Mathematics Research Works

Permanent URI for this collectionhttp://hdl.handle.net/1903/1595

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Analysis and correction of compositional bias in sparse sequencing count data
(Springer Nature, 2018-11-06) Kumar, M. Senthil; Slud, Eric V.; Okrah, Kwame; Hicks, Stephanie C.; Hannenhalli, Sridhar; Bravo, Héctor Corrada
Count data derived from high-throughput deoxy-ribonucliec acid (DNA) sequencing is frequently used in quantitative molecular assays. Due to properties inherent to the sequencing process, unnormalized count data is compositional, measuring relative and not absolute abundances of the assayed features. This compositional bias confounds inference of absolute abundances. Commonly used count data normalization approaches like library size scaling/rarefaction/subsampling cannot correct for compositional or any other relevant technical bias that is uncorrelated with library size. We demonstrate that existing techniques for estimating compositional bias fail with sparse metagenomic 16S count data and propose an empirical Bayes normalization approach to overcome this problem. In addition, we clarify the assumptions underlying frequently used scaling normalization methods in light of compositional bias, including scaling methods that were not designed directly to address it.
Analysis of the spectral vanishing method for periodic conservation laws
(Copyright: Society for Industrial and Applied Mathematics, 1989-08) Maday, Yvon; Tadmor, Eitan
Approximate Isomorphism of Metric Structures
(Wiley, 2023-09-05) Hanson, James E.
We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov [2] and by Ben Yaacov, Doucha, Nies, and Tsankov [6], which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach-Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically by a mild generalization of perturbation systems and semantically by certain elementary classes of two-sorted structures that witness approximate isomorphism. As an application, we show that the theory of any -tree or ultrametric space of finite radius is stable, improving a result of Carlisle and Henson [8].
Better Metrics to Automatically Predict the Quality of a Text Summary
(MDPI, 2012-09-26) Rankel, Peter A.; Conroy, John M.; Schlesinger, Judith D.
In this paper we demonstrate a family of metrics for estimating the quality of a text summary relative to one or more human-generated summaries. The improved metrics are based on features automatically computed from the summaries to measure content and linguistic quality. The features are combined using one of three methods—robust regression, non-negative least squares, or canonical correlation, an eigenvalue method. The new metrics significantly outperform the previous standard for automatic text summarization evaluation, ROUGE.
CENTRAL DISCONTINUOUS GALERKIN METHODS ON OVERLAPPING CELLS WITH A NONOSCILLATORY HIERARCHICAL RECONSTRUCTION
(Copyright: Society for Industrial and Applied Mathematics, 2007) LIU, YINGJIE; SHU, CHI-WANG; TADMOR, EITAN; ZHANG, MENGPING
The central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408–463] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys., 160 (2000), pp. 241–282] employs a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu, J. Comput. Phys., 209 (2005), pp. 82–104]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunov-type finite volume schemes. Overlapping cells lend themselves to the development of a central-type discontinuous Galerkin (DG) method, following the series of works by Cockburn and Shu [J. Comput. Phys., 141 (1998), pp. 199–224] and the references therein. In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities beyond those of Godunov-type schemes. In particular, the central DG is coupled with a novel reconstruction procedure which removes spurious oscillations in the presence of shocks. This reconstruction is motivated by the moments limiter of Biswas, Devine, and Flaherty [Appl. Numer. Math., 14 (1994), pp. 255–283] but is otherwise different in its hierarchical approach. The new hierarchical reconstruction involves a MUSCL or a second order ENO reconstruction in each stage of a multilayer reconstruction process without characteristic decomposition. It is compact, easy to implement over arbitrary meshes, and retains the overall preprocessed order of accuracy while effectively removing spurious oscillations around shocks.
The CFL condition for spectral approximations to hyperbolic initial-boundary value problems.
(American Mathematical Society, 1991-04) Gottlieb, David; Tadmor, Eitan
Complexity-Regularized Regression for Serially-Correlated Residuals with Applications to Stock Market Data
(MDPI, 2014-12-23) Darmon, David; Girvan, Michelle
A popular approach in the investigation of the short-term behavior of a non-stationary time series is to assume that the time series decomposes additively into a long-term trend and short-term fluctuations. A first step towards investigating the short-term behavior requires estimation of the trend, typically via smoothing in the time domain. We propose a method for time-domain smoothing, called complexity-regularized regression (CRR). This method extends recent work, which infers a regression function that makes residuals from a model “look random”. Our approach operationalizes non-randomness in the residuals by applying ideas from computational mechanics, in particular the statistical complexity of the residual process. The method is compared to generalized cross-validation (GCV), a standard approach for inferring regression functions, and shown to outperform GCV when the error terms are serially correlated. Regression under serially-correlated residuals has applications to time series analysis, where the residuals may represent short timescale activity. We apply CRR to a time series drawn from the Dow Jones Industrial Average and examine how both the long-term and short-term behavior of the market have changed over time.
Confidence bands for survival curves from outcome-dependent stratified samples
(Wiley, 2023-12-21) Saegusa, Takumi; Nandori, Peter
We consider the construction of confidence bands for survival curves under the outcome-dependent stratified sampling. A main challenge of this design is that data are a biased dependent sample due to stratification and sampling without replacement. Most literature on regression approximates this design by Bernoulli sampling but variance is generally overestimated. Even with this approximation, the limiting distribution of the inverse probability weighted Kaplan–Meier estimator involves a general Gaussian process, and hence quantiles of its supremum is not analytically available. In this paper, we provide a rigorous asymptotic theory for the weighted Kaplan–Meier estimator accounting for dependence in the sample. We propose the novel hybrid method to both simulate and bootstrap parts of the limiting process to compute confidence bands with asymptotically correct coverage probability. Simulation study indicates that the proposed bands are appropriate for practical use. A Wilms tumor example is presented.
Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems
(American Mathematical Society, 1985-04) Goldberg, Moshe; Tadmor, Eitan
Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems. II
(American Mathematical Society, 1987-04) Goldberg, Moshe; Tadmor, Eitan
Convenient total variation diminishing conditions for nonlinear difference schemes
(Copyright: Society for Industrial and Applied Mathematics, 1988-10) Tadmor, Eitan
Convergence of spectral methods for hyperbolic initial-boundary value systems
(Copyright: Society for Industrial and Applied Mathematics, 1987-06) Gottlieb, David; Lustman, Liviu; Tadmor, Eitan
Convergence of spectral methods for hyperbolic initial-boundary value systems
(Copyright: Society for Industrial and Applied Mathematics, 1987-06) Gottlieb, David; Lustman, Liviu; Tadmor, Eitan
Convergence of spectral methods for nonlinear conservation laws
(Copyright: Society for Industrial and Applied Mathematics, 1989-02) Tadmor, Eitan
The convergence rate of approximate solutions for nonlinear scalar conservation laws
(Copyright: Society for Industrial and Applied Mathematics, 1992-12) Nessyahu, Haim; Tadmor, Eitan
The convergence rate of Godunov type schemes
(Copyright: Society for Industrial and Applied Mathematics, 1994-02) Nessyahu, Haim; Tadmor, Eitan; Tassa, Tamir
Counting Siblings in Universal Theories
(Cambridge University Press, 2022-01-10) Braunfield, Samuel; Laskowski, Michael C.
We show that if a countable structure M in a finite relational language is not cellular, then there is an age-preserving N⊇M such that 2ℵ0 many structures are bi-embeddable with N. The proof proceeds by a case division based on mutual algebraicity.
CRITICAL THRESHOLDS IN 2D RESTRICTED EULER–POISSON EQUATIONS
(Copyright: Society for Industrial and Applied Mathematics, 2003) LIU, HAILIANG; TADMOR, EITAN
We provide a complete description of the critical threshold phenomenon for the twodimensional localized Euler–Poisson equations, introduced by the authors in [Comm. Math. Phys., 228 (2002), pp. 435–466]. Here, the questions of global regularity vs. finite-time breakdown for the two-dimensional (2D) restricted Euler–Poisson solutions are classified in terms of precise explicit formulae, describing a remarkable variety of critical threshold surfaces of initial configurations. In particular, it is shown that the 2D critical thresholds depend on the relative sizes of three quantities: the initial density, the initial divergence, and the initial spectral gap, that is, the difference between the two eigenvalues of the 2 × 2 initial velocity gradient.
CRITICAL THRESHOLDS IN A CONVOLUTION MODEL FOR NONLINEAR CONSERVATION LAWS
(Copyright: Society for Industrial and Applied Mathematics, 2001) LIU, HAILIANG; TADMOR, EITAN
In this work we consider a convolution model for nonlinear conservation laws.Due to the delicate balance between the nonlinear convection and the nonlocal forcing, this model allows for narrower shock layers than those in the viscous Burgers’ equation and yet exhibits the conditional finite time breakdown as in the damped Burgers’ equation.W e show the critical threshold phenomenon by presenting a lower threshold for the breakdown of the solutions and an upper threshold for the global existence of the smooth solution.The threshold condition depends only on the relative size of the minimum slope of the initial velocity and its maximal variation.W e show the exact blow-up rate when the slope of the initial profile is below the lower threshold.W e further prove the L1 stability of the smooth shock profile, provided the slope of the initial profile is above the critical threshold.
A deficiency in SUMOylation activity disrupts multiple pathways leading to neural tube and heart defects in Xenopus embryos
(Springer Nature, 2019-05-17) Bertke, Michelle M.; Dubiak, Kyle M.; Cronin, Laura; Zeng, Erliang; Huber, Paul W.
Adenovirus protein, Gam1, triggers the proteolytic destruction of the E1 SUMO-activating enzyme. Microinjection of an empirically determined amount of Gam1 mRNA into one-cell Xenopus embryos can reduce SUMOylation activity to undetectable, but nonlethal, levels, enabling an examination of the role of this post-translational modification during early vertebrate development.