# Mathematics Research Works

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- ItemMultivariate 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.
- ItemOn 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.
- ItemThe 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.
- ItemCounting 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.
- ItemVALUES OF ZETA FUNCTIONS OF ARITHMETIC SURFACES AT s = 1(Cambridge University Press, 2022-02-28) Lichtenbaum, Stephen; Ramachandran, NiranjanWe 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.