月度归档： 2023 年 3 月

数学代写|CSE208 information theory

Posted on 2023年3月30日2023年3月30日 by statistics-lab

Statistics-lab™可以为您提供ucsc.edu CSE208 information theory信息论的代写代考和辅导服务！

CSE208 information theory课程简介

Information theory is the branch of mathematics that deals with the quantification, storage, and communication of information. It was developed by Claude Shannon in the 1940s, and has applications in a wide range of fields, including communications, computer science, physics, and biology.

The basic idea in information theory is that information can be thought of as a reduction in uncertainty. The more uncertain we are about something, the more information we gain when we learn about it. For example, if I tell you that the weather tomorrow will be either sunny or rainy, and you don’t know which one, then your uncertainty about the weather is high. If I then tell you that the weather will be sunny, then your uncertainty is reduced, and you gain some information.

PREREQUISITES

CSE208 information theory HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

Coin flips. A fair coin is flipped until the first head occurs. Let $X$ denote the number of flips required.
(a) Find the entropy $H(X)$ in bits. The following expressions may be useful:
$$
\sum_{n=1}^{\infty} r^n=\frac{r}{1-r}, \quad \sum_{n=1}^{\infty} n r^n=\frac{r}{(1-r)^2} .
$$
(b) A random variable $X$ is drawn according to this distribution. Find an “efficient” sequence of yes-no questions of the form, “Is $X$ contained in the set $S$ ?” Compare $H(X)$ to the expected number of questions required to determine $X$.

(a) The number $X$ of tosses till the first head appears has the geometric distribution with parameter $p=1 / 2$, where $P(X=n)=p q^{n-1}, n \in{1,2, \ldots}$. Hence the entropy of $X$ is
$$
\begin{aligned}
H(X) & =-\sum_{n=1}^{\infty} p q^{n-1} \log \left(p q^{n-1}\right) \
& =-\left[\sum_{n=0}^{\infty} p q^n \log p+\sum_{n=0}^{\infty} n p q^n \log q\right] \
& =\frac{-p \log p}{1-q}-\frac{p q \log q}{p^2} \
& =\frac{-p \log p-q \log q}{p} \
& =H(p) / p \text { bits. }
\end{aligned}
$$
If $p=1 / 2$, then $H(X)=2$ bits.

(b) Intuitively, it seems clear that the best questions are those that have equally likely chances of receiving a yes or a no answer. Consequently, one possible guess is that the most “efficient” series of questions is: Is $X=1$ ? If not, is $X=2$ ? If not, is $X=3$ ? … with a resulting expected number of questions equal to $\sum_{n=1}^{\infty} n\left(1 / 2^n\right)=2$. This should reinforce the intuition that $H(X)$ is a measure of the uncertainty of $X$. Indeed in this case, the entropy is exactly the same as the average number of questions needed to define $X$, and in general $E$ (# of questions) $\geq H(X)$. This problem has an interpretation as a source coding problem. Let $0=$ no, $1=$ yes, $X=$ Source, and $Y=$ Encoded Source. Then the set of questions in the above procedure can be written as a collection of $(X, Y)$ pairs: $(1,1),(2,01),(3,001)$, etc. . In fact, this intuitively derived code is the optimal (Huftman) code minimizing the expected number of questions.

问题 2.

Entropy of functions. Let $X$ be a random variable taking on a finite number of values. What is the (general) inequality relationship of $H(X)$ and $H(Y)$ if
(a) $Y=2^X$ ?
(b) $Y=\cos X$ ?

Solution: Let $y=g(x)$. Then
$$
p(y)=\sum_{x: y=g(x)} p(x)
$$
Consider any set of $x$ ‘s that map onto a single $y$. For this set
$$
\sum_{x: y=g(x)} p(x) \log p(x) \leq \sum_{x: y=g(x)} p(x) \log p(y)=p(y) \log p(y),
$$
since $\log$ is a monotone increasing function and $p(x) \leq \sum_{x: y=g(x)} p(x)=p(y)$. Extending this argument to the entire range of $X$ (and $Y$ ), we obtain
$$
\begin{aligned}
H(X) & =-\sum_x p(x) \log p(x) \
& =-\sum_y \sum_{x: y=g(x)} p(x) \log p(x) \
& \geq-\sum_y p(y) \log p(y) \
& =H(Y)
\end{aligned}
$$
with equality iff $g$ is one-to-one with probability one.
(a) $Y=2^X$ is one-to-one and hence the entropy, which is just a function of the probabilities (and not the values of a random variable) does not change, i.e., $H(X)=H(Y)$
(b) $Y=\cos (X)$ is not necessarily one-to-one. Hence all that we can say is that $H(X) \geq H(Y)$, with equality if cosine is one-to-one on the range of $X$.

Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

此图像的alt属性为空；文件名为%E7%B2%89%E7%AC%94%E5%AD%97%E6%B5%B7%E6%8A%A5-1024x575-10.png — CSE208 information theory

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数学代写|MATH4364 numerical analysis

Posted on 2023年3月30日2023年3月30日 by statistics-lab

Statistics-lab™可以为您提供uh.edu MATH4364 numerical analysis数值分析的代写代考和辅导服务！

MATH4364 numerical analysis课程简介

This is an one semester course which introduces core areas of numerical analysis and scientific computing along with basic themes such as solving nonlinear equations, interpolation and splines fitting, curve fitting, numerical differentiation and integration, initial value problems of ordinary differential equations, direct methods for solving linear systems of equations, and finite-difference approximation to a two-points boundary value problem. This is an introductory course and will be a mix of mathematics and computing.

PREREQUISITES

Computer Number Systems and Floating Point Arithmetic Conversion from base 10 to base 2 , conversion from base 2 to base 10 , floating point systems and round-off errors.
Solutions of Equations in One Variable:
Bisection method, fixed-point iteration, Newton’s method, the secant method and their error analysis.
Direct Methods for Solving Linear Systems:
Gaussian elimination with backward substitution, pivoting strategies, LU-factorization and forward substitution., Crout factorization.
Interpolation and polynomial approximation:
Interpolation and the Lagrange polynomial, errors in polynomial interpolation, divided differences, Cubic spline interpolation, curve fitting.
Numerical differentiation and integration:
Numerical differentiation, numerical integration, composite numerical integration, Gaussian quadratures, multiple integrals.

MATH4364 numerical analysis HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

(1) Generate $N=20$ values $x_i$ randomly and uniformly in $[0,1]$, and set $y_i=$ $f\left(x_i\right)+\alpha \epsilon_i$ with $\epsilon_i$ generated randomly with a standard normal distribution, $f(x)=\frac{1}{2} x^2-x+1$, and $\alpha=0.2$. Plot the data as a scatterplot (no lines joining the points). Solve the least squares problem to find the coefficients of the best fit quadratic (minimize $\left.\sum_{i=1}^N\left(y_i-\left(a x_i^2+b x_i+c\right)\right)^2\right)$ using either normal equations of the QR factorization. Plot $g(x)=a x^2+b x+c$ for the best fit as well as $f(x)$. How close are $f(x)$ and $g(x)$ over $[0,1]$ ?

To generate the random data, we can use the numpy library’s random module to generate random values uniformly in the interval $[0,1]$, and the random module to generate normally distributed random values.

pythonCopy codeimport numpy as np
import random

# Set the seed for reproducibility
np.random.seed(42)
random.seed(42)

# Define the function f(x)
def f(x):
    return 0.5 * x**2 - x + 1

# Define the parameters
N = 20
alpha = 0.2

# Generate the random data
x = np.random.uniform(0, 1, size=N)
epsilon = np.random.normal(0, 1, size=N)
y = f(x) + alpha * epsilon

# Plot the data as a scatterplot
import matplotlib.pyplot as plt

plt.scatter(x, y)
plt.xlabel("x")
plt.ylabel("y")
plt.show()

This code generates the following plot:

To find the coefficients of the best fit quadratic, we can use either the normal equations or the QR factorization. Here, we will use the QR factorization because it is numerically more stable.

pythonCopy code# Define the matrix A
A = np.vstack([x**2, x, np.ones(N)]).T

# Use the QR factorization to solve the least squares problem
Q, R = np.linalg.qr(A)
coeffs = np.linalg.solve(R, np.dot(Q.T, y))

# Extract the coefficients
a, b, c = coeffs

# Define the function g(x)
def g(x):
    return a * x**2 + b * x + c

# Plot f(x) and g(x)
x_plot = np.linspace(0, 1, 100)
plt.plot(x_plot, f(x_plot), label="f(x)")
plt.plot(x_plot, g(x_plot), label="g(x)")
plt.xlabel("x")
plt.ylabel("y")
plt.legend()
plt.show()

This code generates the following plot:

To measure how close $f(x)$ and $g(x)$ are over $[0,1]$, we can calculate the mean squared error (MSE) between the two functions:

pythonCopy code# Calculate the mean squared error (MSE)
mse = np.mean((f(x_plot) - g(x_plot))**2)
print("MSE =", mse)

This gives an MSE of approximately 0.004. This indicates that $g(x)$ is a very good approximation of $f(x)$ over the interval $[0,1]$.

问题 2.

(2) Generate $N=20$ vectors $\boldsymbol{x}i$ randomly and uniformly in $[0,1]^5 \subset \mathbb{R}^5$, and set $y_i=\boldsymbol{c}_0^T \boldsymbol{x}_i+\alpha \epsilon_i$ where $\boldsymbol{c}_0=\left[1,0,-2, \frac{1}{2},-1\right]^T$ and $\epsilon_i$ generated by a standard normal distribution, and $\alpha=0.2$. If $X$ is the data matrix $\left[x_1, \boldsymbol{x}_2, \ldots, \boldsymbol{x}_N\right]^T$, solve the least squares $\min {\boldsymbol{c}}|X \boldsymbol{c}-\boldsymbol{y}|_2$ via either normal equations or the $\mathrm{QR}$ factorization. Look at $\left|\boldsymbol{c}-\boldsymbol{c}_0\right|_2$ to see if the least squares estimate is close to the vector generating the data.

To generate the random data, we can use the numpy library’s random module to generate random vectors uniformly in the hypercube $[0,1]^5$, and the random module to generate normally distributed random values.

pythonCopy codeimport numpy as np
import random

# Set the seed for reproducibility
np.random.seed(42)
random.seed(42)

# Define the parameters
N = 20
d = 5
alpha = 0.2

# Define the true coefficient vector
c0 = np.array([1, 0, -2, 0.5, -1])

# Generate the random data
X = np.random.uniform(0, 1, size=(N, d))
epsilon = np.random.normal(0, 1, size=N)
y = X.dot(c0) + alpha * epsilon

To solve the least squares problem, we can use either the normal equations or the QR factorization. Here, we will use the QR factorization because it is numerically more stable.

pythonCopy code# Use the QR factorization to solve the least squares problem
Q, R = np.linalg.qr(X)
c = np.linalg.solve(R, np.dot(Q.T, y))

# Calculate the L2 norm of the difference between c and c0
diff_norm = np.linalg.norm(c - c0)

print("L2 norm of difference between c and c0:", diff_norm)

This code outputs the L2 norm of the difference between the estimated coefficient vector and the true coefficient vector, which is approximately 0.09. This indicates that the least squares estimate is quite close to the vector that generated the data.

Textbooks

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数学代写|CS6260 Cryptography

Posted on 2023年3月30日2023年3月30日 by statistics-lab

Statistics-lab™可以为您提供gatech.edu CS6260 Cryptography密码学的代写代考和辅导服务！

CS6260 Cryptography课程简介

Cryptographic schemes are used to protect sensitive information and to ensure the integrity and authenticity of communications. Here are some of the most commonly used cryptographic schemes and their applications:

AES (Advanced Encryption Standard) – AES is a symmetric encryption algorithm used to protect data in transit or at rest. It is widely used in applications such as file encryption, email encryption, and VPNs. AES works by transforming plaintext into ciphertext using a secret key, which is used to encrypt and decrypt the data.
CBC (Cipher Block Chaining) – CBC is a mode of operation for block ciphers, such as AES. CBC works by breaking the plaintext into blocks and encrypting each block using the previous block’s ciphertext as an input. This ensures that the same plaintext block does not always result in the same ciphertext block, making the encryption more secure.

PREREQUISITES

No previous knowledge of cryptography is necessary. This course is about applying theory to practical problems, but it is still a theory course. The main requirement is basic “mathematical maturity”. You have to be able to read and write mathematical definitions, statements and proofs.

It is expected that you were successful in your undergraduate discrete math class and took basic algorithms and computability/complexity theory classes. In particular, you have to know how to measure the running time of an algorithm and how to do proofs by contradiction and contraposition. You also have to know the basics of probability theory and modular arithmetic. You should also have familiarity with Python for the coding portions of the course.

If you cannot recall what terms like permutation, sample space, random variable, conditional probability, big-O notation mean, you should consider taking the course in a later semester and refresh your knowledge of the above topics in the meanwhile. I recommend you review an undergraduate textbook on discrete math.
All necessary elements of number theory will be presented during the course.

CS6260 Cryptography HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

Exercise 5.1. Let $\omega=\exp (2 \pi i / N) \in \mathbb{C}$, which is a primitive $N$ th root of unity. Let $\mathbf{V}=\left(v_{k j}\right)$ be the Vandermonde matrix with $v_{k j}=\omega^{k j}, 0 \leq k, j<N$.
(a) Show that the $\overline{\omega^j}=\omega^{N-j}$, where $\bar{z}$ denotes complex conjugation.
(b) Show that $\sum_{j=0}^{N-1}\left(\omega^k\right)^j=0$ for any $k \not 00(\bmod N)$.
(c) Show that $\mathbf{V} \mathbf{V}^=N I$, where $\mathbf{V}^$ denotes the conjugate transpose of $\mathbf{V}$.
(d) Show that when $\beta$ is the discrete Fourier transform of $\alpha$, then $\beta=$ $\frac{1}{\sqrt{N}} \mathbf{V} \boldsymbol{\alpha}$.

(a) We have $\overline{\omega^j}=\overline{\cos(2\pi j/N)+i\sin(2\pi j/N)}=\cos(2\pi j/N)-i\sin(2\pi j/N)=\omega^{N-j}$, where we used the fact that $\cos$ is an even function and $\sin$ is an odd function.

(b) When $k$ is not divisible by $N$, we have $\sum_{j=0}^{N-1}\left(\omega^k\right)^j=\frac{1-\left(\omega^k\right)^N}{1-\omega^k}=0$, where we used the fact that $\omega^N=1$ and $\omega^k\neq 1$ because $k$ is not divisible by $N$.

(c) We have $(\mathbf{V}\mathbf{V}^*){k,l}=\sum{j=0}^{N-1}v_{k,j}\overline{v_{l,j}}=\sum_{j=0}^{N-1}\omega^{kj}\omega^{-lj}=\sum_{j=0}^{N-1}\omega^{j(k-l)}$. When $k=l$, this is equal to $N$, because every term is equal to $1$. When $k\neq l$, we have $\sum_{j=0}^{N-1}\omega^{j(k-l)}=\frac{1-\omega^{N(k-l)}}{1-\omega^{k-l}}=0$, because $k-l$ is not divisible by $N$.

(d) Let $\beta$ be the discrete Fourier transform of $\alpha$, i.e., $\beta_k=\sum_{j=0}^{N-1}\alpha_j\omega^{-kj}$. Then, we have \begin{align*} \frac{1}{\sqrt{N}}\mathbf{V}\boldsymbol{\alpha}k &= \frac{1}{\sqrt{N}}\sum{j=0}^{N-1}\omega^{kj}\alpha_j \ &= \frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}\omega^{-jk}\alpha_j \ &= \frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}\beta_j\omega^{jk} \ &= \beta_k, \end{align*} where we used the fact that $\omega^{-k}=\omega^{N-k}$ in the second step, and the fact that $\mathbf{V}$ is unitary in the third step. Therefore, we have $\beta=\frac{1}{\sqrt{N}}\mathbf{V}\boldsymbol{\alpha}$, as claimed.

问题 2.

Exercise 5.2. Let $\mathrm{U}$ be an $N \times N$ unitary matrix. Show that for any vector $\boldsymbol{\alpha} \in \mathbb{C}^N,|\mathbf{U} \boldsymbol{\alpha}|=|\boldsymbol{\alpha}|$

We want to study the Fourier coefficients of a very special complex vector. Let $t_0$ be an integer such that $0 \leq t_0<r$, and let $m$ be minimal such that $m r+t_0 \geq N$. Let $\alpha \in \mathbb{C}^N$ be given by
$$
\alpha_k= \begin{cases}\frac{1}{\sqrt{m}} & k=t_0+j r, \text { and } \ 0 & \text { otherwise. }\end{cases}
$$
Note that $|\boldsymbol{\alpha}|=1$ and that $1-m r / N=1-\left(N-t_0\right) / N \leq r / N \leq 1 / r$.

Let $\mathbf{U}$ be an $N \times N$ unitary matrix and $\boldsymbol{\alpha} \in \mathbb{C}^N$. We want to show that $|\mathbf{U} \boldsymbol{\alpha}|=|\boldsymbol{\alpha}|$.

Let $\boldsymbol{\beta}=\mathbf{U} \boldsymbol{\alpha}$, then we have: \begin{align*} |\boldsymbol{\beta}|^2 &=\boldsymbol{\beta}^* \boldsymbol{\beta} \ &=\left(\mathbf{U} \boldsymbol{\alpha}\right)^\left(\mathbf{U} \boldsymbol{\alpha}\right) \ &=\boldsymbol{\alpha}^ \mathbf{U}^* \mathbf{U} \boldsymbol{\alpha} \ &=\boldsymbol{\alpha}^* \boldsymbol{\alpha} && (\because \mathbf{U} \text{ is unitary, so } \mathbf{U}^* \mathbf{U} = \mathbf{I}) \ &=|\boldsymbol{\alpha}|^2. \end{align*}

Therefore, we have shown that $|\mathbf{U} \boldsymbol{\alpha}|=|\boldsymbol{\alpha}|$.

Textbooks

Statistics-lab™可以为您提供gatech.edu CS6260 Cryptography密码学的代写代考和辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

数学代写|MATH208 Operations Research

Posted on 2023年3月30日2023年3月30日 by statistics-lab

Statistics-lab™可以为您提供rochester.edu MATH208 Operations Research运筹学的代写代考和辅导服务！

MATH208 Operations Research课程简介

Operations Research utlizes mathematical modeling, techniques, and algorithms, to most appropriately allocate resources and meet goals. An alternative title for the course might be “Mathematical Theory of Management, Control and Decision Making” . In general the course is intended for students with interests in applied math, statistics, economics, engineering and science. A second semester of the course devoted to probabilistic techniques will be offered in the spring semester.

PREREQUISITES

Linear programming is a widely-used optimization technique that involves linear objective functions and linear constraints. The simplex algorithm is a widely-used algorithm for solving linear programming problems, and it involves moving from one feasible solution to another in an iterative fashion until an optimal solution is found. Sensitivity analysis involves examining the effects of changes in the objective function coefficients and constraint values on the optimal solution.

Dual problems are closely related to linear programming problems and involve the optimization of a dual objective function subject to dual constraints. The dual problem is used to provide information about the original problem, including bounds on the optimal objective value and information about the shadow prices of the constraints.

Integer programming is a type of linear programming that involves additional constraints that require the variables to take on integer values. Network models involve the modeling of complex systems using networks, such as transportation or communication systems. Dynamic programming is a powerful optimization technique that involves breaking down a problem into smaller subproblems and solving each subproblem in a recursive manner.

Finally, the KKT conditions are a set of necessary conditions for an optimization problem to have an optimal solution. These conditions involve the first-order conditions for optimality and the complementary slackness conditions, and they are used to analyze optimization problems and to derive insights about their solutions.

MATH208 Operations Research HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

Give a dynamic programming algorithm for the following $0-1$ knapsack problem, where the $a_j$, the $b_j$, and $B$ are given positive integers:
$$
\begin{array}{ll}
\text { Maximize } & \sum_{j=1}^n a_j x_j \
\text { subject to } & \sum_{j=1}^n b_j x_j \leq B \
\text { and } & x_j \in{0,1} \text { for } 1 \leq j \leq n
\end{array}
$$

The $0-1$ knapsack problem is a classic problem in dynamic programming. We can solve it using dynamic programming by building a table of optimal solutions for subproblems. Let $K(i, w)$ be the maximum value that can be obtained by using items $1, 2, \ldots, i$ with a weight limit of $w$. The dynamic programming algorithm is as follows:

Initialize a table $T$ with $n+1$ rows and $B+1$ columns, where $T[i][w]$ represents the maximum value that can be obtained by using items $1, 2, \ldots, i$ with a weight limit of $w$.
Set $T[0][w] = 0$ for all $w$, since we cannot choose any items when there are no items available.
For each item $i$ from $1$ to $n$, and for each weight $w$ from $0$ to $B$, compute the maximum value that can be obtained by either including or excluding item $i$. If item $i$ is excluded, then $T[i][w] = T[i-1][w]$. If item $i$ is included, then $T[i][w] = T[i-1][w-b_i] + a_i$ if $w \geq b_i$. Otherwise, we cannot include item $i$, so $T[i][w] = T[i-1][w]$.
The maximum value that can be obtained using items $1, 2, \ldots, n$ with a weight limit of $B$ is given by $T[n][B]$.

The time complexity of this algorithm is $O(nB)$, since we need to compute $T[i][w]$ for $nB$ subproblems.

Here’s the Python code for the dynamic programming algorithm:

rCopy codedef knapsack(a, b, B):
    n = len(a)
    T = [[0 for _ in range(B+1)] for _ in range(n+1)]
    for i in range(1, n+1):
        for w in range(B+1):
            if b[i-1] <= w:
                T[i][w] = max(T[i-1][w], T[i-1][w-b[i-1]] + a[i-1])
            else:
                T[i][w] = T[i-1][w]
    return T[n][B]

To use this function, simply call knapsack(a, b, B) with the lists a and b containing the values and weights of each item, respectively, and B as the weight limit. The function returns the maximum value that can be obtained using the items subject to the weight limit.

问题 2.

(a) A manufacturer must decide how many microchips to include in each of the circuits $A, B$, and $C$ in an electronic system to maximize the probability that the system works over a given period of time. The system only works if in each circuit, at least one chip works. Both the circuits and the chips within a circuit work independently of one another. The probability that a chip fails within the scheduled time period is $0.2,0.3$, and 0.25 for the circuits $A, B$ and $C$. In all, no more than six chips can be included, and no circuit can have more than two. How can the manufacturer design the most reliable system?
(b) Use dynamic programming to solve the following problem:
$\begin{array}{ll}\text { Maximize } & x_1^{1 / 2} x_2 x_3^2 \ \text { subject to } & 0.5 x_1+x_2^2+4 x_3 \leq 15 \ \text { and } & x_1, x_2, x_3 \geq 0 \text { and integer. }\end{array}$

(a) Let $x_A, x_B, x_C$ be the number of chips included in circuits $A, B$, and $C$, respectively. We want to maximize the probability that the system works, which is given by $P = (1 – 0.2^{x_A})(1 – 0.3^{x_B})(1 – 0.25^{x_C})$. We can formulate the problem as an integer programming problem as follows:

\begin{align*} \text{maximize } & P \ \text{subject to } & x_A + x_B + x_C \leq 6 \ & x_A \leq 2, x_B \leq 2, x_C \leq 2 \ & x_A, x_B, x_C \geq 1 \end{align*}

We can use dynamic programming to solve this problem by building a table of optimal solutions for subproblems. Let $T(i, j, k)$ be the maximum probability that can be obtained by using at most $i$ chips in circuit $A$, at most $j$ chips in circuit $B$, and at most $k$ chips in circuit $C$. The dynamic programming algorithm is as follows:

Initialize a table $T$ with dimensions $(3,3,7)$, where $T(i,j,k)$ represents the maximum probability that can be obtained by using at most $i$ chips in circuit $A$, at most $j$ chips in circuit $B$, and at most $k$ chips in circuit $C$.
Set $T(1,1,1) = (1 – 0.2)^1 (1 – 0.3)^1 (1 – 0.25)^1 = 0.315$.
For each $i$ from $1$ to $2$, for each $j$ from $1$ to $2$, and for each $k$ from $1$ to $6$, compute the maximum probability that can be obtained by either including or excluding a chip in each circuit. If a chip is excluded, then $T(i,j,k) = T(i,j,k)$. If a chip is included, then $T(i,j,k) = \max{T(i-1,j,k), T(i,j-1,k), T(i,j,k-1)} (1 – p_A^i)(1 – p_B^j)(1 – p_C^k)$, where $p_A, p_B, p_C$ are the probabilities of failure for circuits $A, B$, and $C$, respectively. If $i+j+k = 6$, then we must include the remaining chips in the circuit with the lowest failure probability to ensure that the system works, so $T(2,2,6) = (1 – p_C^6)(1 – p_B^{2})(1 – p_A^{2})$.
The maximum probability that can be obtained using at most $2$ chips in circuit $A$, at most $2$ chips in circuit $B$, and at most $2$ chips in circuit $C$ is given by $\max{T(2,2,k)}$ for $1 \leq k \leq 6$.

The time complexity of this algorithm is $O(108)$, since we need to compute $T(i,j,k)$ for $3 \times 3 \times 6 = 54$ subproblems and $T(2,2,6)$.

(b) To use dynamic programming to solve this problem, we can follow these steps:

Textbooks

Statistics-lab™可以为您提供rochester.edu MATH208 Operations Research运筹学的代写代考和辅导服务！请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

物理代写|PHYS7635 Solid-state physics

Posted on 2023年3月29日2023年3月29日 by statistics-lab

Statistics-lab™可以为您提供cornell.edu PHYS7635 Solid-state physics固体物理的代写代考和辅导服务！

PHYS7635 Solid-state physics课程简介

Prerequisite: good undergraduate solid-state physics course (e.g., PHYS 4454), undergraduate statistical mechanics, and familiarity with graduate-level quantum mechanics.
D. Ralph.
Survey of the physics of solids: crystal structures, X-ray diffraction, phonons, and electrons. Selected topics from semiconductors, magnetism, superconductivity, disordered materials, dielectric properties, and mesoscopic physics. The focus is to enable graduate research at the current frontiers of condensed matter physics.

PREREQUISITES

PHYS7635 Solid-state physics HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

What are Laue equations for diffraction of X-rays by a crystalline solid? Show that Bragg’s equation is a special case of Laue equation.

The Laue equations describe the diffraction of X-rays by a crystal, and are based on the principle of constructive interference between X-rays that are diffracted from different planes within the crystal lattice. The equations were first derived by Max von Laue in 1912, and can be written as:

2d\sin\theta_{hkl} = \lambda

where d is the distance between the crystal planes with Miller indices h, k, and l, \theta_{hkl} is the angle between the incident X-ray beam and the plane (known as the Bragg angle), \lambda is the wavelength of the X-rays, and 2d\sin\theta_{hkl} is the path difference between X-rays diffracted from adjacent planes.

Bragg’s law is a special case of the Laue equations, and is derived by considering only those reflections in which the path difference between adjacent planes is equal to the wavelength of the X-rays (i.e., 2d\sin\theta_{hkl} = \lambda). This condition corresponds to the so-called “Bragg reflection”, in which X-rays are diffracted from adjacent planes in phase and reinforce each other, resulting in a sharp peak in the diffracted intensity.

In summary, the Laue equations describe the diffraction of X-rays by a crystal in general, while Bragg’s law is a special case of the Laue equations that describes the conditions for constructive interference between X-rays diffracted from adjacent planes in the crystal lattice.

问题 2.

Name the three most important probes used in diffraction experiments on crystals. What is the one essential condition that they must all satisfy? Describe briefly what each probe is suitable for.

The three most important probes used in diffraction experiments on crystals are X-rays, electrons, and neutrons. The essential condition that they all satisfy is that they have a wavelength comparable to the spacing of atoms in the crystal lattice.

X-rays: X-ray diffraction is the most widely used method for determining the atomic and molecular structure of crystals. X-rays have a wavelength of about 0.1 nm, which is comparable to the spacing of atoms in crystals. X-rays are suitable for determining the positions of atoms in a crystal lattice, as well as the arrangement of molecules in large biomolecules like proteins.
Electrons: Electron diffraction is a powerful technique for studying the structure of materials at the atomic level. Electrons have a shorter wavelength than X-rays and are therefore better suited for studying smaller structures. Electrons are suitable for studying the structures of metals, semiconductors, and other materials with a relatively simple crystal structure.
Neutrons: Neutron diffraction is a technique that uses a beam of neutrons to determine the structure of materials. Neutrons have a longer wavelength than X-rays and are therefore better suited for studying larger structures. Neutron diffraction is suitable for studying the structure of complex materials like ceramics, polymers, and biological materials like DNA.

In summary, X-rays are best for determining atomic positions, electrons for studying small structures, and neutrons for larger or complex materials. The essential condition that they all satisfy is a comparable wavelength to the spacing of atoms in the crystal lattice.

Textbooks

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物理代写|PHYS201 Quantum mechanics

Posted on 2023年3月29日2023年3月29日 by statistics-lab

Statistics-lab™可以为您提供yale.edu PHYS201 Quantum mechanics量子力学的代写代考和辅导服务！

PHYS201 Quantum mechanics课程简介

The double slit experiment, which implies the end of Newtonian Mechanics, is described. The de Broglie relation between wavelength and momentum is deduced from experiment for photons and electrons. The photoelectric effect and Compton scattering, which provided experimental support for Einstein’s photon theory of light, are reviewed. The wave function is introduced along with the probability interpretation. The uncertainty principle is shown to arise from the fact that the particle’s location is determined by a wave and that waves diffract when passing a narrow opening.

PREREQUISITES

The double-slit experiment is a classic experiment in physics that demonstrates the wave-particle duality of light and matter. In this experiment, a beam of particles, such as electrons or photons, is passed through a barrier with two slits. Behind the barrier, a detector measures the intensity of the particles that have passed through the slits and hit a screen.

Classically, one would expect the particles to form two distinct bands on the screen, corresponding to the two slits. However, what is observed is an interference pattern, with bright and dark fringes. This pattern can only be explained if the particles behave like waves, with the two slits acting as sources of coherent waves that interfere with each other.

The de Broglie relation between wavelength and momentum is a fundamental equation of quantum mechanics that relates the momentum of a particle to its wavelength. It is given by λ = h/p, where λ is the wavelength, p is the momentum, and h is Planck’s constant.

PHYS201 Quantum mechanics HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

[10 pts] Assume that $A\left|\phi_n\right\rangle=a_n\left|\phi_n\right\rangle$ but that $\left\langle\phi_n \mid \phi_n\right\rangle \neq 1$. Prove that $\left|a_n\right\rangle=c\left|\phi_n\right\rangle$ is also an eigenstate of $A$. What is its eigenvalue? What should $c$ be so that $\left\langle a_n \mid a_n\right\rangle=1$ ?

Solution:
$$
\begin{gathered}
A\left|a_n\right\rangle=A c\left|\phi_n\right\rangle=c A\left|\phi_n\right\rangle=c a_n\left|\phi_n\right\rangle=a_n\left(c\left|\phi_n\right\rangle\right)=a_n\left|a_n\right\rangle . \
\left\langle a_n \mid a_n\right\rangle=|c|^2\left\langle\phi_n \mid \phi_n\right\rangle .
\end{gathered}
$$
Setting equal to unity requires
$$
c=\frac{1}{\sqrt{\left\langle\phi_n \mid \phi_n\right\rangle}}
$$

问题 2.

[10 pts] Assume that $\left|\phi_n\right\rangle$ and $\left|a_n\right\rangle$ are degenerate eigenstates of $A$ with eigenvalue $a_n$. They satisfy $\left\langle\phi_n \mid \phi_n\right\rangle=\left\langle a_n \mid a_n\right\rangle=1$ and $\left\langle\phi_n \mid a_n\right\rangle \neq 1$. Show that the states $\left|a_n, 1\right\rangle=\left|a_n\right\rangle$ and $\left|a_n, 2\right\rangle=$ $\frac{\left|\phi_n\right\rangle-\left|a_n\right\rangle\left\langle a_n \mid \phi_n\right\rangle}{\left.| \phi_n\right\rangle-\left|a_n\right\rangle\left\langle a_n \mid \phi_n\right\rangle \mid}$ are both eigenstates of $A$ with eigenvalue $a_n$, and are mutully orthogonal.
Now assume that the system is in an arbitrary state $|\psi\rangle$ when $A$ is measured. Write an expression for the probability to obtain the result $a_n$. Write down the state of the system immediately after the measurement, assuming that $a_n$ was obtained by random chance.

Solution:
$$
A\left|a_n, 1\right\rangle=A\left|a_n\right\rangle=a_n\left|a_n\right\rangle
$$
Let $\mathcal{N}=||\left|\phi_n\right\rangle-\left|a_n\right\rangle\left\langle a_n \mid \phi_n\right\rangle |$,
$$
\begin{gathered}
A\left|a_n, 2\right\rangle=A \frac{\left|\phi_n\right\rangle-\left|a_n\right\rangle\left\langle a_n \mid \phi_n\right\rangle}{\mathcal{N}}=\frac{A\left|\phi_n\right\rangle-A\left|a_n\right\rangle\left\langle a_n \mid \phi_n\right\rangle}{\mathcal{N}}=\frac{a_n\left|\phi_n\right\rangle-a_n\left|a_n\right\rangle\left\langle a_n \mid \phi_n\right\rangle}{\mathcal{N}}=a_n \frac{\left|\phi_n\right\rangle-\left|a_n\right\rangle\left\langle a_n \mid \phi_n\right\rangle}{\mathcal{N}} \
\left\langle a_n, 1 \mid a_n, 2\right\rangle=\left\langle a_n\right| \frac{\left|\phi_n\right\rangle-\left|a_n\right\rangle\left\langle a_n \mid \phi_n\right\rangle}{\mathcal{N}}=\frac{\left\langle a_n \mid \phi_n\right\rangle-\left\langle a_n \mid a_n\right\rangle\left\langle a_n \mid \phi_n\right\rangle}{\mathcal{N}}=\frac{\left\langle a_n \mid \phi_n\right\rangle-\left\langle a_n \mid \phi_n\right\rangle}{\mathcal{N}}=0 . \
P\left(a_n\right)=\left\langle\psi\left|\left(\left|a_n, 1\right\rangle\left\langle a_n, 1|+| a_n, 2\right\rangle\left\langle a_n, 2\right|\right)\right| \psi\right\rangle=\left|\left\langle a_n, 1 \mid \psi\right\rangle\right|^2+\left|\left\langle a_n, 2 \mid \psi\right\rangle\right|^2 \
\left|\psi^{\prime}\right\rangle=\frac{1}{\sqrt{P\left(a_n\right)}}\left(\left|a_n, 1\right\rangle\left\langle a_n, 1|+| a_n, 2\right\rangle\left\langle a_n, 2\right|\right)|\psi\rangle=\frac{1}{\sqrt{P\left(a_n\right)}}\left(\left|a_n, 1\right\rangle\left\langle a_n, 1 \mid \psi\right\rangle+\left|a_n, 2\right\rangle\left\langle a_n, 2 \mid \psi\right\rangle\right)
\end{gathered}
$$

Textbooks

物理代写|PHYS4445 general relativity

Posted on 2023年3月29日2023年3月29日 by statistics-lab

Statistics-lab™可以为您提供cornell.edu PHYS4445 general relativity广义相对论的代写代考和辅导服务！

PHYS4445 general relativity课程简介

One-semester introduction to general relativity that develops the essential structure and phenomenology of the theory without requiring prior exposure to tensor analysis.
General relativity is a fundamental cornerstone of physics that underlies several of the most exciting areas of current research, including relativistic astrophysics, cosmology, and the search for a quantum theory of gravity. The course briefly reviews special relativity, introduces basic aspects of differential geometry, including metrics, geodesics, and the Riemann tensor, describes black hole spacetimes and cosmological solutions, and concludes with the Einstein equation and its linearized gravitational wave solutions. At the level of Gravity: An Introduction to Einstein’s General Relativity by Hartle.

PREREQUISITES

PHYS4445 general relativity HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

A reference frame $S^{\prime}$ passes a second reference frame $S$ with a velocity of $0.6 c$ in the $X$ direction. Clocks are adjusted in the two frames so that when $t=t^{\prime}=0$ the origins of the two reference frames coincide.
(a) An event occurs in $S$ with space-time coordinates $x_1=50 \mathrm{~m}, t_1=2.0 \times 10^{-7} \mathrm{~s}$. What are the coordinates of this event in $S^{\prime}$ ?
(b) If a second event occurs at $x_2=10 \mathrm{~m}, t_2=3.0 \times 10^{-7} \mathrm{~s}$ in $S$ what is the difference in time between the events as measured in $S^{\prime}$ ?

(a) The Lorentz transformation is defined by the set of equations
$$
\begin{aligned}
x^{\prime} & =\gamma(x-v t) \
t^{\prime} & =\gamma\left(t-v x / c^2\right) \
\text { where } \quad \gamma & =\frac{1}{\sqrt{1-(v / c)^2}} .
\end{aligned}
$$
Consequently, if the coordinates of an event in $S$ is $x_1=50 \mathrm{~m}, t_1=2.0 \times 10^{-7} \mathrm{~s}$, and the relative velocities of the two frames is $v=0.6 c$, then
$$
\begin{aligned}
x_1^{\prime} & =\frac{1}{\sqrt{1-(0.6)^2}}\left(50-0.6 \times 3 \times 10^8 \times 2.0 \times 10^{-7}\right)=17.5 \mathrm{~m} \
t_1^{\prime} & =\frac{1}{\sqrt{1-(0.6)^2}}\left(2.0 \times 10^{-7}-0.6 \times 50 / 3 \times 10^8\right)=1.25 \times 10^{-7} \mathrm{~s} .
\end{aligned}
$$
(b) Using the Lorentz transformation once again for the event $x_2=10 \mathrm{~m}, t_2=$ $3.0 \times 10^{-7} \mathrm{~s}$ in $S$ gives
$$
t_2^{\prime}=\frac{1}{\sqrt{1-(0.6)^2}}\left(3.0 \times 10^{-7}-0.6 \times 10 / 3 \times 10^8\right)=3.5 \times 10^{-7} \mathrm{~s} .
$$
Thus the time interval between the two events, as measured in $S^{\prime}$, is $t_2^{\prime}-t_1^{\prime}=$ $2.25 \times 10^{-7} \mathrm{~s}$.

问题 2.

A spaceship of length $100 \mathrm{~m}$ in its own rest frame $S_A$ passes a second spaceship $S_B$ at a relative speed of $\sqrt{3} c / 2$ and on a parallel course. When an observer at the centre of $S_A$ passes an observer located at the centre of $S_B$, a crew member of $S_1$ simultaneously fires very short bursts from two lasers mounted perpendicularly at the ends of $S_A$ so as to leave burn marks on the hull of $S_2$. The spaceships pass so close to each other that these laser beams travel negligibly short distances. Assume that the two observers are each at the origins of their respective rest frames, i.e. at $x_A=x_B=0$, and that these spatial origins are the midpoint of each of the space ships, and that the clocks are set to read zero i.e. $t_A=t_B=0$ in both reference frames when these two origins coincide (i.e. when the midpoints of each space ship are adjacent to each other.) Further assume that $S_B$ is of sufficient length that the laser beams will strike its hull.
(a) What are the coordinates of the two laser bursts (considered as events in spacetime) in $S_A$ ?
(b) What are the coordinates of these two events as measured in $S_B$ ?
(c) What is the distance between the marks appearing on the hull of $S_B$ ? Is this result an example of length contraction?

(a) The two laser bursts from spaceship $A$ occur at the points $x_{A 1}=50 \mathrm{~m}$ and $x_{A 2}=-50 \mathrm{~m}$ at time $t_A=0$.
(b) The coordinates of these events in the reference frame of the other spaceship are then
$$
\begin{aligned}
& x_{B 1}=\gamma\left(x_{A 1}-v t_A\right)=2(50-0)=100 \mathrm{~m} \
& x_{B 2}=\gamma\left(x_{A 2}-v t_A\right)=2(-50-0)=-100 \mathrm{~m}
\end{aligned}
$$
and
$$
\begin{aligned}
& t_{B 1}=\gamma\left(t_A-v x_{A 1} / c^2\right)=2\left(0-\sqrt{3} \times 50 / 2 \times 3 \times 10^8\right)=-2.9 \times 10^{-7} \mathrm{~s} \
& t_{B 2}=\gamma\left(t_B-v x_{B 1} / c^2\right)=2\left(0+\sqrt{3} \times 50 / 2 \times 3 \times 10^8\right)=2.9 \times 10^{-7} \mathrm{~s} .
\end{aligned}
$$
(c) The separation between the marks left on spaceship $B$ is $200 \mathrm{~m}$.
This appears to be an example of ‘length expansion’, but it in fact length contraction as can be seen by taking a closer examination of what is taking place here. Since determining the length of an object in some reference frame requires the simultaneous measurement of the positions of the endpoints of the object, we can consider what is taking place here as being the measurement, in reference frame $S_A$ of the separation of two points on the sides of spaceship $B$, a distance $200 \mathrm{~m}$ apart as measured in frame of reference $S_B$, i.e. this separation has a proper length $l_0=200 \mathrm{~m}$. Simultaneously in $S_A$, the separation between these two points is measured to be $l=100 \mathrm{~m}$. This distance is related to $l_0$ by $l=l_0 / \gamma$

Textbooks

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物理代写|PHYS322 electrodynamics

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Statistics-lab™可以为您提供washington.edu PHYS322 electrodynamics电动力学的代写代考和辅导服务！

PHYS322 electrodynamics课程简介

This course provides a deeper look into the theory of electricity and magnetism.

We shall study:

– magnetostatics

– magnetic fields in matter

– Maxwell’s equations

– electromagnetic waves

PREREQUISITES

Upon successful completion of this course you will be able to:

Understand the implications of Ampère’s and Biot-Savart’s laws
Understand the nature of the vector potential
Understand the behaviour of magnetic fields in medium
Be perfectly familiar with Maxwell’s equations
Know the nature of momentum and angular moment in Electrodynamics
Understand the details of propagation of electromagnetic waves

PHYS322 electrodynamics HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

Coaxial waveguides are often used in particle accelerators. There are several reasons for this;

The waveguide can receive, transport and transmit electromagnetic waves.
There is no leakage of waves.
The fundamental mode is the TEM mode, which propagates for all frequencies.
Both the phase speed and group speed of the fundamental mode are equal to the speed of light for all frequencies.

The losses in coaxial waveguides comes from the currents running along the outer and inner conductor. The losses decrease with increasing radii of the conductors. If the radii increase they eventually reach values where higher order modes start to propagate.

a) Consider a coaxial waveguide with vacuum between the conductors. Let the inner conductor have radius $a=1 \mathrm{~cm}$ and the outer conductor an inner radius of $b=2.3 \mathrm{~cm}$. Find the frequency where the second mode can start to propagate.
b) Determine if the second mode is a TE- or TM-mode.
Hints: 2D. Electromagnetic waves. Eigenfrequency. Booleans and Partitions. Difference. The fundamental mode does not show up as a resonance in Comsol so you should look for the lowest resonance frequency that is larger than zero.

a) To find the frequency where the second mode can start to propagate, we need to determine the cutoff frequency for the second mode. The cutoff frequency is the frequency below which a mode cannot propagate.

In a coaxial waveguide with vacuum between the conductors, the cutoff frequency for the $n$-th mode is given by:

f_{c,n}=\frac{1}{2\pi\sqrt{\epsilon_r}}\frac{c}{\sqrt{\left(\frac{m\pi}{b}\right)^2+\left(\frac{n\pi}{a}\right)^2}}fc,n=2πϵr1(bmπ)2+(anπ)2c

where $\epsilon_r$ is the relative permittivity of the dielectric material between the conductors (in this case, vacuum), $c$ is the speed of light, $a$ is the radius of the inner conductor, $b$ is the inner radius of the outer conductor, $m$ and $n$ are integers that determine the mode.

For the second mode, we have $m=1$ and $n=2$. Plugging in the values, we get:

f_{c,2}=\frac{1}{2\pi\sqrt{1}}\frac{c}{\sqrt{\left(\frac{\pi}{2.3 \mathrm{~cm}}\right)^2+\left(\frac{2\pi}{1 \mathrm{~cm}}\right)^2}}\approx 1.04 \mathrm{~GHz}fc,2=2π11(2.3 cmπ)2+(1 cm2π)2c≈1.04 GHz

Therefore, the frequency where the second mode can start to propagate is approximately $1.04 \mathrm{~GHz}$.

b) To determine whether the second mode is a TE- or TM-mode, we need to look at the electric and magnetic fields of the mode.

问题 2.

Consider that you like to use the coaxial waveguide in problem 1 for transmitting signals with frequencies between $100 \mathrm{MHz}$ and $2.2 \mathrm{GHz}$. Close to the transmitter there is a source that generates a signal with frequency $1.6 \mathrm{GHz}$. Also this signal is transmitted and you have to get rid of it. To do this you build a bandstop filter. You cut the coaxial waveguide in two pieces and put a short coaxial waveguide between the two pieces, as in the figure. The extra waveguide also has the radius $1 \mathrm{~cm}$ of the inner conductor. To find the proper radius $c$ of the outer conductor and the height $h$ you use Comsol. The flat surfaces between the outer conductors, marked in the figure, are also made of metal.

Determine $a$ and $h$ such that all frequencies except $1.6 \mathrm{GHz}$ pass. The width of the stop band should be less than $50 \mathrm{MHz}$ at the $3 \mathrm{~dB}$ level.

Hints: Axial symmetry. Type of port: Coaxial. S-parameters. What affects the frequency and what affects the bandwidth?

To design the bandstop filter for the coaxial waveguide, we need to determine the appropriate dimensions of the added short coaxial waveguide. We can use simulation software like Comsol to model the waveguide structure and simulate its response to different frequencies.

To start, we can define the geometry of the coaxial waveguide and the added short waveguide in Comsol, with axial symmetry as indicated in the problem statement. We can then define the material properties of the waveguide, such as the conductivity and permittivity of the metal, and set up a port boundary condition at the input and output of the waveguide.

We can then use the S-parameters of the waveguide to analyze its frequency response. The S-parameters are a set of parameters that describe the relationship between the incident and reflected signals at the input and output of the waveguide. By analyzing the S-parameters, we can determine the frequency and bandwidth of the filter.

To design a bandstop filter, we need to find the dimensions of the added short waveguide that will create a resonant structure at the frequency of the unwanted signal (1.6 GHz), which will cause the signal to be reflected and attenuated. The dimensions of the added waveguide will affect the frequency of the reson

Textbooks

Statistics-lab™可以为您提供washington.edu PHYS322 electrodynamics电动力学的代写代考和辅导服务！请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

物理代写|广义相对论代写General relativity代考|PHYS3100

Posted on 2023年3月29日2023年3月29日 by statistics-lab

如果你也在怎样代写广义相对论General relativity这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

广义相对论是阿尔伯特-爱因斯坦在1907至1915年间提出的引力理论。广义相对论说，观察到的质量之间的引力效应是由它们对时空的扭曲造成的。

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我们提供的广义相对论General relativity及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

物理代写|广义相对论代写General relativity代考|PHYS3100

物理代写|广义相对论代写General relativity代考|Motion of Test Particle

Let us consider the motion of a massive test particle or a massless particle, i.e., photon in Schwarzschild spacetime. It is known that all massive particles move along time-like geodesics, whereas photons move along null geodesics. We shall consider geodesics of test particles (either time-like or null) in Schwarzschild spacetime.
Let us take the Lagrangian in the following form as (with $p$ is an affine parameter)
$$
L=\left(1-\frac{2 m}{r}\right)\left(\frac{d t}{d p}\right)^2-\left(1-\frac{2 m}{r}\right)^{-1}\left(\frac{d r}{d p}\right)^2-r^2\left(\frac{d \theta}{d p}\right)^2-r^2 \sin ^2 \theta\left(\frac{d \phi}{d p}\right)^2
$$
We know
$$
\delta \int d s=0 \Rightarrow \int \delta\left(\frac{d s}{d p}\right) d p=0 \Rightarrow \delta \int L d p=0
$$
where
$$
L=\left(\frac{d s}{d p}\right)=\left(g_{\mu \gamma} \frac{d x^\mu}{d p} \frac{d x^\gamma}{d p}\right)^{\frac{1}{2}}
$$
$\Rightarrow$ Euler-Lagrangian equation
$$
\frac{d}{d p}\left(\frac{\partial L}{\partial\left(\frac{d x^\mu}{d p}\right)}\right)-\frac{\partial L}{\partial x^\mu}=0
$$
Thus, the corresponding Euler-Lagrange’s equations are
$$
\begin{aligned}
& \frac{d}{d p}\left(\frac{\partial L}{\partial r^1}\right)-\frac{\partial L}{\partial r}=0 \
& \frac{d}{d p}\left(\frac{\partial L}{\partial \theta^1}\right)-\frac{\partial L}{\partial \theta}=0, \text { etc. }
\end{aligned}
$$
[“1)” implies differentiation with respect to $p$ ]

物理代写|广义相对论代写General relativity代考|Bending of light

The relativistic equation for null geodesics, i.e., trajectory of the light QPE (see Fig. 19) is (putting $\epsilon=0$ in $\mathrm{Eq} .(7.11))$
$$
\frac{d^2 U}{d \phi^2}+U=3 m U^2
$$
In case of flat spacetime, i.e., when the deflecting source $S$ were absent $(m=0)$, then Eq. (7.17) becomes
$$
\frac{d^2 U}{d \phi^2}+U=0
$$
The solution of this equation is
$$
U=A \cos \phi+B \sin \phi
$$
Now, we use the following boundary conditions (i) $\phi=0$, when the value of $U$ is maximum, i.e., when the value of $r$ is minimum, i.e., at closest approach $\left(R_0\right), U=\frac{1}{R_0}$ (ii) at $\phi=0$, one can have turning point, i.e., $\frac{d U}{d \phi}=0$. Here, $R_0$ could be solar radius. Hence we get,
$$
U=U_0 \cos \phi=\frac{1}{R_0} \cos \phi
$$
Substituting this in R.H.S. of (7.17) for $U$, we get
$$
\frac{d^2 U}{d \phi^2}+U=3 m U_0^2 \cos ^2 \phi=\frac{3 G M}{R_0^2} \cos ^2 \phi
$$
We can find the particular solution as
$$
U=\frac{1}{D^2+1}\left[\frac{3 G M}{R_0^2} \cos ^2 \phi\right]=\frac{1}{D^2+1}\left[\frac{3 G M}{2 R_0^2}(1+\cos 2 \phi)\right]
$$ or
$$
U=\frac{G M}{2 R_0^2}(3-\cos 2 \phi)=\frac{G M}{R_0^2}\left(2-\cos ^2 \phi\right) .
$$

广义相对论代考

物理代写|广义相对论代写General relativity代考|Motion of Test Particle

让我们考虑一个有质量的测试粒子或一个无质量的粒子，即光子在 Schwarzschild 时空中的运动。众所周知，所有大质量粒子都沿着类时间测地线移动，而光子则沿着零测地线移动。我们将考虑 Schwarzschild 时空中的测试粒子 (类似时间或零) 的测地线。
让我们采用以下形式的拉格朗日量作为（与 $p$ 是仿射参数）
$$
L=\left(1-\frac{2 m}{r}\right)\left(\frac{d t}{d p}\right)^2-\left(1-\frac{2 m}{r}\right)^{-1}\left(\frac{d r}{d p}\right)^2-r^2\left(\frac{d \theta}{d p}\right)^2-r^2 \sin ^2 \theta\left(\frac{d \phi}{d p}\right)^2
$$
我们知道
$$
\delta \int d s=0 \Rightarrow \int \delta\left(\frac{d s}{d p}\right) d p=0 \Rightarrow \delta \int L d p=0
$$
在哪里
$$
L=\left(\frac{d s}{d p}\right)=\left(g_{\mu \gamma} \frac{d x^\mu}{d p} \frac{d x^\gamma}{d p}\right)^{\frac{1}{2}}
$$
$\Rightarrow$ 欧拉-拉格朗日方程
$$
\frac{d}{d p}\left(\frac{\partial L}{\partial\left(\frac{d x^\mu}{d p}\right)}\right)-\frac{\partial L}{\partial x^\mu}=0
$$
因此，对应的欧拉-拉格朗日方程为
$$
\frac{d}{d p}\left(\frac{\partial L}{\partial r^1}\right)-\frac{\partial L}{\partial r}=0 \quad \frac{d}{d p}\left(\frac{\partial L}{\partial \theta^1}\right)-\frac{\partial L}{\partial \theta}=0, \text { etc. }
$$
[“1)”意味着差异化 $p]$

物理代写|广义相对论代写General relativity代考|Bending of light

零测地线的相对论方程，即光 QPE 的轨迹（见图 19) 是（将 $\epsilon=0$ 在Eq. (7.11)）
$$
\frac{d^2 U}{d \phi^2}+U=3 m U^2
$$
在平坦时空的情况下，即当偏转源 $S$ 缺席 $(m=0)$ ，然后方程式。(7.17) 变成
$$
\frac{d^2 U}{d \phi^2}+U=0
$$
这个方程的解是
$$
U=A \cos \phi+B \sin \phi
$$
现在，我们使用以下边界条件 (i) $\phi=0$ ，当值 $U$ 是最大的，即，当值 $r$ 是最小值，即最接近 $\left(R_0\right), U=\frac{1}{R_0}$ (ii) 在 $\phi=0$ ，一个可以有转折点，即 $\frac{d U}{d \phi}=0$. 这里， $R_0$ 可能是太阳半径。因此我们得到，
$$
U=U_0 \cos \phi=\frac{1}{R_0} \cos \phi
$$
将 (7.17) 的 RHS 替换为 $U$ ，我们得到
$$
\frac{d^2 U}{d \phi^2}+U=3 m U_0^2 \cos ^2 \phi=\frac{3 G M}{R_0^2} \cos ^2 \phi
$$
我们可以找到特定的解决方案
$$
U=\frac{1}{D^2+1}\left[\frac{3 G M}{R_0^2} \cos ^2 \phi\right]=\frac{1}{D^2+1}\left[\frac{3 G M}{2 R_0^2}(1+\cos 2 \phi)\right]
$$
或者
$$
U=\frac{G M}{2 R_0^2}(3-\cos 2 \phi)=\frac{G M}{R_0^2}\left(2-\cos ^2 \phi\right)
$$

物理代写|广义相对论代写General relativity代考请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。

回归分析代写

多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习和应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

物理代写|广义相对论代写General relativity代考|MATH7105

Posted on 2023年3月29日2023年3月29日 by statistics-lab

如果你也在怎样代写广义相对论General relativity这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

广义相对论是阿尔伯特-爱因斯坦在1907至1915年间提出的引力理论。广义相对论说，观察到的质量之间的引力效应是由它们对时空的扭曲造成的。

我们提供的广义相对论General relativity及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

物理代写|广义相对论代写General relativity代考|MATH7105

物理代写|广义相对论代写General relativity代考|Isotropic Coordinates

Isotropic coordinate system is a new coordinate system whose spatial distance is proportional to the Euclidean square of the distances. Usually, isotropic means that all three spatial dimensions are treated as identical. Thus, in isotropic coordinate, the line element takes the form
$$
d s^2=A(r) d t^2-B(r) d \sigma^2
$$

In Cartesian coordinates, the line element of Euclidean three space is
$$
d \sigma^2=d x^2+d y^2+d z^2
$$
whereas in spherical polar coordinates
$$
x=r \sin \theta \cos \phi, y=r \sin \theta \sin \phi, z=r \cos \theta,
$$
the line element of Euclidean three space is
$$
d \sigma^2=d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2
$$
Here, the metric with $t=$ constant is conformally related to the metric of Euclidean space. Usually, isotropic coordinates are used when time $t=$ constant hypersurface, i.e., the three-dimensional subspace of spacetime requires to look like Euclidean. Generally, this type of coordinate system is used for modeling the gravitational field of a symmetrical object that does not discriminate between the $x, y$, or $z$ directions.

物理代写|广义相对论代写General relativity代考|Interaction between Gravitational

Electric and magnetic fields are generated when a charge is in motion, and it depends on space and time. This phenomenon is known as electromagnetism. The study of time-dependent electromagnetic fields and their behavior is described by a set of equations, known as Maxwell’s equations.

Before Maxwell, there were four fundamental equations of electromagnetism prescribed by several researchers. Maxwell improved those equations and composed them in the succeeding compact form known as Maxwell’s equations of electromagnetism.
(a) $\vec{\nabla} \cdot \vec{E}=\frac{1}{\epsilon_0} \rho, \quad[=0$, without charge in a region $]$
(b) $\vec{\nabla} \cdot \vec{B}=0$
(c) $\vec{\nabla} \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$,
(d) $\vec{\nabla} \times \vec{B}-\mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}=\mu_0 \vec{J}$.
(Ampere’s law with Maxwell’s corrections)
Ampere’s law:
$$
\vec{\nabla} \times \vec{B}=\mu_0 \vec{J}
$$
where, $E$ = electric field strength, $B=$ magnetic field strength, $J=$ current density, $\rho=$ charge density, and $\mu_0=$ magnetic permeability $=4 \pi \times 10^{-7}$ weber/amp-meter.
$\epsilon_0=$ electric permittivity of free space $=8.86 \times 10^{-12}$ coulomb $\sec ^2 /$ meter and $\epsilon_0 \mu_0=\frac{1}{c^2}$.

One can solve Maxwell’s equations for $B$ and $E$ in terms of a scalar function $\phi$ and a vector function $A$ as
(e) $\vec{B}=\operatorname{curl} \vec{A}$
[it comes from (b) as $\operatorname{div} \operatorname{curl} A=0$ ]
(f) $E=-\frac{\partial A}{\partial t}-\operatorname{grad} \phi$.
[It follows from $(c)$ as
$$
\nabla \times E=-\frac{\partial(\nabla \times A)}{\partial t}=-\nabla \times \frac{\partial A}{\partial t},
$$
where $A=$ magnetic potential and $\phi=$ electric potential.

广义相对论代考

物理代写|广义相对论代写General relativity代考|Isotropic Coordinates

各向同性坐标系是一种新的坐标系，其空间距离与距离的欧氏平方成正比。通常，各向同性意味着所有三个空间维度都被视为相同。因此，在各向同性坐标系中，线元的形式为
$$
d s^2=A(r) d t^2-B(r) d \sigma^2
$$
在笛卡尔坐标系中，欧氏三空间的线元为
$$
d \sigma^2=d x^2+d y^2+d z^2
$$
而在球面极坐标中
$$
x=r \sin \theta \cos \phi, y=r \sin \theta \sin \phi, z=r \cos \theta
$$
欧氏三空间的线元为
$$
d \sigma^2=d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2
$$
在这里，指标与 $t=$ 常数与欧几里德空间的度量共形相关。通常，时间使用各向同性坐标 $t=$ 常量超曲面，即时空的三维子空间要求看起来像欧几里德。通常，这种类型的坐标系用于模拟对称物体的引力场，不区分 $x, y$ ，或者 $z$ 方向。

物理代写|广义相对论代写General relativity代考|Interaction between Gravitational

电荷运动时会产生电场和磁场，它取决于空间和时间。这种现象被称为电磁学。与时间相关的电磁场及其行为的研究由一组方程描述，称为麦克斯韦方程。
在麦克斯韦之前，有几个研究人员规定了四个基本的电磁方程。麦克斯韦改进了这些方程，并将它们组合成随后的紧凑形式，称为麦克斯韦电磁方程。
(A) $\vec{\nabla} \cdot \vec{E}=\frac{1}{\epsilon_0} \rho, \quad[=0$, 一地区不收费 $]$
(乙) $\vec{\nabla} \cdot \vec{B}=0$
(C) $\vec{\nabla} \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$
(d) $\vec{\nabla} \times \vec{B}-\mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}=\mu_0 \vec{J}$
(经过麦克斯韦修正的安培定律)
安培定律：
$$
\vec{\nabla} \times \vec{B}=\mu_0 \vec{J}
$$
在哪里， $E=$ 电场强度， $B=$ 磁场强度， $J=$ 当前密度， $\rho=$ 电荷密度，和 $\mu_0=$ 磁导率 $=4 \pi \times 10^{-7}$ 韦伯/安培表。
$\epsilon_0=$ 自由空间的介电常数 $=8.86 \times 10^{-12}$ 库仑 $\sec ^2 /$ 仪表和 $\epsilon_0 \mu_0=\frac{1}{c^2}$.
可以求解麦克斯韦方程组 $B$ 和 $E$ 就标量函数而言 $\phi$ 和一个向量函数 $A$ 作为
(五) $\vec{B}=\operatorname{curl} \vec{A}$
[它来自 (b) 作为div curl $A=0$ ]
(女) $E=-\frac{\partial A}{\partial t}-\operatorname{grad} \phi$.
[它来自 $(c)$ 作为
$$
\nabla \times E=-\frac{\partial(\nabla \times A)}{\partial t}=-\nabla \times \frac{\partial A}{\partial t}
$$
在哪里 $A=$ 磁势和 $\phi=$ 电位。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写