Fractal dimension of brittle fracture

Summary Using a worldwide catalog of earthquakes we analyze the distribution of distances between pairs of earthquake hypocenters to determine the spatial fractal dimension of an earthquake fracture. As the time span of the catalog increases, asymptotically reaches the value 2.1–2.2 for shallow earthquakes. Approximately the same asymptotic value of dimension is obtained for a catalog of earthquakes with aftershocks removed. This value is compared with other known determinations of for brittle fractures. The fractal dimension declines to 1.8–1.9 for intermediate events (depth interval 71–280 km) and to about 1.5–1.6 for deeper events. Taking into account various possible errors and biases, we conclude that the fractal dimension of brittle shear fracture in rocks is 2.20 ± 0.05.     

The simplified partial digest problem: Approximation and a graph-theoretic model

The goal of the simplified partial digest problem (SPDP) is motivated by the reconstruction of the linear structure of a DNA chain with respect to a given nucleotide pattern, based on the multiset of distances between the adjacent patterns (interpoint distances) and the multiset of distances between each pattern and the two unlabeled endpoints of the DNA chain (end distances). We consider optimization versions of the problem, called SPDP-Min and SPDP-Max. The aim of SPDP-Min (SPDP-Max) is to find a DNA linear structure with the same multiset of end distances and the minimum (maximum) number of incorrect (correct) interpoint distances. Results are presented on the worst-case efficiency of approximation algorithms for these problems. We suggest a graph-theoretic model for SPDP-Min and SPDP-Max, which can be used to reduce the search space for an optimal solution in either of these problems. We also present heuristic polynomial time algorithms based on this model. In computational experiments with randomly generated and real-life input data, our best algorithm delivered an optimal solution in 100% of the instances for a number of restriction sites not greater than 50.     

A multivariate nonparametric test of independence

A new nonparametric approach to the problem of testing the joint independence of two or more random vectors in arbitrary dimension is developed based on a measure of association determined by interpoint distances. The population independence coefficient takes values between 0 and 1, and equals zero if and only if the vectors are independent. We show that the corresponding statistic has a finite limit distribution if and only if the two random vectors are independent; thus we have a consistent test for independence. The coefficient is an increasing function of the absolute value of product moment correlation in the bivariate normal case, and coincides with the absolute value of correlation in the Bernoulli case. A simple modification of the statistic is affine invariant. The independence coefficient and the proposed statistic both have a natural extension to testing the independence of several random vectors. Empirical performance of the test is illustrated via a comparative Monte Carlo study.     

Complex Region Spatial Smoother (CReSS)

Conventional smoothing over complicated coastal and island regions may result in errors across boundaries, due to the use of Euclidean distances to measure interpoint similarity. The new Complex Region Spatial Smoother (CReSS) method presented here uses estimated geodesic distances, model averaging, and a local radial basis function to provide improved smoothing over complex domains. CReSS is compared, via simulation, with recent related smoothing techniques, Thin Plate Splines (TPS), geodesic low rank TPS (GLTPS), and the Soap film smoother (SOAP). The GLTPS method cannot be used in areas with islands and SOAP can be hard to parameterize. CReSS is comparable with, if not better than, all considered methods on a range of simulations. Supplementary materials for this article are available online.     

A space-time clustering model for historical earthquakes

This paper describes a generalization of Hawkes' self-exciting process in which each event creates a process of offspring with conditional intensity governed by a diffusion kernel. The process may be described as a space-time branching process with immigration, the immigration representing a background series of independent events. The model can be fitted by likelihood methods. As an illustration it is fitted to the catalogue of historical Italian earthquakes.     

Time dynamics in the point process modeling of seismicity of Aswan area (Egypt)

The seismicity observed in the Aswan area (Egypt) between 1986 and 2003 was deeply investigated by means of time-fractal methods. The time dynamics of the aftershock-depleted seismicity, investigated by means of the Allan Factor, reveals that the time-clustering behavior for events occurred at shallow depths (down to 12.5 km from the ground) as well as for events occurred at larger depths (from 15 km down to 27.5 km) does not depend on the ordering of the interevent times but mainly on the shape of the probability density functions of the interevent intervals. Moreover, deep seismicity is more compatible with a Poissonian dynamics than shallow seismicity that is definitely more super-Poissonian. Additionally, the set of shallow events shows a periodicity at about 402 days, which could be consistent with the cyclic loading/unloading operations of the Lake Naser Dam. Such findings contribute to better characterize the seismicity of the Aswan area, which is one of the most interesting water reservoirs in the world, featured by reservoir-induced earthquakes.     

Markov modulated Poisson process associated with state-dependent marks and its applications to the deep earthquakes

In this paper, we introduce one type of Markov-Modulated Poisson Process (MMPP) whose arrival times are associated with state-dependent marks. Statistical inference problems including the derivation of the likelihood, parameter estimation through EM algorithm and statistical inference on the state process and the observed point process are addressed. A goodness-of-fit test is proposed for MMPP with state-dependent marks by utilizing the theories of rescaling marked point process. We also perform some numerical simulations to indicate the effects of different marks on the efficiencies and accuracies of MLE. The effects of the attached marks on the estimation tend to be weakened for increasing data sizes. Then we apply these methods to characterize the occurrence patterns of New Zealand deep earthquakes through a second-order MMPP with state-dependent marks. In this model, the occurrence times and magnitudes of the deep earthquakes are associated with two levels of seismicity which evolves in terms of an unobservable two-state Markov chain.     

On 5-isosceles sets in the plane, with linear restriction

A finite planar set is k-isosceles for k≥3 if every k-point subset of the set contains a point equidistant from two others. We show that an 8-set on a line is 5-isosceles if and only if its adjacent interpoint distances are equal to each other, and no 5-isosceles 9-set has 9 points on a line. We also show that the maximum 5-isosceles set with 8 points on a line contains at most 10 points.     

Two-person repeated games with finite automata

We study two-person repeated games in which a player with a restricted set of strategies plays against an unrestricted player. An exogenously given bound on the complexity of strategies, which is measured by the size of the smallest automata that implement them, gives rise to a restriction on strategies available to a player.  We examine the asymptotic behavior of the set of equilibrium payoffs as the bound on the strategic complexity of the restricted player tends to infinity, but sufficiently slowly. Results from the study of zero sum case provide the individually rational payoff levels. Received February 1997/revised version March 2000     

High powers of random elements of compact Lie groups

Summary. If a random unitary matrix is raised to a sufficiently high power, its eigenvalues are exactly distributed as independent, uniform phases. We prove this result, and apply it to give exact asymptotics of the variance of the number of eigenvalues of falling in a given arc, as the dimension of tends to infinity. The independence result, it turns out, extends to arbitrary representations of arbitrary compact Lie groups. We state and prove this more general theorem, paying special attention to the compact classical groups and to wreath products. This paper is excerpted from the author's doctoral thesis, [9]. Received: 15 October 1995 / In revised form: 7 March 1996     

Data Clustering Based on Maximization of Outlier Factor

There exist many data clustering algorithms, but they can not adequately handle the number of clusters or cluster shapes. Their performance mainly depends on a choice of algorithm parameters. Our approach to data clustering and algorithm does not require the parameter choice; it can be treated as a natural adaptation to the existing structure of distances between data points. The outlier factor introduced by the author specifies a degree of being an outlier for each data point. The outlier factor notion is based on the difference between the frequency distribution of interpoint distances in a given dataset and the corresponding distribution of uniformly distributed points. Then data clusters can be determined by maximizing the outlier factor function. The data points in dataset are divided into clusters according to the attractor regions of local optima. An experimental evaluation of the proposed algorithm shows that the proposed method can identify complex cluster shapes. Key advantages of the approach are: good clustering properties for datasets with comparatively large amount of noise (an additional data points), and an absence of important parameters which adequate choice determines the quality of results.     

A lower bound for Hausdorff dimensions of harmonic measures on negatively curved manifolds

Employing the methods of [KL], a lower bound for Hausdorff dimension of harmonic measures on negatively curved manifolds is derived yielding, in particular, that if the curvature tends to a constant then the above Hausdorff dimension tends to the dimension of the sphere at infinity. Supported by U.S.-Israel BSF.     

Asymptotic properties of conditional quantiles of the Cauchy distribution in Hilbert space

In this paper, we consider the conditional distributions that are induced by finite-dimensional projections of a σ-additive Cauchy measure defined in a real Hilbert space. In the case where the dimension of projections tends to infinity, we establish the almost sure convergence of “conforming” sequences of finite-dimensional conditional quantiles and prove the strong law of large numbers for the triangular array scheme applied to a family of conditional distributions. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

Hybrid kernel estimates of space–time earthquake occurrence rates using the epidemic-type aftershock sequence model

The following steps are suggested for smoothing the occurrence patterns in a clustered space–time process, in particular the data from an earthquake catalogue. First, the original data is fitted by a temporal version of the ETAS model, and the occurrence times are transformed by using the cumulative form of the fitted ETAS model. Then the transformed data (transformed times and original locations) is smoothed by a space–time kernel with bandwidth obtained by optimizing a naive likelihood cross-validation. Finally, the estimated intensity for the original data is obtained by back-transforming the estimated intensity for the transformed data. This technique is used to estimate the intensity for earthquake occurrence data for associated with complex sequences of events off the East Coast of Tohoku district, northern Japan. The intensity so obtained is compared to the conditional intensity estimated from a full space–time ETAS model for the same data.     

Pricing public health care services using DEA: Methodology versus politics

The New Zealand public health sector has used DEA since 1997 to identify efficient expenditure levels to set prices for hospital services at the DRG level. Given the size of the expenditure (NZ$ 2.6 billion), considerable robustness was required for the results and sophistication of the models/process. While the model development and application appeared to be successful, politics overturned the results in the short run. In the longer term, the results have been shown to be reasonably robust and have become a base-line reference for future developments. As such, this paper reports a relatively successful transfer of theory into practice.     

Scaling Methods and Approximative Equations for Homogeneous Reaction—Diffusion Systems and Applications to Epidemics

Summary. We consider a reaction-diffusion equation that is homogeneous of degree one. This homogeneity is a symmetry. The dynamics is factorized into trivial evolution due to symmetry and nontrivial behavior by a projection to an appropriate hypermanifold. The resulting evolution equations are rather complex. We examine the bifurcation behavior of a stationary point of the projected system. For these purposes we develop techniques for dimension reduction similar to the Ginzburg-Landau (GL) approximation, the modulation equations. Since we are not in the classical GL situation, the remaining approximative equations have a quadratic nonlinearity and the amplitude does not scale with ε but with ε ² where ε ² denotes the bifurcation parameter. Moreover, the symmetry requires that not only one but two equations are necessary to describe the behavior of the system. We investigate traveling fronts for the modulation equations. This result is used to analyze an epidemic model. Received April 9, 1996; second revision received January 3, 1997; final revision received October 7, 1997; accepted January 19, 1998     

Dynamical Systems and Adaptive Timestepping in ODE Solvers

Initial value problems for ODEs are often solved numerically using adaptive timestepping algorithms. These algorithms are controlled by a user-defined tolerance which bounds from above the estimated error committed at each step. We formulate a large class of such algorithms as discrete dynamical systems which are discontinuous and of higher dimension than the underlying ODE. By assuming sufficiently strong finite-time convergence results on some neighbourhood of an attractor of the ODE we prove existence and upper semicontinuity results for a nearby numerical attractor as the tolerance tends to zero.This assumption of sufficiently strong finite-time convergence results is then examined for adaptive algorithms that use a pair of explicit Runge-Kutta methods of different order to estimate the one-step error. For arbitrary Runge-Kutta pairs the necessary finite-time convergence results fail to hold on a set of points in the phase space that includes all the equilibria of the ODE. Therefore, in general, the asymptotic convergence results cannot be applied to attractors containing equilibria. However, for a particular class of Runge-Kutta pairs, the finite-time convergence results can be strengthened to include neighbourhoods of equilibrium points for which the Jacobian is invertible.     

Landmarks in graphs

Navigation can be studied in a graph-structured framework in which the navigating agent (which we shall assume to be a point robot) moves from node to node of a “graph space”. The robot can locate itself by the presence of distinctively labeled “landmark” nodes in the graph space. For a robot navigating in Euclidean space, visual detection of a distinctive landmark provides information about the direction to the landmark, and allows the robot to determine its position by triangulation. On a graph, however, there is neither the concept of direction nor that of visibility. Instead, we shall assume that a robot navigating on a graph can sense the distances to a set of landmarks.

Evidently, if the robot knows its distances to a sufficiently large set of landmarks, its position on the graph is uniquely determined. This suggests the following problem: given a graph, what are the fewest number of landmarks needed, and where should they be located, so that the distances to the landmarks uniquely determine the robot's position on the graph? This is actually a classical problem about metric spaces. A minimum set of landmarks which uniquely determine the robot's position is called a “metric basis”, and the minimum number of landmarks is called the “metric dimension” of the graph. In this paper we present some results about this problem. Our main new results are that the metric dimension of a graph with n nodes can be approximated in polynomial time within a factor of O(log n), and some properties of graphs with metric dimension two.     

On the metric dimension of HDN

The concept of metric basis is useful for robot navigation. In graph G, a robot is aware of its current location by sending signals to obtain the distances between itself and the landmarks in G. Its position is determined uniquely in G if it knows its distances to sufficiently many landmarks. The metric basis of G is defined as the minimum set of landmarks such that all other vertices in G can be uniquely determined and the metric dimension of G is defined as the cardinality of the minimum set of landmarks. The major contribution of this paper is that we have partly solved the open problem proposed by Manuel et al. [9], by proving that the metric dimension of $\text{HDN1}(n)$ and $\text{HDN2}(n)$ are either 3 or 4. However, the problem of finding the exact metric dimension of HDN networks is still open.