querent

Suppose $𝑅$ is a Dedekind domain with fractional field $𝐾$. Let $𝐼$ and $𝐽$ be nonzero ideals of $𝑅$.

(a) Show that there is some $𝛼\in {𝐾}^{\times }$ so that $𝛼𝐼$ is contained in $𝑅$, and in addition $𝛼𝐼+𝐽=𝑅$. (Hint: First deal with the case $𝐼=𝔭$ is a prime and $𝔭|𝐽$, using Chinese reminder theorem or prime avoidance lemma.)

(b) Show that there is an isomorphism of $𝑅$-modules $𝐼\oplus 𝐽\cong 𝑅\oplus 𝐼𝐽$. (Hint: First do the case $𝐼+𝐽=𝑅$, and then use (a) to reduce to this case.)

Let $𝑀$ be a finitely generated module over a commutative ring $𝑅$ and let $𝑁\subset 𝑀$ be a submodule. For each positive integer $𝑑$, let ${𝑀}^{\otimes 𝑑}$ is the $𝑑$th tensor power of $𝑀$ as an $𝑅$-module. Let $𝐺\subset {GL}_{𝑅}(𝑀)$ be the subgroup of $𝑅$-linear automorphisms of $𝑀$ that preserve $𝑁$. We let $𝑔\in {GL}_{𝑅}(𝑀)$ act on ${𝑀}^{\otimes 𝑑}$ by $𝑔({𝑚}_1\otimes \cdots \otimes {𝑚}_{𝑑})=𝑔({𝑚}_1)\otimes \cdots \otimes 𝑔({𝑚}_{𝑑})$.

(a) Let ${𝑒}_1,{𝑒}_2,\dots ,{𝑒}_{𝑛}\in 𝑁$ generate $𝑁$ as an $𝑅$-module, and let

Here ${𝑆}_{𝑛}$ is the symmetric group on $𝑛$ letters, and $sgn(𝜎)$ denotes the sign of $𝜎\in {𝑆}_{𝑛}$. Show that for $𝑔\in 𝐺$, we have $𝑔𝑡=𝑟𝑡$ for some $𝑟\in 𝑅$.

(b) Suppose $𝑀$ is a free $𝑅$-module with a basis $\{{𝑒}_1,\dots ,{𝑒}_{𝑚}\}$ and suppose $𝑁$ is generated by $\{{𝑒}_1,\dots ,{𝑒}_{𝑛}\}$. Show that $𝑡$ as in (a) can be extended to a basis of ${𝑀}^{\otimes 𝑛}$.

(c) Assumptions are as in (b). Show that $𝐺=\{𝑔\in {GL}_{𝑅}(𝑀)|𝑔𝑡\in 𝑅𝑡\}$.

(a) Suppose that $𝐸$ is a $2\times 2$ matrix over a field $𝐾$ with ${𝐸}^2=𝐸$. Show that either $𝐸=0$ or $𝐸=1$ or there exists a matrix $𝑔\in {GL}_2(𝐾)$ with $𝑔𝐸{𝑔}^{-1}=𝑃$, where $𝑃=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$.

(b) We identify the set ${Mat}_2(ℂ)$ of $2\times 2$ complex matrices with an affine 4-space via matrix entries ${𝑥}_{𝑖𝑗}$. Let $𝑉$ be the subset of all matrices $𝐸$ satisfying ${𝐸}^2=𝐸$. We also identify the subset ${GL}_2(ℂ)\subset {Mat}_2(ℂ)$ of invertible matrices as a subset of an affine 5-space by sending $𝐴=({𝑎}_{𝑖𝑗})$ to $({𝑎}_{11},{𝑎}_{12},{𝑎}_{21},{𝑎}_{22},{det(𝐴)}^{-1})$. Explain why $𝑉$ is a Zariski closed subset of ${𝔸}^4$ and why ${GL}_2(ℂ)$ is a Zariski closed subset of ${𝔸}^5$.

(c) Let $𝐸\in 𝑉$. Explain why the map of sets ${𝜙}_{𝐸}:{GL}_2(ℂ)\to 𝑉$ defined by $𝑔\mapsto 𝑔𝐸{𝑔}^{-1}$ is an algebraic (also called polynomial) mapping of affine algebraic sets.

(d) Let $𝐸=0,1$ and $𝑃$ be as in (a). Show that

is the decomposition of $𝑉$ into irreducible components.

Suppose $𝐵$ is a (commutative) ring, and $𝐴\subset 𝐵$ is a subring.

(a) State what it means for an element $𝑏\in 𝐵$ to be integral over $𝐴$.

(b) Suppose $𝐵-𝐴$ is closed under multiplication. Show that $𝐴$ is integrally closed in $𝐵$, i.e., any element $𝑏\in 𝐵$ that is integral over $𝐴$ is already in $𝐴$.

(c) Suppose $𝐵$ is a field. Show that the pair $𝐴\subset 𝐵$ satisfies the condition in (b) if and only if for every nonzero element $𝑥\in 𝐵$, either $𝑥$ or ${𝑥}^{-1}$ belongs to $𝐴$.

Let $𝐺$ be a finite group and let $𝐻$ be a normal subgroup of index 2. Assume that for every irreducible character $𝜒$ of $𝐺$, $𝜒$ remains irreducible when restricted to $𝐻$.

(a) Let $𝜓$ be an irreducible character of $𝐻$. Prove that the induced character ${𝜓}^{𝐺}$ splits as ${𝜒}_1+{𝜒}_2$, where ${𝜒}_1$ and ${𝜒}_2$ are distinct irreducible characters of $𝐺$, and that the restriction of either ${𝜒}_{𝑖}$ to $𝐻$ agrees with $𝜓$.

(b) Let $𝑥,𝑦\in 𝐻$ be conjugate in $𝐺$. Prove that they are conjugate in $𝐻$.

Let $𝐴$ be an invertible real symmetric $𝑛\times 𝑛$ matrix, and let $𝐵:{ℝ}^{𝑛}\times {ℝ}^{𝑛}\to ℝ$ be the symmetric bilinear form defined by $𝐵(𝑣,𝑤)={𝑣}^{𝑡}𝐴𝑤$. Here we regard $𝑣\in {ℝ}^{𝑛}$ as an $𝑛\times 1$ matrix and let ${𝑣}^{𝑡}$ denote its transpose.

(a) Let ${𝐴}_{𝑖}$ denote the symmetric upper left $𝑖\times 𝑖$ submatrix of $𝐴$. Show that $𝐴$ is positive definite if and only if $det{𝐴}_{𝑖}>0$ for all $𝑖=1,\dots ,𝑛$.

(b) Show that if $𝐴$ is positive definite, then any maximal entry of $𝐴$ is on the diagonal.

(c) Suppose $𝑛=2𝑚$ and there is an $𝑚$-dimensional subspace $𝐻\subset {ℝ}^{𝑛}$ such that $𝐵(𝑣,𝑤)=0$ for every $𝑣,𝑤\in 𝐻$. Show that $𝐴$ can be written as $𝐴={𝐶}^{𝑡}\left(\begin{array}{cc}{𝐼}_{𝑚}& \\ & -{𝐼}_{𝑚}\end{array}\right)𝐶$, where ${𝐼}_{𝑚}$ is the $𝑚\times 𝑚$ identity matrix, $𝐶$ is an invertible $𝑛\times 𝑛$-matrix, and ${𝐶}^{𝑡}$ is the transpose of $𝐶$.

Let $𝑝$ be an odd prime number and $𝑒$ a positive integer. Consider the commutative ring $\mathbb{Z}/{𝑝}^{𝑒}\mathbb{Z}$.

(a) Show that the group ${(\mathbb{Z}/{𝑝}^{𝑒}\mathbb{Z})}^{\times }$ of invertible elements is a cyclic group of order $(𝑝-1){𝑝}^{𝑒-1}$. (The case $𝑒=1$, which states that the multiplicative group of a finite field is cyclic, may be assumed.)

(b) Determine the number of elements in the ring $\mathbb{Z}/{𝑝}^{𝑒}\mathbb{Z}$ that are squares.

Let $𝑘$ be a field. Let $𝑛$ be an odd positive integer. Let $𝑅=𝑘[𝑋,𝑌]/({𝑌}^2-{𝑋}^{𝑛})$. This problem aims to compute ${Tor}_{𝑖}^{𝑅}(𝑘,𝑘)$ for $𝑖\ge 1$.

(a) Show that $𝑅$ is a domain by showing that ${𝑌}^2-𝑓(𝑋)$ is irreducible in $𝑘[𝑋,𝑌]$ for any non-square $𝑓\in 𝑘[𝑋]$.

(b) Let $𝑎,𝑏\ge 1$ be positive integers such that $𝑛=𝑎+𝑏$, and let $𝑐\in {𝑘}^{\times }$. Prove exactness of the $𝑅$-linear complex

for $𝑀=\left(\begin{array}{cc}𝑌& (-1/𝑐){𝑋}^{𝑎}\\ (-𝑐){𝑋}^{𝑏}& 𝑌\end{array}\right)$ and ${𝑀}^{\prime }=\left(\begin{array}{cc}𝑌& (1/𝑐){𝑋}^{𝑎}\\ 𝑐{𝑋}^{𝑏}& 𝑌\end{array}\right)$. (Hint: $𝑅$ has $\{1,𝑌\}$ as a $𝑘[𝑋]$-basis.)

(c) Prove exactness of

for the matrix $𝐿=\left(\begin{array}{cc}𝑌& 𝑋\\ {𝑋}^{𝑛-1}& 𝑌\end{array}\right)$. Using (a), construct a free resolution of $𝑘=𝑅/(𝑋,𝑌)$ as an $𝑅$-module, and compute ${Tor}_{𝑖}^{𝑅}(𝑘,𝑘)$ for all $𝑖\ge 1$.

Consider the following problem on $[0,1]$ with the periodic boundary condition

with $𝑓\in {𝐿}^2$ and $𝜅>0$.

(a) Identify an appropriate Sobolev space for this problem.

(b) Derive the variational formulation of this problem with respect to this space.

(c) Consider a finite element mesh with uniform spacing $ℎ=1/𝑛$ for an integer $𝑛$. Write down explicitly the linear system of the piecewise linear finite element method.

(d) Estimate the condition number of the linear system. How does the condition number depend on $𝑛$ and $𝜅$?

Consider the weight function $𝑤(𝑥)=exp(-{𝑥}^2/2)$ on $ℝ$.

(a) The orthogonal polynomial family with respect to $𝑤(𝑥)$ is the Hermite polynomials. Describe how to construct these orthogonal polynomials from low to high degrees. Please list the first three of them.

(b) Describe how to construct the $𝑛$-point Gaussian quadrature with respect to $𝑤(𝑥)$.

(c) What is the order of this $𝑛$-point Gaussian quadrature?