Pro-Lie Groups: A survey with Open Problems

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally compact group which has a compact quotient group modulo its identity component and thus, in particular, each compact and each connected locally compact group; it also includes all locally compact abelian groups. This paper provides an overview of the structure theory and Lie theory of pro-Lie groups including results more recent than those in the authors' reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function which links the two. (A topological vector space is weakly complete if it is isomorphic to a power $\R^X$ of an arbitrary set of copies of $\R$. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected with pro-Lie groups.Comment: 19 page

The weights of closed subgroups of a locally compact group

Let $G$ be an infinite locally compact group and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le w(G)$ for the weight $w(G)$ of $G$. It is shown that there is a closed subgroup $N$ of $G$ with $w(N)=\aleph$. Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup. (2) For a locally compact group $G$ and $\aleph$ a cardinal satisfying \aleph_0\le\aleph\le \lw(G), where \lw(G) is the local weight of $G$, there are either no infinite compact subgroups at all or there is a compact subgroup $N$ of $G$ with $w(N)=\aleph$. (3) For an infinite abelian group $G$ there exists a properly ascending family of locally quasiconvex group topologies on $G$, say, (\tau_\aleph)_{\aleph_0\le \aleph\le \card(G)}, such that $(G,\tau_\aleph)\hat{\phantom{m}}\cong\hat G$. Items (2) and (3) are shown in Section 5

Nonmeasurable subgroups of compact groups

In 1985 S.~Saeki and K.~Stromberg published the following question: {\it Does every infinite compact group have a subgroup which is not Haar measurable?} An affirmative answer is given for all compact groups with the exception of some metric profinite groups known as strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup

Advances in the theory of compact groups and pro-lie groups in the last quarter century

This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th centuries. Next is from Hilbert’s Fifth Problem in 1900 to its solution in 1952 by Montgomery, Zippin, and Gleason and Yamabe’s important structure theorem on almost connected locally compact groups. This half century included profound contributions by Weyl and Peter, Haar, Pontryagin, van Kampen, Weil, and Iwasawa. The focus in the last quarter century has been structure theory, largely resulting from extending Lie Theory to compact groups and then to pro-Lie groups, which are projective limits of finite-dimensional Lie groups. The category of pro-Lie groups is the smallest complete category containing Lie groups and includes all compact groups, locally compact abelian groups, and connected locally compact groups. Amongst the structure theorems is that each almost connected pro-Lie group G is homeomorphic to RI × C for a suitable set I and some compact subgroup C. Finally, there is a perfect generalization to compact groups G of the age-old natural duality of the group algebra R[G] of a finite group G to its representation algebra R(G, R), via the natural duality of the topological vector space RI to the vector space R(I), for any set I, thus opening a new approach to the Hochschild-Tannaka duality of compact groups. © 2021 by the authors. Licensee MDPI, Basel, Switzerland

A Spitzer Study of the Mass Loss Histories of Three Bipolar Pre-Planetary Nebulae

We present the results of far-infrared imaging of extended regions around three bipolar pre-planetary nebulae, AFGL 2688, OH 231.8+4.2, and IRAS 16342$-$3814, at 70 and 160 $\mu$m with the MIPS instrument on the Spitzer Space Telescope. After a careful subtraction of the point spread function of the central star from these images, we place constraints on the existence of extended shells and thus on the mass outflow rates as a function of radial distance from these stars. We find no apparent extended emission in AFGL 2688 and OH 231.8+4.2 beyond 100 arcseconds from the central source. In the case of AFGL 2688, this result is inconsistent with a previous report of two extended dust shells made on the basis of ISO observations. We derive an upper limit of $2.1\times10^{-7}$ M$_\odot$ yr$^{-1}$ and $1.0\times10^{-7}$ M$_\odot$ yr$^{-1}$ for the dust mass loss rate of AFGL 2688 and OH 231.8, respectively, at 200 arcseconds from each source. In contrast to these two sources, IRAS 16342$-$3814 does show extended emission at both wavelengths, which can be interpreted as a very large dust shell with a radius of $\sim$ 400 arcseconds and a thickness of $\sim$ 100 arcseconds, corresponding to 4 pc and 1 pc, respectively, at a distance of 2 kpc. However, this enhanced emission may also be galactic cirrus; better azimuthal coverage is necessary for confirmation of a shell. If the extended emission is a shell, it can be modeled as enhanced mass outflow at a dust mass outflow rate of $1.5\times10^{-6}$ M$_\odot$ yr$^{-1}$ superimposed on a steady outflow with a dust mass outflow rate of $1.5\times10^{-7}$ M$_\odot$ yr$^{-1}$. It is likely that this shell has swept up a substantial mass of interstellar gas during its expansion, so these estimates are upper limits to the stellar mass loss rate.Comment: 31 pages, 12 figures, accepted to A

Regulation of seed germination in the close Arabidopsis relative Lepidium sativum : a global tissue-specific transcript analysis

The completion of germination in Lepidium sativum and other endospermic seeds (e.g. Arabidopsis [Arabidopsis thaliana]) is regulated by two opposing forces, the growth potential of the radicle (RAD) and the resistance to this growth from the micropylar endosperm cap (CAP) surrounding it. We show by puncture force measurement that the CAP progressively weakens during germination, and we have conducted a time-course transcript analysis of RAD and CAP tissues throughout this process. We have also used specific inhibitors to investigate the importance of transcription, translation, and posttranslation levels of regulation of endosperm weakening in isolated CAPs. Although the impact of inhibiting translation is greater, both transcription and translation are required for the completion of endosperm weakening in the whole seed population. The majority of genes expressed during this process occur in both tissues, but where they are uniquely expressed, or significantly differentially expressed between tissues, this relates to the functions of the RAD as growing tissue and the CAP as a regulator of germination through weakening. More detailed analysis showed that putative orthologs of cell wall-remodeling genes are expressed in a complex manner during CAP weakening, suggesting distinct roles in the RAD and CAP. Expression patterns are also consistent with the CAP being a receptor for environmental signals influencing germination. Inhibitors of the aspartic, serine, and cysteine proteases reduced the number of isolated CAPs in which weakening developed, and inhibition of the 26S proteasome resulted in its complete cessation. This indicates that targeted protein degradation is a major control point for endosperm weakening

A Middleware to Support Services Delivery in a Domain-Specific Virtual Machine

The increasing use of model-driven software development has renewed emphasis on using domain-specific models during application development. More specifically, there has been emphasis on using domain-specific modeling languages (DSMLs) to capture user-specified requirements when creating applications. The current approach to realizing these applications is to translate DSML models into source code using several model-to-model and model-to-code transformations. This approach is still dependent on the underlying source code representation and only raises the level of abstraction during development. Experience has shown that developers will many times be required to manually modify the generated source code, which can be error-prone and time consuming. An alternative to the aforementioned approach involves using an interpreted domain-specific modeling language (i-DSML) whose models can be directly executed using a Domain Specific Virtual Machine (DSVM). Direct execution of i-DSML models require a semantically rich platform that reduces the gap between the application models and the underlying services required to realize the application. One layer in this platform is the domain-specific middleware that is responsible for the management and delivery of services in the specific domain. In this dissertation, we investigated the problem of designing the domain-specific middleware of the DSVM to facilitate the bifurcation of the semantics of the domain and the model of execution (MoE) while supporting runtime adaptation and validation. We approached our investigation by seeking solutions to the following sub-problems: (1) How can the domain-specific knowledge (DSK) semantics be separated from the MoE for a given domain? (2) How do we define a generic model of execution (GMoE) of the middleware so that it is adaptable and realizes DSK operations to support delivery of services? (3) How do we validate the realization of DSK operations at runtime? Our research into the domain-specific middleware was done using an i-DSML for the user-centric communication domain, Communication Modeling Language (CML), and for microgrid energy management domain, Microgrid Modeling Language (MGridML). We have successfully developed a methodology to separate the DSK and GMoE of the middleware of a DSVM that supports specialization for a given domain, and is able to perform adaptation and validation at runtime

Getting HPC into Regional University Curricula with Few Resources

Karl Frinkle is an applied mathematician who earned his PhD from the University of New Mexico. He is deeply interested in numerical simulations, and most recently in parallel programming. Karl joined the SE Mathematics department in 2005, and thoroughly enjoys teaching parallel programming courses with Mike Morris through the CS department. He also can be found teaching physics courses in the Physics department. Mike Morris' degrees are in math, but he has always said he wound up on the business end of a computer. He taught Computer Science (CS) in the early 80s after working as an Operations Research Analyst for Conoco in Ponca City OK. Mike left teaching and spent 15 years doing various things in the CS industry before returning to Southeastern Oklahoma State to once again teach CS, where he remains today.We showcase our successful project of injecting high performance computing (HPC) into the traditional computer science curriculum at Southeastern Oklahoma State University. We incorporated a three-semester sequence of parallel programming courses, with the third course focusing on a research-level mathematical project that was executed on OU's supercomputer. Emphasis was placed on utilizing Open MPI and CUDA libraries along with parallel algorithm analysis and file I/O optimization. We recruited students with varied CS backgrounds for the program, some with only CS-1 in their portfolios. These courses are currently being followed with a more hardware-oriented course this fall, which will be profiled in the talk. We will discuss our approach to making this a successful sequence of courses and divulge many of the tips and tricks we have learned from our efforts.University of Oklahoma (http://ou.edu)OU Supercomputing Center for Education & Research (http://www.oscer.ou.edu)N

Spitzer Space Telescope Spectroscopy of Ices toward Low-Mass Embedded Protostars

Sensitive 5-38 μm Spitzer Space Telescope and ground-based 3-5 μm spectra of the embedded low-mass protostars B5 IRS1 and HH 46 IRS show deep ice absorption bands superposed on steeply rising mid-infrared continua. The ices likely originate in the circumstellar envelopes. The CO_2 bending mode at 15 μm is a particularly powerful tracer of the ice composition and processing history. Toward these protostars, this band shows little evidence for thermal processing at temperatures above 50 K. Signatures of lower temperature processing are present in the CO and OCN^- bands, however. The observed CO2 profile indicates an intimate mixture with H_(2)O, but not necessarily with CH_(3)OH, in contrast to some high-mass protostars. This is consistent with the low CH_(3)OH abundance derived from the ground-based L-band spectra. The CO_2 : H_(2)O column density ratios are high in both B5 IRS1 and HH 46 IRS (~35%). Clearly, the Spitzer spectra are essential for studying ice evolution in low-mass protostellar environments and for eventually determining the relation between interstellar and solar system ices