Multiplicative Functions of Numbers Set and Logarithmic Identities
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Guide for the Use of the International System of Units (SI)
Guide for the Use of the International System of Units (SI) m kg s cd SI mol K A NIST Special Publication 811 2008 Edition Ambler Thompson and Barry N. Taylor NIST Special Publication 811 2008 Edition Guide for the Use of the International System of Units (SI) Ambler Thompson Technology Services and Barry N. Taylor Physics Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899 (Supersedes NIST Special Publication 811, 1995 Edition, April 1995) March 2008 U.S. Department of Commerce Carlos M. Gutierrez, Secretary National Institute of Standards and Technology James M. Turner, Acting Director National Institute of Standards and Technology Special Publication 811, 2008 Edition (Supersedes NIST Special Publication 811, April 1995 Edition) Natl. Inst. Stand. Technol. Spec. Publ. 811, 2008 Ed., 85 pages (March 2008; 2nd printing November 2008) CODEN: NSPUE3 Note on 2nd printing: This 2nd printing dated November 2008 of NIST SP811 corrects a number of minor typographical errors present in the 1st printing dated March 2008. Guide for the Use of the International System of Units (SI) Preface The International System of Units, universally abbreviated SI (from the French Le Système International d’Unités), is the modern metric system of measurement. Long the dominant measurement system used in science, the SI is becoming the dominant measurement system used in international commerce. The Omnibus Trade and Competitiveness Act of August 1988 [Public Law (PL) 100-418] changed the name of the National Bureau of Standards (NBS) to the National Institute of Standards and Technology (NIST) and gave to NIST the added task of helping U.S. -
Frequency Response and Bode Plots
1 Frequency Response and Bode Plots 1.1 Preliminaries The steady-state sinusoidal frequency-response of a circuit is described by the phasor transfer function Hj( ) . A Bode plot is a graph of the magnitude (in dB) or phase of the transfer function versus frequency. Of course we can easily program the transfer function into a computer to make such plots, and for very complicated transfer functions this may be our only recourse. But in many cases the key features of the plot can be quickly sketched by hand using some simple rules that identify the impact of the poles and zeroes in shaping the frequency response. The advantage of this approach is the insight it provides on how the circuit elements influence the frequency response. This is especially important in the design of frequency-selective circuits. We will first consider how to generate Bode plots for simple poles, and then discuss how to handle the general second-order response. Before doing this, however, it may be helpful to review some properties of transfer functions, the decibel scale, and properties of the log function. Poles, Zeroes, and Stability The s-domain transfer function is always a rational polynomial function of the form Ns() smm as12 a s m asa Hs() K K mm12 10 (1.1) nn12 n Ds() s bsnn12 b s bsb 10 As we have seen already, the polynomials in the numerator and denominator are factored to find the poles and zeroes; these are the values of s that make the numerator or denominator zero. If we write the zeroes as zz123,, zetc., and similarly write the poles as pp123,, p , then Hs( ) can be written in factored form as ()()()s zsz sz Hs() K 12 m (1.2) ()()()s psp12 sp n 1 © Bob York 2009 2 Frequency Response and Bode Plots The pole and zero locations can be real or complex. -
On the Irresistible Efficiency of Signal Processing Methods in Quantum
On the Irresistible Efficiency of Signal Processing Methods in Quantum Computing 1,2 2 Andreas Klappenecker∗ , Martin R¨otteler† 1Department of Computer Science, Texas A&M University, College Station, TX 77843-3112, USA 2 Institut f¨ur Algorithmen und Kognitive Systeme, Universit¨at Karlsruhe,‡ Am Fasanengarten 5, D-76 128 Karlsruhe, Germany February 1, 2008 Abstract We show that many well-known signal transforms allow highly ef- ficient realizations on a quantum computer. We explain some elemen- tary quantum circuits and review the construction of the Quantum Fourier Transform. We derive quantum circuits for the Discrete Co- sine and Sine Transforms, and for the Discrete Hartley transform. We show that at most O(log2 N) elementary quantum gates are necessary to implement any of those transforms for input sequences of length N. arXiv:quant-ph/0111039v1 7 Nov 2001 §1 Introduction Quantum computers have the potential to solve certain problems at much higher speed than any classical computer. Some evidence for this statement is given by Shor’s algorithm to factor integers in polynomial time on a quan- tum computer. A crucial part of Shor’s algorithm depends on the discrete Fourier transform. The time complexity of the quantum Fourier transform is polylogarithmic in the length of the input signal. It is natural to ask whether other signal transforms allow for similar speed-ups. We briefly recall some properties of quantum circuits and construct the quantum Fourier transform. The main part of this paper is concerned with ∗e-mail: [email protected] †e-mail: [email protected] ‡research group Quantum Computing, Professor Thomas Beth the construction of quantum circuits for the discrete Cosine transforms, for the discrete Sine transforms, and for the discrete Hartley transform. -
Computational Entropy and Information Leakage∗
Computational Entropy and Information Leakage∗ Benjamin Fuller Leonid Reyzin Boston University fbfuller,[email protected] February 10, 2011 Abstract We investigate how information leakage reduces computational entropy of a random variable X. Recall that HILL and metric computational entropy are parameterized by quality (how distinguishable is X from a variable Z that has true entropy) and quantity (how much true entropy is there in Z). We prove an intuitively natural result: conditioning on an event of probability p reduces the quality of metric entropy by a factor of p and the quantity of metric entropy by log2 1=p (note that this means that the reduction in quantity and quality is the same, because the quantity of entropy is measured on logarithmic scale). Our result improves previous bounds of Dziembowski and Pietrzak (FOCS 2008), where the loss in the quantity of entropy was related to its original quality. The use of metric entropy simplifies the analogous the result of Reingold et. al. (FOCS 2008) for HILL entropy. Further, we simplify dealing with information leakage by investigating conditional metric entropy. We show that, conditioned on leakage of λ bits, metric entropy gets reduced by a factor 2λ in quality and λ in quantity. ∗Most of the results of this paper have been incorporated into [FOR12a] (conference version in [FOR12b]), where they are applied to the problem of building deterministic encryption. This paper contains a more focused exposition of the results on computational entropy, including some results that do not appear in [FOR12a]: namely, Theorem 3.6, Theorem 3.10, proof of Theorem 3.2, and results in Appendix A. -
A Weakly Informative Default Prior Distribution for Logistic and Other
The Annals of Applied Statistics 2008, Vol. 2, No. 4, 1360–1383 DOI: 10.1214/08-AOAS191 c Institute of Mathematical Statistics, 2008 A WEAKLY INFORMATIVE DEFAULT PRIOR DISTRIBUTION FOR LOGISTIC AND OTHER REGRESSION MODELS By Andrew Gelman, Aleks Jakulin, Maria Grazia Pittau and Yu-Sung Su Columbia University, Columbia University, University of Rome, and City University of New York We propose a new prior distribution for classical (nonhierarchi- cal) logistic regression models, constructed by first scaling all nonbi- nary variables to have mean 0 and standard deviation 0.5, and then placing independent Student-t prior distributions on the coefficients. As a default choice, we recommend the Cauchy distribution with cen- ter 0 and scale 2.5, which in the simplest setting is a longer-tailed version of the distribution attained by assuming one-half additional success and one-half additional failure in a logistic regression. Cross- validation on a corpus of datasets shows the Cauchy class of prior dis- tributions to outperform existing implementations of Gaussian and Laplace priors. We recommend this prior distribution as a default choice for rou- tine applied use. It has the advantage of always giving answers, even when there is complete separation in logistic regression (a common problem, even when the sample size is large and the number of pre- dictors is small), and also automatically applying more shrinkage to higher-order interactions. This can be useful in routine data analy- sis as well as in automated procedures such as chained equations for missing-data imputation. We implement a procedure to fit generalized linear models in R with the Student-t prior distribution by incorporating an approxi- mate EM algorithm into the usual iteratively weighted least squares. -
1 1. Data Transformations
260 Archives ofDisease in Childhood 1993; 69: 260-264 STATISTICS FROM THE INSIDE Arch Dis Child: first published as 10.1136/adc.69.2.260 on 1 August 1993. Downloaded from 1 1. Data transformations M J R Healy Additive and multiplicative effects adding a constant quantity to the correspond- In the statistical analyses for comparing two ing before readings. Exactly the same is true of groups of continuous observations which I the unpaired situation, as when we compare have so far considered, certain assumptions independent samples of treated and control have been made about the data being analysed. patients. Here the assumption is that we can One of these is Normality of distribution; in derive the distribution ofpatient readings from both the paired and the unpaired situation, the that of control readings by shifting the latter mathematical theory underlying the signifi- bodily along the axis, and this again amounts cance probabilities attached to different values to adding a constant amount to each of the of t is based on the assumption that the obser- control variate values (fig 1). vations are drawn from Normal distributions. This is not the only way in which two groups In the unpaired situation, we make the further of readings can be related in practice. Suppose assumption that the distributions in the two I asked you to guess the size of the effect of groups which are being compared have equal some treatment for (say) increasing forced standard deviations - this assumption allows us expiratory volume in one second in asthmatic to simplify the analysis and to gain a certain children. -
C-17Bpages Pdf It
SPECIFICATION DEFINITIONS FOR LOGARITHMIC AMPLIFIERS This application note is presented to engineers who may use logarithmic amplifiers in a variety of system applica- tions. It is intended to help engineers understand logarithmic amplifiers, how to specify them, and how the loga- rithmic amplifiers perform in a system environment. A similar paper addressing the accuracy and error contributing elements in logarithmic amplifiers will follow this application note. INTRODUCTION The need to process high-density pulses with narrow pulse widths and large amplitude variations necessitates the use of logarithmic amplifiers in modern receiving systems. In general, the purpose of this class of amplifier is to condense a large input dynamic range into a much smaller, manageable one through a logarithmic transfer func- tion. As a result of this transfer function, the output voltage swing of a logarithmic amplifier is proportional to the input signal power range in dB. In most cases, logarithmic amplifiers are used as amplitude detectors. Since output voltage (in mV) is proportion- al to the input signal power (in dB), the amplitude information is displayed in a much more usable format than accomplished by so-called linear detectors. LOGARITHMIC TRANSFER FUNCTION INPUT RF POWER VS. DETECTED OUTPUT VOLTAGE 0V 50 MILLIVOLTS/DIV. 250 MILLIVOLTS/DIV. 0V 50.0 ns/DIV. 50.0 ns/DIV. There are three basic types of logarithmic amplifiers. These are: • Detector Log Video Amplifiers • Successive Detection Logarithmic Amplifiers • True Log Amplifiers DETECTOR LOG VIDEO AMPLIFIER (DLVA) is a type of logarithmic amplifier in which the envelope of the input RF signal is detected with a standard "linear" diode detector. -
Linear and Logarithmic Scales
Linear and logarithmic scales. Defining Scale You may have thought of a scale as something to weigh yourself with or the outer layer on the bodies of fish and reptiles. For this lesson, we're using a different definition of a scale. A scale, in this sense, is a leveled range of values/numbers from lowest to highest that measures something at regular intervals. A great example is the classic number line that has numbers lined up at consistent intervals along a line. What Is a Linear Scale? A linear scale is much like the number line described above. They key to this type of scale is that the value between two consecutive points on the line does not change no matter how high or low you are on it. For instance, on the number line, the distance between the numbers 0 and 1 is 1 unit. The same distance of one unit is between the numbers 100 and 101, or -100 and -101. However you look at it, the distance between the points is constant (unchanging) regardless of the location on the line. A great way to visualize this is by looking at one of those old school intro to Geometry or mid- level Algebra examples of how to graph a line. One of the properties of a line is that it is the shortest distance between two points. Another is that it is has a constant slope. Having a constant slope means that the change in x and y from one point to another point on the line doesn't change. -
Bits, Bans Y Nats: Unidades De Medida De Cantidad De Información
Bits, bans y nats: Unidades de medida de cantidad de información Junio de 2017 Apellidos, Nombre: Flores Asenjo, Santiago J. (sfl[email protected]) Departamento: Dep. de Comunicaciones Centro: EPS de Gandia 1 Introducción: bits Resumen Todo el mundo conoce el bit como unidad de medida de can- tidad de información, pero pocos utilizan otras unidades también válidas tanto para esta magnitud como para la entropía en el ám- bito de la Teoría de la Información. En este artículo se presentará el nat y el ban (también deno- minado hartley), y se relacionarán con el bit. Objetivos y conocimientos previos Los objetivos de aprendizaje este artículo docente se presentan en la tabla 1. Aunque se trata de un artículo divulgativo, cualquier conocimiento sobre Teo- ría de la Información que tenga el lector le será útil para asimilar mejor el contenido. 1. Mostrar unidades alternativas al bit para medir la cantidad de información y la entropía 2. Presentar las unidades ban (o hartley) y nat 3. Contextualizar cada unidad con la historia de la Teoría de la Información 4. Utilizar la Wikipedia como herramienta de referencia 5. Animar a la edición de la Wikipedia, completando los artículos en español, una vez aprendidos los conceptos presentados y su contexto histórico Tabla 1: Objetivos de aprendizaje del artículo docente 1 Introducción: bits El bit es utilizado de forma universal como unidad de medida básica de canti- dad de información. Su nombre proviene de la composición de las dos palabras binary digit (dígito binario, en inglés) por ser históricamente utilizado para denominar cada uno de los dos posibles valores binarios utilizados en compu- tación (0 y 1). -
The Indefinite Logarithm, Logarithmic Units, and the Nature of Entropy
The Indefinite Logarithm, Logarithmic Units, and the Nature of Entropy Michael P. Frank FAMU-FSU College of Engineering Dept. of Electrical & Computer Engineering 2525 Pottsdamer St., Rm. 341 Tallahassee, FL 32310 [email protected] August 20, 2017 Abstract We define the indefinite logarithm [log x] of a real number x> 0 to be a mathematical object representing the abstract concept of the logarithm of x with an indeterminate base (i.e., not specifically e, 10, 2, or any fixed number). The resulting indefinite logarithmic quantities naturally play a mathematical role that is closely analogous to that of dimensional physi- cal quantities (such as length) in that, although these quantities have no definite interpretation as ordinary numbers, nevertheless the ratio of two of these entities is naturally well-defined as a specific, ordinary number, just like the ratio of two lengths. As a result, indefinite logarithm objects can serve as the basis for logarithmic spaces, which are natural systems of logarithmic units suitable for measuring any quantity defined on a log- arithmic scale. We illustrate how logarithmic units provide a convenient language for explaining the complete conceptual unification of the dis- parate systems of units that are presently used for a variety of quantities that are conventionally considered distinct, such as, in particular, physical entropy and information-theoretic entropy. 1 Introduction The goal of this paper is to help clear up what is perceived to be a widespread arXiv:physics/0506128v1 [physics.gen-ph] 15 Jun 2005 confusion that can found in many popular sources (websites, popular books, etc.) regarding the proper mathematical status of a variety of physical quantities that are conventionally defined on logarithmic scales. -
Shannon's Formula Wlog(1+SNR): a Historical Perspective
Shannon’s Formula W log(1 + SNR): A Historical· Perspective on the occasion of Shannon’s Centenary Oct. 26th, 2016 Olivier Rioul <[email protected]> Outline Who is Claude Shannon? Shannon’s Seminal Paper Shannon’s Main Contributions Shannon’s Capacity Formula A Hartley’s rule C0 = log2 1 + ∆ is not Hartley’s 1 P Many authors independently derived C = 2 log2 1 + N in 1948. Hartley’s rule is exact: C0 = C (a coincidence?) C0 is the capacity of the “uniform” channel Shannon’s Conclusion 2 / 85 26/10/2016 Olivier Rioul Shannon’s Formula W log(1 + SNR): A Historical Perspective April 30, 2016 centennial day celebrated by Google: here Shannon is juggling with bits (1,0,0) in his communication scheme “father of the information age’’ Claude Shannon (1916–2001) 100th birthday 2016 April 30, 1916 Claude Elwood Shannon was born in Petoskey, Michigan, USA 3 / 85 26/10/2016 Olivier Rioul Shannon’s Formula W log(1 + SNR): A Historical Perspective here Shannon is juggling with bits (1,0,0) in his communication scheme “father of the information age’’ Claude Shannon (1916–2001) 100th birthday 2016 April 30, 1916 Claude Elwood Shannon was born in Petoskey, Michigan, USA April 30, 2016 centennial day celebrated by Google: 3 / 85 26/10/2016 Olivier Rioul Shannon’s Formula W log(1 + SNR): A Historical Perspective Claude Shannon (1916–2001) 100th birthday 2016 April 30, 1916 Claude Elwood Shannon was born in Petoskey, Michigan, USA April 30, 2016 centennial day celebrated by Google: here Shannon is juggling with bits (1,0,0) in his communication scheme “father of the information age’’ 3 / 85 26/10/2016 Olivier Rioul Shannon’s Formula W log(1 + SNR): A Historical Perspective “the most important man.. -
Quantum Computing and a Unified Approach to Fast Unitary Transforms
Quantum Computing and a Unified Approach to Fast Unitary Transforms Sos S. Agaiana and Andreas Klappeneckerb aThe City University of New York and University of Texas at San Antonio [email protected] bTexas A&M University, Department of Computer Science, College Station, TX 77843-3112 [email protected] ABSTRACT A quantum computer directly manipulates information stored in the state of quantum mechanical systems. The available operations have many attractive features but also underly severe restrictions, which complicate the design of quantum algorithms. We present a divide-and-conquer approach to the design of various quantum algorithms. The class of algorithm includes many transforms which are well-known in classical signal processing applications. We show how fast quantum algorithms can be derived for the discrete Fourier transform, the Walsh-Hadamard transform, the Slant transform, and the Hartley transform. All these algorithms use at most O(log2 N) operations to transform a state vector of a quantum computer of length N. 1. INTRODUCTION Discrete orthogonal transforms and discrete unitary transforms have found various applications in signal, image, and video processing, in pattern recognition, in biocomputing, and in numerous other areas [1–11]. Well- known examples of such transforms include the discrete Fourier transform, the Walsh-Hadamard transform, the trigonometric transforms such as the Sine and Cosine transform, the Hartley transform, and the Slant transform. All these different transforms find applications in signal and image processing, because the great variety of signal classes occuring in practice cannot be handeled by a single transform. On a classical computer, the straightforward way to compute a discrete orthogonal transform of a signal vector of length N takes in general O(N 2) operations.