Mathematics Research Works

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    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.
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    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.
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    Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State
    (MDPI, 2021-05-27) Zhang, Xuze; Pyne, Saumyadipta; Kedem, Benjamin
    In disease modeling, a key statistical problem is the estimation of lower and upper tail probabilities of health events from given data sets of small size and limited range. Assuming such constraints, we describe a computational framework for the systematic fusion of observations from multiple sources to compute tail probabilities that could not be obtained otherwise due to a lack of lower or upper tail data. The estimation of multivariate lower and upper tail probabilities from a given small reference data set that lacks complete information about such tail data is addressed in terms of pertussis case count data. Fusion of data from multiple sources in conjunction with the density ratio model is used to give probability estimates that are non-obtainable from the empirical distribution. Based on a density ratio model with variable tilts, we first present a univariate fit and, subsequently, improve it with a multivariate extension. In the multivariate analysis, we selected the best model in terms of the Akaike Information Criterion (AIC). Regional prediction, in Washington state, of the number of pertussis cases is approached by providing joint probabilities using fused data from several relatively small samples following the selected density ratio model. The model is validated by a graphical goodness-of-fit plot comparing the estimated reference distribution obtained from the fused data with that of the empirical distribution obtained from the reference sample only.
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    On the Su–Schrieffer–Heeger model of electron transport: Low-temperature optical conductivity by the Mellin transform
    (Wiley, 2023-05-30) Margetis, Dionisios; Watson, Alexander B.; Luskin, Mitchell
    We describe the low-temperature optical conductivity as a function of frequency for a quantum-mechanical system of electrons that hop along a polymer chain. To this end, we invoke the Su–Schrieffer–Heeger tight-binding Hamiltonian for noninteracting spinless electrons on a one-dimensional (1D) lattice. Our goal is to show via asymptotics how the interband conductivity of this system behaves as the smallest energy bandgap tends to close. Our analytical approach includes: (i) the Kubo-type formulation for the optical conductivity with a nonzero damping due to microscopic collisions, (ii) reduction of this formulation to a 1D momentum integral over the Brillouin zone, and (iii) evaluation of this integral in terms of elementary functions via the three-dimensional Mellin transform with respect to key physical parameters and subsequent inversion in a region of the respective complex space. Our approach reveals an intimate connection of the behavior of the conductivity to particular singularities of its Mellin transform. The analytical results are found in good agreement with direct numerical computations.
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    The emergence of lines of hierarchy in collective motion of biological systems
    (Institute of Physics, 2023-06-29) Greene, James M.; Tadmor, Eitan; Zhong, Ming
    The emergence of large-scale structures in biological systems, and in particular the formation of lines of hierarchy, is observed at many scales, from collections of cells to groups of insects to herds of animals. Motivated by phenomena in chemotaxis and phototaxis, we present a new class of alignment models that exhibit alignment into lines. The spontaneous formation of such ‘fingers’ can be interpreted as the emergence of leaders and followers in a system of identically interacting agents. Various numerical examples are provided, which demonstrate emergent behaviors similar to the ‘fingering’ phenomenon observed in some phototaxis and chemotaxis experiments; this phenomenon is generally known to be a challenging pattern for existing models to capture. A novel protocol for pairwise interactions provides a fundamental alignment mechanism by which agents may form lines of hierarchy across a wide range of biological systems.
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    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.
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    (Cambridge University Press, 2022-02-28) Lichtenbaum, Stephen; Ramachandran, Niranjan
    We show that the conjecture of [27] for the special value at s=1 of the zeta function of an arithmetic surface is equivalent to the Birch–Swinnerton–Dyer conjecture for the Jacobian of the generic fibre.
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    Detection of co-eluted peptides using database search methods
    (Springer Nature, 2008-07-02) Alves, Gelio; Ogurtsov, Aleksey Y; Kwok, Siwei; Wu, Wells W; Wang, Guanghui; Shen, Rong-Fong; Yu, Yi-Kuo
    Current experimental techniques, especially those applying liquid chromatography mass spectrometry, have made high-throughput proteomic studies possible. The increase in throughput however also raises concerns on the accuracy of identification or quantification. Most experimental procedures select in a given MS scan only a few relatively most intense parent ions, each to be fragmented (MS2) separately, and most other minor co-eluted peptides that have similar chromatographic retention times are ignored and their information lost. We have computationally investigated the possibility of enhancing the information retrieval during a given LC/MS experiment by selecting the two or three most intense parent ions for simultaneous fragmentation. A set of spectra is created via superimposing a number of MS2 spectra, each can be identified by all search methods tested with high confidence, to mimick the spectra of co-eluted peptides. The generated convoluted spectra were used to evaluate the capability of several database search methods – SEQUEST, Mascot, X!Tandem, OMSSA, and RAId_DbS – in identifying true peptides from superimposed spectra of co-eluted peptides. We show that using these simulated spectra, all the database search methods will gain eventually in the number of true peptides identified by using the compound spectra of co-eluted peptides. Reviewed by Vlad Petyuk (nominated by Arcady Mushegian), King Jordan and Shamil Sunyaev. For the full reviews, please go to the Reviewers' comments section.
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    Simultaneous transcriptional profiling of Leishmania major and its murine macrophage host cell reveals insights into host-pathogen interactions
    (Springer Nature, 2015-12-29) Dillon, Laura A. L.; Suresh, Rahul; Okrah, Kwame; Corrada Bravo, Hector; Mosser, David M.; El-Sayed, Najib M.
    Parasites of the genus Leishmania are the causative agents of leishmaniasis, a group of diseases that range in manifestations from skin lesions to fatal visceral disease. The life cycle of Leishmania parasites is split between its insect vector and its mammalian host, where it resides primarily inside of macrophages. Once intracellular, Leishmania parasites must evade or deactivate the host's innate and adaptive immune responses in order to survive and replicate. We performed transcriptome profiling using RNA-seq to simultaneously identify global changes in murine macrophage and L. major gene expression as the parasite entered and persisted within murine macrophages during the first 72 h of an infection. Differential gene expression, pathway, and gene ontology analyses enabled us to identify modulations in host and parasite responses during an infection. The most substantial and dynamic gene expression responses by both macrophage and parasite were observed during early infection. Murine genes related to both pro- and anti-inflammatory immune responses and glycolysis were substantially upregulated and genes related to lipid metabolism, biogenesis, and Fc gamma receptor-mediated phagocytosis were downregulated. Upregulated parasite genes included those aimed at mitigating the effects of an oxidative response by the host immune system while downregulated genes were related to translation, cell signaling, fatty acid biosynthesis, and flagellum structure. The gene expression patterns identified in this work yield signatures that characterize multiple developmental stages of L. major parasites and the coordinated response of Leishmania-infected macrophages in the real-time setting of a dual biological system. This comprehensive dataset offers a clearer and more sensitive picture of the interplay between host and parasite during intracellular infection, providing additional insights into how pathogens are able to evade host defenses and modulate the biological functions of the cell in order to survive in the mammalian environment.
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    Evolution of transcriptional networks in yeast: alternative teams of transcriptional factors for different species
    (Springer Nature, 2016-11-11) Muñoz, Adriana; Santos Muñoz, Daniella; Zimin, Aleksey; Yorke, James A.
    The diversity in eukaryotic life reflects a diversity in regulatory pathways. Nocedal and Johnson argue that the rewiring of gene regulatory networks is a major force for the diversity of life, that changes in regulation can create new species. We have created a method (based on our new “ping-pong algorithm) for detecting more complicated rewirings, where several transcription factors can substitute for one or more transcription factors in the regulation of a family of co-regulated genes. An example is illustrative. A rewiring has been reported by Hogues et al. that RAP1 in Saccharomyces cerevisiae substitutes for TBF1/CBF1 in Candida albicans for ribosomal RP genes. There one transcription factor substitutes for another on some collection of genes. Such a substitution is referred to as a “rewiring”. We agree with this finding of rewiring as far as it goes but the situation is more complicated. Many transcription factors can regulate a gene and our algorithm finds that in this example a “team” (or collection) of three transcription factors including RAP1 substitutes for TBF1 for 19 genes. The switch occurs for a branch of the phylogenetic tree containing 10 species (including Saccharomyces cerevisiae), while the remaining 13 species (Candida albicans) are regulated by TBF1. To gain insight into more general evolutionary mechanisms, we have created a mathematical algorithm that finds such general switching events and we prove that it converges. Of course any such computational discovery should be validated in the biological tests. For each branch of the phylogenetic tree and each gene module, our algorithm finds a sub-group of co-regulated genes and a team of transcription factors that substitutes for another team of transcription factors. In most cases the signal will be small but in some cases we find a strong signal of switching. We report our findings for 23 Ascomycota fungi species.
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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.
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    A statistical analysis of vaccine-adverse event data
    (Springer Nature, 2019-05-28) Ren, Jian-Jian; Sun, Tingni; He, Yongqun; Zhang, Yuji
    Vaccination has been one of the most successful public health interventions to date, and the U.S. FDA/CDC Vaccine Adverse Event Reporting System (VAERS) currently contains more than 500,000 reports for post-vaccination adverse events that occur after the administration of vaccines licensed in the United States. The VAERS dataset is huge, contains very large dimension nominal variables, and is complex due to multiple listing of vaccines and adverse symptoms in a single report. So far there has not been any statistical analysis conducted in attempting to identify the cross-board patterns on how all reported adverse symptoms are related to the vaccines.
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    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.
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    On The Number of Unlabeled Bipartite Graphs
    (2016) Atmaca, Abdullah; Oruc, Yavuz A
    Let $I$ and $O$ denote two sets of vertices, where $I\cap O =\Phi$, $|I| = n$, $|O| = r$, and $B_u(n,r)$ denote the set of unlabeled graphs whose edges connect vertices in $I$ and $O$. It is shown that the following two-sided equality holds. $\displaystyle \frac{\binom{r+2^{n}-1}{r}}{n!} \le |B_u(n,r)| \le 2\frac{\binom{r+2^{n}-1}{r}}{n!} $
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    On Number Of Partitions Of An Integer Into A Fixed Number Of Positive Integers
    (2015-04) Oruc, A. Yavuz
    This paper focuses on the number of partitions of a positive integer $n$ into $k$ positive summands, where $k$ is an integer between $1$ and $n$. Recently some upper bounds were reported for this number in [Merca14]. Here, it is shown that these bounds are not as tight as an earlier upper bound proved in [Andrews76-1] for $k\le 0.42n$. A new upper bound for the number of partitions of $n$ into $k$ summands is given, and shown to be tighter than the upper bound in [Merca14] when $k$ is between $O(\frac{\sqrt{n}}{\ln n})$ and $n-O(\frac{\sqrt{n}}{\ln n})$. It is further shown that the new upper bound is also tighter than two other upper bounds previously reported in~[Andrews76-1] and [Colman82]. A generalization of this upper bound to number of partitions of $n$ into at most $k$ summands is also presented.
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    (1994-01) Tadmor, Eitan
    We review several topics concerning spectral approximations of time-dependent problems, primarily | the accuracy and stability of Fourier and Chebyshev methods for the approximate solutions of hyperbolic systems. To make these notes self contained, we begin with a very brief overview of Cauchy problems. Thus, the main focus of the first part is on hyperbolic systems which are dealt with two (related) tools: the energy method and Fourier analysis. The second part deals with spectral approximations. Here we introduce the main ingredients of spectral accuracy, Fourier and Chebyshev interpolants, aliasing, differentiation matrices ... The third part is devoted to Fourier method for the approximate solution of periodic systems. The questions of stability and convergence are answered by combining ideas from the first two sections. In this context we highlight the role of aliasing and smoothing; in particular, we explain how the lack of resolution might excite small scales weak instability, which is avoided by high modes smoothing. The forth and final part deals with non-periodic problems. We study the stability of the Chebyshev method, paying special attention to the intricate issue of the CFL stability restriction on the permitted time-step.
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    (Copyright: Society for Industrial and Applied Mathematics, 2008) ENGELBERG, SHLOMO; TADMOR, EITAN
    We consider the problem of detecting edges—jump discontinuities in piecewise smooth functions from their N-degree spectral content, which is assumed to be corrupted by noise. There are three scales involved: the “smoothness” scale of order 1/N, the noise scale of order √η, and the O(1) scale of the jump discontinuities. We use concentration factors which are adjusted to the standard deviation of the noise √η ≫ 1/N in order to detect the underlying O(1)-edges, which are separated from the noise scale √η ≪ 1.
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    LONG-TIME EXISTENCE OF SMOOTH SOLUTIONS FOR THLong time existence of smooth solutions for the rapidly rotating shallow-water and Euler equationsE
    (Copyright: Society for Industrial and Applied Mathematics, 2008) CHENG, BIN; TADMOR, EITAN
    We study the stabilizing effect of rotational forcing in the nonlinear setting of two-dimensional shallow-water and more general models of compressible Euler equations. In [Phys. D, 188 (2004), pp. 262–276] Liu and Tadmor have shown that the pressureless version of these equations admit a global smooth solution for a large set of subcritical initial configurations. In the present work we prove that when rotational force dominates the pressure, it prolongs the lifespan of smooth solutions for t ≲ ln(δ^−1); here δ ≪ 1 is the ratio of the pressure gradient measured by the inverse squared Froude number, relative to the dominant rotational forces measured by the inverse Rossby number. Our study reveals a “nearby” periodic-in-time approximate solution in the small δ regime, upon which hinges the long-time existence of the exact smooth solution. These results are in agreement with the close-to-periodic dynamics observed in the “near-inertial oscillation” (NIO) regime which follows oceanic storms. Indeed, our results indicate the existence of a smooth, “approximate periodic” solution for a time period of days, which is the relevant time period found in NIO observations.
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    (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.
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    Non-oscillatory central schemes for one- and two-dimensional MHD equations. II: high-order semi-discrete schemes.
    (Copyright: Society for Industrial and Applied Mathematics, 2006) Balbas, Jorge; Tadmor, Eitan
    We present a new family of high-resolution, nonoscillatory semidiscrete central schemes for the approximate solution of the ideal magnetohydrodynamics (MHD) equations. This is the second part of our work, where we are passing from the fully discrete staggered schemes in [J. Balb´as, E. Tadmor, and C.-C. Wu, J. Comput. Phys., 201 (2004), pp. 261–285] to the semidiscrete formulation advocated in [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241–282]. This semidiscrete formulation retains the simplicity of fully discrete central schemes while enhancing efficiency and adding versatility. The semidiscrete algorithm offers a wider range of options to implement its two key steps: nonoscillatory reconstruction of point values followed by the evolution of the corresponding point valued fluxes. We present the solution of several prototype MHD problems. Solutions of one-dimensional Brio–Wu shock-tube problems and the two-dimensional Kelvin–Helmholtz instability, Orszag–Tang vortex system, and the disruption of a high density cloud by a strong shock are carried out using third- and fourth-order central schemes based on the central WENO reconstructions. These results complement those presented in our earlier work and confirm the remarkable versatility and simplicity of central schemes as black-box, Jacobian-free MHD solvers. Furthermore, our numerical experiments demonstrate that this family of semidiscrete central schemes preserves the ∇ · B = 0-constraint within machine round-off error; happily, no constrained-transport enforcement is needed.