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We present an alternative 2-parametric deformation $ GL(2)_{h,h'} $ , and construct the differential calculus on the quantum plane on which this quantum group acts. Also we give a new deformation of the two dimensional Heisenberg algebra

Book Source: Digital Library of India Item 2015.77536 dc.contributor.author: Forder, Henry George. dc.date.accessioned: 2015-06-30T15:57:10Z dc.date.available: 2015-06-30T15:57:10Z dc.date.citation: 1941 dc.identifier.barcode: 2030020000536 dc.identifier.origpath: /data7/upload/0184/924 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/77536 dc.description.scannerno: 1 dc.description.scanningcentre: RMSC, IIIT-H dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 516 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: At The University dc.rights: Not Available dc.source.library: Osmania University dc.subject.classification: Natural Sciences dc.subject.classification: Mathematics dc.subject.classification: Fundamental And General Consideration Of Mathematics dc.title: The Calculus Of Extension

Over the last decade, kernel methods for nonlinear processing have successfully been used in the machine learning community. The primary mathematical tool employed in these methods is the notion of the Reproducing Kernel Hilbert Space. However, so far, the emphasis has been on batch techniques. It is only recently, that online techniques have been considered in the context of adaptive signal processing tasks. Moreover, these efforts have only been focussed on real valued data sequences. To the best of our knowledge, no adaptive kernel-based strategy has been developed, so far, for complex valued signals. Furthermore, although the real reproducing kernels are used in an increasing number of machine learning problems, complex kernels have not, yet, been used, in spite of their potential interest in applications that deal with complex signals, with Communications being a typical example. In this paper, we present a general framework to attack the problem of adaptive filtering of complex signals, using either real reproducing kernels, taking advantage of a technique called \textit{complexification} of real RKHSs, or complex reproducing kernels, highlighting the use of the complex gaussian kernel. In order to derive gradients of operators that need to be defined on the associated complex RKHSs, we employ the powerful tool of Wirtinger's Calculus, which has recently attracted attention in the signal processing community. To this end, in this paper, the notion of Wirtinger's calculus is extended, for the first time, to include complex RKHSs and use it to derive several realizations of the Complex Kernel Least-Mean-Square (CKLMS) algorithm. Experiments verify that the CKLMS offers significant performance improvements over several linear and nonlinear algorithms, when dealing with nonlinearities.

"Lagrange's Equation in the Calculus of Variations, and the Extension of a Theorem of Erdmann" is an article from The Annals of Mathematics, Volume 2 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2007190

"Lagrange's Equation in the Calculus of Variations, and the Extension of a Theorem of Erdmann" is an article from The Annals of Mathematics, Volume 2 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2007190

"On a Geometrical Extension of the Calculus of Quaternions" is an article from Proceedings of the Royal Irish Academy (1836-1869), Volume 6 . View more articles from Proceedings of the Royal Irish Academy (1836-1869) . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-20520485

Over the last decade, kernel methods for nonlinear processing have successfully been used in the machine learning community. However, so far, the emphasis has been on batch techniques. It is only recently, that online adaptive techniques have been considered in the context of signal processing tasks. To the best of our knowledge, no kernel-based strategy has been developed, so far, that is able to deal with complex valued signals. In this paper, we take advantage of a technique called complexification of real RKHSs to attack this problem. In order to derive gradients and subgradients of operators that need to be defined on the associated complex RKHSs, we employ the powerful tool ofWirtinger's Calculus, which has recently attracted much attention in the signal processing community. Writinger's calculus simplifies computations and offers an elegant tool for treating complex signals. To this end, in this paper, the notion of Writinger's calculus is extended, for the first time, to include complex RKHSs and use it to derive the Complex Kernel Least-Mean-Square (CKLMS) algorithm. Experiments verify that the CKLMS can be used to derive nonlinear stable algorithms, which offer significant performance improvements over the traditional complex LMS orWidely Linear complex LMS (WL-LMS) algorithms, when dealing with nonlinearities.

Research conducted by The RP Group's Multiple Measures Assessment Project (MMAP) and summarized in "Preparatory Pathways and STEM Calculus Completion: Implications of the AB 1705 Standards" found high rates of attrition along the path to calculus for students pursuing programs in Science, Technology, Engineering and Math (STEM). In this study of over 37,000 STEM students in the California Community Colleges (CCC), 70% began in a transferable, college math course below calculus, yet only 32% of those starting in one of these preparatory courses completed STEM Calculus 1 within two years. This brief provides a follow-up to the initial report with a particular focus on students in biology majors. The goal is to determine if the original report underestimated calculus completion rates for certain biology majors because their programs of study allowed for a different form of calculus. For example, even though STEM Calculus is the gateway requirement for most STEM majors, some California State University (CSU) biology programs allow for applied calculus courses, and a few permit precalculus to satisfy biology degree requirements. This brief provides a short overview of additions to the original methodology, followed by key findings that address questions about the applicability of the original research when we examine biology majors separate from other STEM majors.

For nonsmooth Euler-Lagrange extremals, Noether's conservation laws cease to be valid. We show that Emmy Noether's theorem of the calculus of variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBois-Reymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether's theorem. This is in contrast with the recent developments of Noether's symmetry theorems to the optimal control setting, which give rise to non-proper extensions when specified for the problems of the calculus of variations.

For nonsmooth Euler-Lagrange extremals, Noether's conservation laws cease to be valid. We show that Emmy Noether's theorem of the calculus of variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBois-Reymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether's theorem. This is in contrast with the recent developments of Noether's symmetry theorems to the optimal control setting, which give rise to non-proper extensions when specified for the problems of the calculus of variations.

This work is intended to provide a semantics for a fragment of a programming language described by Gyorgy Revesz, for which no model was known. We begin with a brief presentation of the syntax of the lambda calculus and some relevant extensions. We then describe a class of complete lattices and use them as models for the lambda calculus. We then find specialized sublattices which we use as models for the extensions of the lambda calculus, thus achieving the original goal of finding a semantics for Revesz's language.