The Press-Schechter mass function
|
|
- Norah Cross
- 6 years ago
- Views:
Transcription
1 The Press-Schechter mass function To state the obvious: It is important to relate our theories to what we can observe. We have looke at linear perturbation theory, an we have consiere a simple moel for nonlinear structure formation, the spherical collapse moel. But we cannot observe irectly the process of structure formation. What we can see are the results, for example clusters of galaxies. We can count these objects an fin their ensity, an also further etails about their statistical istribution. One example we have encountere alreay is the galaxy luminosity function, which gives the number ensity of galaxies of a given luminosity. The quantity we will consier in the following is the mass function nm) of cosmic structures, efine by N = nm)m, 1) where N is the number of structures per unit volume with mass between M an M + M. It shoul be fairly obvious how the mass function is measure in principle: Just select a volume in space an count the number of structures of a given mass containe within it. In practice there will, of course, be complications an I will mention a couple of them when we look at an application of the theory that now follows. Having an analytical expression for the mass function woul be goo. In a classic paper publishe in 1974 Press an Schechter set out to provie us with that. We will now look at what they i. Consier the ensity fluctuation fiel δx) I have omitte to make the time epenence explicit, but it is of course there.) We can filter the fiel on a length scale R, which means throwing away information about the etaile behaviour of δx) on scales smaller than R. The filtering can be accomplishe by convolving δ with a winow function W R x x ): δx; R) = 3 x δx )W R x x ). ) A popular choice of W R, an the one we will use, is the top hat function Wx x ) = 1, x x < R, = 0, otherwise. 3) 1
2 The region of raius R contains a mass M = 4π 3 ρ 0R 3, 4) where ρ 0 is the uniform backgroun ensity. We introuce the notation δx; R) δ M. Press an Schechter assume that the ensity fiel has a Gaussian probability istribution: Pδ M )δ M = ) 1 exp δ M δ πσm σm M, 5) where σ M is the variance of the ensity fiel filtere on a scale R enclosing a mass M. The probability that at some point δ M excees some critical value is now given by P >δc M) = Pδ M )δ M. 6) This probability epens on the filter mass M, an also on the reshift z through the variance σm = σr = 1 kk P π m k, z) W Rk), 7) 0 where P m k, z) is the matter power spectrum an W R k) is the Fourier transform of the top hat filter function. It is also the case that P >δc is proportional to the number of cosmic structures characterize by a ensity perturbation >, regarless of whether they are isolate or containe within enser structures which collapse with them. A value of of great interest to us is 1.686, since we have seen that this is the linear-theory ensity contrast which correspons to virialize structures. To fin the number of regions with mass M which are isolate, in other wors surroune by unerense regions, we must subtract P >δc M + M). This ignores to so-calle clou-in-clou problem: At a given instant some object, which is nonlinear on a scale M, can later be containe within another object on a larger mass scale. In essence we assume that the only objects which exist on a given mass scale are those which have just collapse. Another problem is that we cannot treat unerense regions properly those with δ < 0), which means that half the mass is unaccounte for.
3 Presumably these unerensities will accrete onto overensities. Press an Schechter fixe this problem by brute force: They multiplie their mass function by a factor of. We can now set up an expression for the mass function nm). It is proportional to the ifference of the probabilities alreay mentione, but to convert probabilites to a quantity with units of per volume, we nee to multiply by the backgroun ensity ivie by the mass scale M. In aition we have the artificial factor of to account for unerense regions: nm)m = ρ 0 M [P > M) P >δc M + M)] = ρ 0 M = ρ 0 M P >δc We can use the funamental theorem of calculus x x a M M P >δc M. 8) M ft)t = fx) = x to evaluate the erivative, along with the substitution x = P >δc = = 1 π 1 a πσm exp / σ M e x = 1 π e /σ M = Finally we have nm)m = π x x σm ft)t, 9) δ M σ M ) δc σm : δ M 1 δ c e δ π σm c /σ M. 10) M ρ 0 Mσ M exp σ M ) M. 11) This mass function, with = 1.686, gives us the number ensity of collapse objects per unit mass. The normal proceure is to evaluate nm) by calculating σ M an its erivative from the linear theory matter power spectrum. 3
4 An application of Press-Schechter theory to galaxy clusters Determining nm) from galaxy clusters has been a popular job for cosmologists. One thing that allows one to o is to fin the normalization of the matter power spectrum through the parameter sigma 8, that is, σ R evaluate for R = 8 h 1 Mpc at z = 0. But in orer to o this, one must be able to etermine the mass of galaxy clusters. Since the ominant contribution to the mass of a cluster is in the form of ark matter, this is a non-trivial task. At least three ifferent approaches may be taken: 1. Determine the line-of-sight velocity ispersion, assume isotropy, an use the virial theorem.. Determine the temperature of the X-ray emitting hot gas in the cluster. The temperature is relate to the epth of the graviational potential well in the cluster, so there shoul be a corresponence between temperature an mass. The problem here is to moel the gas accurately. 3. Gravitational lensing. This is in principle the least ambigious an most assumption free metho, but it is time-consuming. Increasingly, though, this is the metho of choice. We will consier a simplifie version of how the cluster abunance can be use to fin σ 8, assuming for simplicity that the backgroun universe is escribe by the Einstein-e Sitter moel. The number ensity of clusters at temperature corresponing to k B T = 7 kev at the present epoch has been measure to be n7 kev, z = 0) = h 3 Mpc 3 kev 1. 1) In simulations of structure formation in an ES backgroun one fins that a cluster with X-ray temperature T = 7.5 kev has a mass within a raius 1.5 h 1 Mpc this is often calle the Abell raius) of M Abell = 1.1 ± 0.) h 1 M. 13) So the simulation gives us useful, but not immeiately applicable information. To compare with the theoretical mass function, we nee to know what virial 4
5 mass a temperature of 7 kev correspons to. Fortunately the simulation can also give us the ensity profile of the cluster, allowing us to convert the Abell mass to the mass within the virial raius. An a scaling relation like ) /3 k B T 0.07 kev = M 1 + z vir) 14) 10 1 h 1 M can be use to scale the mass from 7.5 to 7 kev. The en result turns out to be M vir = 1. ± 0.3) h 1 M. 15) With the top-hat filter this correspons to a sphere of raius R = 3Mvir 4πρ c0 ) 1/3 10 h 1 Mpc, 16) where the critical ensity at the present epoch is ρ c0 = h M Mpc 3. We now nee to relate M to k B T) using equation 14) with z vir = 0: k B T) 0.07 kev = 1 3 M ) /3 M M 17) 10 1 h 1 M which can be rewritten as k B T) = k B T M. 18) 3 M The next thing we nee is the erivative of σ M. In accurate work we woul have to evaluate the erivative numerically from the efinition of σ M in terms of the matter power spectrum. Here we will settle for something simpler. A ecent fit to σ M = σ R for the ES moel with the scalar spectral inex n s = 1 is ) 0.8 R σ R = σ 8. 19) 8 h 1 Mpc This an the relation M = 4πρ c0 R 3 /3 between mass an raius allows us to calculate the erivative we nee: M = σ R R 5 R M
6 = 0.8 σ ) 1 R M R R = 0.8 σ ) R 3M 1 R R = 0.8 σ R 3 M. 0) Inserting this in expression 11) for the mass function gives ) ρ c0 0.8 nm)m = exp M. 1) π M 3 σ R σr To get something we can compare with the observe cluster abunance we use equation 18) an nm) = N ) M to go from nm) to nt): so that N M = N k B T) k B T) M = nt)k BT) M, 3) nt) = 3M nm). 4) k B T Inserting equation 0) an substituting M = M vir = h 1 M, k B T = 7 kev finally leaves us with ) nk B T = 7 kev) = h 3 Mpc 3 kev 1 exp. 5) σ R σr Equating this to the observe value h 3 Mpc 3 kev 1 leaves us with the equation ) x exp x = 0.019, 6) with x = /σ R. This equation must be solve numerically. A simple way of oing this is to rewrite it as ) x x = ln 7)
7 an iterate on your calculator. Make a first guess for x an plug it in on the right han sie. Use the output as the input for the next iteration. When the result changes little from one iteration to the next, a solution has been foun. Starting with the guess x =, I foun the solution x = 3.0 after five iterations. So Since = we get x = σ R=10 h 1 Mpc = 3.. 8) σ R=10 h 1 Mpc = ) Finally, we can now use equation 19) to fin σ 8 : ) σ R=10 h 1 Mpc = 0.53 = σ 8, 30) 8 an we receive the fruit of all our labour in the form of the value σ 8 = ) There are several things we have neglecte to o here. We have not taken the uncertainties in the values of n an M into account an seen how they propagate to an uncertainty in σ 8. An we shoul not, unless we have strong reason to o so, assume an Einstein-e Sitter universe, but rather allow Ω m0 to vary. But still this example illustrates an important application of the mass function an some of the steps that must be taken to relate observations to theory. 7
Implicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More informationLINEAR DIFFERENTIAL EQUATIONS OF ORDER 1. where a(x) and b(x) are functions. Observe that this class of equations includes equations of the form
LINEAR DIFFERENTIAL EQUATIONS OF ORDER 1 We consier ifferential equations of the form y + a()y = b(), (1) y( 0 ) = y 0, where a() an b() are functions. Observe that this class of equations inclues equations
More informationThe Principle of Least Action
Chapter 7. The Principle of Least Action 7.1 Force Methos vs. Energy Methos We have so far stuie two istinct ways of analyzing physics problems: force methos, basically consisting of the application of
More informationLecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations
Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:
More informationensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y
Ph195a lecture notes, 1/3/01 Density operators for spin- 1 ensembles So far in our iscussion of spin- 1 systems, we have restricte our attention to the case of pure states an Hamiltonian evolution. Toay
More informationThe Exact Form and General Integrating Factors
7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily
More informationVectors in two dimensions
Vectors in two imensions Until now, we have been working in one imension only The main reason for this is to become familiar with the main physical ieas like Newton s secon law, without the aitional complication
More informationSurvey Sampling. 1 Design-based Inference. Kosuke Imai Department of Politics, Princeton University. February 19, 2013
Survey Sampling Kosuke Imai Department of Politics, Princeton University February 19, 2013 Survey sampling is one of the most commonly use ata collection methos for social scientists. We begin by escribing
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationGeneral relativity, 7
General relativity, 7 The expaning universe The fact that the vast majority of galaxies have a spectral reshift can be interprete as implying that the universe is expaning. This interpretation stems from
More informationThe total derivative. Chapter Lagrangian and Eulerian approaches
Chapter 5 The total erivative 51 Lagrangian an Eulerian approaches The representation of a flui through scalar or vector fiels means that each physical quantity uner consieration is escribe as a function
More informationq = F If we integrate this equation over all the mass in a star, we have q dm = F (M) F (0)
Astronomy 112: The Physics of Stars Class 4 Notes: Energy an Chemical Balance in Stars In the last class we introuce the iea of hyrostatic balance in stars, an showe that we coul use this concept to erive
More informationMA 2232 Lecture 08 - Review of Log and Exponential Functions and Exponential Growth
MA 2232 Lecture 08 - Review of Log an Exponential Functions an Exponential Growth Friay, February 2, 2018. Objectives: Review log an exponential functions, their erivative an integration formulas. Exponential
More informationThe Three-dimensional Schödinger Equation
The Three-imensional Schöinger Equation R. L. Herman November 7, 016 Schröinger Equation in Spherical Coorinates We seek to solve the Schröinger equation with spherical symmetry using the metho of separation
More information18 EVEN MORE CALCULUS
8 EVEN MORE CALCULUS Chapter 8 Even More Calculus Objectives After stuing this chapter you shoul be able to ifferentiate an integrate basic trigonometric functions; unerstan how to calculate rates of change;
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationOptimization of Geometries by Energy Minimization
Optimization of Geometries by Energy Minimization by Tracy P. Hamilton Department of Chemistry University of Alabama at Birmingham Birmingham, AL 3594-140 hamilton@uab.eu Copyright Tracy P. Hamilton, 1997.
More informationSeparation of Variables
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical
More information23 Implicit differentiation
23 Implicit ifferentiation 23.1 Statement The equation y = x 2 + 3x + 1 expresses a relationship between the quantities x an y. If a value of x is given, then a corresponing value of y is etermine. For
More informationAPPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France
APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation
More informationExperiment 2, Physics 2BL
Experiment 2, Physics 2BL Deuction of Mass Distributions. Last Upate: 2009-05-03 Preparation Before this experiment, we recommen you review or familiarize yourself with the following: Chapters 4-6 in Taylor
More informationLecture 2 Lagrangian formulation of classical mechanics Mechanics
Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,
More informationExercise 1. Exercise 2.
Exercise. Magnitue Galaxy ID Ultraviolet Green Re Infrare A Infrare B 9707296462088.56 5.47 5.4 4.75 4.75 97086278435442.6.33 5.36 4.84 4.58 2255030735995063.64.8 5.88 5.48 5.4 56877420209795 9.52.6.54.08
More informationConstruction of the Electronic Radial Wave Functions and Probability Distributions of Hydrogen-like Systems
Construction of the Electronic Raial Wave Functions an Probability Distributions of Hyrogen-like Systems Thomas S. Kuntzleman, Department of Chemistry Spring Arbor University, Spring Arbor MI 498 tkuntzle@arbor.eu
More informationSolution to the exam in TFY4230 STATISTICAL PHYSICS Wednesday december 1, 2010
NTNU Page of 6 Institutt for fysikk Fakultet for fysikk, informatikk og matematikk This solution consists of 6 pages. Solution to the exam in TFY423 STATISTICAL PHYSICS Wenesay ecember, 2 Problem. Particles
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationDiagonalization of Matrices Dr. E. Jacobs
Diagonalization of Matrices Dr. E. Jacobs One of the very interesting lessons in this course is how certain algebraic techniques can be use to solve ifferential equations. The purpose of these notes is
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationDiophantine Approximations: Examining the Farey Process and its Method on Producing Best Approximations
Diophantine Approximations: Examining the Farey Process an its Metho on Proucing Best Approximations Kelly Bowen Introuction When a person hears the phrase irrational number, one oes not think of anything
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical
More informationMathcad Lecture #5 In-class Worksheet Plotting and Calculus
Mathca Lecture #5 In-class Worksheet Plotting an Calculus At the en of this lecture, you shoul be able to: graph expressions, functions, an matrices of ata format graphs with titles, legens, log scales,
More informationMath 1271 Solutions for Fall 2005 Final Exam
Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly
More informationQuantum mechanical approaches to the virial
Quantum mechanical approaches to the virial S.LeBohec Department of Physics an Astronomy, University of Utah, Salt Lae City, UT 84112, USA Date: June 30 th 2015 In this note, we approach the virial from
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationConservation Laws. Chapter Conservation of Energy
20 Chapter 3 Conservation Laws In orer to check the physical consistency of the above set of equations governing Maxwell-Lorentz electroynamics [(2.10) an (2.12) or (1.65) an (1.68)], we examine the action
More informationLectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs
Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent
More informationfv = ikφ n (11.1) + fu n = y v n iσ iku n + gh n. (11.3) n
Chapter 11 Rossby waves Supplemental reaing: Pelosky 1 (1979), sections 3.1 3 11.1 Shallow water equations When consiering the general problem of linearize oscillations in a static, arbitrarily stratifie
More informationHomework 7 Due 18 November at 6:00 pm
Homework 7 Due 18 November at 6:00 pm 1. Maxwell s Equations Quasi-statics o a An air core, N turn, cylinrical solenoi of length an raius a, carries a current I Io cos t. a. Using Ampere s Law, etermine
More informationA. Incorrect! The letter t does not appear in the expression of the given integral
AP Physics C - Problem Drill 1: The Funamental Theorem of Calculus Question No. 1 of 1 Instruction: (1) Rea the problem statement an answer choices carefully () Work the problems on paper as neee (3) Question
More informationCalculus and optimization
Calculus an optimization These notes essentially correspon to mathematical appenix 2 in the text. 1 Functions of a single variable Now that we have e ne functions we turn our attention to calculus. A function
More informationApplications of First Order Equations
Applications of First Orer Equations Viscous Friction Consier a small mass that has been roppe into a thin vertical tube of viscous flui lie oil. The mass falls, ue to the force of gravity, but falls more
More information1 Heisenberg Representation
1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.
More informationarxiv: v1 [physics.class-ph] 20 Dec 2017
arxiv:1712.07328v1 [physics.class-ph] 20 Dec 2017 Demystifying the constancy of the Ermakov-Lewis invariant for a time epenent oscillator T. Pamanabhan IUCAA, Post Bag 4, Ganeshkhin, Pune - 411 007, Inia.
More informationCalculus of Variations
Calculus of Variations Lagrangian formalism is the main tool of theoretical classical mechanics. Calculus of Variations is a part of Mathematics which Lagrangian formalism is base on. In this section,
More informationQubit channels that achieve capacity with two states
Qubit channels that achieve capacity with two states Dominic W. Berry Department of Physics, The University of Queenslan, Brisbane, Queenslan 4072, Australia Receive 22 December 2004; publishe 22 March
More informationBEYOND THE CONSTRUCTION OF OPTIMAL SWITCHING SURFACES FOR AUTONOMOUS HYBRID SYSTEMS. Mauro Boccadoro Magnus Egerstedt Paolo Valigi Yorai Wardi
BEYOND THE CONSTRUCTION OF OPTIMAL SWITCHING SURFACES FOR AUTONOMOUS HYBRID SYSTEMS Mauro Boccaoro Magnus Egerstet Paolo Valigi Yorai Wari {boccaoro,valigi}@iei.unipg.it Dipartimento i Ingegneria Elettronica
More informationNode Density and Delay in Large-Scale Wireless Networks with Unreliable Links
Noe Density an Delay in Large-Scale Wireless Networks with Unreliable Links Shizhen Zhao, Xinbing Wang Department of Electronic Engineering Shanghai Jiao Tong University, China Email: {shizhenzhao,xwang}@sjtu.eu.cn
More informationChapter 2 Lagrangian Modeling
Chapter 2 Lagrangian Moeling The basic laws of physics are use to moel every system whether it is electrical, mechanical, hyraulic, or any other energy omain. In mechanics, Newton s laws of motion provie
More informationinflow outflow Part I. Regular tasks for MAE598/494 Task 1
MAE 494/598, Fall 2016 Project #1 (Regular tasks = 20 points) Har copy of report is ue at the start of class on the ue ate. The rules on collaboration will be release separately. Please always follow the
More informationIntroduction to Markov Processes
Introuction to Markov Processes Connexions moule m44014 Zzis law Gustav) Meglicki, Jr Office of the VP for Information Technology Iniana University RCS: Section-2.tex,v 1.24 2012/12/21 18:03:08 gustav
More informationTEST 2 (PHY 250) Figure Figure P26.21
TEST 2 (PHY 250) 1. a) Write the efinition (in a full sentence) of electric potential. b) What is a capacitor? c) Relate the electric torque, exerte on a molecule in a uniform electric fiel, with the ipole
More informationChapter 6: Energy-Momentum Tensors
49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel.
More informationThis section outlines the methodology used to calculate the wave load and wave wind load values.
COMPUTERS AND STRUCTURES, INC., JUNE 2014 AUTOMATIC WAVE LOADS TECHNICAL NOTE CALCULATION O WAVE LOAD VALUES This section outlines the methoology use to calculate the wave loa an wave win loa values. Overview
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationLecture 2: Correlated Topic Model
Probabilistic Moels for Unsupervise Learning Spring 203 Lecture 2: Correlate Topic Moel Inference for Correlate Topic Moel Yuan Yuan First of all, let us make some claims about the parameters an variables
More informationAP Calculus AB One Last Mega Review Packet of Stuff. Take the derivative of the following. 1.) 3.) 5.) 7.) Determine the limit of the following.
AP Calculus AB One Last Mega Review Packet of Stuff Name: Date: Block: Take the erivative of the following. 1.) x (sin (5x)).) x (etan(x) ) 3.) x (sin 1 ( x3 )) 4.) x (x3 5x) 4 5.) x ( ex sin(x) ) 6.)
More informationCOUPLING REQUIREMENTS FOR WELL POSED AND STABLE MULTI-PHYSICS PROBLEMS
VI International Conference on Computational Methos for Couple Problems in Science an Engineering COUPLED PROBLEMS 15 B. Schrefler, E. Oñate an M. Paparakakis(Es) COUPLING REQUIREMENTS FOR WELL POSED AND
More information05 The Continuum Limit and the Wave Equation
Utah State University DigitalCommons@USU Founations of Wave Phenomena Physics, Department of 1-1-2004 05 The Continuum Limit an the Wave Equation Charles G. Torre Department of Physics, Utah State University,
More informationarxiv:hep-th/ v1 3 Feb 1993
NBI-HE-9-89 PAR LPTHE 9-49 FTUAM 9-44 November 99 Matrix moel calculations beyon the spherical limit arxiv:hep-th/93004v 3 Feb 993 J. Ambjørn The Niels Bohr Institute Blegamsvej 7, DK-00 Copenhagen Ø,
More informationCalculus of Variations
16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t
More informationIntegration Review. May 11, 2013
Integration Review May 11, 2013 Goals: Review the funamental theorem of calculus. Review u-substitution. Review integration by parts. Do lots of integration eamples. 1 Funamental Theorem of Calculus In
More informationcosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) 2t 1 + t 2 cos x = 1 t2 sin x =
6.4 Integration using tan/ We will revisit the ouble angle ientities: sin = sin/ cos/ = tan/ sec / = tan/ + tan / cos = cos / sin / tan = = tan / sec / tan/ tan /. = tan / + tan / So writing t = tan/ we
More informationPHYS 414 Problem Set 2: Turtles all the way down
PHYS 414 Problem Set 2: Turtles all the way own This problem set explores the common structure of ynamical theories in statistical physics as you pass from one length an time scale to another. Brownian
More informationTMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments
Problem F U L W D g m 3 2 s 2 0 0 0 0 2 kg 0 0 0 0 0 0 Table : Dimension matrix TMA 495 Matematisk moellering Exam Tuesay December 6, 2008 09:00 3:00 Problems an solution with aitional comments The necessary
More information. Using a multinomial model gives us the following equation for P d. , with respect to same length term sequences.
S 63 Lecture 8 2/2/26 Lecturer Lillian Lee Scribes Peter Babinski, Davi Lin Basic Language Moeling Approach I. Special ase of LM-base Approach a. Recap of Formulas an Terms b. Fixing θ? c. About that Multinomial
More information6 Wave equation in spherical polar coordinates
6 Wave equation in spherical polar coorinates We now look at solving problems involving the Laplacian in spherical polar coorinates. The angular epenence of the solutions will be escribe by spherical harmonics.
More informationChapter 4. Electrostatics of Macroscopic Media
Chapter 4. Electrostatics of Macroscopic Meia 4.1 Multipole Expansion Approximate potentials at large istances 3 x' x' (x') x x' x x Fig 4.1 We consier the potential in the far-fiel region (see Fig. 4.1
More informationELECTRON DIFFRACTION
ELECTRON DIFFRACTION Electrons : wave or quanta? Measurement of wavelength an momentum of electrons. Introuction Electrons isplay both wave an particle properties. What is the relationship between the
More informationA SIMPLE ENGINEERING MODEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PRODUCTS
International Journal on Engineering Performance-Base Fire Coes, Volume 4, Number 3, p.95-3, A SIMPLE ENGINEERING MOEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PROCTS V. Novozhilov School of Mechanical
More informationarxiv: v1 [physics.flu-dyn] 8 May 2014
Energetics of a flui uner the Boussinesq approximation arxiv:1405.1921v1 [physics.flu-yn] 8 May 2014 Kiyoshi Maruyama Department of Earth an Ocean Sciences, National Defense Acaemy, Yokosuka, Kanagawa
More informationd dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1
Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of
More informationProblem Sheet 2: Eigenvalues and eigenvectors and their use in solving linear ODEs
Problem Sheet 2: Eigenvalues an eigenvectors an their use in solving linear ODEs If you fin any typos/errors in this problem sheet please email jk28@icacuk The material in this problem sheet is not examinable
More informationSection 2.7 Derivatives of powers of functions
Section 2.7 Derivatives of powers of functions (3/19/08) Overview: In this section we iscuss the Chain Rule formula for the erivatives of composite functions that are forme by taking powers of other functions.
More informationSummary: Differentiation
Techniques of Differentiation. Inverse Trigonometric functions The basic formulas (available in MF5 are: Summary: Differentiation ( sin ( cos The basic formula can be generalize as follows: Note: ( sin
More informationFinal Exam: Sat 12 Dec 2009, 09:00-12:00
MATH 1013 SECTIONS A: Professor Szeptycki APPLIED CALCULUS I, FALL 009 B: Professor Toms C: Professor Szeto NAME: STUDENT #: SECTION: No ai (e.g. calculator, written notes) is allowe. Final Exam: Sat 1
More informationA Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential
Avances in Applie Mathematics an Mechanics Av. Appl. Math. Mech. Vol. 1 No. 4 pp. 573-580 DOI: 10.4208/aamm.09-m0946 August 2009 A Note on Exact Solutions to Linear Differential Equations by the Matrix
More informationConvergence of Random Walks
Chapter 16 Convergence of Ranom Walks This lecture examines the convergence of ranom walks to the Wiener process. This is very important both physically an statistically, an illustrates the utility of
More informationImplicit Differentiation. Lecture 16.
Implicit Differentiation. Lecture 16. We are use to working only with functions that are efine explicitly. That is, ones like f(x) = 5x 3 + 7x x 2 + 1 or s(t) = e t5 3, in which the function is escribe
More informationEntanglement is not very useful for estimating multiple phases
PHYSICAL REVIEW A 70, 032310 (2004) Entanglement is not very useful for estimating multiple phases Manuel A. Ballester* Department of Mathematics, University of Utrecht, Box 80010, 3508 TA Utrecht, The
More informationNOTES ON EULER-BOOLE SUMMATION (1) f (l 1) (n) f (l 1) (m) + ( 1)k 1 k! B k (y) f (k) (y) dy,
NOTES ON EULER-BOOLE SUMMATION JONATHAN M BORWEIN, NEIL J CALKIN, AND DANTE MANNA Abstract We stuy a connection between Euler-MacLaurin Summation an Boole Summation suggeste in an AMM note from 196, which
More informationKramers Relation. Douglas H. Laurence. Department of Physical Sciences, Broward College, Davie, FL 33314
Kramers Relation Douglas H. Laurence Department of Physical Sciences, Browar College, Davie, FL 333 Introuction Kramers relation, name after the Dutch physicist Hans Kramers, is a relationship between
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Friday 8 June 2001 1.30 to 4.30 PAPER 41 PHYSICAL COSMOLOGY Answer any THREE questions. The questions carry equal weight. You may not start to read the questions printed on
More informationQuantum Search on the Spatial Grid
Quantum Search on the Spatial Gri Matthew D. Falk MIT 2012, 550 Memorial Drive, Cambrige, MA 02139 (Date: December 11, 2012) This paper explores Quantum Search on the two imensional spatial gri. Recent
More informationConvective heat transfer
CHAPTER VIII Convective heat transfer The previous two chapters on issipative fluis were evote to flows ominate either by viscous effects (Chap. VI) or by convective motion (Chap. VII). In either case,
More informationIntroduction to variational calculus: Lecture notes 1
October 10, 2006 Introuction to variational calculus: Lecture notes 1 Ewin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE-106 91 Stockholm, Sween Abstract I give an informal summary of variational
More informationarxiv:physics/ v2 [physics.ed-ph] 23 Sep 2003
Mass reistribution in variable mass systems Célia A. e Sousa an Vítor H. Rorigues Departamento e Física a Universiae e Coimbra, P-3004-516 Coimbra, Portugal arxiv:physics/0211075v2 [physics.e-ph] 23 Sep
More informationA simple model for the small-strain behaviour of soils
A simple moel for the small-strain behaviour of soils José Jorge Naer Department of Structural an Geotechnical ngineering, Polytechnic School, University of São Paulo 05508-900, São Paulo, Brazil, e-mail:
More informationTIME-DELAY ESTIMATION USING FARROW-BASED FRACTIONAL-DELAY FIR FILTERS: FILTER APPROXIMATION VS. ESTIMATION ERRORS
TIME-DEAY ESTIMATION USING FARROW-BASED FRACTIONA-DEAY FIR FITERS: FITER APPROXIMATION VS. ESTIMATION ERRORS Mattias Olsson, Håkan Johansson, an Per öwenborg Div. of Electronic Systems, Dept. of Electrical
More informationIMPLICIT DIFFERENTIATION
IMPLICIT DIFFERENTIATION CALCULUS 3 INU0115/515 (MATHS 2) Dr Arian Jannetta MIMA CMath FRAS Implicit Differentiation 1/ 11 Arian Jannetta Explicit an implicit functions Explicit functions An explicit function
More informationNoether s theorem applied to classical electrodynamics
Noether s theorem applie to classical electroynamics Thomas B. Mieling Faculty of Physics, University of ienna Boltzmanngasse 5, 090 ienna, Austria (Date: November 8, 207) The consequences of gauge invariance
More informationSYDE 112, LECTURE 1: Review & Antidifferentiation
SYDE 112, LECTURE 1: Review & Antiifferentiation 1 Course Information For a etaile breakown of the course content an available resources, see the Course Outline. Other relevant information for this section
More informationState-Space Model for a Multi-Machine System
State-Space Moel for a Multi-Machine System These notes parallel section.4 in the text. We are ealing with classically moele machines (IEEE Type.), constant impeance loas, an a network reuce to its internal
More informationThe Standard Atmosphere. Dr Andrew French
The Stanar Atmosphere Dr Anrew French 1 The International Stanar Atmosphere (ISA) is an iealize moel of the variation of average air pressure an temperature with altitue. Assumptions: The atmosphere consists
More informationNuclear Physics and Astrophysics
Nuclear Physics an Astrophysics PHY-302 Dr. E. Rizvi Lecture 2 - Introuction Notation Nuclies A Nuclie is a particular an is esignate by the following notation: A CN = Atomic Number (no. of Protons) A
More informationAssignment 1. g i (x 1,..., x n ) dx i = 0. i=1
Assignment 1 Golstein 1.4 The equations of motion for the rolling isk are special cases of general linear ifferential equations of constraint of the form g i (x 1,..., x n x i = 0. i=1 A constraint conition
More information