22

That isn't a conjecture but a routine exercise assigned after the students learn about Bang's solution of the Tarski plank problem. The proof goes in 2 steps: 1) Consider all sums $\sum_j \varepsilon_i u_i$ with $\varepsilon_i=\pm 1$ and choose the longest one. Replacing some $u_j$ with $-u_j$ if necessary, we can assume WLOG that it is $y=\sum_i u_i$. ...


16

For $k=2$ with $n \equiv 2 \mod 4$: $$ \pmatrix{1 & (-1)^{(n-2)/4} 2^{n/2} n^{n-1} \cr 0 & 1}^n + \pmatrix{n & -n\cr n & n\cr}^n = \pmatrix{1 & 0\cr (-1)^{(n-2)/4} 2^{n/2} n^{n-1} & 1\cr}^n $$ An example for $n=4$ is $$ \pmatrix{3 & -2\cr 1 & 2\cr}^4 + \pmatrix{2 & -4\cr 2 & 0\cr}^4 = \pmatrix{1 & 2\cr -1 &...


6

Nice question! More generally, let $V=\mathbb{C}^{2n}$. Consider the $2n\times 2n$ matrix $$ \varepsilon=\begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix} $$ and the symplectic group ${\mathsf{S}\mathsf{p}}_{2n}$ which preserves the fundamental alternating bilinear form with matrix $\varepsilon$. An element $F$ of the symmetric power $S^p(V^{\vee})$ ...


6

My comments converted to an answer: 1st comment. I know no reference but the proof need not be [as per OP] lengthy — one just computes the two scalars by which the map $\mathscr I:$ $$ \textstyle A\mapsto\mathscr I(A)=\int_S h\langle h,Ah\rangle\langle h,\cdot\rangle d\lambda(h) \tag1 $$ acts on the irreducible components of $\mathfrak{gl}(n,\mathbf C)...


5

No way. Let $G={\mathbb Z}$ so that ${\mathbb Z}[G]={\mathbb Z}[x,x^{-1}]$. Then use the complex $$\ldots \rightarrow 0 \rightarrow 0 \rightarrow {\mathbb Z}[x,x^{-1}] \xrightarrow{1-x+x^2} {\mathbb Z}[x,x^{-1}] \rightarrow 0 \rightarrow 0 \rightarrow \ldots$$


5

I'm not very familiar with this book (in particular, I don't know how introductory or not it is), but I think Torsten Wedhorn, Manifolds, Sheaves, and Cohomology. Springer Studium Mathematik—Master. Springer Spektrum, Wiesbaden, 2016. xvi+354 pp. ISBN: 978-3-658-10632-4; 978-3-658-10633-1 would fit your description.


4

Let $\chi$ be a non-principal real Dirichlet character modulo $q$. Let $$\beta_0=1-\frac{1}{\eta\log q}$$ be a real zero of $L(s,\chi)$ satisfying $\eta\geq 100$ for convenience (Heath-Brown's condition is $\eta\geq 3$). Let $\rho=\beta+i\gamma$ be any zero of $L(s,\chi)$ such that $\rho\neq\beta_0$ and $|\gamma|\leq 1$. We strengthen Heath-Brown's claim $...


4

In matrix analysis, the Schur complement is an object that you obtain after eliminating a part of the unknowns. It works that way: you have to solve $Mx=b$ where $M$ is a square, invertible matrix. You write the system in block form $$\begin{pmatrix} A & B \\ C & D \end{pmatrix}\binom{y}{z}=\binom{c}{d},$$ where you are lucky enough that $A$ is ...


4

In Introduction to differential geometry (see the review) by R.Sikorski the author introduces the concept of (what is now called) Sikorski space. Sikorski spaces are "affine, reduced differential spaces" and hence they can be approached algebraically by looking at their coordinate rings. Differentiable manifolds are important examples Sikorski spaces. ...


4

This is an easy consequence of the $k=2$ case of the complex version of the Isserlis-Wick theorem for moments of Gaussian measures, i.e., the identity $$ \int_{\mathbb{C}^n} z_{i_1}\cdots z_{i_k}\ {\bar{z}}_{j_1}\cdots {\bar{z}}_{j_k} \ e^{-|z|^2} \prod_{a=1}^{n}\frac{d(\Re z_a) d(\Im z_a)}{\pi}\ =\ \sum_{\sigma\in\mathfrak{S}_k} \delta_{i_1 j_{\sigma(1)}}\...


4

Jet Nestruev (a collective author, I think) does this in "Smooth Manifolds and Observables".


3

Let $G$ be infinite cyclic, generated by $x$. Let $M_\bullet$ be a free resolution of the $\mathbb{Z}[G]$-module $U=\mathbb{Z}/3\mathbb{Z}$ with $x$ acting by multiplication by $-1$. For example, take $M_\bullet$ to be $$\dots\to0\to\mathbb{Z}[G]\stackrel{\pmatrix{-3\\x+1}}{\longrightarrow}\mathbb{Z}[G]\oplus\mathbb{Z}[G]\stackrel{\pmatrix{x+1&3}}{\...


3

From this recent paper I would conclude the statement is false: the second Neumann eigenfunction of an acute triangle has one non-vertex critical point. This was a Polymath problem.


3

The problem of this question is qualitatively and quantitatively different in some ways from that considered by Andrew Bremner and myself. If we take the cubic, with $N$ as a fixed constant, it is possible to show that the related elliptic curve is \begin{equation*} E_N:G^2=H^3+((35N+18)H+4(1260N+1441))^2 \end{equation*} with the formulae linking $(H,G)$ to ...


3

I'll try to summarize YCor's comments into an answer (using big guns): Let $G$ be the real points of an algebraic group (a restriction by the OP in the comments) and assume $\Gamma$ irreducible. Then Raghunathan shows that the answer is "yes" if $\Gamma$ is arithmetic. Margulis (Discrete subgroups of semisimple Lie groups) says that $\Gamma$ will be ...


3

Ramanan's "Global Calculus" is a very nice introductory text which defines manifolds by their sheaves of differentiable functions. I don't know if this is what you're looking for: I don't see it as very algebraic, and tools from analysis have an important role.


2

Denote by $\mathbb{A}$ the adeles over a number field $F$. An cuspidal automorphic representation $\pi$ of $\text{GL}_2(\mathbb{A})$ now factors by the tensor product theorem as $\pi\cong \prod_v\pi_v$, where $\pi_v$ are smooth irreducible unitary representations of $\text{GL}_2(F_v)$. The Ramanujan-Petersson conjecture now states that the representations $\...


2

The power series $F(x)$ is closely related to the series of the "exponential reversion of Fibonacci numbers" $$R(x)=\sum_{n\ge1}r_n\frac{x^n}{n!}$$ (the $r_n$ are A258943, quoted in a comment). In fact it appears that, again in the notation of Henri Cohen, $$a_{n+1}=nr_n,$$ equivalently $$F'(x)=xR'(x).$$ So if the Fibonacci numbers are encapsulated by $$x=\...


2

Although i do not know much more to say, i recall i have seen this variant of the definition of the comodule you are describing, used in the context of Hopf-von Neumann algebras. See for example: Crossed products of dual operator spaces by locally compact groups, Dimitrios Andreou, arXiv:1910.00433 [math.OA] (see the def in p.3) and also: Masamichi ...


2

Yet another reference is Theorem A on page 105 in: A. Wayne Roberts, Dale E. Varberg, Convex functions. Pure and Applied Mathematics, Vol. 57. Academic Press , New York-London, 1973. This reference provides also an answer to a question in one of the comments: What are some applications which can not be proved with the "sublinear" version? If $p:\...


2

The answer is, that $p_N$ is not necessarily increasing. Not that $\mathbb{P}(\bar X_N > 0) = \mathbb{P}(X_1 + \ldots + X_N > 0)$. Put $\mathbb{P}(X=1) = 0.99$ and $\mathbb{P}(X=-98) = 0.01$. Then $\mathbb{E}X_1 = 0.01$, but $p_2 < p_1$.


2

Speaking as a nonexpert, the enumeration and classification of monotone Boolean functions can give insight into optimization problems in logic, for instance by considering how far off an arbitrary function is from a monotone one. Doing a web search should reveal other motivations for studying Dedekind's problem. Gerhard "Who Doesn't Like Enumeration ...


2

To any skew-pairing $\lambda:U\otimes H\rightarrow k$, one can associate a hopf algebra $D(U, H)$ (built on $U\otimes H$) which is called the generalized quantum double of $U$ and $H$. (If $H$ is finite dimensional, $U=H^{*cop}$ and $\lambda$ is the usual evaluation map, then, this corresponds to the usual quantum double $D(H)$ introduced by Drinfeld). In ...


2

In the paper [MR3135704 Inci, H.; Kappeler, T.; Topalov, P. On the regularity of the composition of diffeomorphisms. Mem. Amer. Math. Soc. 226 (2013), no. 1062, vi+60 pp. ISBN: 978-0-8218-8741-7] you find the following Putting $r=0$ and choosing $s$ such that $C^\alpha\subset H^s$ gives a positive answer under slightly more strict conditions on $\Phi$: ...


1

A group $G$ is not bi-orderable if and only if for some finite subset $J$ of $G\smallsetminus\{1\}$, for every $e\in \{-1,1\}^J$ there exists $n\ge 1$, group elements $c_1,\dots,c_n$, and a function $s:\{1,\dots,n\}\to J$ such that $$\prod_{i=1}^nc_is(i)^{e(s(i))}c_i^{-1}=1_G.$$ Indeed, first clearly if there is such $J$, then if by contradiction there ...


1

An improvement to the theorem from the book Manifolds, Tensor Analysis and Applications of R. Abraham, J.E. Marsden and T. Ratiu mentioned above was achieved by J. Blot The Rank Theorem in Infinite Dimension. The assumption from Abraham et al. is that for every $u$ in a neighborhood of $u_0$ the subspace $DF[u]$ is closed and $DF[u]|_{B_1}$ is an isomorphism ...


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