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Understanding the spread of infectious diseases such as COVID-19 is crucial for informed decision-making and resource allocation. A critical component of disease behavior is the velocity with which disease spreads, defined as the rate of change between time and space. In this paper, we propose a spatio-temporal modeling approach to determine the velocities of infectious disease spread. Our approach assumes that the locations and times of people infected can be considered as a spatio-temporal point pattern that arises as a realization of a spatio-temporal log-Gaussian Cox process. The intensity of this process is estimated using fast Bayesian inference by employing the integrated nested Laplace approximation (INLA) and the Stochastic Partial Differential Equations (SPDE) approaches. The velocity is then calculated using finite differences that approximate the derivatives of the intensity function. Finally, the directions and magnitudes of the velocities can be mapped at specific times to examine better the spread of the disease throughout the region. We demonstrate our method by analyzing COVID-19 spread in Cali, Colombia, during the 2020-2021 pandemic., Comment: 29 pages, 46 figures, Journal American Statistican, submitted on 3rd of September

Ultrafast electron diffraction (UED) instruments typically operate at kHz or lower repetition rates and rely on indirect detection of electrons. However, these experiments encounter limitations because they are required to use electron beams containing a relatively large number of electrons (>>100 electrons/pulse), leading to severe space-charge effects. Consequently, electron pulses with long durations and large transverse diameters are used to interrogate the sample. Here, we introduce a novel UED instrument operating at a high repetition rate and employing direct electron detection. We operate significantly below the severe space-charge regime by using electron beams containing 1 to 140 electrons per pulse at 30-kHz. We demonstrate the ability to detect time-resolved signals from thin film solid samples with a difference contrast signal, ${\Delta}I/I_0$, and an instrument response function as low as $10^{-5}$ and 184-fs (FWHM), respectively, without temporal compression. Overall, our findings underscore the importance of increasing the repetition rate of UED experiments and adopting a direct electron detection scheme, which will be particularly impactful for gas-phase UED. Our newly developed scheme enables more efficient and sensitive investigations of ultrafast dynamics in photoexcited samples using ultrashort electron beams.

We consider all irreducible rank-4 hypergeometric local systems defined over $\mathbb{Q}$ that support a rational one-dimensional variation of Hodge structures of weight 3 and Hodge vector $(1,1,1,1)$. Up to a natural equivalence there are only 47 cases. The first 14 cases have maximally unipotent monodromy at one point and have been extensively studied in the literature. We show that all 47 local systems are associated to families of generically smooth threefolds and we analyze the geometry and arithmetic at their conifold point., Comment: 16 pages. Comments are welcome

We give two constructions of families of elliptic curves over cubic or quartic fields with three, respectively four, `integral' elements in the kernel of the tame symbol on the curves. The fields are in general non-Abelian, and the elements linearly independent. For their integrality, we prove a new criterion that does not ignore any torsion. We also verify Beilinson's conjecture numerically for just over 90 of the curves.

We study a hypergeometric local system that arises from the quantum Chen-Ruan cohomology of a family of weighted del Pezzo hypersurfaces. We prove that it is the anti-invariant variation of a pencil of genus-7 curves with respect to an involution having 4 fixed points., Comment: 14 pages, 3 figures

Survey of hypergeometric motives, with a focus on their source varieties, Hodge numbers, and L-functions.

Let $\Gamma$ be a simple graph and $I_\Gamma(x)$ its multivariate independence polynomial. The main result of this paper is the characterization of chordal graphs as the only $\Gamma$ for which the power series expansion of $I_\Gamma^{-1}(x)$ is Horn hypergeometric., Comment: v2: 15 pages, added references, improved statement of Theorem 1.2

This note is an extended version of the slides for my talk with the same title at the {\it Arithmetic, geometry, and modular forms: a conference in honour of Bill Duke} in June 2019 at the ETH in Z"urich. The results presented concern three geometric criteria for the integrality of factorial ratios, numbers such as (30n)!n!/(6n)!(10n)!(15n)!, which are integral in a non-immediate way for all n. This work is an offshoot of an ongoing project on hypergeometric motives joint with D. Roberts and M. Watkins.

In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t^2). Using these results together with results of Geiss, Leclerc and Schroer we give, when k is algebraically closed, a classification of pairs (Q,R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally, when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra F_q[t]/(t^r). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation., Comment: 45 pages, final version, to appear in Selecta Mathematica

With the objective of exploiting hardware capabilities and preparing the ground for the next-generation X-band synthetic aperture radar (SAR) missions, TerraSAR-X and TanDEM-X are now able to operate in staring spotlight mode, which is characterized by an increased azimuth resolution of approximately 0.24 m compared to 1.1 m of the conventional sliding spotlight mode. In this paper, we demonstrate for the first time its potential for SAR tomography. To this end, we tailored our interferometric and tomographic processors for the distinctive features of the staring spotlight mode, which will be analyzed accordingly. By means of its higher spatial resolution, the staring spotlight mode will not only lead to a denser point cloud, but also to more accurate height estimates due to the higher signal-to-clutter ratio. As a result of a first comparison between sliding and staring spotlight TomoSAR, the following were observed: 1) the density of the staring spotlight point cloud is approximately 5.1--5.5 times as high; 2) the relative height accuracy of the staring spotlight point cloud is approximately 1.7 times as high., Comment: Accepted publication in IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing

We discuss two related principles for hypergeometric supercongrences, one related to accelerated convergence and the other to the vanishing of Hodge numbers. This is an extended abstract of a talk given at the workshop "Hypergeometric motives and Calabi-Yau differential equations" at the Mathematical Research Institute (MATRIX) of The University of Melbourne in Creswick, Australia in January of 2017.

We study classical hypergeometric series as a p-adic function of its parameters inspired by a problem in the American Mathematical Monthly solved by D. Zagier. This is an extended abstract of a talk given at the workshop "Hypergeometric motives and Calabi-Yau differential equations" at the Mathematical Research Institute (MATRIX) of The University of Melbourne in Creswick, Australia in January of 2017.

We study the general properties of certain rank four rigid local systems considered by Goursat. We analyze when they are irreducible, give an explicit integral description as well as the invariant Hermitian form when it exists. By a computer search we find what we expect are all irreducible such systems all whose solutions are algebraic functions and give several explicit examples defined over the rationals. We also exhibit one example with infinite monodromy as arising from a family of genus two curves.

In this paper we survey geometric and arithmetic techniques to study the cohomology of semiprojective hyperkaehler manifolds including toric hyperkaehler varieties, Nakajima quiver varieties and moduli spaces of Higgs bundles on Riemann surfaces. The resulting formulae for their Poincare polynomials are combinatorial and representation theoretical in nature. In particular we will look at their Betti numbers and will establish some results and expectations on their asymptotic shape., Comment: 43 pages, 11 figures

In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.

We extend and reformulate a result of Solomon on the divisibility of the title. We show, for example, that if $\Gamma$ is a finitely generated group, then $|G|$ divides $#\Hom(\Gamma,G)$ for every finite group $G$ if and only if $\Gamma$ has infinite abelianization. As a consequence we obtain some arithmetic properties of the number of subgroups of a given index in such a group $\Gamma$.

We study a refinement of the A-polynomial in the case of the g-loop quiver. We give an explicit formula for its value at q=1. Conjecturally this implies a formula for the middle Betti number of the moduli space of Higgs bundles or equivalently of the character variety of a Riemann surface of genus g., Comment: 8 pages

We describe the structure of all codimension-two lattice configurations $A$ which admit a stable rational $A$-hypergeometric function, that is a rational function $F$ all whose partial derivatives are non zero, and which is a solution of the $A$-hypergeometric system of partial differential equations defined by Gel'fand, Kapranov and Zelevinsky. We show, moreover, that all stable rational $A$-hypergeometric functions may be described by toric residues and apply our results to study the rationality of bivariate series whose coefficients are quotients of factorials of linear forms., Comment: 25 pages, 1 figure. Minor changes

Let $\Gamma$ be a quiver on n vertices $v_1, v_2, ..., v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\alpha \in \N^n$. Hua gave a formula for $A_{\Gamma}(\alpha, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\Gamma$ over the finite field $\F_q$ with dimension vector $\alpha$. Kac showed that $A_{\Gamma}(\bm{\alpha}, q)$ is a polynomial in q with integer coefficients. Using Hua's formula, we show that for each non-negative integer s, the s-th derivative of $A_{\Gamma}(\alpha,q)$ with respect to q, when evaluated at q = 1, is a polynomial in the variables $g_{ij}$, and we compute the highest degree terms in this polynomial. Our formulas for these coefficients depend on the enumeration of certain families of connected graphs., Comment: 16 pages

Sketch of proof of a theorem relating the two subjects of the title. It can be thought as an extension of results of Landau for the classical hypergeometric function. It relies on the characterization of algebraic hypergeometric functions of Beukers and Heckman. In the process we also show that a variant of a classical construction of Bezout (producing a quadratic form, the Bezoutian, out of two polynomials in one variable) gives the Hermitian form fixed by the monodromy group, up to scaling., Comment: Summary of talk at 2005 Explicit Methods in Number Theory, Oberwolfach

This is a mostly expository note concerning the following question: How fast can we compute the value of an L-function at the center of the critical strip? It is the writeup of my talk at the Matrix Ensembles and L-Functions workshop at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK on July 2005.

The goal of this note is to present a combinatorial mechanism for counting certain objects associated to a variety X defined over a finite field. The basic example is that of counting conjugacy classes in GL_n(F_q), where X is the multiplicative group.

The purpose of this short note is to give a proof of the following identity between (logarithmic) Mahler measures m(y^2+2xy+y-x^3-2x^2-x)=5/7*m(y^2+4xy+y-x^3+x^2) which is one of many examples that arise from the comparison of Mahler measures and special values of L-functions.

We outline a general algorithm for computing an explicit model over a number field of any curve of genus 2 whose (unpolarized) Jacobian is isomorphic to the product of two elliptic curves with CM by the same order in an imaginary quadratic field. We give the details and some examples for the case where the order has prime discriminant and class number one

Let P be a simple lattice polytope. We define an action of the Hecke operators on E (P), the Ehrhart polynomial of P, and describe their effect on the coefficients of E (P). We also describe how the Brion-Vergne formula transforms under the Hecke operators for nonsingular lattice polytopes P., Comment: 24 pp

The National Aeronautics and Space Administration (NASA)’s Operation IceBridge (OIB) was a 13-year (2009–2021) airborne mission to survey land and sea ice across the Arctic, Antarctic, and Alaska. Here, we review OIB’s goals, instruments, campaigns, key scientific results, and implications for future investigations of the cryosphere. OIB’s primary goal was to use airborne laser altimetry to bridge the gap in fine-resolution elevation measurements of ice from space between the conclusion of NASA’s Ice, Cloud, and land Elevation Satellite (ICESat; 2003–2009) and its follow-on, ICESat-2 (launched 2018). Additional scientific requirements were intended to contextualize observed elevation changes using a multisensor suite of radar sounders, gravimeters, magnetometers, and cameras. Using 15 different aircraft, OIB conducted 968 science flights, of which 42% were repeat surveys of land ice, 42% were surveys of previously unmapped terrain across the Greenland and Antarctic ice sheets, Arctic ice caps, and Alaskan glaciers, and 16% were surveys of sea ice. The combination of an expansive instrument suite and breadth of surveys enabled numerous fundamental advances in our understanding of the Earth’s cryosphere. For land ice, OIB dramatically improved knowledge of interannual outlet-glacier variability, ice-sheet, and outlet-glacier thicknesses, snowfall rates on ice sheets, fjord and sub-ice-shelf bathymetry, and ice-sheet hydrology. Unanticipated discoveries included a reliable method for constraining the thickness within difficult-to-sound incised troughs beneath ice sheets, the extent of the firn aquifer within the Greenland Ice Sheet, the vulnerability of many Greenland and Antarctic outlet glaciers to ocean-driven melting at their grounding zones, and the dominance of surface-melt-driven mass loss of Alaskan glaciers. For sea ice, OIB significantly advanced our understanding of spatiotemporal variability in sea ice freeboard and its snow cover, especially through combined analysis of fine-resolution altimetry, visible imagery, and snow radar measurements of the overlying snow thickness. Such analyses led to the unanticipated discovery of an interdecadal decrease in snow thickness on Arctic sea ice and numerous opportunities to validate sea ice freeboards from satellite radar altimetry. While many of its data sets have yet to be fully explored, OIB’s scientific legacy has already demonstrated the value of sustained investment in reliable airborne platforms, airborne instrument development, interagency and international collaboration, and open and rapid data access to advance our understanding of Earth’s remote polar regions and their role in the Earth system.