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We establish general sufficient conditions for exact (and global) regularity in the $\bar\partial$-Neumann problem on $(p,q)$-forms, $0 \leq p \leq n$ and $1\leq q \leq n$, on a pseudoconvex domain $\Omega$ with smooth boundary $b\Omega$ in an $n$-dimensional complex manifold $M$. Our hypotheses include two assumptions: 1) $M$ admits a function that is strictly plurisubharmonic acting on $(p_0,q_0)$-forms in a neighborhood of $b\Omega$ for some fixed $0 \leq p_0 \leq n$, $1 \leq q_0 \leq n$, or $M$ is a K\"ahler metric whose holomorphic bisectional curvature acting $(p,q)$-forms is positive; and 2) there exists a family of vector fields $T_\epsilon$ that are transverse to the boundary $b\Omega$ and generate one forms, which when applied to $(p,q)$-forms, $0 \leq p \leq n$ and $q_0 \leq q \leq n$, satisfy a "weak form" of the compactness estimate. We also provide examples and applications of our main theorems., Comment: 28 pages. Comments welcome!

Morphogenesis is the process whereby the body of an organism develops its target shape. The morphogen BMP is known to play a conserved role across bilaterian organisms in determining the dorsoventral (DV) axis. Yet, how BMP governs the spatio-temporal dynamics of cytoskeletal proteins driving morphogenetic flow remains an open question. Here, we use machine learning to mine a morphodynamic atlas of Drosophila development, and construct a mathematical model capable of predicting the coupled dynamics of myosin, E-cadherin, and morphogenetic flow. Mutant analysis shows that BMP sets the initial condition of this dynamical system according to the following signaling cascade: BMP establishes DV pair-rule-gene patterns that set-up an E-cadherin gradient which in turn creates a myosin gradient in the opposite direction through mechanochemical feedbacks. Using neural tube organoids, we argue that BMP, and the signaling cascade it triggers, prime the conserved dynamics of neuroectoderm morphogenesis from fly to humans., Comment: Main text: 22 pages, 7 figures. Supplement: 18 pages, 10 figures

Given a complex manifold containing a relatively compact $Z(q)$ domain, we give sufficient geometric conditions on the domain so that its $L^2$-cohomology in degree $(p,q)$ (known to be finite dimensional) vanishes. The condition consists of the existence of a smooth weight function in a neighborhood of the closure of the domain, where the complex Hessian of the weight has a prescribed number of eigenvalues of a particular sign, along with good interaction at the boundary of the Levi form with the complex Hessian, encoded in a subbundle of common positive directions for the two Hermitian forms., Comment: 29 pages. Comments welcome!

We evaluated the sensitivity of estimated PM2.5 and NO2 health impacts to varying key input parameters and assumptions including: 1) the spatial scale at which impacts are estimated, 2) using either a single concentration-response function (CRF) or using racial/ethnic group specific CRFs from the same epidemiologic study, 3) assigning exposure to residents based on home, instead of home and work locations. This analysis was carried out for the state of Colorado. We found that the spatial scale of the analysis influences the magnitude of NO2, but not PM2.5, attributable deaths. Using county-level predictions instead of 1 km2 predictions of NO2 resulted in a lower estimate of mortality attributable to NO2 by ~ 50% for all of Colorado for each year between 2000-2020. Using an all-population CRF instead of racial/ethnic group specific CRFs results in a higher estimate of annual mortality attributable to PM2.5 by a factor 1.3 for the white population and a lower estimate of mortality attributable to PM2.5 by factors of 0.4 and 0.8 for Black and Hispanic residents, respectively. Using racial/ethnic group specific CRFs did not result in a different estimation of NO2 attributable mortality for white residents, but led to lower estimates of mortality by a factor of ~ 0.5 for Black residents, and by a factor of 2.9 for to Hispanic residents. Using NO2 based on home instead of home and workplace locations results in a smaller estimate of annual mortality attributable to NO2 for all of Colorado by ~0.980 each year and 0.997 for PM2.5., Comment: 24 pages, 6 figures, 2 tables

The problem of foreground material signature extraction in an intimate (nonlinear) mixing setting is considered. It is possible for a foreground material signature to appear in combination with multiple background material signatures. We explore a framework for foreground material signature extraction based on a patch model that accounts for such background variation. We identify data conditions under which a foreground material signature can be extracted up to scaling and elementwise-inverse variations. We present algorithms based on volume minimization and endpoint member identification to recover foreground material signatures under these conditions. Numerical experiments on real and synthetic data illustrate the efficacy of the proposed algorithms., Comment: 41 pages, 5 figures

The Gemini North Adaptive Optics (GNAO) facility is the upcoming AO facility for Gemini North providing a state-of-the-art AO system for surveys and time domain science in the era of JWST and Rubin operations. GNAO will be optimized to feed the Gemini infrared Multi Object Spectrograph (GIRMOS). While GIRMOS is the primary science driver for defining the capabilities of GNAO, any instrument operating with an f/32 beam can be deployed using GNAO. The GNAO project includes the development of a new laser guide star facility which will consist of four side-launched laser beams supporting the two primary AO modes of GNAO: a wide-field mode providing an improved image quality over natural seeing for a 2-arcminute circular field-of-view and a narrow-field mode providing near diffraction-limited performance over a 20x20 arcsecond square field-of-view. The GNAO wide field mode will enable GIRMOS's multi-IFU configuration in which the science beam to each individual IFU will be additionally corrected using multi-object AO within GIRMOS. The GNAO narrow field mode will feed the GIRMOS tiled IFU configuration in which all IFUs are combined into a "super"-IFU in the center of the field. GNAO also includes the development of a new Real Time Controller, a new GNAO Facility System Controller and finally the development of a new AO Bench. We present in this paper an overview of the GNAO facility and provide a status update of each product., Comment: SPIE conference 2022 Montreal

The unit-modulus least squares (UMLS) problem has a wide spectrum of applications in signal processing, e.g., phase-only beamforming, phase retrieval, radar code design, and sensor network localization. Scalable first-order methods such as projected gradient descent (PGD) have recently been studied as a simple yet efficient approach to solving the UMLS problem. Existing results on the convergence of PGD for UMLS often focus on global convergence to stationary points. As a non-convex problem, only a sublinear convergence rate has been established. However, these results do not explain the fast convergence of PGD frequently observed in practice. This manuscript presents a novel analysis of convergence of PGD for UMLS, justifying the linear convergence behavior of the algorithm near the solution. By exploiting the local structure of the objective function and the constraint set, we establish an exact expression for the convergence rate and characterize the conditions for linear convergence. Simulations show that our theoretical analysis corroborates numerical examples. Furthermore, variants of PGD with adaptive step sizes are proposed based on the new insight revealed in our convergence analysis. The variants show substantial acceleration in practice., Comment: 16 pages

The second African Conference of Fundamental and Applied Physics (ACP2021) took place in the week of March 7-11, 2022. During this conference, all the African Strategy for Fundamental and Applied Physics (ASFAP) working groups had been reserved specials sessions to discuss their scope, activities (past & current) and topics of common interests. The aim of this report is to summarize the discussion of the ASFAP working groups in societal engagements, namely Physics Education, Community Engagement, Young Physicists and Women in Physics. The recommendations for future activities in societal engagements are summarised in the report as well., Comment: 7 pages, 2 figures Submitted to the Proceedings of the African Conference on Fundamental and Applied Physics Second Edition, ACP2021, March 7-11, 2022 - Virtual Event

It is well known that elliptic estimates fail for the $\bar\partial$-Neumann problem. Instead, the best that one can hope for is that derivatives in every direction but one can be estimated by the associated Dirichlet form, and when this happens, we say that the $\bar\partial$-Neumann problem satisfies maximal estimates. In the pseudoconvex case, a necessary and sufficient geometric condition for maximal estimates has been derived by Derridj (for $(0,1)$-forms) and Ben Moussa (for $(0,q)$-forms when $q\geq 1$). In this paper, we explore necessary conditions and sufficient conditions for maximal estimates in the non-pseudoconvex case. We also discuss when the necessary conditions and sufficient conditions agree and provide examples. Our results subsume the earlier known results from the pseudoconvex case., Comment: 35 pages

Iterative hard thresholding (IHT) has gained in popularity over the past decades in large-scale optimization. However, convergence properties of this method have only been explored recently in non-convex settings. In matrix completion, existing works often focus on the guarantee of global convergence of IHT via standard assumptions such as incoherence property and uniform sampling. While such analysis provides a global upper bound on the linear convergence rate, it does not describe the actual performance of IHT in practice. In this paper, we provide a novel insight into the local convergence of a specific variant of IHT for matrix completion. We uncover the exact linear rate of IHT in a closed-form expression and identify the region of convergence in which the algorithm is guaranteed to converge. Furthermore, we utilize random matrix theory to study the linear rate of convergence of IHTSVD for large-scale matrix completion. We find that asymptotically, the rate can be expressed in closed form in terms of the relative rank and the sampling rate. Finally, we present various numerical results to verify the aforementioned theoretical analysis., Comment: 13 pages (plus Supplementary Material)

Many recent problems in signal processing and machine learning such as compressed sensing, image restoration, matrix/tensor recovery, and non-negative matrix factorization can be cast as constrained optimization. Projected gradient descent is a simple yet efficient method for solving such constrained optimization problems. Local convergence analysis furthers our understanding of its asymptotic behavior near the solution, offering sharper bounds on the convergence rate compared to global convergence analysis. However, local guarantees often appear scattered in problem-specific areas of machine learning and signal processing. This manuscript presents a unified framework for the local convergence analysis of projected gradient descent in the context of constrained least squares. The proposed analysis offers insights into pivotal local convergence properties such as the conditions for linear convergence, the region of convergence, the exact asymptotic rate of convergence, and the bound on the number of iterations needed to reach a certain level of accuracy. To demonstrate the applicability of the proposed approach, we present a recipe for the convergence analysis of projected gradient descent and demonstrate it via a beginning-to-end application of the recipe on four fundamental problems, namely, linear equality-constrained least squares, sparse recovery, least squares with the unit norm constraint, and matrix completion., Comment: 21 pages including Supplementary Material

In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the asymptotic linear rate of convergence, it does not explain the non-linear behavior of these algorithms in the non-asymptotic regime. In this letter, we take into account the effect of the first-order approximation error and present a closed-form bound on the convergence in terms of the number of iterations required for the distance between the iterate and the limit point to reach an arbitrarily small fraction of the initial distance. Our bound includes two terms: one corresponds to the number of iterations required for the linearized version of the fixed-point operator and the other corresponds to the overhead associated with the approximation error. With a focus on the convergence in the scalar case, the tightness of the proposed bound is proven for positively quadratic first-order difference equations., Comment: 14 pages - final version accepted to OPTL22

In multiple instance multiple label learning, each sample, a bag, consists of multiple instances. To alleviate labeling complexity, each sample is associated with a set of bag-level labels leaving instances within the bag unlabeled. This setting is more convenient and natural for representing complicated objects, which have multiple semantic meanings. Compared to single instance labeling, this approach allows for labeling larger datasets at an equivalent labeling cost. However, for sufficiently large datasets, labeling all bags may become prohibitively costly. Active learning uses an iterative labeling and retraining approach aiming to provide reasonable classification performance using a small number of labeled samples. To our knowledge, only a few works in the area of active learning in the MIML setting are available. These approaches can provide practical solutions to reduce labeling cost but their efficacy remains unclear. In this paper, we propose a novel bag-class pair based approach for active learning in the MIML setting. Due to the partial availability of bag-level labels, we focus on the incomplete-label MIML setting for the proposed active learning approach. Our approach is based on a discriminative graphical model with efficient and exact inference. For the query process, we adapt active learning criteria to the novel bag-class pair selection strategy. Additionally, we introduce an online stochastic gradient descent algorithm to provide an efficient model update after each query. Numerical experiments on benchmark datasets illustrate the robustness of the proposed approach., Comment: Machine learning, Multiple instance multiple label learning, Active learning, incomplete label learning

Factorization-based gradient descent is a scalable and efficient algorithm for solving low-rank matrix completion. Recent progress in structured non-convex optimization has offered global convergence guarantees for gradient descent under certain statistical assumptions on the low-rank matrix and the sampling set. However, while the theory suggests gradient descent enjoys fast linear convergence to a global solution of the problem, the universal nature of the bounding technique prevents it from obtaining an accurate estimate of the rate of convergence. In this paper, we perform a local analysis of the exact linear convergence rate of gradient descent for factorization-based matrix completion for symmetric matrices. Without any additional assumptions on the underlying model, we identify the deterministic condition for local convergence of gradient descent, which only depends on the solution matrix and the sampling set. More crucially, our analysis provides a closed-form expression of the asymptotic rate of convergence that matches exactly with the linear convergence observed in practice. To the best of our knowledge, our result is the first one that offers the exact rate of convergence of gradient descent for matrix factorization in Euclidean space for matrix completion., Comment: Full version

This paper is the fourth of a multi-part series in which we study the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of $\mathbb{C}^n\times\mathbb{C}^m$. The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We 1) investigate the geometric conditions on $M$ which the eigenvalue condition forces, 2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, 3) explore the $L^p$ and $L^p$-Sobolev mapping properties of the associated kernels, and 4) provide examples., Comment: 34 pages. Comments welcome!

This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of $\mathbb{C}^n\times\mathbb{C}^m$. In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator $N$ and the projection onto the nullspace of $\Box_b$. The main application of our formulas is the critical case of codimension two quadrics in $\mathbb{C}^4$ where we discuss the known solvability and hypoellipticity criteria of Peloso and Ricci \cite{PeRi03}. We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups., Comment: 15 pages. Comments welcome!

Truncated singular value decomposition is a reduced version of the singular value decomposition in which only a few largest singular values are retained. This paper presents a novel perturbation analysis for the truncated singular value decomposition for real matrices. First, we describe perturbation expansions for the singular value truncation of order $r$. We extend perturbation results for the singular subspace decomposition to derive the first-order perturbation expansion of the truncated operator about a matrix with rank greater than or equal to $r$. Observing that the first-order expansion can be greatly simplified when the matrix has exact rank $r$, we further show that the singular value truncation admits a simple second-order perturbation expansion about a rank-$r$ matrix. Second, we introduce the first-known error bound on the linear approximation of the truncated singular value decomposition of a perturbed rank-$r$ matrix. Our bound only depends on the least singular value of the unperturbed matrix and the norm of the perturbation matrix. Intriguingly, while the singular subspaces are known to be extremely sensitive to additive noises, the newly established error bound holds universally for perturbations with arbitrary magnitude. Finally, we demonstrate an application of our results to the analysis of the mean squared error associated with the TSVD-based matrix denoising solution., Comment: Accepted to Linear Algebra and Its Applications

In this paper we introduce a new type of restriction problem, called the \textit{restriction problem with moments}. We show that the surface area measure of the sphere satisfies the $L^p$-$L^2$ restriction problem with moments if $1 \leq p < \frac{2(d+2)}{d+3}$ and that the Frostman measure constructed by Salem satisfies the $L^p$-$L^2$ restriction problem with moments if $1 \leq p < \frac{2(2-2\alpha+\beta)}{4(1-\alpha)+\beta}$ for certain values of $\alpha$ and $\beta$. The main tool to obtain these new type of restriction phenomenon is the notion of distributions with decay in connection with classes of global $L^q$ ultradifferentiable functions. We develop the notion of distributions with decay and use it to define global wavefront sets of classes of function spaces, including $L^p$-Sobolev spaces on \mathbb{R}^d$ as well as global $L^q$-Denjoy Carleman functions. We also introduce the corresponding notion of microglobal regularity. We prove a characterization of distributions (in a given function space) with decay in terms of microglobal regularity in every direction of their Fourier transforms., Comment: 14 pages. Comments welcome!

The purpose of this paper is to establish sufficient conditions for closed range estimates on $(0,q)$-forms, for some fixed $q$, $1 \leq q \leq n-1$, for $\bar\partial_b$ in both $L^2$ and $L^2$-Sobolev spaces in embedded, not necessarily pseudoconvex CR manifolds of hypersurface type. The condition, named weak $Y(q)$, is both more general than previously established sufficient conditions and easier to check. Applications of our estimates include estimates for the Szeg\"o projection as well as an argument that the harmonic forms have the same regularity as the complex Green operator. We use a microlocal argument and carefully construct a norm that is well-suited for a microlocal decomposition of form. We do not require that the CR manifold is the boundary of a domain. Finally, we provide an example that demonstrates that weak $Y(q)$ is an easier condition to verify than earlier, less general conditions., Comment: 24 pages

The $L^2$ theory of the $\bar\partial$ operator on domains in $\mathbb{C}^n$ is predicated on establishing a good basic estimate. Typically, one proves not a single basic estimate but a family of basic estimates that we call a family of strong closed range estimates. Using this family of estimates on $(0,q)$-forms as our starting point, we establish necessary geometric and potential theoretic conditions. The paper concludes with several applications. We investigate the consequences for compactness estimates for the $\bar\partial$-Neumann problem, and we also establish a generalization of Kohn's weighted theory via elliptic regularization. Since our domains are not necessarily pseudoconvex, we must take extra care with the regularization., Comment: 26 pages

This paper provides a connection between two distinct branches of research in CR geometry -- namely, analytic and geometric conditions that suffice to establish the closed range of the Cauchy-Riemann operator and CR invariants on CR manifolds. Specifically, we work on not necessarily pseudoconvex domains $\Omega\subset\mathbb{C}^n$ and define third and fourth order CR invariants on $\bd\Om$ and show that these invariants provide enough information to establish closed range for the $\bar\partial$-Laplacian in $L^2_{(0,q)}(\Omega)$ for a given, fixed $q$. The closed range estimates follow from our previously defined weak $Z(q)$ condition. We also develop powerful linear algebra machinery to translate the information from the invariants into information about the Levi form and its eigenvalues. We conclude with several examples that demonstrate the usefulness and ease of use of the new conditions., Comment: 23 pages. Comments welcome!

We present a probabilistic modeling and inference framework for discriminative analysis dictionary learning under a weak supervision setting. Dictionary learning approaches have been widely used for tasks such as low-level signal denoising and restoration as well as high-level classification tasks, which can be applied to audio and image analysis. Synthesis dictionary learning aims at jointly learning a dictionary and corresponding sparse coefficients to provide accurate data representation. This approach is useful for denoising and signal restoration, but may lead to sub-optimal classification performance. By contrast, analysis dictionary learning provides a transform that maps data to a sparse discriminative representation suitable for classification. We consider the problem of analysis dictionary learning for time-series data under a weak supervision setting in which signals are assigned with a global label instead of an instantaneous label signal. We propose a discriminative probabilistic model that incorporates both label information and sparsity constraints on the underlying latent instantaneous label signal using cardinality control. We present the expectation maximization (EM) procedure for maximum likelihood estimation (MLE) of the proposed model. To facilitate a computationally efficient E-step, we propose both a chain and a novel tree graph reformulation of the graphical model. The performance of the proposed model is demonstrated on both synthetic and real-world data.

The goal of this note is to explore the Bergman projection on forms. In particular, we show that some of most basic facts used to construct the Bergman kernel on functions, such as pointwise evaluation in $L^2_{0,q}(\Omega)\cap\ker\bar\partial_q$, fail for $(0,q)$-forms, $q \geq 1$. We do, however, provide a careful construction of the Bergman kernel and explicitly compute the Bergman kernel on $(0,n-1)$-forms. For the ball in $\mathbb{C}^2$, we also show that the size of the Bergman kernel on $(0,1)$-forms is not governed by the control metric, in stark contrast to Bergman kernel on functions., Comment: 10 pages. Comments welcome!

Let $\Omega\subset\mathbb{C}^n$ be a domain and $1 \leq q \leq n-1$ fixed. Our purpose in this article is to establish a general sufficient condition for the closed range of the Cauchy-Riemann operator $\bar\partial$ in appropriately weighted $L^2$-Sobolev spaces on $(0,q)$-forms. The domains we consider may be neither bounded nor pseudoconvex, and our condition is a generalization of the classical $Z(q)$ condition that we call weak $Z(q)$. We provide examples that explain the necessity of working in weighted spaces both for closed range in $L^2$ and, even more critically, in $L^2$-Sobolev spaces., Comment: 23 pages. Comments welcome! arXiv admin note: substantial text overlap with arXiv:1507.06211

Let $M$ be a compact, pseudoconvex-oriented, $(2n+1)$-dimensional, abstract CR manifold of hypersurface type, $n\geq 2$. We prove the following: (i) If $M$ admits a strictly CR-plurisubharmonic function on $(0,q_0)$-forms, then the complex Green operator $G_q$ exists and is continuous on $L^2_{0,q}(M)$ for degrees $q_0\le q\le n-q_0$. In the case that $q_0=1$, we also establish continuity for $G_0$ and $G_n$. Additionally, the $\bar\partial_b$-equation on $M$ can be solved in $C^\infty(M)$. (ii) If $M$ satisfies "a weak compactness property" on $(0,q_0)$-forms, then $G_q$ is a continuous operator on $H^s_{0,q}(M)$ and is therefore globally regular on $M$ for degrees $q_0\le q\le n-q_0$; and also for the top degrees $q=0$ and $q=n$ in the case $q_0=1$. We also introduce the notion of a "plurisubharmonic CR manifold" and show that it generalizes the notion of "plurisubharmonic defining function" for a a domain in $\mathbb C^N$ and implies that $M$ satisfies the weak compactness property., Comment: 29 pages

The purpose of this paper is to characterize the zero sets of holomorphic functions in the Nevanlinna class on a class of convex domains of infinite type in $\mathbb{C}^2$. Moreover, we also obtain $L^p$ estimates, $1 \leq p \leq \infty$, for a particular solution of the tangential Cauchy-Riemann equation on the boundaries of these domains., Comment: 15 pages

Multi-instance data, in which each object (bag) contains a collection of instances, are widespread in machine learning, computer vision, bioinformatics, signal processing, and social sciences. We present a maximum entropy (ME) framework for learning from multi-instance data. In this approach each bag is represented as a distribution using the principle of ME. We introduce the concept of confidence-constrained ME (CME) to simultaneously learn the structure of distribution space and infer each distribution. The shared structure underlying each density is used to learn from instances inside each bag. The proposed CME is free of tuning parameters. We devise a fast optimization algorithm capable of handling large scale multi-instance data. In the experimental section, we evaluate the performance of the proposed approach in terms of exact rank recovery in the space of distributions and compare it with the regularized ME approach. Moreover, we compare the performance of CME with Multi-Instance Learning (MIL) state-of-the-art algorithms and show a comparable performance in terms of accuracy with reduced computational complexity.

The goal of this note is to explore the relationship between the Folland-Kohn basic estimate and the $Z(q)$-condition. In particular, on unbounded pseudoconvex (resp., pseudoconcave) domains, we prove that the Folland-Kohn basic estimate is equivalent to uniform strict pseudoconvexity (resp., pseudoconcavity). As a corollary, we observe that despite the Siegel upper half space being strictly pseudoconvex and biholomorphic to a the unit ball, it fails to satisfy uniform strict pseudoconvexity and hence the Folland-Kohn basic estimate fails. On unbounded non-pseudoconvex domains, we show that the Folland-Kohn basic estimate on $(0,q)$-forms implies a uniform $Z(q)$ condition, and conversely, that a uniform $Z(q)$ condition with some additional hypotheses implies the Folland-Kohn basic estimate for $(0,q)$-forms., Comment: 6 pages

Within the framework of the LHC Injector Upgrade (LIU), the new LINAC4 is currently being commissioned to replace the existing LINAC2 proton source at CERN. After the expected completion at the end of 2016, the LINAC4 will accelerate H- ions to 160 MeV. To measure the transverse emittance of the H- beam, a method based on photo-detachment is proposed. This system will operate using a pulsed laser with light delivered via an optical fibre and subsequently focused through a thin slice of the H- beam. The laser photons have sufficient energy to detach the outer electron and create H0/e- pairs. In a downstream dipole, the created H0 particles are separated from the unstripped H- ions and their distribution is measured with a dedicated detector. By scanning the focused laser across the H- beam, the transverse emittance of the H- beam can be reconstructed. This paper will first discuss the concept, design and simulations of the laser, Comment: Presented at the International Conference on Laser Applications at Accelerators, LA3NET 2015; Submitted to Nucl. Instr. and Meth. in Phys. Res. section A

We present new examples of complexes of differential operators of order $k$ (any given positive integer) that satisfy div-curl and/or $L^1$-duality estimates., Comment: This is an electronic reprint of the original article that appeared in {\em Advances in Analysis: the legacy of E. M. Stein}, Princeton U. Press (2013) ISBN: 9780691159416. This reprint differs from the original in pagination and typographical detail

We study the heat kernel $p(x,y,t)$ associated to the real Schr\"odinger operator $H = -\Delta + V$ on $L^2(\mathbb{R}^n)$, $n \geq 1$. Our main result is a pointwise upper bound on $p$ when the potential $V \in A_\infty$. In the case that $V\in RH_\infty$, we also prove a lower bound. Additionally, we compute $p$ explicitly when $V$ is a quadratic polynomial., Comment: 15 pages. Comments welcome!

Precise asymptotics known for the Green function of the Laplacian have found their analogs for bounded below periodic elliptic operators of the second-order below and at the bottom of the spectrum. Due to the band-gap structure of the spectra of such operators, the question arises whether similar results can be obtained near or at the edges of spectral gaps. In a previous work, two of the authors considered the case of a spectral edge. The main result of this article is finding such asymptotics near a gap edge, for "generic" periodic elliptic operators of second-order with real coefficients in dimension $d \geq 2$, when the gap edge occurs at a symmetry point of the Brillouin zone., Comment: In v3, a couple of misprints are corrected

A non-invasive, compact laserwire system has been developed to measure the transverse emittance of an H- beam and has been demonstrated at the new LINAC4 injector for the LHC at CERN. Light from a low power, pulsed laser source is conveyed via fibre to collide with the H- beam, a fraction of which is neutralized and then intercepted by a downstream diamond detector. Scanning the focused laser across the H- beam and measuring the distribution of the photo-neutralized particles enables the transverse emittance to be reconstructed. The vertical phase-space distribution of a 3 MeV beam during LINAC4 commissioning has been measured by the laserwire and verified with a conventional slit and grid method., Comment: 10 pages, 13 figures, to be published in Physical Review Special Topics - Accelerators and Beams

In this article, we establish a general sufficient condition for closed range of the Cauchy-Riemann operator $\bar\partial$ in appropriately weighted $L^2$ and $L^2$-Sobolev spaces on $(0,q)$-forms for a fixed $q$ on domains in $\mathbb{C}^n$. The domains we consider may be neither bounded nor pseudoconvex, and our condition is a generalization of the classical $Z(q)$ condition that we call weak $Z(q)$. We provide examples that explain the necessity of working in weighted spaces both for closed range in $L^2$ and even more critically, in $L^2$-Sobolev spaces., Comment: 19 pages. We split the original paper into two, smaller papers

Labeling data for classification requires significant human effort. To reduce labeling cost, instead of labeling every instance, a group of instances (bag) is labeled by a single bag label. Computer algorithms are then used to infer the label for each instance in a bag, a process referred to as instance annotation. This task is challenging due to the ambiguity regarding the instance labels. We propose a discriminative probabilistic model for the instance annotation problem and introduce an expectation maximization framework for inference, based on the maximum likelihood approach. For many probabilistic approaches, brute-force computation of the instance label posterior probability given its bag label is exponential in the number of instances in the bag. Our key contribution is a dynamic programming method for computing the posterior that is linear in the number of instances. We evaluate our methods using both benchmark and real world data sets, in the domain of bird song, image annotation, and activity recognition. In many cases, the proposed framework outperforms, sometimes significantly, the current state-of-the-art MIML learning methods, both in instance label prediction and bag label prediction.

We generalize Nagel's formula for the Szeg\"o kernel and use it to compute the Szeg\"o kernel on a class of noncompact CR manifolds whose tangent space decomposes into one complex direction and several totally real directions. We also discuss the control metric on these manifolds and relate it to the size of the Szeg\"o kernel., Comment: 7 pages

The purpose of this paper is to prove that if a pseudoconvex domains $\Omega\subset\mathbb{C}^n$ satisfies Bell-Ligocka's Condition R and admits a ``good" dilation, then the Bergman projection has local $L^p$-Sobolev and H\"older estimates. The good dilation structure is phrased in terms of uniform $L^2$ pseudolocal estimates for the Bergman projection on a family of anisotropic scalings. We conclude the paper by showing that $h$-extendible domains satisfy our hypotheses., Comment: 25 pages. This version is a major overhaul from the previous versions

As soon as the first particles emerge from an ion source, the source characteristics need to be determined. The total beam intensity, the transverse particle distributions, the beam divergence and emittance as well as the longitudinal parameters of the beam must be measured. This chapter provides an overview of typical measurement methods and the instruments used, and shows the results obtained., Comment: 12 pages, contribution to the CAS-CERN Accelerator School: Ion Sources, Senec, Slovakia, 29 May - 8 June 2012, edited by R. Bailey, CERN-2013-007

Novelty detection plays an important role in machine learning and signal processing. This paper studies novelty detection in a new setting where the data object is represented as a bag of instances and associated with multiple class labels, referred to as multi-instance multi-label (MIML) learning. Contrary to the common assumption in MIML that each instance in a bag belongs to one of the known classes, in novelty detection, we focus on the scenario where bags may contain novel-class instances. The goal is to determine, for any given instance in a new bag, whether it belongs to a known class or a novel class. Detecting novelty in the MIML setting captures many real-world phenomena and has many potential applications. For example, in a collection of tagged images, the tag may only cover a subset of objects existing in the images. Discovering an object whose class has not been previously tagged can be useful for the purpose of soliciting a label for the new object class. To address this novel problem, we present a discriminative framework for detecting new class instances. Experiments demonstrate the effectiveness of our proposed method, and reveal that the presence of unlabeled novel instances in training bags is helpful to the detection of such instances in testing stage.

Our work is focused on the joint sparsity recovery problem where the common sparsity pattern is corrupted by Poisson noise. We formulate the confidence-constrained optimization problem in both least squares (LS) and maximum likelihood (ML) frameworks and study the conditions for perfect reconstruction of the original row sparsity and row sparsity pattern. However, the confidence-constrained optimization problem is non-convex. Using convex relaxation, an alternative convex reformulation of the problem is proposed. We evaluate the performance of the proposed approach using simulation results on synthetic data and show the effectiveness of proposed row sparsity and row sparsity pattern recovery framework.

In this paper we prove that on a CR manifold of hypersurface type that satisfies the weak $Y(q)$ condition, the complex Green operator $G_q$ is exactly (globally) regular if and only if the Szeg\"o projections $S_{q-1}, S_q$ and a third orthogonal projection $S'_{q+1}$ are exactly (globally) regular. The projection $S'_{q+1}$ is closely related to the Szeg\"o projection $S_{q+1}$ and actually coincides with it if the space of harmonic $(0,q+1)$-forms is trivial. This result extends the important and by now classical result by H. Boas and E. Straube on the equivalence of the regularity of the $\bar\partial$-Neumann operator and the Bergman projections on a smoothly bounded pseudoconvex domain. We also prove an extension of this result to the case of bounded smooth domains satisfying the weak $Z(q)$ condition on a Stein manifold., Comment: 15 pages

In this article, we develop the theory of weighted $L^2$ Sobolev spaces on unbounded domains in $\mathbb R^n$. As an application, we establish the elliptic theory for elliptic operators and prove trace and extension results analogous to the bounded, unweighted case., Comment: 47 pages

The Kerzman-Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on a rectifiable curve. If the curve is continuously differentiable, the Kerzman-Stein operator is compact on the Hilbert space of square integrable functions; when there is a corner, the operator is noncompact. Here we give a complete description of the spectrum for a finite symmetric wedge and we show how this reveals the essential spectrum for curves that are piecewise continuously differentiable. We also give an explicit construction for a smooth curve whose Kerzman-Stein operator has large norm, and we demonstrate the variation in norm with respect to a continuously differentiable perturbation., Comment: 18 pages

For a domain $\Omega\subset\mathbb R^n$, we introduce the concept of a uniformly $C^m$ defining function. We characterize uniformly $C^m$ defining functions in terms of the signed distance function for the boundary and provide a large class of examples of unbounded domains with uniformly $C^m$ defining functions. Some of our results extend results from the bounded case., Comment: 14 pages

The purpose of this article is to expand the number of examples for which the complex Green operator, that is, the fundamental solution to the Kohn Laplacian, can be computed. We use the Lie group structure of quadric submanifolds of $\mathbb C^n\times\mathbb C^m$ and the group Fourier transform to reduce the $\Box_b$ equation to ones that can be solved using modified Hermite functions. We use Mehler's formula and investigate 1) quadric hypersurfaces, where the eigenvalues of the Levi form are not identical (including possibly zero eigenvalues), and 2) the canonical quadrics in $\mathbb C^4$ of codimension two., Comment: 20 pages

Precise asymptotics known for the Green's function of the Laplace operator have found their analogs for periodic elliptic operators of the second order at and below the bottom of the spectrum. Due to the band-gap structure of the spectra of such operators, the question arises whether similar results can be obtained near or at the edges of spectral gaps. As the result of this work shows, this is possible at a spectral edge in dimensions d>2.

We define weak $Z(q)$, a generalization of $Z(q)$ on bounded domains $\Omega$ in a Stein manifold $M^n$ that suffices to prove closed range of $\bar\partial$. Under the hypothesis of weak $Z(q)$, we also show (i) that harmonic $(0,q)$-forms are trivial and (ii) if $\partial\Omega$ satisfies weak $Z(q)$ and weak $Z(n-1-q)$, then $\dbar_b$ has closed range on $(0,q)$-forms on $\partial\Omega$. We provide examples to show that our condition contains examples that are excluded from $(q-1)$-pseudoconvexity and the authors' previous notion of weak $Z(q)$., Comment: 29 pages

This paper introduces a class of k-nearest neighbor ($k$-NN) estimators called bipartite plug-in (BPI) estimators for estimating integrals of non-linear functions of a probability density, such as Shannon entropy and R\'enyi entropy. The density is assumed to be smooth, have bounded support, and be uniformly bounded from below on this set. Unlike previous $k$-NN estimators of non-linear density functionals, the proposed estimator uses data-splitting and boundary correction to achieve lower mean square error. Specifically, we assume that $T$ i.i.d. samples ${X}_i \in \mathbb{R}^d$ from the density are split into two pieces of cardinality $M$ and $N$ respectively, with $M$ samples used for computing a k-nearest-neighbor density estimate and the remaining $N$ samples used for empirical estimation of the integral of the density functional. By studying the statistical properties of k-NN balls, explicit rates for the bias and variance of the BPI estimator are derived in terms of the sample size, the dimension of the samples and the underlying probability distribution. Based on these results, it is possible to specify optimal choice of tuning parameters $M/T$, $k$ for maximizing the rate of decrease of the mean square error (MSE). The resultant optimized BPI estimator converges faster and achieves lower mean squared error than previous $k$-NN entropy estimators. In addition, a central limit theorem is established for the BPI estimator that allows us to specify tight asymptotic confidence intervals., Comment: Version 3 changes : Additional results on bias correction factors for specifically estimating Shannon and Renyi entropy in Section 5

We study $C^2$ extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a `localized' form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.

In this article, we establish Gaussian decay for the Box_b-heat kernel on polynomial models in C^2. Our technique attains the exponential decay via a partial Fourier transform. On the transform side, the problem becomes finding quantitative smoothness estimates on a heat kernel associated to the weighted dbar-operator on L^2(C). The bounds are established with Duhamel's formula and careful estimation., Comment: 25 pages