Integration of Truncated Distributions in Exponential Family with Simulation Models of Logistics and Supply Chain Management

Truncated distributions of the exponential family have great influence in the simulation models. This paper discusses the truncated Weibull distribution specifically. The truncation of the distribution is achieved by the Maximum Likelihood Estimation method or combined with the expectation and variance expressions. After the fitting of distribution, the goodness-of-fit tests (the Chi-Square test and the Kolmogorov-Smirnov test) are executed to rule out the rejected hypotheses. Finally the distributions are integrated in various simulation models, e. g. shipment consolidation model, to compare the influence of truncated and original versions of Weibull distribution on the model.

hp-dGFEM for Second-Order Elliptic Problems in Plyhedra II: Exponential Convergence

The goal of this paper is to establish exponential convergence of $hp$-version interior penalty (IP) discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with homogeneous Dirichlet boundary conditions and piecewise analytic data in three-dimensional polyhedral domains. More precisely, we shall analyze the convergence of the $hp$-IP dG methods considered in [D. Schötzau, C. Schwab, T. P. Wihler, SIAM J. Numer. Anal., 51 (2013), pp. 1610--1633] based on axiparallel $\sigma$-geometric anisotropic meshes and $\bm{s}$-linear anisotropic polynomial degree distributions.

Non-exponential return time distributions for vorticity extremes explained by fractional Poisson processes

Serial correlation of extreme midlatitude cyclones observed at the storm track exits is explained by deviations from a Poisson process. To model these deviations, we apply fractional Poisson processes (FPPs) to extreme midlatitude cyclones, which are defined by the 850 hPa relative vorticity of the ERA interim reanalysis during boreal winter (DJF) and summer (JJA) seasons. Extremes are defined by a 99% quantile threshold in the grid-point time series. In general, FPPs are based on long-term memory and lead to non-exponential return time distributions. The return times are described by a Weibull distribution to approximate the Mittag–Leffler function in the FPPs. The Weibull shape parameter yields a dispersion parameter that agrees with results found for midlatitude cyclones. The memory of the FPP, which is determined by detrended fluctuation analysis, provides an independent estimate for the shape parameter. Thus, the analysis exhibits a concise framework of the deviation from Poisson statistics (by a dispersion parameter), non-exponential return times and memory (correlation) on the basis of a single parameter. The results have potential implications for the predictability of extreme cyclones.

Quasi-Self-Dual Exponential Lévy Processes

The important application of semistatic hedging in financial markets naturally leads to the notion of quasi--self-dual processes. The focus of our study is to give new characterizations of quasi--self-duality. We analyze quasi--self-dual Lévy driven markets which do not admit arbitrage opportunities and derive a set of equivalent conditions for the stochastic logarithm of quasi--self-dual martingale models. Since for nonvanishing order parameter two martingale properties have to be satisfied simultaneously, there is a nontrivial relation between the order and shift parameter representing carrying costs in financial applications. This leads to an equation containing an integral term which has to be inverted in applications. We first discuss several important properties of this equation and, for some well-known Lévy-driven models, we derive a family of closed-form inversion formulae.

Random Sampling with Interspike-Intervals of the Exponential Integrate and Fire Neuron: A Computational Interpretation of UP-States

Oscillations between high and low values of the membrane potential (UP and DOWN states respectively) are an ubiquitous feature of cortical neurons during slow wave sleep and anesthesia. Nevertheless, a surprisingly small number of quantitative studies have been conducted only that deal with this phenomenon’s implications for computation. Here we present a novel theory that explains on a detailed mathematical level the computational benefits of UP states. The theory is based on random sampling by means of interspike intervals (ISIs) of the exponential integrate and fire (EIF) model neuron, such that each spike is considered a sample, whose analog value corresponds to the spike’s preceding ISI. As we show, the EIF’s exponential sodium current, that kicks in when balancing a noisy membrane potential around values close to the firing threshold, leads to a particularly simple, approximative relationship between the neuron’s ISI distribution and input current. Approximation quality depends on the frequency spectrum of the current and is improved upon increasing the voltage baseline towards threshold. Thus, the conceptually simpler leaky integrate and fire neuron that is missing such an additional current boost performs consistently worse than the EIF and does not improve when voltage baseline is increased. For the EIF in contrast, the presented mechanism is particularly effective in the high-conductance regime, which is a hallmark feature of UP-states. Our theoretical results are confirmed by accompanying simulations, which were conducted for input currents of varying spectral composition. Moreover, we provide analytical estimations of the range of ISI distributions the EIF neuron can sample from at a given approximation level. Such samples may be considered by any algorithmic procedure that is based on random sampling, such as Markov Chain Monte Carlo or message-passing methods. Finally, we explain how spike-based random sampling relates to existing computational theories about UP states during slow wave sleep and present possible extensions of the model in the context of spike-frequency adaptation.

Mixed effects model in negative exponential curve

Despite many researches on development in education and psychology, not often is the methodology tested with real data. A major barrier to test the growth model is that the design of study includes repeated observations and the nature of the growth is nonlinear. The repeat measurements on a nonlinear model require sophisticated statistical methods. In this study, we present mixed effects model in a negative exponential curve to describe the development of children's reading skills. This model can describe the nature of the growth on children's reading skills and account for intra-individual and inter-individual variation. We also apply simple techniques including cross-validation, regression, and graphical methods to determine the most appropriate curve for data, to find efficient initial values of parameters, and to select potential covariates. We illustrate with an example that motivated this research: a longitudinal study of academic skills from grade 1 to grade 12 in Connecticut public schools. ^

A SIMULATION STUDY OF THE SIZE AND POWER OF SOME TWO-SAMPLE TESTS FOR UNEQUALLY CENSORED SURVIVAL DATA (RANK TESTS, LOGRANK, PROPORTIONAL HAZARDS, WILCOXON TESTS, EXPONENTIAL DISTRIBUTION)

Sizes and power of selected two-sample tests of the equality of survival distributions are compared by simulation for small samples from unequally, randomly-censored exponential distributions. The tests investigated include parametric tests (F, Score, Likelihood, Asymptotic), logrank tests (Mantel, Peto-Peto), and Wilcoxon-Type tests (Gehan, Prentice). Equal sized samples, n = 18, 16, 32 with 1000 (size) and 500 (power) simulation trials, are compared for 16 combinations of the censoring proportions 0%, 20%, 40%, and 60%. For n = 8 and 16, the Asymptotic, Peto-Peto, and Wilcoxon tests perform at nominal 5% size expectations, but the F, Score and Mantel tests exceeded 5% size confidence limits for 1/3 of the censoring combinations. For n = 32, all tests showed proper size, with the Peto-Peto test most conservative in the presence of unequal censoring. Powers of all tests are compared for exponential hazard ratios of 1.4 and 2.0. There is little difference in power characteristics of the tests within the classes of tests considered. The Mantel test showed 90% to 95% power efficiency relative to parametric tests. Wilcoxon-type tests have the lowest relative power but are robust to differential censoring patterns. A modified Peto-Peto test shows power comparable to the Mantel test. For n = 32, a specific Weibull-exponential comparison of crossing survival curves suggests that the relative powers of logrank and Wilcoxon-type tests are dependent on the scale parameter of the Weibull distribution. Wilcoxon-type tests appear more powerful than logrank tests in the case of late-crossing and less powerful for early-crossing survival curves. Guidelines for the appropriate selection of two-sample tests are given. ^

A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates

We present a compact formula for the derivative of a 3-D rotation matrix with respect to its exponential coordinates. A geometric interpretation of the resulting expression is provided, as well as its agreement with other less-compact but better-known formulas. To the best of our knowledge, this simpler formula does not appear anywhere in the literature. We hope by providing this more compact expression to alleviate the common pressure to reluctantly resort to alternative representations in various computational applications simply as a means to avoid the complexity of differential analysis in exponential coordinates.

The Malthusian parameter of ascents: What prevents the exponential increase of one’s ancestors?

The reason that the indefinite exponential increase in the number of one’s ancestors does not take place is found in the law of sibling interference, which can be expressed by the following simple equation:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}\begin{matrix}{\mathit{N}}_{{\mathit{n}}} \enskip & \\ {\mathit{{\blacksquare}}} \enskip & \\ {\mathit{ASZ}} \enskip & \end{matrix} {\mathrm{\hspace{.167em}{\times}\hspace{.167em}2\hspace{.167em}=\hspace{.167em}}}{\mathit{N_{n+1},}}\end{equation*}\end{document} where Nn is the number of ancestors in the nth generation, ASZ is the average sibling size of these ancestors, and Nn+1 is the number of ancestors in the next older generation (n + 1). Accordingly, the exponential increase in the number of one’s ancestors is an initial anomaly that occurs while ASZ remains at 1. Once ASZ begins to exceed 1, the rate of increase in the number of ancestors is progressively curtailed, falling further and further behind the exponential increase rate. Eventually, ASZ reaches 2, and at that point, the number of ancestors stops increasing for two generations. These two generations, named AN SA and AN SA + 1, are the most critical in the ancestry, for one’s ancestors at that point come to represent all the progeny-produced adults of the entire ancestral population. Thereafter, the fate of one’s ancestors becomes the fate of the entire population. If the population to which one belongs is a successful, slowly expanding one, the number of ancestors would slowly decline as you move toward the remote past. This is because ABZ would exceed 2. Only when ABZ is less than 2 would the number of ancestors increase beyond the AN SA and AN SA + 1 generations. Since the above is an indication of a failing population on the way to extinction, there had to be the previous AN SA involving a far greater number of individuals for such a population. Simulations indicated that for a member of a continuously successful population, the AN SA ancestors might have numbered as many as 5.2 million, the AN SA generation being the 28th generation in the past. However, because of the law of increasingly irrelevant remote ancestors, only a very small fraction of the AN SA ancestors would have left genetic traces in the genome of each descendant of today.

Exponential increases of RNA virus fitness during large population transmissions.

The great adaptability shown by RNA viruses is a consequence of their high mutation rates. Here we investigate the kinetics of virus fitness gains during repeated transfers of large virus populations in cell culture. Results always show that fitness increases exponentially. Low fitness clones exhibit regular increases observed as biphasic periods of exponential evolutionary improvement, while neutral clones show monophasic kinetics. These results are significant for RNA virus epidemiology, optimal handling of attenuated live virus vaccines, and routine laboratory procedures.

Exponential decay of spin-spin correlation between distant defect states produced by contour hydrogenation of polycyclic aromatic hydrocarbon molecules

The first few low-lying spin states of alternant polycyclic aromatic hydrocarbon (PAH) molecules of several shapes showing defect states induced by contour hydrogenation have been studied both by ab initio methods and by a precise numerical solution of Pariser-Parr-Pople (PPP) interacting model. In accordance with Lieb's theorem, the ground state shows a spin multiplicity equal to one for balanced molecules, and it gets larger values for imbalanced molecules (that is, when the number of π electrons on both subsets is not equal). Furthermore, we find a systematic decrease of the singlet-triplet splitting as a function of the distance between defects, regardless of whether the ground state is singlet or triplet. For example, a splitting smaller than 0.001 eV is obtained for a medium size C46H28 PAH molecule (di-hydrogenated [11]phenacene) showing a singlet ground state. We conclude that π electrons unbound by lattice defects tend to remain localized and unpaired even when long-range Coulomb interaction is taken into account. Therefore they show a biradical character (polyradical character for more than two defects) and should be studied as two or more local doublets. The implications for electron transport are potentially important since these unpaired electrons can trap traveling electrons or simply flip their spin at a very small energy cost.

On the existence of exponential polynomials with prefixed gaps

This paper shows that the conjecture of Lapidus and Van Frankenhuysen on the set of dimensions of fractality associated with a nonlattice fractal string is true in the important special case of a generic nonlattice self-similar string, but in general is false. The proof and the counterexample of this have been given by virtue of a result on exponential polynomials P(z), with real frequencies linearly independent over the rationals, that establishes a bound for the number of gaps of RP, the closure of the set of the real projections of its zeros, and the reason for which these gaps are produced.

On the Result of Invariance of the Closure Set of the Real Projections of the Zeros of an Important Class of Exponential Polynomials

In this paper we provide the proof of a practical point-wise characterization of the set RP defined by the closure set of the real projections of the zeros of an exponential polynomial P(z) = Σn j=1 cjewjz with real frequencies wj linearly independent over the rationals. As a consequence, we give a complete description of the set RP and prove its invariance with respect to the moduli of the c′ js, which allows us to determine exactly the gaps of RP and the extremes of the critical interval of P(z) by solving inequations with positive real numbers. Finally, we analyse the converse of this result of invariance.