Tag Archives: Bath代写

有限群的代表理论|MA40054 Representation theory of finite groups代写

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will be able to state and prove the main theorems of Maschke and Schur and be conversant with their many applications in representation theory and character theory. Moreover, they will be able to apply these results to problems in group theory.

这是一份Bath巴斯大学MA40054作业代写的成功案

有限群的代表理论|MA40054 Representation theory of finite groups代写

Let $n \geq 0, m \geq 1$. Let $\lambda \vdash n$.
(i) If $\lambda$ is an $m$-core, it is an $m^{\prime}$-core for any $m^{\prime} \geq 1$.
(ii) For any sequence $\gamma_{1}, \ldots, \gamma_{t}$ such that each $\gamma_{i}$ is an $m$-hook of $\lambda * \gamma_{1} *$ $\ldots * \gamma_{i-1}$ and $\lambda * \gamma_{1} * \ldots$ * $\gamma_{t}$ has no $m$-hook, the outcome $\lambda * \gamma_{1} * \ldots * \gamma_{t}$ is independent of the sequence $\gamma_{1}, \ldots, \gamma_{t}$. It is called the $m$-core of $\lambda$
(iii) (Iterated version of Murnaghan-Nakayama formula) With notation as above, let $x \in \mathcal{S}{n}$, write $x^{\prime} c{1} \ldots c_{t}$ with $x^{\prime} \in \mathcal{S}{n-t m}$ and $c{i}$ ‘s being disjoint cycles of order $m$ in $\mathcal{S}{[n-t m+1, \ldots, n]}$ Then $$ \chi^{2}\left(x^{\prime} c{1} \ldots c_{t}\right)=N_{\lambda, m} \chi^{\lambda^{\prime}}\left(x^{\prime}\right)
$$


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MA40054 COURSE NOTES ：

$F_{0}^{(\Sigma)}$ is therefore
$$
F_{0}^{(\Sigma)}(t)=\prod_{j \geq 1} \frac{\left(1-t^{2 j}\right)^{2}}{1-q t^{t^{j}}} .
$$
A similar argument in the full linear group, omitting the condition $\tilde{g}=g$, gives
$$
\Phi(t):=\prod_{d \geq 1} \mathcal{P}{d} 2 M{d}+N_{d}=\prod_{j \geq 1} \frac{\left(1-t^{j}\right)^{3}}{1-q t^{j}}
$$
hence
$$
F_{0}^{(\Sigma)}(t)=\Phi\left(t^{2}\right) \mathcal{P}{2}=\mathcal{P}{2} \prod_{d \geq 1} \mathcal{P}{2 d^{2 M{d}+N_{d}}} .
$$




数值优化和大规模系统|MA40050 Numerical optimisation and large-scale systems代写

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To provide a unified introduction to the matUnconstrained Optimisation: Line search vs trust region methods. Quasi-Newton and nonlinear conjugate gradient. Linear and nonlinear least squares.

这是一份Bath巴斯大学MA40050作业代写的成功案

数值优化和大规模系统|MA40050 Numerical optimisation and large-scale systems代写

The robust optimization formutation of the diet prob. lem can be stated as follows.
$\underset{x_{f}}{\operatorname{Minimize}} \sum_{f} x_{f}$
$$
+\omega \frac{1}{|S|} \sum_{s \in 0}\left[b_{e}-\sum_{f=N} a_{f c} x_{f}-a_{N_{c} c} x_{N c}\right]^{2}
$$
subject to
$\sum_{f} a_{f n} x_{f} \geqslant b_{n}, \quad$ for all $n$.


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MA40050 COURSE NOTES ：

$$
\int f_{\Delta 1}\left(\boldsymbol{x}{m}\right) P(d \omega) $$ subject to $x{a} \in X_{a}$ with probability 1 , and $x \in \mathbb{M}$.
A particularly interesting and important case arises when the decision problem has a dynamic structure with time stages $t=1, \ldots, T$ and
$$
x_{\theta}=\left(x_{\omega}(1), x_{\theta}(2), \ldots, x_{\omega}(T)\right)
$$




弹性|MA40049 Elasticity代写

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To provide a unified introduction to the mathematical modelling of elastic materials.

这是一份Bath巴斯大学MA40049作业代写的成功案

弹性|MA40049 Elasticity代写

Note the symmetries:
$$
g_{i j}^{\varepsilon}=g_{j i}^{\varepsilon}, \quad g^{i j, e}=g^{j i, \varepsilon}, \quad \Gamma_{i j}^{p, \varepsilon}=\Gamma_{j i}^{p, \varepsilon} .
$$
The volume element in the set $\Theta\left(\bar{\Omega}^{\varepsilon}\right)$ is $\sqrt{g^{\varepsilon}} d x^{\varepsilon}$, where
$$
g^{e}:=\operatorname{det}\left(g_{i j}^{\varepsilon}\right)
$$


英国论文代写Viking Essay为您提供实分析作业代写Real anlysis代考服务

MA40049 COURSE NOTES ：

We record in passing the symmetries
$$
A^{i j k l, \varepsilon}=A^{j i k l, \varepsilon}=A^{k l i j, \varepsilon}
$$
and the relations (satisfied because the mapping $\boldsymbol{\Theta}$ is of the special form given earlier)
$$
\Gamma_{\alpha 3}^{3, \epsilon}=\Gamma_{33}^{p, \varepsilon}=0 \text { and } A^{\alpha \beta \sigma 3, \varepsilon}=A^{\alpha 333, \varepsilon}=0 \text { in } \bar{\Omega}^{\varepsilon} .
$$




微分的分析和几何理论|MA40048 Analytical & geometrical theory of differential代写

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To give a thorough treatment of the fundamental theory of Galois on solvability of polynomials and the subtle interplay between group theory and field theory that arises in this context.

这是一份Bath巴斯大学MA40037作业代写的成功案

微分的分析和几何理论|MA40048 Analytical & geometrical theory of differential代写

Let $b \in \mathcal{G}$ be such that the integrals on the right hand sides of the equalifies
$$
\begin{aligned}
P(b) &=\int_{M} \exp (-\langle J, b\rangle) \mathrm{d} \lambda_{\omega} \text { and } \
E_{J}(b) &=\mathcal{E}{\rho{b}}(J)=\frac{1}{P(b)} \int_{M} J \exp (-\langle J, b\rangle) \mathrm{d} \lambda_{\omega}
\end{aligned}
$$
converge. The smooth probabilify measure on $M$ with density (with respect fo the Liouville measure $\lambda_{\omega}$ on $M$ )
$$
\rho_{b}=\frac{1}{P(b)} \exp (-\langle J, b\rangle)
$$
is called the generalized Gibbs statistical state associated to $b$. The functions $b \mapsto P(b)$ and $b \mapsto E_{J}(b)$ so defined an the subset of $\mathcal{G}$ made by elements $b$ for which the integrals defining $P(b)$ and $E_{J}(b)$ converge are called the partition function associated to the momentum map I and the mean value of J at generalized Gibbs states.


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MA40037 COURSE NOTES ：

The equalify
$$
E_{J}\left(\operatorname{Ad}{g} b\right)=\mathrm{Ad}{g^{-1}}^{} E_{J}(b)+\theta(g) $$ means that the map $E_{J}: \Omega \rightarrow \mathcal{G}^{}$ is equivariant with respect to the adjoint action of $G$ on the open subset $\Omega$ of its Liealgebra $\mathcal{G}$ and ifs affine action on the left on $\mathcal{G}^{}$ $$ (g, \zeta) \mapsto \mathrm{Ad}{g^{-1}} \zeta+\theta(g), \quad g \in G, \quad \xi \in \mathcal{G}^{}
$$
Proposition 17. The assumptions and notations are the same as those in Proposition 14. For each $b \in \Omega$ and each $X \in \mathcal{G}$, we have
$$
\begin{aligned}
\left\langle E{J}(b),[X, b]\right\rangle &=\langle\Theta(X), b\rangle \
D E_{J}(b)([X, b]) &=-\mathrm{ad}{X}^{} E{J}(b)+\Theta(X)
\end{aligned}
$$
where $\Theta=T_{e} \theta: \mathcal{G} \rightarrow \mathcal{G}^{}$ is the 1-cocycle of the Lie algebra $\mathcal{G}$ associated to the 1-cocycle $\theta$ of the Lie group $G$.




伽罗瓦理论|MA40037 Galois Theory代写

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To give a thorough treatment of the fundamental theory of Galois on solvability of polynomials and the subtle interplay between group theory and field theory that arises in this context.

这是一份Bath巴斯大学MA40037作业代写的成功案

伽罗瓦理论|MA40037 Galois Theory代写

Now $Q$ is a group of order $p^{n-1}$, so by the inductive hypothesis there is a sequence of subgroups $Q=Q_{0} \supset Q_{1} \supset \cdots \supset Q_{n-1}={1}$ with $Q_{i}$ a normal subgroup of $Q$ of index $p^{i}$ for each $i=1, \ldots, n-1$. Let $G_{i}=\pi^{-1}\left(Q_{i}\right)$ for $i=1, \ldots, n-1$ and $G_{n}={1}$. Then $G=G_{0} \supset G_{1} \supset \cdots \supset G_{n}={1}$ form a sequence as claimed. (Clearly $G_{i}$ has index $p^{i}$. Also, $G_{i}$ is a normal subgroup of $\mathrm{G}$ as if $g_{0} \in G_{i}$ and $g \in G$, then $g g_{0} g^{-1} \in G_{i}$ as $\pi\left(g g_{0} g^{-1}\right)=$ $\pi(g) \pi\left(g_{0}\right)(\pi(g))^{-1} \in Q_{i}$ as $Q_{i}$ is a normal subgroup of $\left.Q .\right)$

英国论文代写Viking Essay为您提供实分析作业代写Real anlysis代考服务

MA40037 COURSE NOTES ：

By renumbering if necessary, we may assume the transposition is $\tau=$ (12).

Since $G$ acts transitively on ${1, \ldots, p}$, it has order divisible by $p$ and hence it has an element $\sigma_{0}$ of order $p$. Since $p$ is a prime, $\sigma_{0}$ is a p-cycle. Thus there is some power $\sigma=\sigma_{0}^{k}$ of $\sigma_{0}$ with $\sigma(1)=2$. By renumbering if necessary, we may then assume $\sigma=\left(\begin{array}{llll}1 & 2 & \cdots & p\end{array}\right)$. Now direct calculation shows that
$$
\sigma^{j} \tau \sigma^{-j}=(12 \cdots p)^{j}(12)(12 \cdots p)^{-j}=((j+1)(j+2))
$$
for $j=0, \ldots, p-2$. Direct calculation then shows that
$$
\begin{aligned}
(23)(12)(23) &=(13), \
(34)(13)(34) &=(14), \
\vdots & \
((p-1) p)(1(p-1))((p-1) p) &=(1 p)
\end{aligned}
$$




图形和网络|MA30245 Graphs & Networks代写

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T

  • Problem sheet nn will be released digitally (on Moodle) on Wednesday of week nn. You can hand in your work during the problems class in week n+1n+1, when the solutions will be discussed. Marked work will be returned in week n+2n+2.

这是一份Bath巴斯大学MA30245作业代写的成功案

图形和网络|MA30245 Graphs & Networks代写


The degree sequence of a graph $G$ is the list
$$
\boldsymbol{d}(G)=\left(d_{1}, \ldots, d_{|V|}\right)
$$
of degrees of vertices in $G$, written in descending order
$$
d_{1} \geq d_{2} \geq \ldots \geq d_{|V|}
$$
Note that $d_{1}=\Delta(G)$ and $d_{|V|}=\delta(G)$.
An arbitrary sequence $\boldsymbol{d} \in \mathbb{N}^{n}$ is said to be graphical if there exists a graph $G$ for which $\boldsymbol{d}$ is the degree sequence.

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MA30245 COURSE NOTES ：

Consider the star graph again with $n$ vertices attached to a single central vertex. The outer vertices $2, \ldots, n+1$ are all identical, so $\kappa_{i}=\kappa_{2}$ for $i=3, \ldots, n$. We therefore have to solve
$$
\kappa_{1}=\alpha n \kappa_{2}+1, \quad \kappa_{2}=\alpha \kappa_{1}+1
$$
The solution is
$$
\kappa_{1}=\frac{1+n \alpha}{1-n \alpha^{2}}, \quad \kappa_{2}=\frac{1+\alpha}{1-n \alpha^{2}}
$$
So the central vertex has a higher score than the outer vertices. For example, if $n=10$ and $\alpha=0.2$ then $\kappa_{1}=5, \kappa_{2}=2$.




拖影几何|MA30231 Projective geometry代写

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This course introduces basic notions in projective geometry using linear algebra. It aims to strengthen understanding of linear algebra by demonstrating its geometrical significance, while also pointing towards more advanced algebraic geometry.

这是一份Bath巴斯大学MA30231作业代写的成功案

拖影几何|MA30231 Projective geometry代写


we have
$$
T^{\prime}\left(v_{i}\right)=\lambda_{i} T\left(v_{i}\right)
$$
for some non-zero scalars $\lambda_{i}$ and also
$$
T^{\prime}\left(\sum_{i=0}^{n} v_{i}\right)=\lambda T\left(\sum_{i=0}^{n} v_{i}\right)
$$
for some non-zero $\lambda$. But then
$$
\sum_{i=0}^{n} \lambda T\left(v_{i}\right)=\lambda T\left(\sum_{i=0}^{n} v_{i}\right)=T^{\prime}\left(\sum_{i=0}^{n} v_{i}\right)=\sum_{i=0}^{n} \lambda_{i} T\left(v_{i}\right) .
$$

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MA30231 COURSE NOTES ：

$$
\tau\left(\left[z_{0}, z_{1}\right]\right)=\left(\left[a z_{0}+b z_{1}, c z_{0}+d z_{1}\right]\right)
$$
where $a d-b c \neq 0$. This corresponds to the invertible linear transformation
$$
T=\left(\begin{array}{ll}
a & b \
c & d
\end{array}\right)
$$
It is convenient to write $P^{1}(\mathbf{C})=\mathbf{C} \cup{\infty}$ where the point $\infty$ is now the 1-dimensional space $z_{1}=0$. Then if $z_{1} \neq 0,\left[z_{0}, z_{1}\right]=[z, 1]$ and
$$
\tau([z, 1])=[a z+b, c z+d]
$$
and if $c z+d \neq 0$ we can write
$$
\tau([z, 1])=\left[\frac{a z+b}{c z+d}, 1\right]
$$
which is the usual form of a Möbius transformation, i.e.
$$
z \mapsto \frac{a z+b}{c z+d} .
$$




PDEs的数值解法|MA30170 Numerical Solution of PDEs代写

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The course will teach you how to derive and implement the finite element method for a range of standard elliptic and parabolic partial differential equations in one and several space dimensions.

这是一份Bath巴斯大学MA30170作业代写的成功案

PDEs的数值解法|MA30170 Numerical Solution of PDEs代写

Conservation of mass:
$$
\begin{array}{ll}
\text { (water) } & \frac{\partial}{\partial t}\left(\rho_{W} \phi S_{W}\right)+\operatorname{div}\left(\rho_{W} \phi S_{W} \vec{V}{W}\right)=\rho{W} q_{W} \
\text { (oil) } & \frac{\partial}{\partial t}\left(\rho_{O} \phi S_{O}\right)+\operatorname{div}\left(\rho_{O} \phi S_{O} \vec{V}{O}\right)=\rho{O} q_{O}
\end{array}
$$
Conservation of momentum (1D vertical):
$$
\frac{\partial \sigma_{z}}{\partial z}=\left(\phi\left(S_{W} \rho_{W}+S_{O} \rho_{O}\right)+(1-\phi) \rho_{S}\right) g
$$
Rheology (elastoviscoplastic law) ${ }^{\dagger}$ :
$$
\frac{d_{s} \phi}{d_{s} t}=-\beta\left(\phi, \sigma_{e f f}\right) \frac{d_{s} \sigma_{e f f}}{d_{s} t}-\alpha\left(\phi, \sigma_{e f f}\right) \sigma_{e f f} \quad \text { with } \quad \sigma_{e f f}:=\sigma_{z}-P_{f}
$$

英国论文代写Viking Essay为您提供实分析作业代写Real anlysis代考服务

MA30170 COURSE NOTES ：

A bilinear form is called an inner product on $V$ if it satisfies “symmetry”
$$
a(v, w)=a(w, v) \quad \forall v, w \in V
$$
and if
$$
a(v, v) \geq 0 \quad \forall v \in V
$$
with equality iff $v=0$.
If $a$ is an inner product, we can define the induced norm
$$
|v|_{a}:=a(v, v)^{\frac{1}{2}}
$$
and the Cauchy-Schwarz inequality
$$
|a(v, w)| \leq|v|_{a}|w|_{a} \quad \forall v, w \in V
$$




随机过程与金融|MA30089 Stochastic processes & finance代写

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To perform simple calculations to compute certain quantities relating to Brownian motion, and to understand how these quantities can be important in pricing financial derivatives.

这是一份Bath巴斯大学MA30089作业代写的成功案

随机过程与金融|MA30089 Stochastic processes & finance代写

Let $(P(\cdot \mid X=x)){x \in E}$ be a regular version of the conditional probability with respect to $X$. Then, for any $Y \in \mathcal{L}^{1}(\Omega, \mathcal{F}, P)$ : $$ \int Y(\omega) d P(\omega \mid X=x)=E[Y \mid X=x], \quad P{X}-a . s .
$$
Proof: First, we observe that $Y$, being a random variable, is measurable. ${ }^{3}$ Now from (1.6) it follows that
$$
E\left[I_{F} \mid X=x\right]=P(F \mid X=x)=\int I_{F}(\omega) P(d \omega \mid X=x)
$$
for every $x \in E, P_{X}(x) \neq 0$. Now let $Y$ be an elementary function so that
$$
Y=\sum_{i=1}^{n} \lambda_{i} I_{F_{i}}
$$

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MA30089 COURSE NOTES ：

If $Y:(\Omega, \mathcal{F}) \rightarrow\left(\mathbb{R}, \mathcal{B}{\mathbb{R}}\right)$ is a positive random variable, then $$ E[Y \mid X=x] \geq 0, \quad P{X}-a . s .
$$
Proof: Since
$$
\int_{[X \in B]} Y d P \geq 0 \quad \forall B \in \mathcal{B}
$$
it follows that
$$
\int_{B} E[Y \mid X=x] P_{X}(d x) \geq 0 \quad \forall B \in \mathcal{B}
$$
and therefore $E[Y \mid X=x] \geq 0$, almost surely with respect to $P_{X}$.




时间序列|MA30085 Time series代写

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To introduce a variety of statistical models for time series, cover the main methods for analysis and give practical experience in fitting such models.

这是一份Bath巴斯大学MA30085作业代写的成功案

时间序列|MA30085 Time series代写

so that if
$$
\begin{gathered}
\nabla^{d} Y_{t}=y_{t} \text { and } \nabla^{d} X_{t}=x_{t} \
y_{t}=\delta^{-1}(B) \omega(B) x_{t-b}+\phi^{-1}(B) \theta(B) a_{t}
\end{gathered}
$$
Also, we shall assume that an adequate stochastic model for the leading series is
$$
X_{t}=\varphi_{x}^{-1}(B) \theta_{x}(B) \alpha_{t}
$$
so that with
$$
\begin{aligned}
\varphi_{x}(B) &=\phi_{x}(B) \nabla^{d} \
x_{t} &=\phi_{x}^{-1}(B) \theta_{x}(B) \alpha_{t}
\end{aligned}
$$

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MA30085 COURSE NOTES ：

Using a similar approach, the autocovariance generating function of the noise $n_{t}$ is
$$
\gamma^{n n}(B)=\gamma^{y y}(B)-\frac{\gamma^{x y}(B) \gamma^{x y}(F)}{\gamma^{x x}(B)}
$$
On substituting $B=e^{-i 2 \pi f}$ , we obtain the expression
$$
p_{n n}(f)=p_{y y}(f)\left[1-\kappa_{x y}^{2}(f)\right]
$$
for the spectrum of the noise, where
$$
\kappa_{x y}^{2}(f)=\frac{\left|p_{x y}(f)\right|^{2}}{p_{x x}(f) p_{y y}(f)}
$$