The hardness of approximating the boxicity, cubicity and threshold dimension of a graph

A k-dimensional box is the Cartesian product R-1 X R-2 X ... X R-k where each R-i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R-1 X R-2 X ... X R-k where each R-i is a closed interval oil the real line of the form a(i), a(i) + 1]. The cubicity of G, denoted as cub(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-cubes. The threshold dimension of a graph G(V, E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. In this paper we will show that there exists no polynomial-time algorithm for approximating the threshold dimension of a graph on n vertices with a factor of O(n(0.5-epsilon)) for any epsilon > 0 unless NP = ZPP. From this result we will show that there exists no polynomial-time algorithm for approximating the boxicity and the cubicity of a graph on n vertices with factor O(n(0.5-epsilon)) for any epsilon > 0 unless NP = ZPP. In fact all these hardness results hold even for a highly structured class of graphs, namely the split graphs. We will also show that it is NP-complete to determine whether a given split graph has boxicity at most 3. (C) 2010 Elsevier B.V. All rights reserved.

Computing Fractal Dimension of Signals using Multiresolution Box-counting Method

In this paper, we have developed a method to compute fractal dimension (FD) of discrete time signals, in the time domain, by modifying the box-counting method. The size of the box is dependent on the sampling frequency of the signal. The number of boxes required to completely cover the signal are obtained at multiple time resolutions. The time resolutions are made coarse by decimating the signal. The loglog plot of total number of boxes required to cover the curve versus size of the box used appears to be a straight line, whose slope is taken as an estimate of FD of the signal. The results are provided to demonstrate the performance of the proposed method using parametric fractal signals. The estimation accuracy of the method is compared with that of Katz, Sevcik, and Higuchi methods. In ddition, some properties of the FD are discussed.

Constraining the low-energy pion electromagnetic form factor with space-like and phase of time-like data

The Taylor coefficients c and d of the EM form factor of the pion are constrained using analyticity, knowledge of the phase of the form factor in the time-like region, 4m(pi)(2) <= t <= t(in) and its value at one space-like point, using as input the (g - 2) of the muon. This is achieved using the technique of Lagrange multipliers, which gives a transparent expression for the corresponding bounds. We present a detailed study of the sensitivity of the bounds to the choice of time-like phase and errors present in the space-like data, taken from recent experiments. We find that our results constrain c stringently. We compare our results with those in the literature and find agreement with the chiral perturbation-theory results for c. We obtain d similar to O(10) GeV-6 when c is set to the chiral perturbation-theory values.

Theoretical Description of Time-Resolved Photoemission Spectroscopy: Application to Pump-Probe Experiments

The theory for time-resolved, pump-probe, photoemission spectroscopy and other pump-probe experiments is developed. The formal development is completely general, incorporating all of the nonequilibrium effects of the pump pulse and the finite time width of the probe pulse, and including possibilities for taking into account band structure and matrix element effects, surface states, and the interaction of the photoexcited electrons with the system leading to corrections to the sudden approximation. We also illustrate the effects of windowing that arise from the finite width of the probe pulse in a simple model system by assuming the quasiequilibrium approximation.

Influence of time step in the simulation modelling of evapotranspiration

Three simulations of evapotranspiration were done with two values of time step,viz 10 min and one day. Inputs to the model were weather data, including directly measured upward and downward radiation, and soil characteristics. Three soils were used for each simulation. Analysis of the results shows that the time step has a direct influence on the prediction of potential evapotranspiration, but a complex interaction of this effect with the soil moisture characteristic, rate of increase of ground cover and bare soil evaporation determines the actual transpiration predicted. The results indicate that as small a time step as possible should be used in the simulation.

Theoretical description of time-resolved pump/probe photoemission in TaS2: a single-band DFT+DMFT(NRG) study within the quasiequilibrium approximation

In this work, we theoretically examine recent pump/probe photoemission experiments on the strongly correlated charge-density-wave insulator TaS2.We describe the general nonequilibrium many-body formulation of time-resolved photoemission in the sudden approximation, and then solve the problem using dynamical mean-field theory with the numerical renormalization group and a bare density of states calculated from density functional theory including the charge-density-wave distortion of the ion cores and spin-orbit coupling. We find a number of interesting results: (i) the bare band structure actually has more dispersion in the perpendicular direction than in the two-dimensional planes; (ii) the DMFT approach can produce upper and lower Hubbard bands that resemble those in the experiment, but the upper bands will overlap in energy with other higher energy bands; (iii) the effect of the finite width of the probe pulse is minimal on the shape of the photoemission spectra; and (iv) the quasiequilibrium approximation does not fully describe the behavior in this system.

Monte-Carlo estimation of time-dependent statistical characteristics of random dynamical systems

The problem of estimating the time-dependent statistical characteristics of a random dynamical system is studied under two different settings. In the first, the system dynamics is governed by a differential equation parameterized by a random parameter, while in the second, this is governed by a differential equation with an underlying parameter sequence characterized by a continuous time Markov chain. We propose, for the first time in the literature, stochastic approximation algorithms for estimating various time-dependent process characteristics of the system. In particular, we provide efficient estimators for quantities such as the mean, variance and distribution of the process at any given time as well as the joint distribution and the autocorrelation coefficient at different times. A novel aspect of our approach is that we assume that information on the parameter model (i.e., its distribution in the first case and transition probabilities of the Markov chain in the second) is not available in either case. This is unlike most other work in the literature that assumes availability of such information. Also, most of the prior work in the literature is geared towards analyzing the steady-state system behavior of the random dynamical system while our focus is on analyzing the time-dependent statistical characteristics which are in general difficult to obtain. We prove the almost sure convergence of our stochastic approximation scheme in each case to the true value of the quantity being estimated. We provide a general class of strongly consistent estimators for the aforementioned statistical quantities with regular sample average estimators being a specific instance of these. We also present an application of the proposed scheme on a widely used model in population biology. Numerical experiments in this framework show that the time-dependent process characteristics as obtained using our algorithm in each case exhibit excellent agreement with exact results. (C) 2010 Elsevier Inc. All rights reserved.

Post-seizure EEG fractal dimension of first ECT predicts antidepressant response at two weeks

Background: Duration of seizure by itself is an insufficient criterion for a therapeutically adequate seizure in ECT. Therefore, measures of seizure EEG other than its duration need to be explored as indices of seizure adequacy and predictors of treatment response. We measured the EEG seizure using a geometrical method-fractal dimension (FD) and examined if this measure predicted remission. Methods: Data from an efficacy study on melancholic depressives (n = 40) is used for the present exploration. They received thrice or once weekly ECTs, each schedule at two energy levels - high or low energy level. FD was computed for early-, mid- and post-seizure phases of the ictal EEG. Average of the two channels was used for analysis. Results: Two-thirds of the patients (n = 25) were remitted at the end of 2 weeks. As expected, a significantly higher proportion of patients receiving thrice weekly ECT remitted than in patients receiving once weekly ECT. Smaller post-seizure FD at first ECT is the only variable which predicted remission status after six ECTs. within the once weekly ECT group too, smaller post-seizure FD was associated with remission status. Conclusions: Post-seizure FD is proposed as a novel measure of seizure adequacy and predictor of treatment response. Clinical implications: Seizure measures at first ECT may guide selection of ECT schedule to optimize ECT. Limitations: The study examined short term antidepressant effects only. The results may not be generalized to medication-resistant depressives. (C) 1999 Elsevier Science B.V. All rights reserved.

Estimating the Hausdorff-Besicovitch dimension of boundary of basin of attraction in helicopter trim

Helicopter trim involves solution of nonlinear force equilibrium equations. As in many nonlinear dynamic systems, helicopter trim problem can show chaotic behavior. This chaotic behavior is found in the basin of attraction of the nonlinear trim equations which have to be solved to determine the main rotor control inputs given by the pilot. This study focuses on the boundary of the basin of attraction obtained for a set of control inputs. We analyze the boundary by considering it at different magnification levels. The magnified views reveal intricate geometries. It is also found that the basin boundary exhibits the characteristic of statistical self-similarity, which is an essential property of fractal geometries. These results led the authors to investigate the fractal dimension of the basin boundary. It is found that this dimension is indeed greater than the topological dimension. From all the observations, it is evident that the boundary of the basin of attraction for helicopter trim problem is fractal in nature. (C) 2012 Elsevier Inc. All rights reserved.

On the SIG-dimension of trees under the L (a)-metric

Let where be a set of points in d-dimensional space with a given metric rho. For a point let r (p) be the distance of p with respect to rho from its nearest neighbor in Let B(p,r (p) ) be the open ball with respect to rho centered at p and having the radius r (p) . We define the sphere-of-influence graph (SIG) of as the intersection graph of the family of sets Given a graph G, a set of points in d-dimensional space with the metric rho is called a d-dimensional SIG-representation of G, if G is isomorphic to the SIG of It is known that the absence of isolated vertices is a necessary and sufficient condition for a graph to have a SIG-representation under the L (a)-metric in some space of finite dimension. The SIG-dimension under the L (a)-metric of a graph G without isolated vertices is defined to be the minimum positive integer d such that G has a d-dimensional SIG-representation under the L (a)-metric. It is denoted by SIG (a)(G). We study the SIG-dimension of trees under the L (a)-metric and almost completely answer an open problem posed by Michael and Quint (Discrete Appl Math 127:447-460, 2003). Let T be a tree with at least two vertices. For each let leaf-degree(v) denote the number of neighbors of v that are leaves. We define the maximum leaf-degree as leaf-degree(x). Let leaf-degree{(v) = alpha}. If |S| = 1, we define beta(T) = alpha(T) - 1. Otherwise define beta(T) = alpha(T). We show that for a tree where beta = beta (T), provided beta is not of the form 2 (k) - 1, for some positive integer k a parts per thousand yen 1. If beta = 2 (k) - 1, then We show that both values are possible.

Product Dimension of Forests and Bounded Treewidth Graphs

The product dimension of a graph G is defined as the minimum natural number l such that G is an induced subgraph of a direct product of l complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and k-degenerate graphs. We show that every forest on n vertices has product dimension at most 1.441 log n + 3. This improves the best known upper bound of 3 log n for the same due to Poljak and Pultr. The technique used in arriving at the above bound is extended and combined with a well-known result on the existence of orthogonal Latin squares to show that every graph on n vertices with treewidth at most t has product dimension at most (t + 2) (log n + 1). We also show that every k-degenerate graph on n vertices has product dimension at most inverted right perpendicular5.545 k log ninverted left perpendicular + 1. This improves the upper bound of 32 k log n for the same by Eaton and Rodl.

Solution of Time Dependent Joule Heat Equation for a Graphene Sheet Under Thomson Effect

We address a physics-based solution of joule heating phenomenon in a single-layer graphene (SLG) sheet under the presence of Thomson effect. We demonstrate that the temperature in an isotopically pure (containing only C-12) SLG sheet attains its saturation level quicker than when doped with its isotopes (C-13). From the solution of the joule heating equation, we find that the thermal time constant of the SLG sheet is in the order of tenths of a nanosecond for SLG dimensions of a few micrometers. These results have been formulated using the electron interactions with the inplane and flexural phonons to demonstrate a field-dependent Landauer transmission coefficient. We further develop an analytical model of the SLG specific heat using the quadratic (out of plane) phonon band structure over the room temperature. Additionally, we show that a cooling effect in the SLG sheet can be substantially enhanced with the addition of C-13. The methodologies as discussed in this paper can be put forward to analyze the graphene heat spreader theory.

Perception of time-varying signals: timbre and phonetic JND of diphthong

In this paper we propose a linear time-varying model for diphthong synthesis based on linear interpolation of formant frequencies. We, thence, determine the timbre just-noticeable difference (JND) for diphthong /a I/ (as in ‘buy’) with a constant pitch excitation through perception experiment involving four listeners and explore the phonetic JND of the diphthong. Their JND responses are determined using 1-up-3-down procedure. Using the experimental data, we map the timbre JND and phonetic JND onto a 2-D region of percentage change of formant glides. The timbre and phonetic JND contours for constant pitch show that the phonetic JND region encloses timbre JND region and also varies across listeners. The JND is observed to be more sensitive to ending vowel /I/ than starting vowel /a/ in some listeners and dependent on the direction of perturbation of starting and ending vowels.

Estimation of time variant reliability of randomly parametered non-linear vibrating systems

The problem of time variant reliability analysis of randomly parametered and randomly driven nonlinear vibrating systems is considered. The study combines two Monte Carlo variance reduction strategies into a single framework to tackle the problem. The first of these strategies is based on the application of the Girsanov transformation to account for the randomness in dynamic excitations, and the second approach is fashioned after the subset simulation method to deal with randomness in system parameters. Illustrative examples include study of single/multi degree of freedom linear/non-linear inelastic randomly parametered building frame models driven by stationary/non-stationary, white/filtered white noise support acceleration. The estimated reliability measures are demonstrated to compare well with results from direct Monte Carlo simulations. (C) 2014 Elsevier Ltd. All rights reserved.