Stokes matrices for confluent hypergeometric equations

We apply the method of [arXiv:1705.07610] to compute the Stokes matrices of non-resonant confluent hypergeometric differential equations. We discuss the ambiguity of the presentation of the Stokes matrices regarding different choices. The results rely on an explicit description of the perverse sheaf associated to the non-confluent regular singular hypergeometric system arising via Fourier-Laplace transform. We give assumptions on the parameter such that the Stokes matrices have rational or real values. Under some more restrictive conditions, the Stokes matrices had been computed by Duval-Mitschi before. We compare our results with their formulae in the unramified case.Comment: 34 pages, 5 figures v4: section on the ramified case added, will be published in International Mathematics Research Notice

Periods for rank 1 irregular singular connections on surfaces

We define a period pairing for flat, irregular singular, rank one connections, satisfying a technical condition regarding its stationary set, on complex surfaces between de Rham cohomology of the connection and a modified singular homology, the rapid decay homology. We prove that this gives a perfect duality. A 2-dimensional version of confluent hypergeometrics are contructed as an example.Comment: 28 pages; previous version partly incorrect (for higher rank case), new version contains case of line bundles and a new example. v3: technical assumption inserted (Definition 1.1

Integral representations for solutions of exponential Gauss-Manin systems

Let f,g be two algebraically independent regular functions from the smooth affine complex variety U to the affine line. The associated exponential Gauss-Manin systems on the affine line are defined to be the cohomology sheaves of the direct image of the exponential differential system $\mathcal{O}_U e^g$ with respect to f. We prove that its holomorphic solutions admit representations in terms of period integrals over topological chains with possibly closed support and with rapid decay condition.Comment: 16 page

The local Laplace transform of an elementary irregular meromorphic connection

We give a definition of the topological local Laplace transformation for a Stokes-filtered local system on the complex affine line and we compute in a topological way the Stokes data of the Laplace transform of a differential system of elementary type.Comment: 56 pages, 21 figures. V2: Final version to appear in Rend. Sem. Mat. Univ. Padov

Topological computation of some Stokes phenomena on the affine line

Let $\mathcal M$ be a holonomic algebraic $\mathcal D$-module on the affine line, regular everywhere including at infinity. Malgrange gave a complete description of the Fourier-Laplace transform $\widehat{\mathcal M}$, including its Stokes multipliers at infinity, in terms of the quiver of $\mathcal M$. Let $F$ be the perverse sheaf of holomorphic solutions to $\mathcal M$. By the irregular Riemann-Hilbert correspondence, $\widehat{\mathcal M}$ is determined by the enhanced Fourier-Sato transform $F^\curlywedge$ of $F$. Our aim here is to recover Malgrange's result in a purely topological way, by computing $F^\curlywedge$ using Borel-Moore cycles. In this paper, we also consider some irregular $\mathcal M$'s, like in the case of the Airy equation, where our cycles are related to steepest descent paths.Comment: 50 pages, to appear at Annales de l'Institut Fourier, v3: some minor (editorial) correction

Periods for flat algebraic connections

In previous work, we established a duality between the algebraic de Rham cohomology of a flat algebraic connection on a smooth quasi-projective surface over the complex numbers and the rapid decay homology of the dual connection relying on a conjecture by C. Sabbah, which has been proved recently by T. Mochizuki for algebraic connections in any dimension. In the present article, we verify that Mochizuki's results allow to generalize these duality results to arbitrary dimensions also

Analyzing immune cell infiltration of cancer spheroids in a 3D cell culture platform

The adoption of in vitro 3D cell culture models as a bridge between conventional 2D cell culture models and the complex in vivo animal models has been increasing. A carefully designed 3D model can run biological assays with animal model-like complexities but with the simplicity and affordability of traditional cell culture. For example, traditional 2D and transwell assays of immune cell infiltration efficiency are unrealistic, as cell migration is gravity-driven. AIM 3D cell culture chips offer a more realistic immune cell infiltration model by compartmentalizing immune cells and cancer spheroids in parallel channels. The chips consist of a 3D hydrogel channel and two flanking media channels. Cancer spheroids are cultured within the hydrogel channel while immune cells are seeded in one of the media channels. The seeded immune cells are required to actively invade and seek out target cancer cells in 3D hydrogel in AIM chips before they can infiltrate the cancer spheroids. This is a process similar to immune cell infiltration in vivo. By utilizing high content confocal imaging, fluorescent-labeled immune cells that migrate into the 3D hydrogel can be visualized and quantified. This assay, in combination with adoptive T cell therapy or immune checkpoint blockade, can determine the roles of tumor infiltrated lymphocytes in immunotherapy through quantifying the live: dead ratio of cancer spheroids in the chips. This is particularly useful as a quality control tool for cellular adoptive immunotherapy where the infiltration efficiency and tumoricidal activity of engineered immune cells can be accessed in vitro. In summary, AIM 3D cell culture chips create a more physiologically relevant 3D microenvironment for visualizing immune cell infiltration of cancer spheroids

Distinct firing activities of the hypothalamic arcuate nucleus neurons to appetite hormones

The hypothalamic arcuate nucleus (Arc) is a central unit that controls the appetite through the integration of metabolic, hormonal, and neuronal afferent inputs. Agouti-related protein (AgRP), proopiomelanocortin (POMC), and dopaminergic neurons in the Arc differentially regulate feeding behaviors in response to hunger, satiety, and appetite, respectively. At the time of writing, the anatomical and electrophysiological characterization of these three neurons has not yet been intensively explored. Here, we interrogated the overall characterization of AgRP, POMC, and dopaminergic neurons using genetic mouse models, immunohistochemistry, and whole-cell patch recordings. We identified the distinct geographical location and intrinsic properties of each neuron in the Arc with the transgenic lines labelled with cell-specific reporter proteins. Moreover, AgRP, POMC, and dopaminergic neurons had different firing activities to ghrelin and leptin treatments. Ghrelin led to the increased firing rate of dopaminergic and AgRP neurons, and the decreased firing rate of POMC. In sharp contrast, leptin resulted in the decreased firing rate of AgRP neurons and the increased firing rate of POMC neurons, while it did not change the firing rate of dopaminergic neurons in Arc. These findings demonstrate the anatomical and physiological uniqueness of three hypothalamic Arc neurons to appetite control

Distinct Firing Activities of the Hypothalamic Arcuate Nucleus Neurons to Appetite Hormones

The hypothalamic arcuate nucleus (Arc) is a central unit that controls the appetite through the integration of metabolic, hormonal, and neuronal afferent inputs. Agouti-related protein (AgRP), proopiomelanocortin (POMC), and dopaminergic neurons in the Arc differentially regulate feeding behaviors in response to hunger, satiety, and appetite, respectively. At the time of writing, the anatomical and electrophysiological characterization of these three neurons has not yet been intensively explored. Here, we interrogated the overall characterization of AgRP, POMC, and dopaminergic neurons using genetic mouse models, immunohistochemistry, and whole-cell patch recordings. We identified the distinct geographical location and intrinsic properties of each neuron in the Arc with the transgenic lines labelled with cell-specific reporter proteins. Moreover, AgRP, POMC, and dopaminergic neurons had different firing activities to ghrelin and leptin treatments. Ghrelin led to the increased firing rate of dopaminergic and AgRP neurons, and the decreased firing rate of POMC. In sharp contrast, leptin resulted in the decreased firing rate of AgRP neurons and the increased firing rate of POMC neurons, while it did not change the firing rate of dopaminergic neurons in Arc. These findings demonstrate the anatomical and physiological uniqueness of three hypothalamic Arc neurons to appetite control

Discovery of Two Distant Type Ia Supernovae in the Hubble Deep Field North with the Advanced Camera for Surveys

We present observations of the first two supernovae discovered with the recently installed Advanced Camera for Surveys (ACS) on the Hubble Space Telescope. The supernovae were found in Wide Field Camera images of the Hubble Deep Field North taken with the F775W, F850LP, and G800L optical elements as part of the ACS guaranteed time observation program. Spectra extracted from the ACS G800L grism exposures confirm that the objects are Type Ia supernovae (SNe Ia) at redshifts z=0.47 and z=0.95. Follow-up HST observations have been conducted with ACS in F775W and F850LP and with NICMOS in the near-infrared F110W bandpass, yielding a total of 9 flux measurements in the 3 bandpasses over a period of 50 days in the observed frame. We discuss many of the important issues in doing accurate photometry with the ACS. We analyze the multi-band light curves using two different fitting methods to calibrate the supernovae luminosities and place them on the SNe Ia Hubble diagram. The resulting distances are consistent with the redshift-distance relation of the accelerating universe model, although evolving intergalactic grey dust remains as a less likely possibility. The relative ease with which these SNe Ia were found, confirmed, and monitored demonstrates the potential ACS holds for revolutionizing the field of high-redshift SNe Ia, and therefore of testing the accelerating universe cosmology and constraining the "epoch of deceleration".Comment: 11 pages, 8 embedded figures. Accepted for publication in Ap