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David Anderson

Publications and source records attributed to David Anderson.

At least 55 records · Page 3Linked to original sources
arXiv2019

Chern class formulas for classical-type degeneracy loci

In previous work, we employed a geometric method of Kazarian to prove Pfaffian formulas for a certain class of degeneracy loci in types B, C, and D. Here we refine that approach to obtain formulas for more general loci, including those coming from all isotropic Grassmannians. In these cases, the formulas recover the remarkable theta- and eta-polynomials of Buch, Kresch, Tamvakis, and Wilson. The streamlined geometric approch yields simple and direct proofs, which proceed in parallel for all four classical types. In an appendix, we develop some foundational algebra and prove several Pfaffian identities. Another appendix establishes a basic formula for classes in quadric bundles.

math.AG↗
arXiv2019

K-theoretic Chern class formulas for vexillary degeneracy loci

Using raising operators and geometric arguments, we establish formulas for the K-theory classes of degeneracy loci in classical types. We also find new determinantal and Pfaffian expressions for classical cases considered by Giambelli: the loci where a generic matrix drops rank, and where a generic symmetric or skew-symmetric matrix drops rank. In an appendix, we construct a K-theoretic Euler class for even-rank vector bundles with quadratic form, refining the Chow-theoretic class introduced by Edidin and Graham. We also establish a relation between top Chern classes of maximal isotropic subbundles, which is used in proving the type D degeneracy locus formulas.

math.AG↗
arXiv2019

Scalar Charges and Scaling Relations in Massless Scalar-Tensor Theories

The timing of binary pulsars allows us to place some of the tightest constraints on modified theories of gravity. Perhaps some of the most interesting and well-motivated extensions to General Relativity are scalar-tensor theories, in which gravity is mediated by the metric tensor and a scalar field. These theories predict large deviations from General Relativity in the presence of neutron stars through a phenomenon known as scalarization. Neutron stars in scalar-tensor theories develop scalar charges, which directly enter the timing model for binary pulsars. In this paper, we calculate and tabulate these scalar charges in two popular, massless scalar tensor theories for a collection of neutron star equations of state that are compatible with constraints placed by the recent, gravitational wave observations of a binary neutron star coalescence. We then study these scalar charges and explore analytic scaling relations that allow us to predict their value in a large region of parameter space. Our results allow for the quick evaluation of the scalar charge in a large region of scalar-tensor theory parameter space, which has applications for gravitational wave tests of scalar-tensor theories, as well as binary pulsar experiments.

gr-qc↗
arXiv2019

Binary Pulsar constraints on massless scalar-tensor theories using Bayesian statistics

Binary pulsars provide some of the tightest current constraints on modified theories of gravity and these constraints will only get tighter as radio astronomers continue timing these systems. These binary pulsars are particularly good at constraining scalar-tensor theories in which gravity is mediated by a scalar field in addition to the metric tensor. Scalar-tensor theories can predict large deviations from General Relativity due to the fact that they allow for violation of the strong-equivalence principle through a phenomenon known as scalarization. This effect appears directly in the timing model for binary pulsars, and as such, it can be tightly constrained through precise timing. In this paper, we investigate these constraints for two scalar-tensor theories and a large set of realistic equations of state. We calculate the constraints that can be placed by saturating the current $1σ$ bounds on single post-Keplerian parameters, as well as employing Bayesian methods through Markov-Chain-Monte-Carlo simulations to explore the constraints that can be achieved when one considers all measured parameters simultaneously. Our results demonstrate that both methods are able to place similar constraints and that they are both indeed dominated by the measurements of the orbital period decay. The Bayesian approach, however, allows one to simultaneously explore the posterior distributions of not only the theory parameters but of the masses as well.

gr-qc↗
arXiv2018

Stochastic Chemical Reaction Networks for Robustly Approximating Arbitrary Probability Distributions

We show that discrete distributions on the $d$-dimensional non-negative integer lattice can be approximated arbitrarily well via the marginals of stationary distributions for various classes of stochastic chemical reaction networks. We begin by providing a class of detailed balanced networks and prove that they can approximate any discrete distribution to any desired accuracy. However, these detailed balanced constructions rely on the ability to initialize a system precisely, and are therefore susceptible to perturbations in the initial conditions. We therefore provide another construction based on the ability to approximate point mass distributions and prove that this construction is capable of approximating arbitrary discrete distributions for any choice of initial condition. In particular, the developed models are ergodic, so their limit distributions are robust to a finite number of perturbations over time in the counts of molecules.

cs.CC↗
arXiv2018

Discrepancies between extinction events and boundary equilibria in reaction networks

Reaction networks are mathematical models of interacting chemical species that are primarily used in biochemistry. There are two modeling regimes that are typically used, one of which is deterministic and one that is stochastic. In particular, the deterministic model consists of an autonomous system of differential equations, whereas the stochastic system is a continuous time Markov chain. Connections between the two modeling regimes have been studied since the seminal paper by Kurtz (1972), where the deterministic model is shown to be a limit of a properly rescaled stochastic model over compact time intervals. Further, more recent studies have connected the long-term behaviors of the two models when the reaction network satisfies certain graphical properties, such as weak reversibility and a deficiency of zero. These connections have led some to conjecture a link between the long-term behavior of the two models exists, in some sense. In particular, one is tempted to believe that positive recurrence of all states for the stochastic model implies the existence of positive equilibria in the deterministic setting, and that boundary equilibria of the deterministic model imply the occurrence of an extinction event in the stochastic setting. We prove in this paper that these implications do not hold in general, even if restricting the analysis to networks that are bimolecular and that conserve the total mass. In particular, we disprove the implications in the special case of models that have absolute concentration robustness, thus answering in the negative a conjecture stated in the literature in 2014.

q-bio.MN↗
arXiv2018

Vexillary signed permutations revisited

We study the combinatorial properties of vexillary signed permutations, which are signed analogues of the vexillary permutations first considered by Lascoux and Schützenberger. We give several equivalent characterizations of vexillary signed permutations, including descriptions in terms of essential sets and pattern avoidance, and we relate them to the vexillary elements introduced by Billey and Lam.

math.CO↗
arXiv2017

Solar System Constraints on Scalar-Tensor Gravity with Positive Coupling Constant upon Cosmological Evolution of the Scalar Field

Scalar-tensor theories of gravity modify General Relativity by introducing a scalar field that couples non-minimally to the metric tensor, while satisfying the weak-equivalence principle. These theories are interesting because they have the potential to simultaneously suppress modifications to Einstein's theory on Solar System scales, while introducing large deviations in the strong field of neutron stars. Scalar-tensor theories can be classified through the choice of conformal factor, a scalar that regulates the coupling between matter and the metric in the Einstein frame. The class defined by a Gaussian conformal factor with negative exponent has been studied the most because it leads to spontaneous scalarization (i.e. the sudden activation of the scalar field in neutron stars), which consequently leads to large deviations from General Relativity in the strong field. This class, however, has recently been shown to be in conflict with Solar System observations when accounting for the cosmological evolution of the scalar field. We study whether this remains the case when the exponent of the conformal factor is positive, as well as in another class of theories defined by a hyperbolic conformal factor. We find that in both of these scalar-tensor theories, Solar System tests are passed only in a very small subset of parameter space, for a large set of initial conditions compatible with Big Bang Nucleosynthesis. However, while we find that it is possible for neutron stars to scalarize, one must carefully select the coupling parameter to do so, and even then, the scalar charge is typically two orders of magnitude smaller than in the negative exponent case. Our study suggests that future work on scalar-tensor gravity, for example in the context of tests of General Relativity with gravitational waves from neutron star binaries, should be carried out within the positive coupling parameter class.

gr-qc↗
arXiv2016

Diagrams and essential sets for signed permutations

We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations. In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation. Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.

math.CO↗
arXiv2016

Long-term variability of T Tauri stars using WASP

We present a reference study of the long-term optical variability of young stars using data from the WASP project. Our primary sample is a group of well-studied classical T Tauri stars (CTTS), mostly in Taurus-Auriga. WASP lightcurves cover timescales up to 7 years and typically contain 10000-30000 datapoints. We quantify the variability as function of timescale using the time-dependent standard deviation 'pooled sigma'. We find that the overwhelming majority of CTTS has low-level variability with sigma<0.3mag dominated by timescales of a few weeks, consistent with rotational modulation. Thus, for most young stars monitoring over a month is sufficient to constrain the total amount of variability over timescales up to a decade. The fraction of stars with strong optical variability (sigma>0.3mag) is 21% in our sample and 21% in an unbiased control sample. An even smaller fraction (13% in our sample, 6% in the control) show evidence for an increase in variability amplitude as a function of timescale from weeks to months or years. The presence of long-term variability correlates with the spectral slope at 3-5mu, which is an indicator of inner disk geometry, and with the U-B band slope, which is an accretion diagnostics. This shows that the long-term variations in CTTS are predominantly driven by processes in the inner disk and in the accretion zone. Four of the stars with long-term variations show periods of 20-60d, significantly longer than the rotation periods and stable over months to years. One possible explanation are cyclic changes in the interaction between the disk and the stellar magnetic field.

astro-ph.SR↗
arXiv2016

The Infrared Imaging Spectrograph (IRIS) for TMT: Instrument Overview

IRIS is a near-infrared (0.84 to 2.4 microns) integral field spectrograph and wide-field imager being developed for first light with the Thirty Meter (TMT). It mounts to the advanced optics (AO) system NFIRAOS and has integrated on-instrument wavefront sensors (OIWFS) to achieve diffraction-limited spatial resolution at wavelengths longer than 1 micron. With moderate spectral resolution (R ~4,000 - 8,000) and large bandpass over a continuous field of view, IRIS will open new opportunities in virtually every area of astrophysical science. It will be able to resolve surface features tens of kilometers across Titan, while also mapping the distant galaxies at the scale of an individual star forming region. This paper summarizes the entire design and capabilities, and includes the results from the nearly completed preliminary design phase.

astro-ph.IM↗
arXiv2016

The Effect of Cosmological Evolution on Solar System Constraints and on the Scalarization of Neutron Stars in Massless Scalar-Tensor Theories

Certain scalar-tensor theories of gravity that generalize Jordan-Fierz-Brans-Dicke theory are known to predict non-trivial phenomenology for neutron stars. In these theories, first proposed by Damour and Esposito-Farèse, the scalar field has a standard kinetic term, and couples conformally to the matter fields. The weak equivalence principle is therefore satisfied, but scalar effects may arise in strong-field regimes, e.g. allowing for violations of the strong equivalence principle in neutron stars ("spontaneous scalarization") or in sufficiently tight binary neutron-star systems ("dynamical/induced scalarization"). The original scalar-tensor theory proposed by Damour and Esposito-Farèse is in tension with solar-system constraints (for couplings that lead to scalarization), if one accounts for cosmological evolution of the scalar field and no mass term is included in the action. We here extend the conformal coupling of that theory, in order to ascertain if, in this way, solar-system tests can be passed, while retaining a non-trivial phenomenology for neutron stars. We find that even with this generalized conformal coupling, it is impossible to construct a theory that passes both Big-Bang nucleosynthesis and solar-system constraints, while simultaneously allowing for scalarization in isolated/binary neutron stars.

gr-qc↗
arXiv2016

Pattern Avoiding Linear Extensions of Rectangular Posets

Inspired by Yakoubov's 2015 investigation of pattern avoiding linear extensions of the posets called combs, we study pattern avoiding linear extensions of rectangular posets. These linear extensions are closely related to standard tableaux. For positive integers $s$ and $t$ we consider two natural rectangular partial orders on $\{1,2,\ldots,s t\}$, which we call the NE rectangular order and the EN rectangular order. First we enumerate linear extensions of both rectangular orders avoiding most sets of patterns of length three. Then we use both a generating tree and a bijection to show that the linear extensions of the EN rectangular order which avoid 1243 are counted by the Fuss-Catalan numbers. Next we use the transfer matrix method to enumerate linear extensions of the EN rectangular order which avoid 2143. Finally, we open an investigation of the distribution of the inversion number on pattern avoiding linear extensions.

math.CO↗
arXiv2015

Finding Temporally Consistent Occlusion Boundaries in Videos using Geometric Context

We present an algorithm for finding temporally consistent occlusion boundaries in videos to support segmentation of dynamic scenes. We learn occlusion boundaries in a pairwise Markov random field (MRF) framework. We first estimate the probability of an spatio-temporal edge being an occlusion boundary by using appearance, flow, and geometric features. Next, we enforce occlusion boundary continuity in a MRF model by learning pairwise occlusion probabilities using a random forest. Then, we temporally smooth boundaries to remove temporal inconsistencies in occlusion boundary estimation. Our proposed framework provides an efficient approach for finding temporally consistent occlusion boundaries in video by utilizing causality, redundancy in videos, and semantic layout of the scene. We have developed a dataset with fully annotated ground-truth occlusion boundaries of over 30 videos ($5000 frames). This dataset is used to evaluate temporal occlusion boundaries and provides a much needed baseline for future studies. We perform experiments to demonstrate the role of scene layout, and temporal information for occlusion reasoning in dynamic scenes.

cs.CV↗
arXiv2015

The infrared dielectric function of solid para-hydrogen

We report laboratory measurements of the absorption coefficient of solid para-H2, within the wavelength range from 1 to 16.7 micron, at high spectral resolution. In addition to the narrow rovibrational lines of H2 which are familiar from gas phase spectroscopy, the data manifest double transitions and broad phonon branches that are characteristic specifically of hydrogen in the solid phase. These transitions are of interest because they provide a spectral signature which is independent of the impurity content of the matrix. We have used our data, in combination with a model of the ultraviolet absorptions of the H2 molecule, to construct the dielectric function of solid para-H2 over a broad range of frequencies. Our results will be useful in determining the electromagnetic response of small particles of solid hydrogen. The dielectric function makes it clear that pure H2 dust would contribute to IR extinction predominantly by scattering starlight, rather than absorbing it, and the characteristic IR absorption spectrum of the hydrogen matrix itself will be difficult to observe.

astro-ph.GA↗
arXiv2013

Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

The saturation theorem of [Knutson-Tao '99] concerns the nonvanishing of Littlewood-Richardson coefficients. In combination with work of [Klyachko '98], it implies [Horn '62]'s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization [Friedland '00] to majorized sums of Hermitian matrices. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland's problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aformentioned work together with [Thomas-Yong '12].

math.CO↗
arXiv2012

Degeneracy Loci, Pfaffians, and Vexillary Signed Permutations in Types B, C, and D

We define a notion of vexillary signed permutation in types B, C, and D, corresponding to natural degeneracy loci for vector bundles with symmetries of those types. We show that the classes of these loci are given by explicit Pfaffian formulas. The Grassmannian formulas of Kazarian are important special cases, and the corresponding double Schubert polynomials of Ikeda, Mihalcea, and Naruse are shown to be equal to these Pfaffians.

math.AG↗