Properties of the Spin-flip Amplitude of Hadron Elastic Scattering and Possible Polarization Effects at RHIC
|
|
|
- Claire Crawford
- 5 years ago
- Views:
Transcription
1 Properties of te Spin-flip Amplitude of Hadron Elastic Scattering and Possible Polarization Effects at RHIC arxiv:ep-p/ v1 30 Oct 2002 O. V. Selyugin 1 Joint Institute for Nuclear Researc, Dubna, Russia Abstract. Wit relation to te RHIC spin program we researc te polarization effects in elastic proton-proton scattering at small momentum transfer and in te diffraction dip region. Te calculations take into account te Coulomb-adron interference effects including te additional Coulomb-adron pase. In particular we sow te impact of te form of te adron potential at large distances on te beavior of te adron spin-flip amplitude at small angles. Te t-dependence of te spin-flip amplitude of ig energy adron elastic scattering is analyzed under different assumptions on te adron interaction. I INTRODUCTION Several attempts to extract te spin-flip amplitude from te experimental data sow tat te ratio of spin-flip to spin-nonflip amplitudes can be non-negligible and may be only sligtly dependent on energy [1,2]. For te definition of new effects at small angles and especially in te region of te diffraction minimum one must know te effects of te Coulomb-adron interference wit sufficiently ig accuracy. Te Coulomb-adron pase was calculated in te entire diffraction domain taking into account te form factors of te nucleons [3]. Some polarization effects connected wit te Coulomb adron interference, including some possible odderon contribution, were also calculated [4]. Te model-dependent analysis based on all te existing experimental data of te spin-correlation parameters above p L 6 GeV allows us to determine te structure of te adron spin-flip amplitude at ig energies and to predict its beavior at superig energies [6]. Tis analysis sows tat te ratios Re φ 5(s,t)/( t Re φ 1(s,t)) and Im φ 5(s,t)/( t Im φ 1(s,t)) depend on s and t (see Fig.1 a,b). At small momentum transfers, it was found tat te slope of te 1) selugin@tsun1.jinr.ru
2 FIGURE 1. (a ) and (b)) Ratio of te imaginary (a) and real (b) part of te residual F +. to te imaginary F ++ (c) Contribution to te pure CNI effect from te model F + (te our calculations at t max, t = GeV 2 t = 0.01 GeV 2, t = 0.1 GeV 2 are sown te full line, te long dased line, te dased line and te doted line respectively. residual spin-flip amplitudes is approximately twice te slope of te spin-non flip amplitude. Te obtained spin-flip amplitude leads to te additional contribution to te pure CNI effect at small t (Fig. 1 c). Te dependence of te adron spin-flip amplitude on t at small angles is closely related wit te basic structure of adrons at large distances. We sow tat te slope of te so-called reduced adron spin-flip amplitude (te adron spin-flip amplitude witout te kinematic factor t ) can be larger tan te slope of te adron spin-non-flip amplitude, as was observed long ago [5]. II THE SLOPE OF THE HADRON AMPLITUDES For an exponential form of te amplitudes tis coincides wit te usual slope of te differential cross sections divided by 2. At small t ( GeV 2 ), practically all semipenomenological analyses assume: B + 1 B + 2 B 1 B 2. If te potentials V ++ and V + are assumed to ave a Gaussian form in te first Born approximation φ 1 and ˆφ 5 will ave te same form φ 1 (s,t) exp( B 2 ), φ 5 (s,t) = q B exp( B 2 ). In tis special case, terefore, te slopes of te spin-flip and residual spin-non-flip amplitudes are indeed te same. A Gaussian form of te potential is adequate to represent te central part of te adronic interaction. Te form cuts off te Bessel function and te contributions at large distances. If, owever, te potential (or te corresponding eikonal) as a long tail (exponential or power) in te impact parameter, te Bessel functions can not be taken in te approximation form and te full integration leads to different results. If we take χ i (b,s) H e a ρ, we obtain F nf (s,t) = a/[(a 2 +q 2 ) 3/2 ] 1/[a a 2 +q 2 ] exp( Bq 2 ) (1)
3 wit B = 1/a 2. For te residual spin-flip amplitude, on te oter and, we obtain [8] t F sf (s,t) = (3 a q)/[(a 2 +q 2 ) 5/2 ] (3 aq B 2 )/( a 2 +q 2 ) exp( 2 Bq 2 ). (2) In tis case, terefore, te slope of te residual spin-flip amplitude exceeds te slope of te spin-non-flip amplitudes by a factor of two. A similar beaviour can be obtained wit te standard dipole form factor [8]. III THE DETERMINATION OF THE STRUCTURE OF THE HADRON SPIN-FLIP AMPLITUDE Note tat if te reduced spin-flip amplitude is not small, te impact of a large B will reflect in te beavior of te differential cross section at small angles [7]. Te metod gives only te absolute value of te coefficient of te spin-flip amplitude. Te imaginary and real parts of te spin-flip amplitude can be found only from te measurements of te spin correlation coefficient. Let us take te spin nonflip amplitude in te standard exponential form wit definite parameters: slope B +, σ tot and ρ +. For te residual spin-flip amplitude, on te oter and, we consider two possibilities: equal slopes B = B + and B = 2B +. Te results of tese two different calculations are sown in Fig.2. It is clear tat around te maximum of te Coulomb-adron interference, te difference between te two variants is very small. But wen t > 0.01 GeV 2, tis difference grows. So, if we try to find te contribution of te pomeron spin-flip, we sould take into account tis effect. As te value of A N depends on te determination of te beam polarization, let us calculate te derivative of A N wit respect to t, for example, at s = 500 GeV. If we know te parameters of te adron spin non-flip amplitude, te measurement of te analyzing power at small transfer momenta elps us to find te structure of te adron spin-flip amplitude. Tere is a specific point of te differential cross sections and of A N on te axis of te momentum transfer, - t re, were ReF ++ c = ReF ++. Tis point t re can be found from te measurement of te differential cross sections [9]. At ig energies at te point t re [8] we obtain for pp-scattering ReF sf (s,t) = 1 2(ImFnf (s,t)+imfc nf (t))a N(s,t) dσ dt ReFsf c (t). (3) We can again take te adron spin-nonflip and spin-flip amplitudes wit definite parameters and calculate te magnitude of A N by te usual complete form wile te real part of te adron spin-flip amplitude is given by (3). Our calculation by tis formula and te input real part of te spin-flip amplitude are sown in Fig. 2 c. At te point t re bot curves coincide. So if we obtain from te accurate measurement of te differential cross sections te value of t re, we can find from A N
4 FIGURE 2. (a) A N at s = 50 GeV (b) δa N /δt at s = 500 GeV (te solid line is wit te slope B1 of F sf equal to te slope B 1 + of F nf; te dased line is wit te B1 = 2 B 1 +. (c) Te form of Re(F sf ) : solid and long-dased lines are calculations by (3); sort-dased and dottes lines are model amplitudes wit B1 = B+ 1 and B 1 = 2 B+ 1, respectively. te value of te real part of te adron spin-flip amplitude at te same point of momentum transfer. IV THE MODEL PREDICTIONS Te model [10] takes into account te contribution of te adron interaction at large distances and leads to te ig-energy spin-flip amplitude. Te model gives te large spin effects in te -elastic scattering and predicts non-small effects for te PP2PP experiment at RHIC especially in te diffraction dip domain [11]. Te additional pure CNI effects can be calculated using te Coulomb-nuclear pase [3]. Tese polarization effects will be present at RHIC energy, even toug F + 0 at ig energy. Our model calculations sow on Fig.3 for bot cases. Te model gives te standard t-dependence of ReF ++ and ImF ++. Instead of it, in a convenient parameterization of bot te modulus and te pase one can obtain te alternative case, in wic ImF (s,t) as te zero at small t (for details, (a) (b) (c) FIGURE 3. (a) and (b) A N at s = 50 GeV (a) te full line is te total A N ; te dused line is te A CN N. (c). A NN at s = 500 GeV in te domail of te dip ; te full line is te total A NN ; te dused line is te A CN NN.
5 FIGURE 4. (a) A CN N at s = 50 GeV and small t for two models. (b) and (c) A CN N at s = 50 GeV and s = 500 GeV in te region of te dip (te solid line corresponds to te model I wit zero of IF at dip; te dased line sows te variant II, wit te zero of te ReF at te dip). see [12]). Suc an approac enables one to specify te elastic adron scattering amplitude F (s,t)directly fromte elastic scattering data. Te difference between te pases leads eiter to central or periperal distributions of elastic adron scattering in te impact parameter space. Te obtained form of A CN N at small momentum transfers differs for te two variants beginning at t > 0.05 GeV 2 (Fig.4 a). Te difference reaces 2% at t = 0.15 GeV 2 and, in principle, can be measured in an accurate experiment. Now let us calculate te Coulomb-adron interference effect - A CN N in te two alternatives for iger t : (i) te diffraction dip is created by te zero of te ImF (s,t) part of te scattering amplitude and ReF (s,t) fills it; (ii) te diffraction dip is created by te zero of te ReF (s,t) part of te scattering amplitude and ImF (s,t) fills it. Te results are sown in Fig. 4 (b) for s = 50 GeV and in Fig. 4 (c) at s = 500 GeV. REFERENCES 1. Akcurin N., Buttimore N.H. and Penzo A., Pys. Rev. D 51, 3944 (1995). 2. Selyugin O.V., Pys. Lett. B333, 245 (1993). 3. Selyugin O.V., Pys.Rev. D (1999). 4. Selyugin O.V., in Proc. New Trends in Hig Energy Pysics, Crimea, 2000, ed. P. Bogolyubov, L. Jenkovszky, Kiev, Predazzi E., Soliani G., Nuovo Cim. A (1967); Hinotani K., Neal H.A., Predazzi E. and Walters G., Nuovo Cim., A (1979). 6. Selyugin O.V., Pys. of Atomic Nuclei (1999). 7. Selyugin O.V., Mod. Pys. Lett., A (1994). 8. Predazzzi E. and Selyugin O.V., Eur. Pys. J. A 13, 471 (2002). 9. O.V. Selyugin, Nucl.Pys. B (Proc.Suppl.) 99A (2001) Goloskokov S.V., Kulesov S.P., Selyugin O.V., Z. Pys. C 50, 455 (1991). 11. N.Akcurin, S.V.Goloskokov, O.V.Selyugin, Int.J. of Mod.Pys. A 14 (1999) V. Kundrát and M. Lokajíček, Z. Pys. C 63, 619 (1994).
232 Calculus and Structures
3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE
arxiv:hep-ph/ v1 1 Mar 2001
NYU-TH/01/02/02 ep-p/0103001 arxiv:ep-p/0103001v1 1 Mar 2001 Scale-Independent Calculation of sin 2 θ lept eff. A. Ferroglia 1, G. Ossola 2, and A. Sirlin 3. Department of Pysics, New York University,
The Verlet Algorithm for Molecular Dynamics Simulations
Cemistry 380.37 Fall 2015 Dr. Jean M. Standard November 9, 2015 Te Verlet Algoritm for Molecular Dynamics Simulations Equations of motion For a many-body system consisting of N particles, Newton's classical
Polynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximatinga function fx, wose values at a set of distinct points x, x, x,, x n are known, by a polynomial P x suc
Section 3.1: Derivatives of Polynomials and Exponential Functions
Section 3.1: Derivatives of Polynomials and Exponential Functions In previous sections we developed te concept of te derivative and derivative function. Te only issue wit our definition owever is tat it
4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these.
Mat 11. Test Form N Fall 016 Name. Instructions. Te first eleven problems are wort points eac. Te last six problems are wort 5 points eac. For te last six problems, you must use relevant metods of algebra
Order of Accuracy. ũ h u Ch p, (1)
Order of Accuracy 1 Terminology We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, wic can be for instance te grid size or time step in a numerical
MVT and Rolle s Theorem
AP Calculus CHAPTER 4 WORKSHEET APPLICATIONS OF DIFFERENTIATION MVT and Rolle s Teorem Name Seat # Date UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR ANY OF THESE QUESTIONS In problems 1 and, state
Firooz Arash (a,b) , which have caused confusion in the normalization of ZEUS data is discussed and resolved. PCAC I. INTRODUCTION
PION STRUCTURE FUNCTION F π 2 In te Valon Model Firooz Aras (a,b) (a) Center for Teoretical Pysics and Matematics, AEOI P.O.Box 365-8486, Teran, Iran (b) Pysics Department, AmirKabir University (Tafres
Numerical Differentiation
Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function
Polynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximating a function f(x, wose values at a set of distinct points x, x, x 2,,x n are known, by a polynomial P (x
1. For d=3,2 from ε<< ε F it follows that ετ >> e-e h, i.e.,
Quasiparticle decay rate at T = 0 in a clean Fermi Liquid. ω +ω Fermi Sea τ e e ( ) F ( ) log( ) Conclusions:. For d=3, from > e-e, i.e., tat te qusiparticles are well determined
2.8 The Derivative as a Function
.8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open
A Helium-3 polarimeter using electromagnetic interference. Nigel Buttimore
A Helium-3 polarimeter using electromagnetic interference Nigel Buttimore Trinity College Dublin 12 September 2013 PSTP 2013 09 University of Virginia OUTLINE Need a source of polarized down quarks to
Fractional momentum correlations in multiple production of W bosons and of b b pairs in high energy pp collisions
SMR.1751-43 Fift International Conference on PERSPECTIVES IN HARONIC PHYSICS Particle-Nucleus and Nucleus-Nucleus Scattering at Relativistic Energies 22-26 May 26 Fractional momentum correlations in multiple
(4.2) -Richardson Extrapolation
(.) -Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Suppose tat lim G 0 and lim F L. Te function F is said to converge to L as
Quantum Theory of the Atomic Nucleus
G. Gamow, ZP, 51, 204 1928 Quantum Teory of te tomic Nucleus G. Gamow (Received 1928) It as often been suggested tat non Coulomb attractive forces play a very important role inside atomic nuclei. We can
3.1 Extreme Values of a Function
.1 Etreme Values of a Function Section.1 Notes Page 1 One application of te derivative is finding minimum and maimum values off a grap. In precalculus we were only able to do tis wit quadratics by find
Notes on wavefunctions II: momentum wavefunctions
Notes on wavefunctions II: momentum wavefunctions and uncertainty Te state of a particle at any time is described by a wavefunction ψ(x). Tese wavefunction must cange wit time, since we know tat particles
The derivative function
Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative
Physics with Tagged Forward Protons using the STAR Detector at RHIC. The Relativistic Heavy Ion Collider The pp2pp Experiment STAR 2009
Physics with Tagged Forward Protons using the STAR Detector at RHIC The Relativistic Heavy Ion Collider The pp2pp Experiment 2002 2003 STAR 2009 Elastic and Diffractive Processes Elastic scattering Detect
1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point
MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note
LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION
LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION LAURA EVANS.. Introduction Not all differential equations can be explicitly solved for y. Tis can be problematic if we need to know te value of y
Relativistic nuclear matter in generalized thermo-statistics
Relativistic nuclear matter in generalized termo-statistics K. Miyazaki E-mail: miyazakiro@rio.odn.ne.jp Abstract Te generalized Fermi-Dirac termo-statistics is developed for relativistic nuclear matter.
Excluded Volume Effects in Gene Stretching. Pui-Man Lam Physics Department, Southern University Baton Rouge, Louisiana
Excluded Volume Effects in Gene Stretcing Pui-Man Lam Pysics Department, Soutern University Baton Rouge, Louisiana 7083 Abstract We investigate te effects excluded volume on te stretcing of a single DNA
How to Find the Derivative of a Function: Calculus 1
Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te
Applied Linear Statistical Models. Simultaneous Inference Topics. Simultaneous Estimation of β 0 and β 1 Issues. Simultaneous Inference. Dr.
Applied Linear Statistical Models Simultaneous Inference Dr. DH Jones Simultaneous Inference Topics Simultaneous estimation of β 0 and β 1 Bonferroni Metod Simultaneous estimation of several mean responses
Polarizing Helium-3 for down quark spin enrichment. Nigel Buttimore
Polarizing Helium-3 for down quark spin enrichment Nigel Buttimore Trinity College Dublin 12 September 2012 Diffraction 2012 Polarized Helium-3 OUTLINE Introduction to the spin structure of polarized protons
Solution for the Homework 4
Solution for te Homework 4 Problem 6.5: In tis section we computed te single-particle translational partition function, tr, by summing over all definite-energy wavefunctions. An alternative approac, owever,
2.11 That s So Derivative
2.11 Tat s So Derivative Introduction to Differential Calculus Just as one defines instantaneous velocity in terms of average velocity, we now define te instantaneous rate of cange of a function at a point
Lecture 15. Interpolation II. 2 Piecewise polynomial interpolation Hermite splines
Lecture 5 Interpolation II Introduction In te previous lecture we focused primarily on polynomial interpolation of a set of n points. A difficulty we observed is tat wen n is large, our polynomial as to
Reflection of electromagnetic waves from magnetic having the ferromagnetic spiral
Reflection of electromagnetic waves from magnetic aving te ferromagnetic spiral Igor V. Bycov 1a Dmitry A. Kuzmin 1b and Vladimir G. Savrov 3 1 Celyabins State University 51 Celyabins Br. Kasiriny Street
The total error in numerical differentiation
AMS 147 Computational Metods and Applications Lecture 08 Copyrigt by Hongyun Wang, UCSC Recap: Loss of accuracy due to numerical cancellation A B 3, 3 ~10 16 In calculating te difference between A and
Pumping Heat with Quantum Ratchets
Pumping Heat wit Quantum Ratcets T. E. Humprey a H. Linke ab R. Newbury a a Scool of Pysics University of New Sout Wales UNSW Sydney 5 Australia b Pysics Department University of Oregon Eugene OR 9743-74
Math 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006
Mat 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 f(x+) f(x) 10 1. For f(x) = x 2 + 2x 5, find ))))))))) and simplify completely. NOTE: **f(x+) is NOT f(x)+! f(x+) f(x) (x+) 2 + 2(x+) 5 ( x 2
Introduction to Derivatives
Introduction to Derivatives 5-Minute Review: Instantaneous Rates and Tangent Slope Recall te analogy tat we developed earlier First we saw tat te secant slope of te line troug te two points (a, f (a))
A = h w (1) Error Analysis Physics 141
Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.
(Experimental) Soft Diffraction at LHC. Jan Kašpar. ISMD2017, Tlaxcala, Mexico 15 September, 2017
(Experimental) Soft Diffraction at LHC Jan Kašpar ISMD2017, Tlaxcala, Mexico 15 September, 2017 Introduction selected processes: Elastic scattering p + p p + p Single diffraction p + p p + X Double diffraction
Quantum Numbers and Rules
OpenStax-CNX module: m42614 1 Quantum Numbers and Rules OpenStax College Tis work is produced by OpenStax-CNX and licensed under te Creative Commons Attribution License 3.0 Abstract Dene quantum number.
Chapter 4 Derivatives [ ] = ( ) ( )= + ( ) + + = ()= + ()+ Exercise 4.1. Review of Prerequisite Skills. 1. f. 6. d. 4. b. lim. x x. = lim = c.
![Chapter 4 Derivatives [ ] = ( ) ( )= + ( ) + + = ()= + ()+ Exercise 4.1. Review of Prerequisite Skills. 1. f. 6. d. 4. b. lim. x x. = lim = c.](/thumbs/92/109490812.jpg "Chapter 4 Derivatives [ ] = ( ) ( )= + ( ) + + = ()= + ()+ Exercise 4.1. Review of Prerequisite Skills. 1. f. 6. d. 4. b. lim. x x. = lim = c.")
Capter Derivatives Review of Prerequisite Skills. f. p p p 7 9 p p p Eercise.. i. ( a ) a ( b) a [ ] b a b ab b a. d. f. 9. c. + + ( ) ( + ) + ( + ) ( + ) ( + ) + + + + ( ) ( + ) + + ( ) ( ) ( + ) + 7
AN ANALYSIS OF AMPLITUDE AND PERIOD OF ALTERNATING ICE LOADS ON CONICAL STRUCTURES
Ice in te Environment: Proceedings of te 1t IAHR International Symposium on Ice Dunedin, New Zealand, nd t December International Association of Hydraulic Engineering and Researc AN ANALYSIS OF AMPLITUDE
1. State whether the function is an exponential growth or exponential decay, and describe its end behaviour using limits.
Questions 1. State weter te function is an exponential growt or exponential decay, and describe its end beaviour using its. (a) f(x) = 3 2x (b) f(x) = 0.5 x (c) f(x) = e (d) f(x) = ( ) x 1 4 2. Matc te
4.2 - Richardson Extrapolation
. - Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Definition Let x n n converge to a number x. Suppose tat n n is a sequence
Differential Calculus (The basics) Prepared by Mr. C. Hull
Differential Calculus Te basics) A : Limits In tis work on limits, we will deal only wit functions i.e. tose relationsips in wic an input variable ) defines a unique output variable y). Wen we work wit
Practice Problem Solutions: Exam 1
Practice Problem Solutions: Exam 1 1. (a) Algebraic Solution: Te largest term in te numerator is 3x 2, wile te largest term in te denominator is 5x 2 3x 2 + 5. Tus lim x 5x 2 2x 3x 2 x 5x 2 = 3 5 Numerical
Ballistic electron transport in quantum point contacts
Ballistic electron transport in quantum point contacts 11 11.1 Experimental observation of conductance quantization Wen we discussed te self-consistent calculation of te potential and te modes in an infinite
Exam 1 Review Solutions
Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),
Investigating Euler s Method and Differential Equations to Approximate π. Lindsay Crowl August 2, 2001
Investigating Euler s Metod and Differential Equations to Approximate π Lindsa Crowl August 2, 2001 Tis researc paper focuses on finding a more efficient and accurate wa to approximate π. Suppose tat x
Logarithmic functions
Roberto s Notes on Differential Calculus Capter 5: Derivatives of transcendental functions Section Derivatives of Logaritmic functions Wat ou need to know alread: Definition of derivative and all basic
Recall from our discussion of continuity in lecture a function is continuous at a point x = a if and only if
Computational Aspects of its. Keeping te simple simple. Recall by elementary functions we mean :Polynomials (including linear and quadratic equations) Eponentials Logaritms Trig Functions Rational Functions
MATH1151 Calculus Test S1 v2a
MATH5 Calculus Test 8 S va January 8, 5 Tese solutions were written and typed up by Brendan Trin Please be etical wit tis resource It is for te use of MatSOC members, so do not repost it on oter forums
Calculus I Practice Exam 1A
Calculus I Practice Exam A Calculus I Practice Exam A Tis practice exam empasizes conceptual connections and understanding to a greater degree tan te exams tat are usually administered in introductory
Nuclear attenuation 2 dimensional dependences at HERMES
Nuclear attenuation dimensional dependences at HERMES Inti Lemann University of Glasgow for te HERMES Collaboration DIS, Florence, /4/ Experimental Metod Semi-Inclusive DIS (SIDIS) on te Nucleon Access
Quantization of electrical conductance
1 Introduction Quantization of electrical conductance Te resistance of a wire in te classical Drude model of metal conduction is given by RR = ρρρρ AA, were ρρ, AA and ll are te conductivity of te material,
1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t).
. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd, periodic function tat as been sifted upwards, so we will use
Problem Set 4 Solutions
University of Alabama Department of Pysics and Astronomy PH 253 / LeClair Spring 2010 Problem Set 4 Solutions 1. Group velocity of a wave. For a free relativistic quantum particle moving wit speed v, te
Chapter 1D - Rational Expressions
- Capter 1D Capter 1D - Rational Expressions Definition of a Rational Expression A rational expression is te quotient of two polynomials. (Recall: A function px is a polynomial in x of degree n, if tere
Quasiperiodic phenomena in the Van der Pol - Mathieu equation
Quasiperiodic penomena in te Van der Pol - Matieu equation F. Veerman and F. Verulst Department of Matematics, Utrect University P.O. Box 80.010, 3508 TA Utrect Te Neterlands April 8, 009 Abstract Te Van
Precalculus Test 2 Practice Questions Page 1. Note: You can expect other types of questions on the test than the ones presented here!
Precalculus Test 2 Practice Questions Page Note: You can expect oter types of questions on te test tan te ones presented ere! Questions Example. Find te vertex of te quadratic f(x) = 4x 2 x. Example 2.
Click here to see an animation of the derivative
Differentiation Massoud Malek Derivative Te concept of derivative is at te core of Calculus; It is a very powerful tool for understanding te beavior of matematical functions. It allows us to optimize functions,
Problem Set 4: Whither, thou turbid wave SOLUTIONS
PH 253 / LeClair Spring 2013 Problem Set 4: Witer, tou turbid wave SOLUTIONS Question zero is probably were te name of te problem set came from: Witer, tou turbid wave? It is from a Longfellow poem, Te
LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT
LIMITS AND DERIVATIVES Te limit of a function is defined as te value of y tat te curve approaces, as x approaces a particular value. Te limit of f (x) as x approaces a is written as f (x) approaces, as
Graviton Induced Nuclear Fission through Electromagnetic Wave Flux Phil Russell, * Jerry Montgomery
Graviton Induced Nuclear Fission troug Electromagnetic Wave Flux Pil Russell, * Jerry Montgomery Nort Carolina Central University, Duram, NC 27707 Willowstick Tecnologies LLC, Draper, UT 84020 (Dated:
Mathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative
Matematics 5 Workseet 11 Geometry, Tangency, and te Derivative Problem 1. Find te equation of a line wit slope m tat intersects te point (3, 9). Solution. Te equation for a line passing troug a point (x
Digital Filter Structures
Digital Filter Structures Te convolution sum description of an LTI discrete-time system can, in principle, be used to implement te system For an IIR finite-dimensional system tis approac is not practical
2.3 Product and Quotient Rules
.3. PRODUCT AND QUOTIENT RULES 75.3 Product and Quotient Rules.3.1 Product rule Suppose tat f and g are two di erentiable functions. Ten ( g (x)) 0 = f 0 (x) g (x) + g 0 (x) See.3.5 on page 77 for a proof.
Solution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4.
December 09, 20 Calculus PracticeTest s Name: (4 points) Find te absolute extrema of f(x) = x 3 0 on te interval [0, 4] Te derivative of f(x) is f (x) = 3x 2, wic is zero only at x = 0 Tus we only need
Math 34A Practice Final Solutions Fall 2007
Mat 34A Practice Final Solutions Fall 007 Problem Find te derivatives of te following functions:. f(x) = 3x + e 3x. f(x) = x + x 3. f(x) = (x + a) 4. Is te function 3t 4t t 3 increasing or decreasing wen
Test on Nuclear Physics
Test on Nuclear Pysics Examination Time - 40 minutes Answer all questions in te spaces provided Tis wole test involves an imaginary element called Bedlum wic as te isotope notation sown below: 47 11 Bd
lecture 26: Richardson extrapolation
43 lecture 26: Ricardson extrapolation 35 Ricardson extrapolation, Romberg integration Trougout numerical analysis, one encounters procedures tat apply some simple approximation (eg, linear interpolation)
3.4 Worksheet: Proof of the Chain Rule NAME
Mat 1170 3.4 Workseet: Proof of te Cain Rule NAME Te Cain Rule So far we are able to differentiate all types of functions. For example: polynomials, rational, root, and trigonometric functions. We are
Math 1210 Midterm 1 January 31st, 2014
Mat 110 Midterm 1 January 1st, 01 Tis exam consists of sections, A and B. Section A is conceptual, wereas section B is more computational. Te value of every question is indicated at te beginning of it.
Lecture 21. Numerical differentiation. f ( x+h) f ( x) h h
Lecture Numerical differentiation Introduction We can analytically calculate te derivative of any elementary function, so tere migt seem to be no motivation for calculating derivatives numerically. However
Exercises for numerical differentiation. Øyvind Ryan
Exercises for numerical differentiation Øyvind Ryan February 25, 2013 1. Mark eac of te following statements as true or false. a. Wen we use te approximation f (a) (f (a +) f (a))/ on a computer, we can
Consider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.
Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions
Average Rate of Change
Te Derivative Tis can be tougt of as an attempt to draw a parallel (pysically and metaporically) between a line and a curve, applying te concept of slope to someting tat isn't actually straigt. Te slope
Te comparison of dierent models M i is based on teir relative probabilities, wic can be expressed, again using Bayes' teorem, in terms of prior probab
To appear in: Advances in Neural Information Processing Systems 9, eds. M. C. Mozer, M. I. Jordan and T. Petsce. MIT Press, 997 Bayesian Model Comparison by Monte Carlo Caining David Barber D.Barber@aston.ac.uk
Some Review Problems for First Midterm Mathematics 1300, Calculus 1
Some Review Problems for First Midterm Matematics 00, Calculus. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd,
HOW TO DEAL WITH FFT SAMPLING INFLUENCES ON ADEV CALCULATIONS
HOW TO DEAL WITH FFT SAMPLING INFLUENCES ON ADEV CALCULATIONS Po-Ceng Cang National Standard Time & Frequency Lab., TL, Taiwan 1, Lane 551, Min-Tsu Road, Sec. 5, Yang-Mei, Taoyuan, Taiwan 36 Tel: 886 3
Material for Difference Quotient
Material for Difference Quotient Prepared by Stepanie Quintal, graduate student and Marvin Stick, professor Dept. of Matematical Sciences, UMass Lowell Summer 05 Preface Te following difference quotient
Time (hours) Morphine sulfate (mg)
Mat Xa Fall 2002 Review Notes Limits and Definition of Derivative Important Information: 1 According to te most recent information from te Registrar, te Xa final exam will be eld from 9:15 am to 12:15
Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is
Mat 180 www.timetodare.com Section.7 Derivatives and Rates of Cange Part II Section.8 Te Derivative as a Function Derivatives ( ) In te previous section we defined te slope of te tangent to a curve wit
The Electron in a Potential
Te Electron in a Potential Edwin F. Taylor July, 2000 1. Stopwatc rotation for an electron in a potential For a poton we found tat te and of te quantum stopwatc rotates wit frequency f given by te equation:
MAT244 - Ordinary Di erential Equations - Summer 2016 Assignment 2 Due: July 20, 2016
MAT244 - Ordinary Di erential Equations - Summer 206 Assignment 2 Due: July 20, 206 Full Name: Student #: Last First Indicate wic Tutorial Section you attend by filling in te appropriate circle: Tut 0
(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a?
Solutions to Test 1 Fall 016 1pt 1. Te grap of a function f(x) is sown at rigt below. Part I. State te value of eac limit. If a limit is infinite, state weter it is or. If a limit does not exist (but is
The Priestley-Chao Estimator
Te Priestley-Cao Estimator In tis section we will consider te Pristley-Cao estimator of te unknown regression function. It is assumed tat we ave a sample of observations (Y i, x i ), i = 1,..., n wic are
2016 PRELIM 2 PAPER 2 MARK SCHEME
06 River Valley Hig Scool Prelim Paper Mark Sceme 06 PRELIM PAPER MARK SCHEME (a) V 5.00 X 85. 9V 3 I.7 0 X V I X V I X 0.03 0. 85.9 5.00.7 X 48.3 00 X X 900 00 [A0] Anomalous data can be identified. Systematic
ESTIMATION OF TIME-DOMAIN FREQUENCY STABILITY BASED ON PHASE NOISE MEASUREMENT
ESTIMATION OF TIME-DOMAIN FREQUENCY STABILITY BASED ON PHASE NOISE MEASUREMENT P. C. Cang, H. M. Peng, and S. Y. Lin National Standard Time & Frequenc Laborator, TL, Taiwan, Lane 55, Min-Tsu Road, Sec.
NUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,
NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing
Poisson Equation in Sobolev Spaces
Poisson Equation in Sobolev Spaces OcMountain Dayligt Time. 6, 011 Today we discuss te Poisson equation in Sobolev spaces. It s existence, uniqueness, and regularity. Weak Solution. u = f in, u = g on
Fractional Derivatives as Binomial Limits
Fractional Derivatives as Binomial Limits Researc Question: Can te limit form of te iger-order derivative be extended to fractional orders? (atematics) Word Count: 669 words Contents - IRODUCIO... Error!
Math 31A Discussion Notes Week 4 October 20 and October 22, 2015
Mat 3A Discussion Notes Week 4 October 20 and October 22, 205 To prepare for te first midterm, we ll spend tis week working eamples resembling te various problems you ve seen so far tis term. In tese notes
Continuous Stochastic Processes
Continuous Stocastic Processes Te term stocastic is often applied to penomena tat vary in time, wile te word random is reserved for penomena tat vary in space. Apart from tis distinction, te modelling
Reggeization of the Phillips-Barger model of high-energy hadron scattering
IL NUOVO CIMENTO Vol. C, N. Marzo-Aprile 0 DOI.9/ncc/i0-- Colloquia: LC Reggeization of the Phillips-Barger model of high-energy hadron scattering L. Jenkovszky BITP, National Academy of Sciences of Ukraine
University Mathematics 2
University Matematics 2 1 Differentiability In tis section, we discuss te differentiability of functions. Definition 1.1 Differentiable function). Let f) be a function. We say tat f is differentiable at
Copyright c 2008 Kevin Long
Lecture 4 Numerical solution of initial value problems Te metods you ve learned so far ave obtained closed-form solutions to initial value problems. A closedform solution is an explicit algebriac formula
3. Semiconductor heterostructures and nanostructures
3. Semiconductor eterostructures and nanostructures We discussed before ow te periodicity of a crystal results in te formation of bands. or a 1D crystal, we obtained: a (x) x In 3D, te crystal lattices
Combining functions: algebraic methods
Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)
Geneviève BELANGÉR, Kristjan KANNIKE, Alexander PUKHOV and Martti RAIDAL
Geneviève BELANGÉR,, Alexander PUKHOV and Martti RAIDAL cuola Normale uperiore di Pisa E-mail: belanger@lapt.cnrs.fr, kristjan.kannike@cern.c, pukov@lapt.cnrs.fr, martti.raidal@cern.c calar dark matter
Numerical Analysis MTH603. dy dt = = (0) , y n+1. We obtain yn. Therefore. and. Copyright Virtual University of Pakistan 1
Numerical Analysis MTH60 PREDICTOR CORRECTOR METHOD Te metods presented so far are called single-step metods, were we ave seen tat te computation of y at t n+ tat is y n+ requires te knowledge of y n only.