High Performance Computing and Communications Glossary Release
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Quaternions and Cli Ord Geometric Algebras
Quaternions and Cliord Geometric Algebras Robert Benjamin Easter First Draft Edition (v1) (c) copyright 2015, Robert Benjamin Easter, all rights reserved. Preface As a rst rough draft that has been put together very quickly, this book is likely to contain errata and disorganization. The references list and inline citations are very incompete, so the reader should search around for more references. I do not claim to be the inventor of any of the mathematics found here. However, some parts of this book may be considered new in some sense and were in small parts my own original research. Much of the contents was originally written by me as contributions to a web encyclopedia project just for fun, but for various reasons was inappropriate in an encyclopedic volume. I did not originally intend to write this book. This is not a dissertation, nor did its development receive any funding or proper peer review. I oer this free book to the public, such as it is, in the hope it could be helpful to an interested reader. June 19, 2015 - Robert B. Easter. (v1) [email protected] 3 Table of contents Preface . 3 List of gures . 9 1 Quaternion Algebra . 11 1.1 The Quaternion Formula . 11 1.2 The Scalar and Vector Parts . 15 1.3 The Quaternion Product . 16 1.4 The Dot Product . 16 1.5 The Cross Product . 17 1.6 Conjugates . 18 1.7 Tensor or Magnitude . 20 1.8 Versors . 20 1.9 Biradials . 22 1.10 Quaternion Identities . 23 1.11 The Biradial b/a . -
1 Parity 2 Time Reversal
Even Odd Symmetry Lecture 9 1 Parity The normal modes of a string have either even or odd symmetry. This also occurs for stationary states in Quantum Mechanics. The transformation is called partiy. We previously found for the harmonic oscillator that there were 2 distinct types of wave function solutions characterized by the selection of the starting integer in their series representation. This selection produced a series in odd or even powers of the coordiante so that the wave function was either odd or even upon reflections about the origin, x = 0. Since the potential energy function depends on the square of the position, x2, the energy eignevalue was always positive and independent of whether the eigenfunctions were odd or even under reflection. In 1-D parity is the symmetry operation, x → −x. In 3-D the strong interaction is invarient under the symmetry of parity. ~r → −~r Parity is a mirror reflection plus a rotation of 180◦, and transforms a right-handed coordinate system into a left-handed one. Our Macroscopic world is clearly “handed”, but “handedness” in fundamental interactions is more involved. Vectors (tensors of rank 1), as illustrated in the definition above, change sign under Parity. Scalars (tensors of rank 0) do not. One can then construct, using tensor algebra, new tensors which reduce the tensor rank and/or change the symmetry of the tensor. Thus a dual of a symmetric tensor of rank 2 is a pseudovector (cross product of two vectors), and a scalar product of a pseudovector and a vector creates a pseudoscalar. We will construct bilinear forms below which have these rotational and reflection char- acteristics. -
Automatic Temperature Controls
230900S03 INSTRUMENTATION AND CONTROL FOR HVAC UK Controls Standard SECTION 230900S03 - AUTOMATIC TEMPERATURE CONTROLS PART 1 - GENERAL RELATED DOCUMENTS: Drawings and general provisions of the Contract, including General and Supplementary Conditions, General Mechanical Provisions and General Requirements, Division 1 Specification Sections apply to the work specified in this section. DESCRIPTION OF WORK: Furnish a BACnet system compatible with existing University systems. All building controllers, application controllers, and all input/output devices shall communicate using the protocols and network standards as defined by ANSI/ASHRAE Standard 135-2001, BACnet. This system shall communicate with the University of Kentucky Facility Management’s existing BACnet head-end software using BACnet/IP at the tier 1 level and BACnet/MSTP at the tier 2 level. No gateways shall be used for communication to controllers installed under section. BACnet/MSTP or BACnet/IP shall be used for all other tiers of communication. No servers shall be used for communication to controllers installed under this section. If servers are required, all hardware and operating systems must be approved by the Facilities Management Controls Engineering Manager and/or the Facilities Management Information Technology Manager. All Building Automation Devices should be located behind the University firewall, but outside of the Medical Center Firewall and on the environmental VLAN. Provide all necessary hardware and software to meet the system’s functional specifications. Provide Protocol Implementation Conformance Statement (PICS) for Windows-based control software and every controller in system, including unitary controllers. These must be in compliance with Front End systems PICS and BIBBS and attached Tridium PICS and BIBBS. -
Symmetry and Tensors Rotations and Tensors
Symmetry and tensors Rotations and tensors A rotation of a 3-vector is accomplished by an orthogonal transformation. Represented as a matrix, A, we replace each vector, v, by a rotated vector, v0, given by multiplying by A, 0 v = Av In index notation, 0 X vm = Amnvn n Since a rotation must preserve lengths of vectors, we require 02 X 0 0 X 2 v = vmvm = vmvm = v m m Therefore, X X 0 0 vmvm = vmvm m m ! ! X X X = Amnvn Amkvk m n k ! X X = AmnAmk vnvk k;n m Since xn is arbitrary, this is true if and only if X AmnAmk = δnk m t which we can rewrite using the transpose, Amn = Anm, as X t AnmAmk = δnk m In matrix notation, this is t A A = I where I is the identity matrix. This is equivalent to At = A−1. Multi-index objects such as matrices, Mmn, or the Levi-Civita tensor, "ijk, have definite transformation properties under rotations. We call an object a (rotational) tensor if each index transforms in the same way as a vector. An object with no indices, that is, a function, does not transform at all and is called a scalar. 0 A matrix Mmn is a (second rank) tensor if and only if, when we rotate vectors v to v , its new components are given by 0 X Mmn = AmjAnkMjk jk This is what we expect if we imagine Mmn to be built out of vectors as Mmn = umvn, for example. In the same way, we see that the Levi-Civita tensor transforms as 0 X "ijk = AilAjmAkn"lmn lmn 1 Recall that "ijk, because it is totally antisymmetric, is completely determined by only one of its components, say, "123. -
High Performance Platforms
High Performance Platforms Salvatore Orlando High Perfoamnce Computing - S. Orlando 1 Scope of Parallelism: the roadmap towards HPC • Conventional architectures are coarsely comprised of – processor, memory system, I/O • Each of these components presents significant performance bottlenecks. – It is important to understand each of these performance bottlenecks to reduce their effects • Parallelism addresses each of these components in significant ways to improve performance and reduce the bottlenecks • Applications utilize different aspects of parallelism, e.g., – data intensive applications utilize high aggregate memory bandwidth – server applications utilize high aggregate network bandwidth – scientific applications typically utilize high processing and memory system performance High Perfoamnce Computing - S. Orlando 2 Implicit Parallelism: Trends in Microprocessor Architectures • Microprocessor clock speeds have posted impressive gains over the past two decades (two to three orders of magnitude) – there are limits to this increase, also due to power consumption and heat dissipation • Higher levels of device integration have made available a large number of transistors – the issue is how best to transform these large amount of transistors into computational power • Single-core processors use these resources in multiple functional units and execute multiple instructions in the same cycle. • The precise manner in which these instructions are selected and executed provides impressive diversity in architectures. High Perfoamnce Computing - S. Orlando 3 ILP High Perfoamnce Computing - S. Orlando 4 Instruction Level Parallelism • Pipelining overlaps various stages of instruction execution to achieve performance • At a high level of abstraction, an instruction can be executed while the next one is being decoded and the next one is being fetched. • This is akin to an assembly line for manufacture of cars. -
Multiprocessing Contents
Multiprocessing Contents 1 Multiprocessing 1 1.1 Pre-history .............................................. 1 1.2 Key topics ............................................... 1 1.2.1 Processor symmetry ...................................... 1 1.2.2 Instruction and data streams ................................. 1 1.2.3 Processor coupling ...................................... 2 1.2.4 Multiprocessor Communication Architecture ......................... 2 1.3 Flynn’s taxonomy ........................................... 2 1.3.1 SISD multiprocessing ..................................... 2 1.3.2 SIMD multiprocessing .................................... 2 1.3.3 MISD multiprocessing .................................... 3 1.3.4 MIMD multiprocessing .................................... 3 1.4 See also ................................................ 3 1.5 References ............................................... 3 2 Computer multitasking 5 2.1 Multiprogramming .......................................... 5 2.2 Cooperative multitasking ....................................... 6 2.3 Preemptive multitasking ....................................... 6 2.4 Real time ............................................... 7 2.5 Multithreading ............................................ 7 2.6 Memory protection .......................................... 7 2.7 Memory swapping .......................................... 7 2.8 Programming ............................................. 7 2.9 See also ................................................ 8 2.10 References ............................................. -
Darvas, G.: Quaternion/Vector Dual Space Algebras Applied to the Dirac
Bulletin of the Transilvania University of Bra¸sov • Vol 8(57), No. 1 - 2015 Series III: Mathematics, Informatics, Physics, 27-42 QUATERNION/VECTOR DUAL SPACE ALGEBRAS APPLIED TO THE DIRAC EQUATION AND ITS EXTENSIONS Gy¨orgyDARVAS 1 Comunicated to the conference: Finsler Extensions of Relativity Theory, Brasov, Romania, August 18 - 23, 2014 Abstract The paper re-applies the 64-part algebra discussed by P. Rowlands in a series of (FERT and other) papers in the recent years. It demonstrates that the original introduction of the γ algebra by Dirac to "the quantum the- ory of the electron" can be interpreted with the help of quaternions: both the α matrices and the Pauli (σ) matrices in Dirac's original interpretation can be rewritten in quaternion forms. This allows to construct the Dirac γ matrices in two (quaternion-vector) matrix product representations - in accordance with the double vector algebra introduced by P. Rowlands. The paper attempts to demonstrate that the Dirac equation in its form modified by P. Rowlands, essentially coincides with the original one. The paper shows that one of these representations affects the γ4 and γ5 matrices, but leaves the vector of the Pauli spinors intact; and the other representation leaves the γ matrices intact, while it transforms the spin vector into a quaternion pseudovector. So the paper concludes that the introduction of quaternion formulation in QED does not provide us with additional physical informa- tion. These transformations affect all gauge extensions of the Dirac equation, presented by the author earlier, in a similar way. This is demonstrated by the introduction of an additional parameter (named earlier isotopic field-charge spin) and the application of a so-called tau algebra that governs its invariance transformation. -
Pseudotensors
Pseudotensors Math 1550 lecture notes, Prof. Anna Vainchtein 1 Proper and improper orthogonal transfor- mations of bases Let {e1, e2, e3} be an orthonormal basis in a Cartesian coordinate system and suppose we switch to another rectangular coordinate system with the orthonormal basis vectors {¯e1, ¯e2, ¯e3}. Recall that the two bases are related via an orthogonal matrix Q with components Qij = ¯ei · ej: ¯ei = Qijej, ei = Qji¯ei. (1) Let ∆ = detQ (2) and recall that ∆ = ±1 because Q is orthogonal. If ∆ = 1, we say that the transformation is proper orthogonal; if ∆ = −1, it is an improper orthogonal transformation. Note that the handedness of the basis remains the same under a proper orthogonal transformation and changes under an improper one. Indeed, ¯ V = (¯e1 ׯe2)·¯e3 = (Q1mem ×Q2nen)·Q3lel = Q1mQ2nQ3l(em ×en)·el, (3) where the cross product is taken with respect to some underlying right- handed system, e.g. standard basis. Note that this is a triple sum over m, n and l. Now, observe that the terms in the above some where (m, n, l) is a cyclic (even) permutation of (1, 2, 3) all multiply V = (e1 × e2) · e3 because the scalar triple product is invariant under such permutations: (e1 × e2) · e3 = (e2 × e3) · e1 = (e3 × e1) · e2. Meanwhile, terms where (m, n, l) is a non-cyclic (odd) permutations of (1, 2, 3) multiply −V , e.g. for (m, n, l) = (2, 1, 3) we have (e2 × e1) · e3 = −(e1 × e2) · e3 = −V . All other terms in the sum in (3) (where two or more of the three indices (m, n, l) are the same) are zero (why?). -
Introduction to Vector and Tensor Analysis
Introduction to vector and tensor analysis Jesper Ferkinghoff-Borg September 6, 2007 Contents 1 Physical space 3 1.1 Coordinate systems . 3 1.2 Distances . 3 1.3 Symmetries . 4 2 Scalars and vectors 5 2.1 Definitions . 5 2.2 Basic vector algebra . 5 2.2.1 Scalar product . 6 2.2.2 Cross product . 7 2.3 Coordinate systems and bases . 7 2.3.1 Components and notation . 9 2.3.2 Triplet algebra . 10 2.4 Orthonormal bases . 11 2.4.1 Scalar product . 11 2.4.2 Cross product . 12 2.5 Ordinary derivatives and integrals of vectors . 13 2.5.1 Derivatives . 14 2.5.2 Integrals . 14 2.6 Fields . 15 2.6.1 Definition . 15 2.6.2 Partial derivatives . 15 2.6.3 Differentials . 16 2.6.4 Gradient, divergence and curl . 17 2.6.5 Line, surface and volume integrals . 20 2.6.6 Integral theorems . 24 2.7 Curvilinear coordinates . 26 2.7.1 Cylindrical coordinates . 27 2.7.2 Spherical coordinates . 28 3 Tensors 30 3.1 Definition . 30 3.2 Outer product . 31 3.3 Basic tensor algebra . 31 1 3.3.1 Transposed tensors . 32 3.3.2 Contraction . 33 3.3.3 Special tensors . 33 3.4 Tensor components in orthonormal bases . 34 3.4.1 Matrix algebra . 35 3.4.2 Two-point components . 38 3.5 Tensor fields . 38 3.5.1 Gradient, divergence and curl . 39 3.5.2 Integral theorems . 40 4 Tensor calculus 42 4.1 Tensor notation and Einsteins summation rule . -
Octonic Electrodynamics
Octonic Electrodynamics Victor L. Mironov* and Sergey V. Mironov Institute for physics of microstructures RAS, 603950, Nizhniy Novgorod, Russia (Submitted: 18 February 2008; Revised: 26 June 2014) In this paper we present eight-component values “octons”, generating associative noncommutative algebra. It is shown that the electromagnetic field in a vacuum can be described by a generalized octonic equation, which leads both to the wave equations for potentials and fields and to the system of Maxwell’s equations. The octonic algebra allows one to perform compact combined calculations simultaneously with scalars, vectors, pseudoscalars and pseudovectors. Examples of such calculations are demonstrated by deriving the relations for energy, momentum and Lorentz invariants of the electromagnetic field. The generalized octonic equation for electromagnetic field in a matter is formulated. PACS numbers: 03.50 De, 02.10 De. I. INTRODUCTION Hypercomplex numbers1-4 especially quaternions are widely used in relativistic mechanics, electrodynamics, quantum mechanics and quantum field theory3-10 (see also the bibliographical review Ref. 11). The structure of quaternions with four components (scalar and vector) corresponds to the relativistic space-time structure, which allows one to realize the quaternionic generalization of quantum mechanics5-8. However quaternions do not include pseudoscalar and pseudovector components. Therefore for describing all types of physical values the eight-component hypercomplex numbers enclosing scalars, vectors, pseudoscalars -
Quaternions Multivariate Vectors
A Hierarchy of Symmetries Peter Rowlands Symmetry We are not very good observers – science is a struggle for us. But we have developed one particular talent along with our evolution that serves us well. This is pattern recognition. This is fortunate, for, everywhere in Nature, and especially in physics, there are hints that symmetry is the key to deeper understanding. And physics has shown that the symmetries are often ‘broken’, that is disguised or hidden. A classic example is that between space and time, which are combined in relativity, but which remain obstinately different. Some questions Which are the most fundamental symmetries? Where does symmetry come from? How do the most fundamental symmetries help to explain the subject? Why are some symmetries broken and what does broken symmetry really mean? Many symmetries are expressed in some way using integers. Which are the most important? I would like to propose here that there is a hierarchy of symmetries, emerging at a very fundamental level, all of which are interlinked. The Origin of Symmetry A philosophical starting-point. The ultimate origin of symmetry in physics is zero totality. The sum of every single thing in the universe is precisely nothing. Nature as a whole has no definable characteristic. Zero is the only logical starting-point. If we start from anywhere else we have to explain it. Zero is the only idea we couldn’t conceivably explain. Duality and anticommutativity Where do we go from zero? I will give a semi-empirical answer, though it is possible to do it more fundamentally. The major symmetries in physics begin with just two ideas: duality and anticommutativity There are only two fundamental numbers or integers: 2 and 3 Everything else is a variation of these. -
K1DM3 Control System Software User's Guide
K1DM3 Control System Software User's Guide William Deich University of California Observatories 07 Jan 2020 ver 3.7b Contents 1 Overview 6 1.1 Active Components of K1DM3..........................6 1.1.1 Drum....................................6 1.1.2 Swingarm..................................9 1.2 Position Summary................................. 13 1.3 Motion Controllers................................. 13 1.4 Software....................................... 14 1.4.1 Back-end Services............................. 14 1.4.2 End-User Applications........................... 15 1.5 K1DM3 Private Network.............................. 15 2 Starting, Stopping, Suspending Services 17 2.1 Starting/Stopping................................. 17 2.2 Suspending Dispatchers.............................. 19 2.3 Special No-Dock Engineering Version....................... 19 2.4 Periodic Restarts.................................. 20 3 Components, Assemblies, and Sequencers 21 3.1 Elementary Components.............................. 21 3.2 Compound Keywords............................... 22 3.3 Assemblies...................................... 22 3.4 Sequencers..................................... 23 3.5 The ACTIVATE Sequencer............................ 24 3.6 The M3AGENT Sequencer............................ 27 3.7 Monitoring Sequencer Execution......................... 27 4 Dispatcher Interfaces for Routine Operations 29 4.1 TCS Operations.................................. 31 4.2 M3AGENT..................................... 32 4.3 Hand Paddle...................................