This document summarizes Haar wavelet transforms and Daubechies wavelet transforms, including their forward and inverse transforms. It provides MATLAB code implementations for 1D and 2D transforms using Haar, Daub4, Daub6, Daub5/3 and Daub4 wavelets. The transforms can be applied for multiple levels (layers) to decompose signals into different frequency subbands.
This MATLAB code solves for the displacements, reactions, and member forces of beams with 'N' spans using the stiffness method. It begins by inputting node coordinates, member properties, boundary conditions, and loads. It then calculates member lengths, stiffness matrices, and assembles the total stiffness matrix. The code solves for displacements, reacts displacements back to member forces, and outputs the results.
This document provides formulas and properties for the Laplace transform and its inverse. It lists 10 formulas for common functions and their Laplace transforms, including 1, t, tn, sinat, cosat, sinhat, and coshat. It also lists 7 properties of the Laplace transform, such as how it is affected by scaling, derivatives, division by t, and multiplication by t. Finally, it lists 6 properties of the inverse Laplace transform, such as how it is affected by derivatives, division by s, and the convolution theorem.
This document contains MATLAB code for analyzing the time response (step response, impulse response, and ramp response) of various transfer functions and control systems. It includes examples of open-loop and closed-loop transfer functions, uses the step, impulse, and lsim functions to obtain responses, and plots the results. Exercises are provided to analyze transient and steady-state response characteristics for different systems.
Hyperbola as an-example-learning-shifts-on-internetDanut Dragoi
This document discusses properties of hyperbolas using Dandelin spheres. It shows that for any point P on a hyperbola, the distance PF to the focus F is equal to the distance PC to the directrix. This property does not change as P moves along the hyperbola. The document also derives formulas for the external and internal tangents of two circles based on their radii and distance between centers. These concepts are then applied to define the eccentricity of a hyperbola in terms of the radii of the Dandelin spheres. The method shown can also be used to describe other conic sections like parabolas.
Control system concepts by using matlabCharltonInao1
This document introduces control system concepts and how they can be analyzed using MATLAB. It discusses open and closed loop systems, Laplace transforms, state variable approaches, and MATLAB commands to analyze systems. Examples are provided on determining transfer functions, computing step and frequency responses, plotting root loci and Bode diagrams, and performing time and frequency domain analyses. Block diagrams can be reduced and overall transfer functions obtained through series and parallel combinations.
The document contains MATLAB code for digital signal processing programs including:
1) Bandpass filters, Kaiser window functions, time domain windows, DFT of square waves with different duties, notch filters, and resonators.
2) Comb filters and the Welch method for calculating the power spectral density of a noisy signal.
3) A discrete Fourier transform program that calculates the forward and inverse DFT using twiddle factors.
MATLAB programs Power System Simulation lab (Electrical Engineer)Mathankumar S
The document contains MATLAB code for calculating line constants (inductance L and capacitance C) for overhead transmission lines with different configurations (single-circuit, single-circuit with multiple subconductors, and double-circuit). It requests user input of various line parameters and geometric mean distances and then calculates L and C values. Additional code calculates the network bus admittance matrix and transmission line losses.
This MATLAB code solves for the displacements, reactions, and member forces of beams with 'N' spans using the stiffness method. It begins by inputting node coordinates, member properties, boundary conditions, and loads. It then calculates member lengths, stiffness matrices, and assembles the total stiffness matrix. The code solves for displacements, reacts displacements back to member forces, and outputs the results.
This document provides formulas and properties for the Laplace transform and its inverse. It lists 10 formulas for common functions and their Laplace transforms, including 1, t, tn, sinat, cosat, sinhat, and coshat. It also lists 7 properties of the Laplace transform, such as how it is affected by scaling, derivatives, division by t, and multiplication by t. Finally, it lists 6 properties of the inverse Laplace transform, such as how it is affected by derivatives, division by s, and the convolution theorem.
This document contains MATLAB code for analyzing the time response (step response, impulse response, and ramp response) of various transfer functions and control systems. It includes examples of open-loop and closed-loop transfer functions, uses the step, impulse, and lsim functions to obtain responses, and plots the results. Exercises are provided to analyze transient and steady-state response characteristics for different systems.
Hyperbola as an-example-learning-shifts-on-internetDanut Dragoi
This document discusses properties of hyperbolas using Dandelin spheres. It shows that for any point P on a hyperbola, the distance PF to the focus F is equal to the distance PC to the directrix. This property does not change as P moves along the hyperbola. The document also derives formulas for the external and internal tangents of two circles based on their radii and distance between centers. These concepts are then applied to define the eccentricity of a hyperbola in terms of the radii of the Dandelin spheres. The method shown can also be used to describe other conic sections like parabolas.
Control system concepts by using matlabCharltonInao1
This document introduces control system concepts and how they can be analyzed using MATLAB. It discusses open and closed loop systems, Laplace transforms, state variable approaches, and MATLAB commands to analyze systems. Examples are provided on determining transfer functions, computing step and frequency responses, plotting root loci and Bode diagrams, and performing time and frequency domain analyses. Block diagrams can be reduced and overall transfer functions obtained through series and parallel combinations.
The document contains MATLAB code for digital signal processing programs including:
1) Bandpass filters, Kaiser window functions, time domain windows, DFT of square waves with different duties, notch filters, and resonators.
2) Comb filters and the Welch method for calculating the power spectral density of a noisy signal.
3) A discrete Fourier transform program that calculates the forward and inverse DFT using twiddle factors.
MATLAB programs Power System Simulation lab (Electrical Engineer)Mathankumar S
The document contains MATLAB code for calculating line constants (inductance L and capacitance C) for overhead transmission lines with different configurations (single-circuit, single-circuit with multiple subconductors, and double-circuit). It requests user input of various line parameters and geometric mean distances and then calculates L and C values. Additional code calculates the network bus admittance matrix and transmission line losses.
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively. poles and zeros define the performance of a system.
Power Systems Engineering - Matlab programs for Power system Simulation Lab -...Mathankumar S
This MATLAB code calculates line constants and impedances for single and double circuit transmission lines. It inputs parameters like conductor spacing, diameter, and distances between conductors and calculates the series inductance L and shunt capacitance C per unit length. It also forms the bus admittance matrix Ybus for a power system network and calculates real and reactive power flows and losses for a two-bus system.
This document discusses analyzing control systems using MATLAB. It contains two tasks:
1) Analyze five transfer functions to find pole-zero locations, rise times, and settling times. MATLAB is used to calculate these values.
2) Analyze a circuit transfer function to find its pole-zero location, rise time, and settling time in response to a step input voltage. The circuit equation is derived and MATLAB is used to calculate the results.
1) The document describes an experiment on amplitude modulation where different signals are generated and plotted in the time and frequency domains.
2) Signals like the message signal, carrier signal, modulated signal, and integrated message signal are generated and their representations are shown.
3) The experiment compares amplitude modulation (AM) and frequency modulation (FM) by generating AM and FM signals and observing the differences between their time domain and frequency domain representations.
The document contains code that calculates various parameters for a satellite mission at different altitudes (500km, 600km, 700km). It determines the solar panel area, battery mass, and power input over time needed to support a 300W payload. The solar panel area needed is 4.692 sqm for all altitudes in Part A, and smaller individual values for each altitude are calculated in Part B (1.8628 sqm at 500km, 1.8329 sqm at 600km, 1.8069 sqm at 700km). Plots of battery mass and solar panel area over the mission lifetime are generated, as well as power input over time.
The document describes a simulation of a PMSM motor control system for electric power steering controllers. It includes:
1) A system block diagram showing the main components of an EPS system including a PMSM motor, steering mechanism, and EPS control unit.
2) Simulink models of the key system elements - the PMSM motor, position sensor, current sensing, PI controller, and inverse Park and space vector modulation models.
3) Simulation and experimental results showing the effects of position sensor resolution and current sensing errors on torque ripple, and validating the simulated d-axis step response with experimental measurements.
4) A conclusion that the complete PMSM drive model and experimental validation can
Abstract
キーの値による範囲検索が可能なキー順序保存型構造化オーバレイネットワークは多くの応用があり,重要性が高い.本研究では,新しいキー順序保存型構造化オーバレイネットワークSuzakuを提案する.Suzakuは,(1)Churn時でも最大検索ホップ数がlog_2 n程度に収まる(nはノード数),(2)キーが大小どちらの方向でも近傍ノードの検索は高速に行える,(3)構造は単純で実装が容易,といった特徴を備える.本稿ではSuzakuの詳細について述べ,シミュレーションによって既存のChord#およびSkip Graphと比較する.
A ``key-order preserving structured overlay network,'' which enables
range queries, has various applications and thus be important. In
this study, we propose a novel key-order preserving structured
overlay network ``Suzaku,'' which has the following properties: (1)
maximum lookup hops is almost log_2 n even in churn situations,
where $n$ is the number of nodes, (2) neighbor search is fast
regardless of the direction of their keys, (3) the structure is
simple and easy to implement. In this paper, we describe the
principles and detailed algorithm of Suzaku. We also show
simulation results comparing Suzaku with existing Chord# and Skip
Graph.
This document contains Matlab code that is modeling the effective material properties of a composite material made up of a matrix and fiber materials. It defines material properties, volume fractions, orientations, and calculates effective properties like elastic modulus, thermal expansion coefficient, and Poisson's ratio for the composite over a range of fiber volume fractions and orientations. It then plots the results.
This document discusses methods for calculating and comparing investment returns and wealth over time. It defines formulas for calculating periodic returns, cumulative returns, expected returns, variance of returns, and wealth as an exponential function of cumulative returns. It also discusses how to calculate ratios comparing the terminal wealth of different investments and percentile distributions of terminal wealth.
The document summarizes digital modulation and detection theory. It discusses the error probability analysis for various memoryless modulation schemes including PAM, PSK, and QAM. Key points covered include:
1) The symbol error probability expressions for M-ary PAM and PSK based on the minimum distance between signal points and the Q-function.
2) The distribution of the phase of the received signal for PSK modulation and how this relates to the error probability.
3) How QAM signal constellations are represented and the dominance of error probability by the minimum distance between signal points.
4) Expressions for the symbol error probability of rectangular QAM schemes with even
The Laplace transform is defined as the integral of a function F(t) multiplied by e−st from 0 to infinity. This transform L{F(t)} provides a new function f(s) of the parameter s. The Laplace transform can be used to find the transforms of elementary functions like ekt and sin(kt). For a function F(t) to have a Laplace transform, it must be sectionally continuous over intervals and of exponential order as t approaches infinity. The Laplace transform of derivatives of F(t) can be found from the transform of F(t) itself using formulas involving s. Periodic functions and the derivatives of transforms can also be analyzed using the Laplace transform.
Hilbert transforms are useful for bandpass signal processing like ultrasound. They provide a mathematical basis for representing bandpass signals. The Hilbert transform allows easy determination of a signal's envelope. It may also reduce required ADC sampling rates. The Hilbert transform is defined as the convolution of the signal with 1/πt. In the frequency domain, its transfer function is -j when f>0, +j when f<0, and 0 when f=0. Applying the Hilbert transform to a bandpass signal yields its analytic signal, from which the original signal's envelope can be extracted.
#Import standard math functions from math import import .docxgertrudebellgrove
#Import standard math functions
from math import *
import numpy as np
import matplotlib.pyplot as plt
#Prandtl-Meyer expansion angle function:
def nu(M,y):
return sqrt((y+1)/(y-1))*atan(sqrt((y-1)*(M**2-1)/(y+1)))-
atan(sqrt(M**2-1))
#Prandtl-Meyer Expansion angle for numeric extraction of Mach
number
def nu2(M,nu,y):
return sqrt((y+1)/(y-1))*atan(sqrt((y-1)*(M**2-1)/(y+1)))-
atan(sqrt(M**2-1))-nu
#Derivative of Prandtl-Meyer expansion angle function
def nudif(M, y = 1.25):
return sqrt(M**2-1)/(M*(1+(y-1)*M**2/2))
#Stagnation pressure from freestream pressure.
def Pstag(Pe,M,y):
return Pe*(1+(y-1)/2*M**2)**(y/(y-1))
#Stagnation pressure ratio across a shockwave with P01
calculated from Pstat and M.
def Pstagratio(M,P02,P1,y):
P01 = Pstag(P1,M,y)
f1 = 2/((y+1)*(y*M**2-(y-1)/2)**(1/(y-1)))
f2 = (((y+1)/2*M)**2/(1+(y-1)/2*M**2))**(y/(y-1))
return f1*f2-P02/P01
#Stagnation pressure ratio across a shockwave with P01
calculated from Pstat and M.
def Pstagshock(M,P1,y):#returns P02
P01 = Pstag(P1,M,y)
f1 = 2/((y+1)*(y*M**2-(y-1)/2)**(1/(y-1)))
f2 = (((y+1)/2*M)**2/(1+(y-1)/2*M**2))**(y/(y-1))
return f1*f2*P01
#Pressure across an oblique shockwave as a function of upstream
pressure, gamma, and Mach.
def Pstatshock(P,M,y,theta):
beta = shockangle(M,y,theta)[1]
return P*(1+2*y/(y+1)*((M*sin(beta))**2-1))
#Numerical derivative for this particular function with only x
(which is M) varying.
def centraldiff(func, x, A, Ast, y, h = 0.0000001):
return (func(x+h,A,Ast,y)-func(x-h,A,Ast,y))/(2*h)
#Newton's method for four-term fuction.
def Newts(func, fin,y,A,Ast, err, iMax = 100000):
x1 = fin
it = 0
ea = np.Infinity
while ea > err and it < iMax:
x0 = x1
x1 = x0 - func(x0,A,Ast,y)/
(centraldiff(func,x0,A,Ast,y))
#x1 = x0 - func(x0,A,Ast,y)/(RPdif(x0,y))
it +=1
ea = abs((x0-x1)/x1)
return x1
#Newton's method
def Newt(Mguess,nu,gamma,err,iMax=1000):
x1 = Mguess
it = 0
ea = 100000
while ea > err and it < iMax:
x0 = x1
x1 = x0 - nu2(x0,nu,gamma)/nudif(x0,gamma)
it +=1
ea = abs((x0-x1)/x1)
return x1
#Pressure across expansion fan
def Pexpand(M1, M2, P1, y):
top = 1+(y-1)/2*M1**2
bot = 1+(y-1)/2*M2**2
return P1*(top/bot)**(y/(y-1))
#Explicit solution for beta angle. Requires theta input in
degrees.
def shockangle(M,y,thd):
th = thd*pi/180
lt1 = (M**2-1)**2
lt2 = (1+(y-1)/2*M**2)
lt3 = (1+(y+1)/2*M**2)
l = (lt1-3*lt2*lt3*tan(th)**2)**.5
xt1 = (M**2-1)**3
xt2 = lt2
xt3 = (1+(y-1)/2*M**2+(y+1)/4*M**4)
x = (xt1-9*xt2*xt3*tan(th)**2)/l**3
tanb = []
for i in range(2):
tt1 = (M**2-1)
tt2 = 2*l*cos((4*pi*float(i)+acos(x))/3)
tt3 = 3*(1+(y-1)/2*M**2)*tan(th)
tanb.append((tt1+tt2)/tt3)
return (atan(tanb[0]),atan(tanb[1]))
#Ma ...
This document describes a numerical model for solving the steady-state thickness profile of a glacier. It defines parameters, numerical solution techniques using ODE solvers, and plots the steady-state solution along with inner and outer solutions. The model solves the boundary value problem for glacier thickness given an ice flux defined by a power law.
This document provides chapter summaries and example problems from a solutions manual for a textbook on electromagnetism. It includes 20 chapters that cover topics like vector calculus, electrostatics, magnetostatics, and Maxwell's equations. The document also provides notes on mathematical expressions and references for additional resources related to the textbook.
This document provides an overview of Laplace transforms. Key points include:
- Laplace transforms convert differential equations from the time domain to the algebraic s-domain, making them easier to solve. The process involves taking the Laplace transform of each term in the differential equation.
- Common Laplace transforms of functions are presented. Properties such as linearity, differentiation, integration, and convolution are also covered.
- Partial fraction expansion is used to break complex fractions in the s-domain into simpler forms with individual terms that can be inverted using tables of transforms.
- Solving differential equations using Laplace transforms follows a standard process of taking the Laplace transform of each term, rewriting the equation in the s-domain, solving
This document provides a table of commonly used Laplace transform pairs. There are 37 entries in the table that list various functions of t and their corresponding Laplace transforms F(s). Each entry is of the form f(t) = L-1{F(s)}, which relates a function f(t) to its Laplace transform F(s). Notes are provided to explain details like hyperbolic functions, the Gamma function, and limitations of the table.
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively. poles and zeros define the performance of a system.
Power Systems Engineering - Matlab programs for Power system Simulation Lab -...Mathankumar S
This MATLAB code calculates line constants and impedances for single and double circuit transmission lines. It inputs parameters like conductor spacing, diameter, and distances between conductors and calculates the series inductance L and shunt capacitance C per unit length. It also forms the bus admittance matrix Ybus for a power system network and calculates real and reactive power flows and losses for a two-bus system.
This document discusses analyzing control systems using MATLAB. It contains two tasks:
1) Analyze five transfer functions to find pole-zero locations, rise times, and settling times. MATLAB is used to calculate these values.
2) Analyze a circuit transfer function to find its pole-zero location, rise time, and settling time in response to a step input voltage. The circuit equation is derived and MATLAB is used to calculate the results.
1) The document describes an experiment on amplitude modulation where different signals are generated and plotted in the time and frequency domains.
2) Signals like the message signal, carrier signal, modulated signal, and integrated message signal are generated and their representations are shown.
3) The experiment compares amplitude modulation (AM) and frequency modulation (FM) by generating AM and FM signals and observing the differences between their time domain and frequency domain representations.
The document contains code that calculates various parameters for a satellite mission at different altitudes (500km, 600km, 700km). It determines the solar panel area, battery mass, and power input over time needed to support a 300W payload. The solar panel area needed is 4.692 sqm for all altitudes in Part A, and smaller individual values for each altitude are calculated in Part B (1.8628 sqm at 500km, 1.8329 sqm at 600km, 1.8069 sqm at 700km). Plots of battery mass and solar panel area over the mission lifetime are generated, as well as power input over time.
The document describes a simulation of a PMSM motor control system for electric power steering controllers. It includes:
1) A system block diagram showing the main components of an EPS system including a PMSM motor, steering mechanism, and EPS control unit.
2) Simulink models of the key system elements - the PMSM motor, position sensor, current sensing, PI controller, and inverse Park and space vector modulation models.
3) Simulation and experimental results showing the effects of position sensor resolution and current sensing errors on torque ripple, and validating the simulated d-axis step response with experimental measurements.
4) A conclusion that the complete PMSM drive model and experimental validation can
Abstract
キーの値による範囲検索が可能なキー順序保存型構造化オーバレイネットワークは多くの応用があり,重要性が高い.本研究では,新しいキー順序保存型構造化オーバレイネットワークSuzakuを提案する.Suzakuは,(1)Churn時でも最大検索ホップ数がlog_2 n程度に収まる(nはノード数),(2)キーが大小どちらの方向でも近傍ノードの検索は高速に行える,(3)構造は単純で実装が容易,といった特徴を備える.本稿ではSuzakuの詳細について述べ,シミュレーションによって既存のChord#およびSkip Graphと比較する.
A ``key-order preserving structured overlay network,'' which enables
range queries, has various applications and thus be important. In
this study, we propose a novel key-order preserving structured
overlay network ``Suzaku,'' which has the following properties: (1)
maximum lookup hops is almost log_2 n even in churn situations,
where $n$ is the number of nodes, (2) neighbor search is fast
regardless of the direction of their keys, (3) the structure is
simple and easy to implement. In this paper, we describe the
principles and detailed algorithm of Suzaku. We also show
simulation results comparing Suzaku with existing Chord# and Skip
Graph.
This document contains Matlab code that is modeling the effective material properties of a composite material made up of a matrix and fiber materials. It defines material properties, volume fractions, orientations, and calculates effective properties like elastic modulus, thermal expansion coefficient, and Poisson's ratio for the composite over a range of fiber volume fractions and orientations. It then plots the results.
This document discusses methods for calculating and comparing investment returns and wealth over time. It defines formulas for calculating periodic returns, cumulative returns, expected returns, variance of returns, and wealth as an exponential function of cumulative returns. It also discusses how to calculate ratios comparing the terminal wealth of different investments and percentile distributions of terminal wealth.
The document summarizes digital modulation and detection theory. It discusses the error probability analysis for various memoryless modulation schemes including PAM, PSK, and QAM. Key points covered include:
1) The symbol error probability expressions for M-ary PAM and PSK based on the minimum distance between signal points and the Q-function.
2) The distribution of the phase of the received signal for PSK modulation and how this relates to the error probability.
3) How QAM signal constellations are represented and the dominance of error probability by the minimum distance between signal points.
4) Expressions for the symbol error probability of rectangular QAM schemes with even
The Laplace transform is defined as the integral of a function F(t) multiplied by e−st from 0 to infinity. This transform L{F(t)} provides a new function f(s) of the parameter s. The Laplace transform can be used to find the transforms of elementary functions like ekt and sin(kt). For a function F(t) to have a Laplace transform, it must be sectionally continuous over intervals and of exponential order as t approaches infinity. The Laplace transform of derivatives of F(t) can be found from the transform of F(t) itself using formulas involving s. Periodic functions and the derivatives of transforms can also be analyzed using the Laplace transform.
Hilbert transforms are useful for bandpass signal processing like ultrasound. They provide a mathematical basis for representing bandpass signals. The Hilbert transform allows easy determination of a signal's envelope. It may also reduce required ADC sampling rates. The Hilbert transform is defined as the convolution of the signal with 1/πt. In the frequency domain, its transfer function is -j when f>0, +j when f<0, and 0 when f=0. Applying the Hilbert transform to a bandpass signal yields its analytic signal, from which the original signal's envelope can be extracted.
#Import standard math functions from math import import .docxgertrudebellgrove
#Import standard math functions
from math import *
import numpy as np
import matplotlib.pyplot as plt
#Prandtl-Meyer expansion angle function:
def nu(M,y):
return sqrt((y+1)/(y-1))*atan(sqrt((y-1)*(M**2-1)/(y+1)))-
atan(sqrt(M**2-1))
#Prandtl-Meyer Expansion angle for numeric extraction of Mach
number
def nu2(M,nu,y):
return sqrt((y+1)/(y-1))*atan(sqrt((y-1)*(M**2-1)/(y+1)))-
atan(sqrt(M**2-1))-nu
#Derivative of Prandtl-Meyer expansion angle function
def nudif(M, y = 1.25):
return sqrt(M**2-1)/(M*(1+(y-1)*M**2/2))
#Stagnation pressure from freestream pressure.
def Pstag(Pe,M,y):
return Pe*(1+(y-1)/2*M**2)**(y/(y-1))
#Stagnation pressure ratio across a shockwave with P01
calculated from Pstat and M.
def Pstagratio(M,P02,P1,y):
P01 = Pstag(P1,M,y)
f1 = 2/((y+1)*(y*M**2-(y-1)/2)**(1/(y-1)))
f2 = (((y+1)/2*M)**2/(1+(y-1)/2*M**2))**(y/(y-1))
return f1*f2-P02/P01
#Stagnation pressure ratio across a shockwave with P01
calculated from Pstat and M.
def Pstagshock(M,P1,y):#returns P02
P01 = Pstag(P1,M,y)
f1 = 2/((y+1)*(y*M**2-(y-1)/2)**(1/(y-1)))
f2 = (((y+1)/2*M)**2/(1+(y-1)/2*M**2))**(y/(y-1))
return f1*f2*P01
#Pressure across an oblique shockwave as a function of upstream
pressure, gamma, and Mach.
def Pstatshock(P,M,y,theta):
beta = shockangle(M,y,theta)[1]
return P*(1+2*y/(y+1)*((M*sin(beta))**2-1))
#Numerical derivative for this particular function with only x
(which is M) varying.
def centraldiff(func, x, A, Ast, y, h = 0.0000001):
return (func(x+h,A,Ast,y)-func(x-h,A,Ast,y))/(2*h)
#Newton's method for four-term fuction.
def Newts(func, fin,y,A,Ast, err, iMax = 100000):
x1 = fin
it = 0
ea = np.Infinity
while ea > err and it < iMax:
x0 = x1
x1 = x0 - func(x0,A,Ast,y)/
(centraldiff(func,x0,A,Ast,y))
#x1 = x0 - func(x0,A,Ast,y)/(RPdif(x0,y))
it +=1
ea = abs((x0-x1)/x1)
return x1
#Newton's method
def Newt(Mguess,nu,gamma,err,iMax=1000):
x1 = Mguess
it = 0
ea = 100000
while ea > err and it < iMax:
x0 = x1
x1 = x0 - nu2(x0,nu,gamma)/nudif(x0,gamma)
it +=1
ea = abs((x0-x1)/x1)
return x1
#Pressure across expansion fan
def Pexpand(M1, M2, P1, y):
top = 1+(y-1)/2*M1**2
bot = 1+(y-1)/2*M2**2
return P1*(top/bot)**(y/(y-1))
#Explicit solution for beta angle. Requires theta input in
degrees.
def shockangle(M,y,thd):
th = thd*pi/180
lt1 = (M**2-1)**2
lt2 = (1+(y-1)/2*M**2)
lt3 = (1+(y+1)/2*M**2)
l = (lt1-3*lt2*lt3*tan(th)**2)**.5
xt1 = (M**2-1)**3
xt2 = lt2
xt3 = (1+(y-1)/2*M**2+(y+1)/4*M**4)
x = (xt1-9*xt2*xt3*tan(th)**2)/l**3
tanb = []
for i in range(2):
tt1 = (M**2-1)
tt2 = 2*l*cos((4*pi*float(i)+acos(x))/3)
tt3 = 3*(1+(y-1)/2*M**2)*tan(th)
tanb.append((tt1+tt2)/tt3)
return (atan(tanb[0]),atan(tanb[1]))
#Ma ...
This document describes a numerical model for solving the steady-state thickness profile of a glacier. It defines parameters, numerical solution techniques using ODE solvers, and plots the steady-state solution along with inner and outer solutions. The model solves the boundary value problem for glacier thickness given an ice flux defined by a power law.
This document provides chapter summaries and example problems from a solutions manual for a textbook on electromagnetism. It includes 20 chapters that cover topics like vector calculus, electrostatics, magnetostatics, and Maxwell's equations. The document also provides notes on mathematical expressions and references for additional resources related to the textbook.
This document provides an overview of Laplace transforms. Key points include:
- Laplace transforms convert differential equations from the time domain to the algebraic s-domain, making them easier to solve. The process involves taking the Laplace transform of each term in the differential equation.
- Common Laplace transforms of functions are presented. Properties such as linearity, differentiation, integration, and convolution are also covered.
- Partial fraction expansion is used to break complex fractions in the s-domain into simpler forms with individual terms that can be inverted using tables of transforms.
- Solving differential equations using Laplace transforms follows a standard process of taking the Laplace transform of each term, rewriting the equation in the s-domain, solving
This document provides a table of commonly used Laplace transform pairs. There are 37 entries in the table that list various functions of t and their corresponding Laplace transforms F(s). Each entry is of the form f(t) = L-1{F(s)}, which relates a function f(t) to its Laplace transform F(s). Notes are provided to explain details like hyperbolic functions, the Gamma function, and limitations of the table.
This document provides a table of commonly used Laplace transform pairs. There are 37 entries in the table that list various functions of t and their corresponding Laplace transforms F(s). Each entry is of the form f(t) = L-1{F(s)}, which relates a function of time f(t) to its Laplace transform F(s). Notes are provided to explain concepts like hyperbolic functions and the Gamma function used in some of the entries.
This document provides a table of commonly used Laplace transform pairs. There are 38 entries in total, each providing the Laplace transform of a specific function. For example, entry 1 gives the Laplace transform of a constant function f(t) as F(s)=1/s. The table also includes brief explanatory notes about properties of Laplace transforms and related functions like the Gamma function.
Trial penang 2014 spm matematik tambahan k1 k2 skema [scan]Cikgu Pejal
1. This document contains the mark scheme for the Additional Mathematics Paper 1 and Paper 2 for the 2014 SPM examination in Malaysia. It provides the solutions and marks awarded for each question.
2. The mark scheme is divided into sections for Paper 1 and Paper 2. For each question, it lists the sub-marks and total marks, along with the key steps and solutions required to earn the marks. Graphs and diagrams are included for questions involving graphical representations.
3. The document serves as a guide for teachers and examiners to consistently apply the marking criteria for the Additional Mathematics SPM papers in order to fairly and accurately assess students' work.
1) The document discusses directional derivatives and the gradient of functions of several variables. It defines the directional derivative Duf(c) as the slope of the function f in the direction of the unit vector u at the point c.
2) It shows that the partial derivatives of f can be computed by treating all but one variable as a constant. The gradient of f is defined as the vector of its partial derivatives.
3) It derives an expression for the directional derivative Duf(c) in terms of the partial derivatives of f and the components of the unit vector u, showing the relationship between directional derivatives and the gradient.
This document contains exercises involving the Laplace transform and inverse Laplace transform of various functions. It provides 9 functions to take the Laplace transform of and 9 functions to take the inverse Laplace transform of. The functions include combinations of step functions, sinusoids, polynomials and exponential terms. It also reminds the reader of the time shifting property of the Laplace transform.
This document is a Master's thesis that applies the technique of differential reduction to the pole (DRTP) to aeromagnetic anomaly data from Iran. DRTP transforms magnetic anomalies measured at various latitudes and orientations to how they would appear if measured at the magnetic pole. The thesis applies this technique using MATLAB programs to iteratively calculate the DRTP correction. It then uses the corrected DRTP map of Iran's magnetic anomalies to better delineate geological structures like fold belts and delineate boundaries between volcanic and metamorphic zones.
This document discusses the inverse Laplace transform. It defines the inverse Laplace transform and provides some key properties and theorems. It then gives examples of taking the inverse Laplace transform of various functions and using Laplace transforms to solve initial value problems involving differential equations. It also defines the unit step function and discusses its properties. Finally, it presents a convolution theorem relating the inverse Laplace transform of a product of functions to a convolution integral.
This document contains a chapter about mathematical descriptions of continuous-time signals. It includes examples of signal functions, operations like shifting and scaling on signals, derivatives and integrals of signals, properties of even and odd signals, and exercises with answers related to these topics. The exercises involve graphing signals, finding signal values at times, manipulating signals using operations, and identifying signal properties.
MATLAB/Assignment 2/Bracketing (Multiple Roots) (4)/Bisection method bracketing/Thumbs.db
MATLAB/Assignment 2/Bracketing (Multiple Roots) (4)/Newton method bracketing/Thumbs.db
MATLAB/Assignment 2/Bracketing (Multiple Roots) (4)/Thumbs.db
MATLAB/Assignment 2/Thumbs.db
MATLAB/Assignment 2/Newton method - intersection of two functions (3)/Thumbs.db
MATLAB/Assignment 2/Bisecting method (1)/Thumbs.db
MATLAB/Assignment 2/Newton Method (2)/Thumbs.db
MATLAB/Assignment 1/Images/Thumbs.db
MATLAB/Assignment 1/Nano Sim Assignment 1.docx
Simulation on the Nanoscale ENG-M03: Assignment 1
Using a script file (I)
clear
format short g
N = 100;
x = linspace(-pi,pi,N)';
y = sin(x);
figure(1), plot(x,y)
Defining a function (II)
function y = sin_scale (x,sf)
y = sin(x./sf);
end
sf = 0.1;
y = sin_scale(x,sf);
figure(2), plot(x,y)
sf=[0.1 0.5 2];
N_sf = numel(sf);
sin_array = zeros(N,N_sf);
for ndx = 1: N_sf
sin_array(:,ndx) = sin_scale(x,sf(ndx));
end
figure(3),plot(x,sin_array)
Numerical Differentiation
Function
function dydx = Deriv(x,y)
N = numel(x);
dydx = zeros(N-1,1);
h= x(2)-x(1);
for ndx = 1 : N-1
dydx(ndx,:) =(y(ndx+1)-y(ndx))/h;
end
Script
clear
format short g
N=100;
x = linspace (-pi,pi,N)';
y = cos(x);
dydx = Deriv(x,y);
xd = x(1:N-1);
figure(4), plot(x,y,xd,dydx,x,-sin(x))
Numerical Integration
1. Trapezoidal Method
a) Integral
b) g=integral(@f,a,b,'ArrayValued',true)
Error = ((I-g)/g).*100 = 23.37%
2. Composite Trapezium Rule
a) function [ aproxInt ] = aproxInt(n,a,b)
function y = sumFx
y = 0;
for k = 1:n-1
y = y + feval(@f,(a+(k.*((b-a)/n))));
end
end
Totx=sumFx;
aproxInt = 0.5.*((b-a)/n).*((feval(@f,a))+(feval(@f,b))+(2.*(Totx)));
end
b) I= aproxInt(n,a,b)= 1.00002056192951
Error = ((I-g)/g).*100 = 0.00205619295078341
c) I1 = aproxInt2(5,1,2) = 0.695634920634921
I2 = aproxInt2(50,1,2) = 0.693172179310195
I3 = aproxInt2(100,1,2) = 0.693153430481824
d) g1 = log(2);
e1 = ((I1-g1)/g1).*100 = 0.358905026918763
e2 = ((I2-g1)/g1).*100 = 0.00360655730140557
e3 = ((I3-g1)/g1).*100 = 0.000901673130050152
e) number of sub-intervals needed to achieve an accuracy to 6 d.p., n=277 (adjusted manually)
3. Simpson’s Rule
MATLAB/Assignment 2/Nanosim assignment 2.docx
Simulation on the Nanoscale ENG-M03: Assignment 2
1. Bisection Method
2. Newton Method
3. Newton Method – Intersection of Two Functions
This script only takes 3 iterations to calculate the root. Newton’s method by intersection of two functions is much faster than the bisection method, but encounters problems unless the root is well defined (e.g. the wrong root may be found).
4. Bracketing (Newton Method)
4. Bracketing (Bisection Method)
MATLAB/Assignment 1/Composite Trapezium Rule(5)/aproxInt.m
%5a)
function [ aproxInt ] = aproxInt(n,a,b)
function y = sumFx
y = 0;
for k = 1:n-1
y = y + feval(@f,(a+(k.*((b-a)/n))) ...
The document provides an introduction to Laplace transforms. Key points:
- Laplace transforms are a mathematical tool that converts differential equations in the time domain to algebraic equations in the complex frequency (s) domain, making problems easier to solve.
- Common transforms include impulse, step, ramp, and exponential functions.
- Properties and theorems allow transforming derivatives, integrals, shifts, and scaling.
- Tables provide standard transforms to convert between time and s domains.
- Solving problems involves taking the Laplace transform of equations, using properties to solve for the transform in s domain, then applying the inverse transform.
- Partial fraction expansions break complex fractions into simpler forms for applying transforms.
The document appears to be lyrics from multiple songs written or co-written by Miranda Cosgrove and others. The lyrics describe themes of new romantic love and relationships, including the euphoric feelings of kissing someone for the first time, obsessive thoughts about a new partner, and struggling with intense romantic feelings.
The document introduces the basic interface of Multisim 8 simulation software. It includes menus, toolbars, circuit window, component library, and instruments. The menus provide most functions of the software. The toolbars offer quick access to common tools like open, save, zoom etc. The component library contains both ideal and real world components categorized in libraries. The instruments toolbar contains various tools to test circuit operation.
No document was provided to summarize. A summary requires source text to extract the key points and essential information from. Without a document, it is not possible to generate an accurate 3 sentence summary.
No document was provided to summarize. A summary requires source text to extract the key points and essential information from. Without a document, it is not possible to generate an accurate 3 sentence summary.
This document summarizes Haar wavelet transforms and Daubechies wavelet transforms, including their forward and inverse transforms. It provides MATLAB code implementations for 1D and 2D transforms using Haar, Daub4, Daub6, Daub5/3 and Daub4 wavelets. The transforms can be applied for multiple levels (levels 1 through k) to decompose signals into approximation and detail coefficients.
No document was provided to summarize. A summary requires source text to extract the key points and essential information from. Without a document, it is not possible to generate an accurate 3 sentence summary.
No document was provided to summarize. A summary requires source text to extract the key points and essential information from. Without a document, it is not possible to generate an accurate 3 sentence summary.
1. 以下是小波变换的相关程序 2.Daub4 变换
function y = Daub4(f)
1.Haar 变换
% Level 1 Daub4 transform
function h = Haar(f)
r = f(2:2:end)-sqrt(3)*f(1:2:end);
s = f(1:2:end)+sqrt(3)/4*r+(sqrt(3)-
% N = length(f);
2)/4*[r(2:end), r(1)];
% a = zeros(1,N/2);
a = (1+sqrt(3))/sqrt(2)*s;
% d = zeros(1,N/2);
d = (1-sqrt(3))/sqrt(2)*(s+[r(2:end), r(1)]);
% for m = 1:N/2
y = [a, d];
% a(m) = (f(2*m-1)+f(2*m))/sqrt(2);
% d(m) = (f(2*m-1)-f(2*m))/sqrt(2);
% end Daub4 逆变换
function f = Daub4I(y)
% a = (f(1:2:end-1)+f(2:2:end))/sqrt(2); % Level 1 Inverse Daub4 Transform
% d = (f(1:2:end-1)-f(2:2:end))/sqrt(2); s = sqrt(2)/(1+sqrt(3))*y(1:end/2);
% h = [a d]; r = sqrt(2)/(1-sqrt(3))*[y(end),y(end/2+1:end-
1)]...
h = [(f(1:2:end-1)+f(2:2:end)), ...
-[s(end),s(1:end-1)];
(f(1:2:end-1)-f(2:2:end))]/sqrt(2); f = y;
f(1:2:end) = s-sqrt(3)/4*r-(sqrt(3)-
2)/4*[r(2:end),r(1)];
Haar 逆变换
f(2:2:end) = r + sqrt(3)*f(1:2:end);
function f = HaarI(h)
Daub4 K 层变换
% M = length(h)/2; function y = Daub4K(x, k)
% f = zeros(1,2*M);
% for i=1:M y = x;
% f(2*i-1:2*i) = [h(i)+h(M+i), h(i)-h(M+i)]; for i=1:k
% end
y(1:end/2^(i-1)) = Daub4(y(1:end/2^(i-1)));
end
a = h(1:end/2);
d = h(end/2+1:end); Daub4 K 层逆变换
f = reshape([a+d; a-d],1,[])/sqrt(2); function y = Daub4KI(x, k)
% Level K Inverse Daub4 Transform
K 层 Haar 变换
y = x;
function y = HaarK(f, k)
for i=k:-1:1
% k-level Haar transform
y = f; y(1:end/2^(i-1)) = Daub4I(y(1:end/2^(i-1)));
for i=1:k end
y(1:end/2^(i-1)) = Haar(y(1:end/2^(i-1)));
end 3.Daub6 变换
function y = Daub6(f)
K 层 Haar 逆变换
% level 1 Daub6 transform
function f = HaarKI(y, k)
a1 = (1+sqrt(10)+sqrt(5+2*sqrt(10)))*sqrt(2)/32;
% k-level Inverse Haar transform
a2 =
f = y;
(5+sqrt(10)+3*sqrt(5+2*sqrt(10)))*sqrt(2)/32;
for i=k:-1:1
a3 = (10-
f(1:end/2^(i-1)) = HaarI(f(1:end/2^(i-1))); 2*sqrt(10)+2*sqrt(5+2*sqrt(10)))*sqrt(2)/32;
end a4 = (10-2*sqrt(10)-
2*sqrt(5+2*sqrt(10)))*sqrt(2)/32;
a5 = (5+sqrt(10)-
3*sqrt(5+2*sqrt(10)))*sqrt(2)/32;
3. f(2:2:end) = y(end/2+1:end)+floor((f(1:2:end)+ y = f;
[f(3:2:end),f(end-1)])/2+1/2); for i = 1:k
for j = 1:2^(i-1)
5.Daub4 2D 变换
for p = 1:2^(i-1)
function y = Daub4_2D(f)
y((j-1)*m/2^(i-1)+1:j*m/2^(i-1),(p-
[m,n] = size(f); 1)*n/2^(i-1)+1:p*n/2^(i-1)) = Daub4_2D(y((j-
y = f; 1)*m/2^(i-1)+1:j*m/2^(i-1),(p-1)*n/2^(i-
for i = 1:m 1)+1:p*n/2^(i-1)));
y(i,:) = Daub4(y(i,:)); end
end
end
for i = 1:n
end
y(:,i) = Daub4(y(:,i)')';
逆变换
end
function f = Daub4p_2D_K_I(y,k)
Daub4 2D 逆变换 [m,n] = size(y);
function f = Daub4_2D_I(y) f = y;
for i = k:-1:1
[m,n] = size(y);
for j = 1:2^(i-1)
f = y;
for i = 1:m for p = 1:2^(i-1)
f(i,:) = Daub4I(f(i,:)); f((j-1)*m/2^(i-1)+1:j*m/2^(i-1),(p-
end 1)*n/2^(i-1)+1:p*n/2^(i-1)) = Daub4_2D_I(f((j-
for i = 1:n 1)*m/2^(i-1)+1:j*m/2^(i-1),(p-1)*n/2^(i-
f(:,i) = Daub4I(f(:,i)')'; 1)+1:p*n/2^(i-1)));
end end
end
Daub4 2D K 层变换 end
function y = Daub4_2D_K(f,k)
7 图片保存信息量为原来的 99.99%
y = f; function f = Daub4p_2D_K_I(y,k)
for i = 1:k [m,n] = size(y);
f = y;
y(1:end/2^(i-1),1:end/2^(i-1)) = for i = k:-1:1
Daub4_2D(y(1:end/2^(i-1),1:end/2^(i-1)));
for j = 1:2^(i-1)
end
for p = 1:2^(i-1)
Daub4 2D K 层逆变换 f((j-1)*m/2^(i-1)+1:j*m/2^(i-1),(p-
function f = Daub4_2D_K_I(y,k) 1)*n/2^(i-1)+1:p*n/2^(i-1)) = Daub4_2D_I(f((j-
1)*m/2^(i-1)+1:j*m/2^(i-1),(p-1)*n/2^(i-
f = y; 1)+1:p*n/2^(i-1)));
for i = k:-1:1
end
f(1:end/2^(i-1),1:end/2^(i-1)) =
end
Daub4_2D_I(f(1:end/2^(i-1),1:end/2^(i-1)));
end end
6.Daub4 2D P 变换
function y = Daub4p_2D_K(f,k)
[m,n] = size(f);