every hyperplane has measure $0$, there is a partition $\mathbb{R}^d=A_1 \cup \ldots \cup A_{k}$ into $k$ measurable parts of equal measure such that all but at most an $\epsilon$-fraction of the $(d+1)$-tuples $A_{i_1},\ldots,A_{i_{d+1}}$ have the property that either all simplices with one vertex in each $A_{i_j}$ contain $q$ or none of these simplices contain $q$

Probabilistic models of information retrieval based on measuring the divergence from randomness

We introduce and create a framework for deriving probabilistic models of Information Retrieval. The models are nonparametric models of IR obtained in the language model approach. We derive term-weighting models by measuring the divergence of the actual term distribution from that obtained under a random process. Among the random processes we study the binomial distribution and Bose--Einstein statistics. We define two types of term frequency normalization for tuning term weights in the document--query matching process. The first normalization assumes that documents have the same length and measures the information gain with the observed term once it has been accepted as a good descriptor of the observed document. The second normalization is related to the document length and to other statistics. These two normalization methods are applied to the basic models in succession to obtain weighting formulae. Results show that our framework produces different nonparametric models forming baseline alternatives to the standard tf-idf model

Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compared with various geometric quantities such as volume, diameter and length of the shortest periodic geodesic of the manifolds. The deviation of low-lying eigenvalue spectra of manifolds converging to a cusped hyperbolic manifold from the asymptotic distribution has been measured by $\zeta-$ function and spectral distance.Comment: 19 pages, 18 EPS figures and 2 GIF figures (fig.10) Description of cusped manifolds in section 2 is correcte

Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that

We report on some recent developments in the search for optimal network topologies. First we review some basic concepts on spectral graph theory, including adjacency and Laplacian matrices, and paying special attention to the topological implications of having large spectral gaps. We also introduce related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we discuss two different dynamical feautures of networks: synchronizability and flow of random walkers and so that they are optimized if the corresponding Laplacian matrix have a large spectral gap. From this, we show, by developing a numerical optimization algorithm that maximum synchronizability and fast random walk spreading are obtained for a particular type of extremely homogeneous regular networks, with long loops and poor modular structure, that we call entangled networks. These turn out to be related to Ramanujan and Cage graphs. We argue also that these graphs are very good finite-size approximations to Bethe lattices, and provide almost or almost optimal solutions to many other problems as, for instance, searchability in the presence of congestion or performance of neural networks. Finally, we study how these results are modified when studying dynamical processes controlled by a normalized (weighted and directed) dynamics; much more heterogeneous graphs are optimal in this case. Finally, a critical discussion of the limitations and possible extensions of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted for pub. in JSTA

Regular graphs of large girth and arbitrary degree

For every integer d > 9, we construct infinite families {G_n}_n of d+1-regular graphs which have a large girth > log_d |G_n|, and for d large enough > 1,33 log_d |G_n|. These are Cayley graphs on PGL_2(q) for a special set of d+1 generators whose choice is related to the arithmetic of integral quaternions. These graphs are inspired by the Ramanujan graphs of Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is prime. When d is not equal to the power of an odd prime, this improves the previous construction of Imrich in 1984 where he obtained infinite families {I_n}_n of d+1-regular graphs, realized as Cayley graphs on SL_2(q), and which are displaying a girth > 0,48 log_d |I_n|. And when d is equal to a power of 2, this improves a construction by Morgenstern in 1994 where certain families {M_n}_n of 2^k+1-regular graphs were shown to have a girth > 2/3 log_d |M_n|.Comment: (15 pages) Accepted at Combinatorica. Title changed following referee's suggestion. Revised version after reviewing proces

Blood ties: ABO is a trans-species polymorphism in primates

The ABO histo-blood group, the critical determinant of transfusion incompatibility, was the first genetic polymorphism discovered in humans. Remarkably, ABO antigens are also polymorphic in many other primates, with the same two amino acid changes responsible for A and B specificity in all species sequenced to date. Whether this recurrence of A and B antigens is the result of an ancient polymorphism maintained across species or due to numerous, more recent instances of convergent evolution has been debated for decades, with a current consensus in support of convergent evolution. We show instead that genetic variation data in humans and gibbons as well as in Old World Monkeys are inconsistent with a model of convergent evolution and support the hypothesis of an ancient, multi-allelic polymorphism of which some alleles are shared by descent among species. These results demonstrate that the ABO polymorphism is a trans-species polymorphism among distantly related species and has remained under balancing selection for tens of millions of years, to date, the only such example in Hominoids and Old World Monkeys outside of the Major Histocompatibility Complex.Comment: 45 pages, 4 Figures, 4 Supplementary Figures, 5 Supplementary Table

Chaotic maps and flows: Exact Riemann-Siegel lookalike for spectral fluctuations

To treat the spectral statistics of quantum maps and flows that are fully chaotic classically, we use the rigorous Riemann-Siegel lookalike available for the spectral determinant of unitary time evolution operators $F$. Concentrating on dynamics without time reversal invariance we get the exact two-point correlator of the spectral density for finite dimension $N$ of the matrix representative of $F$, as phenomenologically given by random matrix theory. In the limit $N\to\infty$ the correlator of the Gaussian unitary ensemble is recovered. Previously conjectured cancellations of contributions of pseudo-orbits with periods beyond half the Heisenberg time are shown to be implied by the Riemann-Siegel lookalike

Structure and stability of finite gold nanowires

Finite gold nanowires containing less than 1000 atoms are studied using the molecular dynamics simulation method and embedded atom potential. Nanowires with the face-centered cubic structure and the (111) oriented cross-section are prepared at T=0 K. After annealing and quenching the structure and vibrational properties of nanowires are studied at room temperature. Several of these nanowires form multi-walled structures of lasting stability. They consist of concentrical cylindrical sheets and resemble multi-walled carbon nanotubes. Vibrations are investigated by diagonalization of the dynamical matrix. It was found that several percents of vibrational modes are unstable because of uncompleted restructuring of initial fcc nanowires.Comment: 4 figures in gif forma

On multiplicities in length spectra of arithmetic hyperbolic three-orbifolds

Asymptotic laws for mean multiplicities of lengths of closed geodesics in arithmetic hyperbolic three-orbifolds are derived. The sharpest results are obtained for non-compact orbifolds associated with the Bianchi groups SL(2,o) and some congruence subgroups. Similar results hold for cocompact arithmetic quaternion groups, if a conjecture on the number of gaps in their length spectra is true. The results related to the groups above give asymptotic lower bounds for the mean multiplicities in length spectra of arbitrary arithmetic hyperbolic three-orbifolds. The investigation of these multiplicities is motivated by their sensitive effect on the eigenvalue spectrum of the Laplace-Beltrami operator on a hyperbolic orbifold, which may be interpreted as the Hamiltonian of a three-dimensional quantum system being strongly chaotic in the classical limit.Comment: 29 pages, uuencoded ps. Revised version, to appear in NONLINEARIT

Emergent percutaneous cardiopulmonary bypass in patients having cardiovascular collapse in the cardiac catheterization laboratory

Percutaneous cardiopulmonary bypass (PCB) was instituted in 30 initially stable patients who developed either cardiac arrest refractory to resuscitation (n = 7) or cardiogenic shock (mean arterial blood pressure <50 mm Hg unresponsive to fluid resuscitation or vasopressors) (n = 23) after a cathetertzation laboratory complication. Events leading to collapse included abrupt closure during percutaneous transluminal coronary angioplasty (PTCA) (n = 22), complications from diagnostic cardiac catheterization (n = 6), left ventricular perforation during mural valvuloplasty (n = 1), and right ventricular perforation during pericardiocentesis (n = 1). PCB was initiated within 20 minutes of cardiovascular collapse in 83% of patients (arrest: 21 +/- 13 minutes [range 10 to 50]; and shock: 17 +/- 6 minutes [range 10 to 30]). Mean arterial blood pressure increased on PCB from 0 to 56 mm Hg in patients with cardiac arrest and from 37 to 63 mm Hg in those with cardiogenic shock at mean PCB flow rates of 2.5 to 5.0 liters/min. Subsequent therapy on PCB included emergent cardiac surgery (n = 14), PTCA (n = 13) and medical therapy (n = 3). Six patients (20%) survived to hospital discharge (3 with cardiac surgery, 2 with PTCA, and 1 with medical therapy). All 7 patients with refractory cardiac arrest died despite further interventions on PCB, whereas 6 of 23 (26%) with cardiogenic shock survived to hospital discharge. Thus, in response to cardiovascular collapse in the catheterization laboratory, PCB does not salvage patients who do not regain a stable cardiac rhythm. PCB can stabilize patients who develop cardiogenic shock for further interventions which are lifesaving in only a minority of patients.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31621/1/0000554.pd