Vector valued logarithmic residues and the extraction of elementary factors

An analysis is presented of the circumstances under which, by the extraction of elementary factors, an analytic Banach algebra valued function can be transformed into one taking invertible values only. Elementary factors are generalizations of the simple scalar expressions λ – α, the building blocks of scalar polynomials. In the Banach algebra situation they have the form e – p + (λ – α)p with p an idempotent. The analysis elucidates old results (such as on Fredholm operator valued functions) and yields new insights which are brought to bear on the study of vector-valued logarithmic residues. These are contour integrals of logarithmic derivatives of analytic Banach algebra valued functions. Examples illustrate the subject matter and show that new ground is covered. Also a long standing open problem is discussed from a fresh angle.analytic vector-valued function;annihilating family of idempotents;elementary factor;generalizations of analytic functions;idempotent;integer combination of idempotents;logarithmic residue;plain function;resolving family of traces;topological algebras

Exact semi-relativistic model for ionization of atomic hydrogen by electron impact

We present a semi-relativistic model for the description of the ionization process of atomic hydrogen by electron impact in the first Born approximation by using the Darwin wave function to describe the bound state of atomic hydrogen and the Sommerfeld-Maue wave function to describe the ejected electron. This model, accurate to first order in $Z/c$ in the relativistic correction, shows that, even at low kinetic energies of the incident electron, spin effects are small but not negligible. These effects become noticeable with increasing incident electron energies. All analytical calculations are exact and our semi-relativistic results are compared with the results obtained in the non relativistic Coulomb Born Approximation both for the coplanar asymmetric and the binary coplanar geometries.Comment: 8 pages, 6 figures, Revte

Network Modularity in the Presence of Covariates

We characterize the large-sample properties of network modularity in the presence of covariates, under a natural and flexible null model. This provides for the first time an objective measure of whether or not a particular value of modularity is meaningful. In particular, our results quantify the strength of the relation between observed community structure and the interactions in a network. Our technical contribution is to provide limit theorems for modularity when a community assignment is given by nodal features or covariates. These theorems hold for a broad class of network models over a range of sparsity regimes, as well as for weighted, multiedge, and power-law networks. This allows us to assign p-values to observed community structure, which we validate using several benchmark examples from the literature. We conclude by applying this methodology to investigate a multiedge network of corporate email interactions

Open-source Tools for Dense Facial Tissue Depth Mapping (FTDM) of Computed Tomography Models

Computed tomography (CT) scans provide anthropologists with a resource to generate three- dimensional (3D) digital skeletal material to expand quantification methods and build more standardized reference collections. The ability to visualize and manipulate the bone and skin of the face simultaneously in a 3D digital environment introduces a new way for forensic facial approximation practitioners to access and study the face. Craniofacial relationships can be quantified with landmarks or with surface processing software that can quantify the geometric properties of the entire 3D facial surface. This paper describes tools for the generation of dense facial tissue depth maps (FTDMs) using de-identified head CT scans of modern Americans from the public repository, The Cancer Imaging Archives (TCIA), and the open-source program Meshlab. CT scans of 43 females and 63 males from TCIA were segmented and converted to 3D skull and face models using Mimics and exported as stereolithography (STL) files. All subsequent processing steps were performed in Meshlab. Heads were transformed to a common orientation and coordinate system using the coordinates of nasion, left orbitale, and left and right porion. Dense FTDMs were generated on hollowed, cropped face shells using the Hausdorff sampling filter. Two new point clouds consisting of the 3D coordinates for both skull and face were colorized on an RGB scale from 0.0 (red) to 40.0 mm (blue) depth values and exported as polygon file format (PLY) models with tissue depth values saved in the “vertex quality” field. FTDMs were also split into 1.0 mm increments to facilitate viewing of common depths across all faces. In total, 112 FTDMs were generated for 106 individuals. Minimum depth values ranged from 1.2 mm to 3.4 mm, indicating a common range of starting depths for most faces regardless of weight, as well as common locations for these values over the nasal bones, lateral orbital margins, and forehead superior to the supraorbital border. Maximum depths were found in the buccal region and neck, excluding the nose. Individuals with multiple scans at visibly different weights presented the greatest differences within larger depth areas such as the cheeks and neck, with little to no difference in the thinnest areas. A few individuals with minimum tissue depths at the lateral orbital margins and thicker tissues over the nasal bones (\u3e 3.0 mm) suggested the potential influence of nasal bone morphology on tissue depths. This study produced visual quantitative representations of the face and skull for forensic facial approximation research and practice that can be further analyzed or interacted with using free software. The presented tools can be applied to pre-existing CT scans, traditional or cone-beam, adult or subadult individuals, with or without landmarks, and regardless of head orientation, for forensic applications as well as for studies of facial variation and facial growth. In contrast with other facial mapping studies, this method produced both skull and face points based on replicable geometric relationships producing multiple data outputs that are easily readable and software that is openly accessible

Logarithmic residues, Rouché’s theorem, and spectral regularity: The C∗-algebra case

AbstractUsing families of irreducible Hilbert space representations as a tool, the theory of analytic Fredholm operator valued function is extended to a C∗-algebra setting. This includes a C∗-algebra version of Rouché’s Theorem known from complex function theory. Also, criteria for spectral regularity of C∗-algebras are developed. One of those, involving the (generalized) Calkin algebra, is applied to C∗-algebras generated by a non-unitary isometry

Faster PET reconstruction with non-smooth priors by randomization and preconditioning

Uncompressed clinical data from modern positron emission tomography (PET) scanners are very large, exceeding 350 million data points (projection bins). The last decades have seen tremendous advancements in mathematical imaging tools many of which lead to non-smooth (i.e. non-differentiable) optimization problems which are much harder to solve than smooth optimization problems. Most of these tools have not been translated to clinical PET data, as the state-of-the-art algorithms for non-smooth problems do not scale well to large data. In this work, inspired by big data machine learning applications, we use advanced randomized optimization algorithms to solve the PET reconstruction problem for a very large class of non-smooth priors which includes for example total variation, total generalized variation, directional total variation and various different physical constraints. The proposed algorithm randomly uses subsets of the data and only updates the variables associated with these. While this idea often leads to divergent algorithms, we show that the proposed algorithm does indeed converge for any proper subset selection. Numerically, we show on real PET data (FDG and florbetapir) from a Siemens Biograph mMR that about ten projections and backprojections are sufficient to solve the MAP optimisation problem related to many popular non-smooth priors; thus showing that the proposed algorithm is fast enough to bring these models into routine clinical practice

Logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity.

A logarithmic residue is a contour integral of the (left or right) logarithmic derivative of an analytic Banach algebra valued function. Logarithmic residues are intimately related to sums of idempotents. The present paper is concerned with logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity in the case when the underlying Banach space is infinite dimensional. The situation is more complex than encoutered in previous investigations. Logarithmic derivatives may have essential singularities and the geometric properties of the Banach space play a role. The set of sums of idempotens and the set of logarithmic residues have an intriguing topological structure.Banach algebra;Logarithmic residues;sums of idempotents

Logarithmic residues of analytic Banach algebra valued functions possessing a simply meromorphic inverse

A logarithmic residue is a contour integral of a logarithmic derivative (left or right) of an analytic Banach algebra valued function. For functions possessing a meromorphic inverse with simple poles only, the logarithmic residues are identified as the sums of idempotents. With the help of this observation, the issue of left versus right logarithmic residues is investigated, both for connected and nonconnected underlying Cauchy domains. Examples are given to elucidate the subject matter.Logarithmic residues;Cauchy domains;analytic Banach algebra valued function;meromorphic inverse

On the convergence and sampling of randomized primal-dual algorithms and their application to parallel MRI reconstruction

Stochastic Primal-Dual Hybrid Gradient (SPDHG) is an algorithm to efficiently solve a wide class of nonsmooth large-scale optimization problems. In this paper we contribute to its theoretical foundations and prove its almost sure convergence for convex but neither necessarily strongly convex nor smooth functionals. We also prove its convergence for any sampling. In addition, we study SPDHG for parallel Magnetic Resonance Imaging reconstruction, where data from different coils are randomly selected at each iteration. We apply SPDHG using a wide range of random sampling methods and compare its performance across a range of settings, including mini-batch size and step size parameters. We show that the sampling can significantly affect the convergence speed of SPDHG and for many cases an optimal sampling can be identified

Convergence Properties of a Randomized Primal-Dual Algorithm with Applications to Parallel MRI

The Stochastic Primal-Dual Hybrid Gradient (SPDHG) was proposed by Chambolle et al. (2018) and is an efficient algorithm to solve some nonsmooth large-scale optimization problems. In this paper we prove its almost sure convergence for convex but not necessarily strongly convex functionals. We also look into its application to parallel Magnetic Resonance Imaging reconstruction in order to test performance of SPDHG. Our numerical results show that for a range of settings SPDHG converges significantly faster than its deterministic counterpart