Young-Capelli bitableaux, Capelli immanants in U(gl(n)) and the Okounkov quantum immanants

We propose a new approach to a unified study of determinants, permanents, immanants, (determinantal) bitableaux and symmetrized bitableaux in the polynomial algebra $C[M_{n, n}]$ as well as of their Lie analogues in the enveloping algebra $U(gl(n))$. This leads to new relevant classes of elements in $U(gl(n))$: Capelli bitableaux, right Young-Capelli bitableaux and Capelli immanants. The set of standard Capelli bitableaux and the set of standard right Young-Capelli bitableaux are bases of $U(gl(n))$, whose action on the Gordan-Capelli basis of $C[M_{n, n}]$ have remarkable properties. Capelli immanants can be efficiently computed and provide a system of generators of $U(gl(n))$. The Okounkov quantum immanants are proved to be simple linear combinations of Capelli immanants. Several examples are provided throughout the paper.Comment: arXiv admin note: text overlap with arXiv:1608.0678

Discrete Mathematics

The purpose of the present work is to provide short and supple teaching notes for a $30$ hours introductory course on elementary \textit{Enumerative Algebraic Combinatorics}. We fully adopt the \textit{Rota way} (see, e.g. \cite{KY}). The themes are organized into a suitable sequence that allows us to derive any result from the preceding ones by elementary processes. Definitions of \textit{combinatorial coefficients} are just by their \textit{combinatorial meaning}. The derivation techniques of formulae/results are founded upon constructions and two general and elementary principles/methods: - The \textit{bad element} method (for \textit{recursive} formulae). As the reader should recognize, the bad element method might be regarded as a combinatorial companion of the idea of \textit{conditional probability}. - The \textit{overcounting} principle (for \textit{close form} formulae). Therefore, \textit{no computation} is required in \textit{proofs}: \textit{computation formulae are byproducts of combinatorial constructions}. We tried to provide a self-contained presentation: the only prerequisite is standard high school mathematics. We limited ourselves to the \textit{combinatorial point of view}: we invite the reader to draw the (obvious) \textit{probabilistic interpretations}

Quantum immanants, double Young-Capelli bitableaux and Schur shifted symmetric functions

We propose a new method for a unified study of some of the main features of the theory of the center $\boldsymbol{\zeta}(n)$ of the enveloping algebra U(gl(n)) and of the algebra $\Lambda^*(n)$ of shifted symmetric polynomials, that allows the whole theory to be developed, in a transparent and concise way, from the representation-theoretic point of view, that is entirely in the center of U(gl(n)). Our methodological innovation is the systematic use of the superalgebraic method of virtual variables for gl(n), which is, in turn, an extension of Capelli's method of ``variabili ausiliarie''. The passage $n \rightarrow \infty$ for the algebras $\boldsymbol{\zeta}(n)$ and $\Lambda^*(n)$ is here obtained both as direct and inverse limit in the category of filtered algebras. The present approach leads to proofs that are almost direct consequences of the definitions and constructions: they often reduce to a few lines computation

Capelli-Deruyts bitableaux and the classical Capelli generators of the center of the enveloping algebra U(gl(n))U(gl(n))

In this paper, we consider a special class of Capelli bitableaux, namely the Capelli-Deruyts bitableaux. The main results we prove are the hook coefficient lemma and the expansion theorem. Capelli-Deruyts bitableaux of rectangular shape are of particular interest since they are central elements in the enveloping algebra. The expansion theorem implies that these central element is explicitely described as a polynomial in the classical Capelli central elements. The hook coefficient lemma implies that the Capelli-Deruyts bitableaux are (canonically) expressed as the products of column determinants.Comment: arXiv admin note: text overlap with arXiv:2107.1020

Dione and Rhea seasonal exospheres revealed by Cassini CAPS and INMS

AbstractA Dione O2 and CO2 exosphere of similar composition and density to Rhea's is confirmed by Cassini spacecraft Ion Neutral Mass Spectrometer (INMS) flyby data. INMS results from three Dione and two Rhea flybys show exospheric spatial and temporal variability indicative of seasonal exospheres, modulated by winter polar gas adsorption and desorption at the equinoxes. Cassini Plasma Spectrometer (CAPS) pickup ion fluxes also show exospheric structure and evolution at Rhea consistent with INMS, after taking into consideration the anticipated charge exchange, electron impact, and photo-ionization rates. Data-model comparisons show the exospheric evolution to be consistent with polar frost diffusion into the surface regolith, which limits surface exposure and loss of the winter frost cap by sputtering. Implied O2 source rates of ∼45(7)×1021s−1 at Dione(Rhea) are ∼50(300) times less than expected from known O2 radiolysis yields from ion-irradiated pure water ice measured in the laboratory, ruling out secondary sputtering as a major exospheric contributor, and implying a nanometer scale surface refractory lag layer consisting of concentrated carbonaceous impurities. We estimate ∼30:1(2:1) relative O2:CO2 source rates at Dione(Rhea), consistent with a stoichiometric bulk composition below the lag layer of 0.01(0.13) C atoms per H2O molecule, deriving from endogenic constituents, implanted micrometeoritic organics, and (in particular at Dione) exogenous H2O delivery by E-ring grains. Impact deposition, gardening and vaporization may thereby control the global O2 source rates by fresh H2O ice exposure to surface radiolysis and trapped oxidant ejection

Robust H-infinity Output Feedback Control for Nonlinear Systems

The study of robust nonlinear control has attracted increasing interest over the last few years. Progress has been aided by the recent extension of the linear quadratic results which links the theories of L2 gain control (nonlinear H∞ control), differential games, and stochastic risk sensitive control. In fact, significant advances in both linear and nonlinear H∞ theory have drawn upon results from the theories of differential games and stochastic risk sensitive control. Despite these advances in H∞ control theory, practical controllers for complex nonlinear systems which operate on basic H∞ principles have not been realized to date. Issues of importance to the design of a practical controller include (i) computational complexity, (ii) operation solely with observable quantities, and (iii) implementability in finite time. In this dissertation we offer a design procedure which yields, practical and implementable H∞ controllers and meets the, mandate of the above issues for general nonlinear systems. In particular, we develop a well defined and realistically implementable procedure for designing robust output feedback controllers for a large class of nonlinear systems. We analyze this problem in both continuous time and discrete time settings. The robust output feedback control problem is formulated as a dynamic game problem. The solution to the game is obtained by transforming the problem into an equivalent full state feedback problem where the new state is called the information state. The information state method provides a separated control policy which involves the solution of a forward and a backward dynamic programming equation. Obtained from the forward equation is the information state, and from the backward equation is the value function of the game and the optimal information state control. The computer implementation of the information state controller is addressed and several approximations are introduced. The approximations are designed to decrease the online computational complexity of controller

Converting a Small Online Catalog - Improving Service and Satisfaction

Converting the online catalog to a cloud-based system. The library’s software-based catalog experienced technical issues whenever the hospital’s Information Technology department performed updates, and remote access to the system was not available. The authors describe the process of successfully converting the online catalog to a cloud-based system with remote access. It was critical for the library staff to preserve data from the former system. The initial step was to download the statistics from the former system to preserve them for future use. While the library staff weeded both the collection and the online records, they checked to ensure the existing catalog reflected the actual holdings. Patron records older than five years were deleted. Prior to importing the records, the library staff determined which fields, branch locations, and spine label abbreviations to use. Although the new system automatically provided barcodes during cataloging, the library staff chose to keep the existing barcodes from the former system. Circulation polices were established for each book and AV category. The Library staff exported the records to the new vendor and notified the previous vendor of the libraries’ decision to select a new product. Once the catalog and patron records were imported, use of the software-based system was discontinued. Emails were sent to patrons asking them to return circulating books. Outstanding materials were signed out using the cloud-based system. Some of the users commented that the cloud-based interface was easier to use. After the staff shifted to telework due to Covid-19, the catalog remained available remotely. The catalog’s conversion eliminated many of the technical problems encountered with the previous system. Users preferred the catalog’s simplified interface. Remote catalog access has allowed the library staff to continue circulation and catalog services while teleworking during COVID-19

Physicians\u27 use of the personal digital assistant (PDA) in clinical decision making

Purpose: This study examined how frequently attending physicians and physicians in training used personal digital assistants (PDAs) for patient care and explored physicians\u27 perceptions of the impact of PDA use on several areas of clinical decision making. Setting/Subjects: The 108 participants included 59 attending physicians and 49 physicians in training from teaching hospitals in Tennessee, Florida, Alabama, Kentucky, and Pennsylvania. Methodology: Respondents completed a questionnaire designed to explore PDA use in a clinical setting. Results: Eighty-seven percent of the respondents reported PDA use for patient encounters. Fifty-five percent of respondents reported frequent use, and 32% reported occasional use of a PDA for patient care. Of the frequent PDA users, 85% said PDA use had influenced their overall clinical decision making and 73% mentioned treatment alterations specifically. Approximately 60% of the participants reporting occasional PDA use indicated that the PDA had influenced their overall clinical decision making, while 54% specifically mentioned a change to their patient\u27s treatment plan. Discussion/Conclusion: Attending physicians and physicians in training who used a PDA during patient encounters perceived that even occasional PDA use had an impact on their clinical decision making and treatment choices. Health sciences librarians are perfectly positioned to provide PDA training and assistance not only to physicians who are frequent PDA users, but also to those who are occasional users