We introduce the notions of $(G,q)$-opers and Miura $(G,q)$-opers, where $G$ is a simply-connected complex simple Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of $(G,q)$-opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. This may be viewed as a $q$DE/IM correspondence between the spectra of a quantum integrable model (IM) and classical geometric objects ($q$-differential equations). If $\mathfrak{g}$ is simply-laced, the Bethe Ansatz equations we obtain coincide with the equations that appear in the quantum integrable model of XXZ-type associated to the quantum affine algebra $U_q \widehat{\mathfrak{g}}$. However, if $\mathfrak{g}$ is non-simply laced, then these equations correspond to a different integrable model, associated to $U_q {}^L\widehat{\mathfrak{g}}$ where $^L\widehat{\mathfrak{g}}$ is the Langlands dual (twisted) affine algebra. A key element in this $q$DE/IM correspondence is the $QQ$-system that has appeared previously in the study of the ODE/IM correspondence and the Grothendieck ring of the category ${\mathcal O}$ of the relevant quantum affine algebra.

A meromorphic connection on the complex projective line induces formal connections at each singular point, and these formal connections constitute the local behavior at the singularities. In this primarily expository paper, we discuss the extent to which specified local behavior at singular points determines the global connection. In particular, given a finite set of points and a collection of ``formal types'' at these points, does there exist a moduli space of meromorphic connections with this local behavior, and if so, when is this moduli space nonempty or a singleton? In this paper, we discuss variants of these problems (for example, the Deligne--Simpson and rigidity problems) as the allowed singularities get progressively more complicated: first connections with only regular singularities, next connections with additional unramified irregular singularities allowed, and finally the general case.

It is well-known that the spectra of the Gaudin model may be described in terms of solutions of the Bethe Ansatz equations. A conceptual explanation for the appearance of the Bethe Ansatz equations is provided by appropriate $G$-opers: $G$-connections on the projective line with extra structure. In fact, solutions of the Bethe Ansatz equations are parameterized by an enhanced version of opers called Miura opers; here, the opers appearing have only regular singularities. Feigin, Frenkel, Rybnikov, and Toledano Laredo have introduced an inhomogeneous version of the Gaudin model; this model incorporates an additional twist factor, which is an element of the Lie algebra of $G$. They exhibited the Bethe Ansatz equations for this model and gave a geometric interpretation of the spectra in terms of opers with an irregular singularity. In this paper, we consider a new approach to the study of the spectra of the inhomogeneous Gaudin model in terms of a further enhancement of opers called twisted Miura-Pl\"ucker opers and a certain system of nonlinear differential equations called the $qq$-system. We show that there is a close relationship between solutions of the inhomogeneous Bethe Ansatz equations and polynomial solutions of the $qq$-system and use this fact to construct a bijection between the set of solutions of the inhomogeneous Bethe Ansatz equations and the set of nondegenerate twisted Miura-Pl\"ucker opers. We further prove that as long as certain combinatorial conditions are satisfied, nondegenerate twisted Miura-Pl\"ucker opers are in fact Miura opers.

The differential Galois group of an $n^\mathrm{th}$ order linear differential equation is the symmetry group of its solutions; it is an algebraic subgroup of $\mathrm{GL}_n(\mathbb{C})$. More generally, if $G$ is a simple complex algebraic group, the differential Galois group of a $G$-connection is an algebraic subgroup of $G$. A connected reductive subgroup of $G$ is said to have maximal degree if it has a fundamental degree equal to the Coxeter number of $G$. We give a complete classification of these subgroups and generalise a theorem of Katz on linear differential equations by giving a criterion for the differential Galois group of a $G$-connection to be reductive of maximal degree. As an application, we determine the differential Galois groups of certain $G$-connections that play an important role in recent work on the geometric Langlands program: connections on $\mathbb{G}_m$ with an (irregular) "Coxeter" singularity and possibly an additional regular singular point.

esearch evaluation which attempts to measure the quality of scientific articles has a significant impact on judgements concerning research being carried out by scientific institutions, universities, individual scientists, and research groups.Previously, these research evaluations factored in expert opinions.Now, however, such judgements instead commonly rely on citation counts or indicators based upon them.For example, The National Research Council, the operating arm of the National Academies of Sciences, Engineering, and Medicine, carries out evaluations of graduate departments in the US.In 2010, the rankings put emphasis on data including citation counts, whereas previous rankings stressed the importance of the surveys of scholars.Likewise, a research journal's ranking is now measured almost entirely by citation count data.

The classical additive Deligne–Simpson problem is the existence problem for Fuchsian connections with residues at the singular points in specified adjoint orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem in terms of quiver varieties. A more general version of this problem, solved by Hiroe, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne–Simpson problem in which certain ramified singularities are allowed. These allowed singular points are called toral singularities; they are singularities whose leading term with respect to a lattice chain filtration is regular semisimple. We solve this problem in the special case of connections on Gm with a maximally ramified singularity at 0 and possibly an additional regular singular point at infinity. Examples of such connections arise from Airy, Bessel, and Kloosterman differential equations. They play an important role in recent work in the geometric Langlands program. We also give a complete characterization of all such connections which are rigid, under the additional hypothesis of unipotent monodromy at infinity.

The variety of bioimage data and their quality have dramatically increased over the last decade. In parallel, the number of proposed deep learning (DL) models for their analysis grows by the day. Yet, the adequate reuse of published tools by practitioners without DL expertise still raises many practical questions. In this article, we explore four categories of challenges faced by researchers when using supervised DL models in bioimaging applications. We provide examples in which each challenge arises and review the consequences that inadequate decisions may have. We then outline good practices that can be implemented to address the challenges of each category in a scientifically sound way. We provide pointers to the resources that are already available or in active development to help in this endeavor and advocate for the development of further community-driven standards. While primarily intended as a practical tutorial for life scientists, this article also aims at fostering discussions among method developers around the formulation of guidelines for the adequate deployment of DL, with the ultimate goal of accelerating the adoption of novel DL technologies in the biology community.

Peer review and citation metrics are two means of gauging the value of scientific research, but the lack of publicly available peer review data makes the comparison of these methods difficult. Mathematics can serve as a useful laboratory for considering these questions because as an exact science, there is a narrow range of reasons for citations. In mathematics, virtually all published articles are post-publication reviewed by mathematicians in Mathematical Reviews (MathSciNet) and so the data set was essentially the Web of Science mathematics publications from 1993 to 2004. For a decade, especially important articles were singled out in Mathematical Reviews for featured reviews. In this study, we analyze the bibliometrics of elite articles selected by peer review and by citation count. We conclude that the two notions of significance described by being a featured review article and being highly cited are distinct. This indicates that peer review and citation counts give largely independent determinations of highly distinguished articles. We also consider whether hiring patterns of subfields and mathematicians' interest in subfields reflect subfields of featured review or highly cited articles. We reexamine data from two earlier studies in light of our methods for implications on the peer review/citation count relationship to a diversity of disciplines.

We apply the theory of fundamental strata of Bremer and Sage to find cohomologically rigid G ‐connections on the projective line, generalising the work of Frenkel and Gross. In this theory, one studies the leading term of a formal connection with respect to the Moy–Prasad filtration associated to a point in the Bruhat–Tits building. If the leading term is regular semisimple with centraliser a (not necessarily split) maximal torus S , then we have an S ‐toral connection. In this language, the irregular singularity of the Frenkel–Gross connection gives rise to the homogeneous toral connection of minimal slope associated to the Coxeter torus C . In the present paper, we consider connections on Gm which have an irregular homogeneous C ‐toral singularity at zero of slope i/h , where h is the Coxeter number and i is a positive integer coprime to h , and a regular singularity at infinity with unipotent monodromy. Our main result is the characterisation of all such connections which are rigid.

A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers-connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for SL(N). We introduce a difference equation version of opers called q-opers and prove a q-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted q-opers with regular singularities on the projective line. We show that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars-Schneider model may be viewed as a special case of the q-Langlands correspondence. We also describe an application of q-opers to the equivariant quantum K-theory of the cotangent bundles to partial flag varieties.

special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers-connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for SL (N) . We introduce a difference equation version of opers called q -opers and prove a q -Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted q -opers with regular singularities on the projective line. We show that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars–Schneider model may be viewed as a special case of the q -Langlands correspondence. We also describe an application of q -opers to the equivariant quantum K -theory of the cotangent bundles to partial flag varieties.

Let G be a simple complex algebraic group. We prove that the irregularity of the adjoint connection of an irregular flat G-bundle on the formal punctured disk is always greater than or equal to the rank of G. This can be considered as a geometric analogue of a conjecture of Gross and Reeder. We will also show that the irregular connections with minimum adjoint irregularity are precisely the (formal) Frenkel-Gross connections. As a corollary, we establish the de Rham analogue of a conjecture of Heinloth, Ngo, and Yun for G = SLn.

We present a new active contour to segment cell aggregates. We describe it by a smooth tessellation that is attracted toward the cell membranes. Our approach relies on subdivision schemes that are tightly linked to the theory of wavelets. The shape is encoded by control points grouped in tiles. The smooth and continuously defined boundary of each tile is generated by recursively applying a refinement process to its control points. We deform the smooth tessellation in a global manner using a ridge-based energy that we have designed for that purpose. By construction, cells are segmented without overlap and the tessellation structure is maintained even on dim membranes. Leakage, which afflicts usual image-processing methods (e.g., watershed), is thus prevented. We validate our framework on both synthetic and real microscopy images, showing that the proposed method is robust to membrane gaps and to high levels of noise.

The theory of minimal K-types for p-adic reductive groups was developed in part to classify irreducible admissible representations with wild ramification. An important observation was that minimal K-types associated to such representations correspond to fundamental strata. These latter objects are triples (x, r, beta), where x is a point in the Bruhat-Tits building of the reductive group G, r is a nonnegative real number, and beta is a semistable functional on the degree r associated graded piece of the Moy-Prasad filtration corresponding to x. Recent work on the wild ramification case of the geometric Langlands conjectures suggests that fundamental strata also play a role in the geometric setting. In this paper, we develop a theory of minimal K-types for formal flat G-bundles. We show that any formal flat G-bundle contains a fundamental stratum; moreover, all such strata have the same rational depth. We thus obtain a new invariant of a flat G-bundle called the slope, generalizing the classical definition for flat connections. The slope can also be realized as the minimum depth of a stratum contained in the flat G-bundle, and in the case of positive slope, all such minimal depth strata are fundamental. Finally, we show that a flat G-bundle is irregular singular if and only if it has positive slope.

ABSTRACT One-dimensional local mode XH-stretching (X = O, F and Cl) vibrational transition frequencies and intensities have been calculated for six small atmospheric relevant molecules. The calculations are done using Morse, Deng–Fan and numeric potentials at the CCSD(T)/aug-cc-pVTZ level of theory. Parameters of the Morse potential are found from the derivatives of the potential energy surface, with respect to the internal XH-stretching displacement coordinate, evaluated at the equilibrium geometry. Parameters of the Deng–Fan potential are obtained using the first two transition frequencies of the Morse potential. The dipole moment functions (DMFs) are represented by a sixth-order polynomial in the displacement coordinate fitted to dipole moment single points calculated with a finite field approach. Analytical matrix elements are derived and used to calculate oscillator strengths for the Morse and Deng–Fan potential. We compare calculated and experimentally determined oscillator strengths and transition frequencies, as well as displacement matrix elements of the different potentials. For the XH-stretching vibrations considered, the Deng–Fan potential predicts absorption intensities considerably better than the Morse potential. GRAPHICAL ABSTRACT

In joint work with C. Bremer, the author has developed a geometric theory of fundamental strata which provides a new approach to the study of meromorphic G-connections on curves (for complex reductive G). In this theory, a fundamental stratum associated to a connection at a singular point plays the role of the local leading term of the connection. In this paper, we illustrate this theory for G = gl(2) C/ (i.e. for connections on rank two vector bundles). In particular, we show how this approach can be used to construct explicit moduli spaces of irregular singular connections on the projective line with specified singularities and formal types.

A critical role of circadian oscillators in orchestrating insulin secretion and islet gene transcription has been demonstrated recently. However, these studies focused on whole islets and did not explore the interplay between α-cell and β-cell clocks. We performed a parallel analysis of the molecular properties of α-cell and β-cell oscillators using a mouse model expressing three reporter genes: one labeling α cells, one specific for β cells, and a third monitoring circadian gene expression. Thus, phase entrainment properties, gene expression, and functional outputs of the α-cell and β-cell clockworks could be assessed in vivo and in vitro at the population and single-cell level. These experiments showed that α-cellular and β-cellular clocks are oscillating with distinct phases in vivo and in vitro. Diurnal transcriptome analysis in separated α and β cells revealed that a high number of genes with key roles in islet physiology, including regulators of glucose sensing and hormone secretion, are differentially expressed in these cell types. Moreover, temporal insulin and glucagon secretion exhibited distinct oscillatory profiles both in vivo and in vitro. Altogether, our data indicate that differential entrainment characteristics of circadian α-cell and β-cell clocks are an important feature in the temporal coordination of endocrine function and gene expression.

Images in fluorescence microscopy are inherently blurred due to the limit of diffraction of light. The purpose of deconvolution microscopy is to compensate numerically for this degradation. Deconvolution is widely used to restore fine details of 3D biological samples. Unfortunately, dealing with deconvolution tools is not straightforward. Among others, end users have to select the appropriate algorithm, calibration and parametrization, while potentially facing demanding computational tasks. To make deconvolution more accessible, we have developed a practical platform for deconvolution microscopy called DeconvolutionLab. Freely distributed, DeconvolutionLab hosts standard algorithms for 3D microscopy deconvolution and drives them through a user-oriented interface. In this paper, we take advantage of the release of DeconvolutionLab2 to provide a complete description of the software package and its built-in deconvolution algorithms. We examine several standard algorithms used in deconvolution microscopy, notably: Regularized inverse filter, Tikhonov regularization, Landweber, Tikhonov-Miller, Richardson-Lucy, and fast iterative shrinkage-thresholding. We evaluate these methods over large 3D microscopy images using simulated datasets and real experimental images. We distinguish the algorithms in terms of image quality, performance, usability and computational requirements. Our presentation is completed with a discussion of recent trends in deconvolution, inspired by the results of the Grand Challenge on deconvolution microscopy that was recently organized.

Classically, the exponent of a group is the least common multiple of the orders of its elements. This notion was generalized by Etingof and Gelaki to the context of Hopf algebras. Kashina, Sommerhauser and Zhu later observed that there is a strong connection between exponents and Frobenius-Schur indicators. In this paper, we introduce the notion of twisted exponents and show that there is a similar relationship between the twisted exponent and the twisted Frobenius-Schur indicators defined in previous work of the authors. In particular, we exhibit a new formula for the twisted Frobenius-Schur indicators and use it to prove periodicity and rationality statements for the twisted indicators.