Mathematics Research Works
Permanent URI for this collectionhttp://hdl.handle.net/1903/1595
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Item Confidence bands for survival curves from outcome-dependent stratified samples(Wiley, 2023-12-21) Saegusa, Takumi; Nandori, PeterWe 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.Item The anisotropic min-max theory: Existence of anisotropic minimal and CMC surfaces(Wiley, 2023-12-01) De Philippis, Guido; De Rosa, AntonioWe prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3–dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension 3.Item Tableau formulas for skew Schubert polynomials(Wiley, 2023-03-24) Tamvakis, HarryThe skew Schubert polynomials are those that are indexed by skew elements of the Weyl group, in the sense of Tamvakis [J. reine angew. Math. 652 (2011), 207–244]. We obtain tableau formulas for the double versions of these polynomials in all four classical Lie types, where the tableaux used are fillings of the associated skew Young diagram. These are the first such theorems for symplectic and orthogonal Schubert polynomials, even in the single case. We also deduce tableau formulas for double Schur, double theta, and double eta polynomials, in their specializations as double Grassmannian Schubert polynomials. The latter results generalize the tableau formulas for symmetric (and single) Schubert polynomials due to Littlewood (in type A) and the author (in types B, C, and D).Item 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].Item 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.Item Complexity-Regularized Regression for Serially-Correlated Residuals with Applications to Stock Market Data(MDPI, 2014-12-23) Darmon, David; Girvan, MichelleA 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.Item Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State(MDPI, 2021-05-27) Zhang, Xuze; Pyne, Saumyadipta; Kedem, BenjaminIn 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.Item 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, MitchellWe 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.Item The emergence of lines of hierarchy in collective motion of biological systems(Institute of Physics, 2023-06-29) Greene, James M.; Tadmor, Eitan; Zhong, MingThe 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.Item 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.