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7

We claim that given the conditions set forth in the problem, $\text{Ind}_{Z(G)}^G \rho$ is always not irreducible. Set $Z = Z(G)$. Let $W$ be an irreducible $Z$-representation. We would like to compute$$\langle\text{Ind}\,W, \text{Ind}\,W\rangle_G = \langle W, \text{Res}\,\text{Ind}\,W\rangle_H$$by Frobenius reciprocity. Since $Z$ is abelian, we know $W$ is ...


4

There's also a well known overpowered proof of the infinitude of primes: If $P$ is the set of primes, the Euler product formula tells us $$\prod\limits_{p\in P}\frac{1}{1-\frac{1}{p^2}}=\zeta(2)=\frac{\pi^2}{6}$$ However, $\pi$ is transendental, so $|P|$ cannot be finite (otherwise the product would be rational).


4

Let $\{V_j\to S\}$ be a Zariski cover of $S$. If $f:X\to S$ is proper, then the base-change $f_j=f|_{f^{-1}V_j} : f^{-1} V_j = X \times_S V_j \to V_j$ is proper. Conversely, assume that $f_j: f^{-1}V_j \to V_j$ is proper for all $j$. Then $f_j$ is finite type and separated. In particular, since finite type and separated are local in the Zariski topology, ...


4

Here are two candidates providing completely different methods of guided discovery. The first one with focus on visual aha experience: Visual Complex Analysis by T. Needham is a guided tour through Complex Analysis with plenty of illuminating pictures providing additional insight (and additional aesthetic pleasure). From the preface: In the hope of ...


3

My answer is quite philosophical and the books concerned with guided discovery, which I read, are less concrete than proposed by the others: George Polya, “Mathematical discovery: on understanding, learning and teaching” Henri Poincaré, “Mathematical Creation”. Imre Lakatos, “Proofs and refutations. The Logic of Matematical Discovery” Jacques Hadamard, ...


3

The set of functions from a domain $X$ to a field $R$ form a vector space of dimension $|X|$ over $R$. If $X$ is infinite then the vector space is infinite dimensional. The usual spaces $L^p(X,\mu)$ are, however, defined slightly differently, by taking equivalence classes of functions equal up to measure zero (and, of course, having finite $L^p$ norm under ...


3

In my old paper I used Four Colors Theorem to prove a two dimensional case of the following Proposition. Every open subset of the space $\Bbb R^m$ can be partitioned into $n$ homeomorphic parts if $n\ge 2^{m+1}-1$ or $m=2$ and $n\ge 4$. Proof. For every positive integer $k$ let $A_k$ be the set of all points $x\in\Bbb R^m$ such that all coordinates of the ...


3

No, in general this won't be irreducible: for example, if $G$ is a non-trivial finite group with $Z(G)$ trivial, then $\text{Ind}_{Z(G)}^G\text{triv}$ is the regular $G$-representation on ${\mathbb C}[G]$, which contains the proper non-trivial submodule ${\mathbb C}\cdot\sum_g g$.


3

What you call radicals are called square free numbers. Some of your questions have been studied.


2

Without knowing exactly what properties you would like to test I hope the following will be of some help. The best reference for families of high rank is the tables by Andrej Dujella. The tables are available from his web page here and are very much up-to-date as he is himself a researcher of the subject. If you dig into the very extensive bibliography ...


2

If $J$ is only locally small, the definition of $\mathrm{Lan}(f)(F)$ still makes sense even if $f^\ast$ is not definable without jumping to a bigger universe. It satisfies the 'local' adjointness property : there is a natural transformation $\eta \colon F \to \mathrm{Lan}(f)(F) \circ f$ universal in the sense that any $F \to G \circ f$ factors through ...


2

In Algebraic Topology by Allen Hatcher: Theorem 3.26. Let $M$ be a closed, connected $n$-manifold. Then: (a) If $M$ is $R$-orientable, the map $H_n(M;R) \to H_n(M | x; R) \approx R$ is an isomorphism for all $x \in M$. (b) If $M$ is not $R$-orientable, the map $H_n(M; R) \to H_n(M | x; R) \approx R$ is injective with image $\{ r \in R | 2r = 0 ...


2

The classic Galois Theory from 1942 by Emil Artin. Not modern but clear.


2

I prefer "Galios Theory" by Ian Stewert, and "Galios Theory of equations" by David A Cox. Cox is not so bad. Stewert is also good, rather different and interesting.


2

If you google "Galois theory pdf" amongst the first four alone are excellent notes by Milne, Baker, and Reid. These are all excellent teachers and the notes are actually complete texts that are generously made available for free.


2

There is a proof and it's pretty quick. Below, $\mathbb{F}$ is used to denote the trivial representation of the trivial group. Let $R$ be the regular representation of a group G. Then, $R = \mathrm{Ind}_{1}^{G}(\mathbb{F})$. Let $V$ be an irreducible representation of $G$. Then, by Frobenius reciprocity $$\dim \mathrm{Hom}_{G}(R, V) = \dim ...


2

There is definitely some general relation between the two subjects within complexity theory, and as likely one of the few people who know what both fields entail, I've not come across anything specific or even general linking the two fields in a direct manner. I've seen some research that ties the two fields separately to other areas in an indirect manner, ...


2

Combinatorial Problems and Exercises by Laszlo Lovasz is a good book to learn advanced Combinatorial Techniques through guided discovery. The book is divided into three parts: One consisting of problems, the second of hints and the third of solutions. The book is not suitable for a beginner though. Numbers and Functions: Steps into Analysis by R. P. Burn is ...


2

(2) If $G$ is an {\em arbitrary} infinite amenable countable group, then $H^1(G,\ell^2(G))$ is nonzero (because it is not Hausdorff!: the subspace of coboundaries $B^1(G,\ell^2(G))$ is not closed in the space of cocycles $Z^1(G,\ell^2(G))$. Hence if $G$ is a non-virtually-cyclic amenable countable group, e.g. $\mathbf{Z}^2$, it answers (2). (1) The surface ...


2

Another way of looking at this is through the Lebesgue Differentiation theorem, that says: If $v\in L^p_{loc}(\Omega)$ then for a.e. $x\in \Omega$ we have $$ g_v(x,r):= \frac{1}{|B(x,r)|}\int_{B(x,r)} |v|^p dy \to |v(x)|^p \qquad \text{ as } r\to 0. $$ Your hypothesis implies that $g_{\nabla u} (x,r)\leq M$ and so $|\nabla u(x)|^p \leq M$ for a.e. $x$. ...


2

This is not quite true, since, for example, the open unit disk is embedded in $\mathbb{R}^2$ by the inclusion, and the inverse image of the closed unit disk is the open unit disk. What we will show is that embeddings with closed image are the same thing as proper injective immersions. We note that the property of being closed in a topological space is a ...


1

A Primer on Mapping Class Groups by Benson Farb and Dan Margalit is a good bet. You can also take a look at the answers to a similar question on MathOverflow.


1

As far as I concerned ,this property of embedding requires that $f(X)$ is closed in Y. There is a counter-example .(When $f(X)$ is not closed in Y) Let Y = $\{k_\lambda : \lambda \in A\} $ where A is a uncountable set. Let X be $\{k_1,\cdots,k_n,\cdots\}$ , an infinite countable subset of A. Topology of X is discrete topology , hence X is not compact in ...


1

Closed embeddings (equivalently, embeddings with closed image) are precisely proper injective immersions. $(\Rightarrow)$ If $f : M \to N$ is an embedding with $f(M)$ closed, then it is obviously proper (since the intersection of a compact set with $f(M)$ is compact). $(\Leftarrow)$ On the other hand, if $f : M \to N$ is a proper, injective immersion, it ...


1

Jones' index rigidity formula comes to mind. In fact, there was no way to "observe" it in the special cases. It was widely believed that the index could take all values between $1$ and $\infty$- so the theorem came as a surprise. (The theorem states that the index takes values in a discrete series between $1$ and $4$).


1

I would be very surprised if this would not be in Gilbarg/Trudinger "Elliptic Partial Differential Equations of Second Order" - I would start looking in Chapter 6 "Classical Solutions" (in fact they distinguish between "classical solutions" that are $C^2$ and "strong solutions" that are $W^{2,p}$). An example added by another user: The following result is ...


1

I think you need to at least combine the result of existence & uniqueness with the result of regularity. For existence & uniqueness w.r.t a solution $u\in H_0^1(\Omega)$, I would suggest the first existence theorem in Evans book, chapter 6.2, look for Lax-Milgram. But I think you may need to assume in addition that $0<\alpha\leq \gamma$, that is, ...


1

You can find a lecture on Hyperbolic Conservation Laws, given by Constantine Dafermos here. And here a course on unique continuation and nonlinear dispersive equations, given by Gustavo Ponce.


1

Here is answer to your Question .Complete course of PDE are available there http://nptel.ac.in/courses.php THese lectures are there in youtube channel 'nptel" but contents and syllabus can be seen from link above



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