This tag is for questions seeking external references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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Curiosities about the content of a rare book: Topological Vector Spaces by A. Grothendieck

The book is a celebrated and highly influential book by A. Grothendeck, which was published in 1954, in French and for various reasons, it has been out of print since 1973. I am very much interested ...
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149 views

Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p} $ been discussed anywhere?

In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
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170 views

Some rare binomial identities

Long ago , I once saw a nontrivial appealing binomial type of identity that I never saw again. It was something along the line of $\Sigma$$\binom{a(x)}{b(y)}$= where $a$ and $b$ where polynomials not ...
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159 views

Are there asymptotic expressions for multiple zetas $\small \zeta(s),\zeta(s,s),\zeta(s,s,s),\ldots$ where $\small s=1+\delta, \delta\to 0$?

Playing around with elementary symmetric functions I tried to generalize that to infinite series and arrived at the well known concept of MZV ("multiple zeta values"). At the moment I'm only ...
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409 views

German Analysis Texts

My question is somewhat related to this one but is somewhat more specific. Since a lot of good mathematics is written in German, I have decided to start developing my German reading abilities. So far, ...
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127 views

Direct limits of completely positive maps on $C^*$-algebras vs. operator systems

I believe I've heard, as part of the "lore," that the category (operator systems, completely positive maps) has direct limits, whereas the category ($C^*$-algebras, completely positive maps) does not. ...
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173 views

Determinant expression for the power sum

Let $S_{n,r} := \sum_{k=1}^{n} k^r$ be the power sum. On the homepage by W. Hecht (link) I have found the following determinant expression: $$S_{n,r} = (-1)^{r-1} \frac{n(n+1)}{(r+1)!} \det ...
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165 views

Non-closed subgroups of Lie groups

The following are my impressions from playing with a line of irrational slope $\gamma$ in the standard torus $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$, $\mathbb{R}\hookrightarrow ...
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737 views

What is a linear resolution?

Can anyone tell me where I may find an introduction to linear resolutions (of a $k[x_1,\ldots,x_n]$-module or ideal) including, of course, the standard definition of such a resolution, as well as its ...
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235 views

Differential geometry text with categorial flavor

I am blown away by exposition of analytic manifolds in Serre's Lie algebras and Lie groups, I want more! Is there a text that treats classic topics of differential geometry like connections, ...
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222 views

Reference: Geometric group theory

H. Bass has studied existence of lattices on trees. Can someone suggest a (readable) reference for lattices on graphs?
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347 views

How to use Galerkin method to prove uniqueness of solutions of hyperbolic equations?

Galerkin method is used heavily in finite element method, which can conveniently convert continuous problems to discrete ones. Particularly, Galerkin method can be used to prove uniqueness existence ...
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196 views

Dual modules and first cohomology

Let $G$ be a finite group, $K$ a characteristic-$p$ algebraically closed field (say $p$ divides $|G|$), and let $M$ be a finite-dimensional $KG$-module. What hypotheses are needed on $G$, $M$ to ...
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184 views

Is there a notion of *p-adic Dedekind Domains*?

As we all know, the ring $Z_p$ can be constructed as the projective limit of the rings $Z/p^{n}Z$. Now is there any generalization such as the p-adic completions of a Dedekind Domain? This might be ...
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59 views

Kazhdan's property (T) vs residual finiteness

There is a theorem that states that a discrete group $G$ with Kazhdan's Property $(T)$ and Property $(F)$ (so called factorisation property) is residually finite (see Kirchberg, Discrete groups with ...
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55 views

Hilbert's Inequality - improved???

Assume for convenience that $a_n\ge0$ (this also clarifies why certain inequalities below are in fact stronger than certain other inequalities below). Of the various inequalities Hilbert proved, I'm ...
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27 views

References for actions of infinite-dimensional Banach-Lie groups on infinite-dimensional Banach manifolds

I am starting to study infinite-dimensional manifolds, specifically, Banach manifolds. I found some interesting introductory texts in which the mathematical background is developed with some detail. ...
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38 views

Source request for $H^*(B\mathrm{TOP},\mathbb{Q})\cong H^*(BO,\mathbb{Q})$

Let $B\mathrm{TOP}$ denote the classifying space for microbundles, i.e. $B\operatorname{Homeo}(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $B\mathrm{TOP}$ via the inclusion. Let $f$ denote the ...
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95 views

Notebooks used by Paul Erdős

Paul Erdős was known for living out of two half-full (or half-empty) suitcases; one had a few clothes and the other had mathematical papers. Some of these papers were probably referring to his ...
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46 views

Has it been proved that odd perfect numbers cannot be triangular?

(Note: This question has been cross-posted to MO.) Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime ...
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40 views

Determinacy and uniformization

It's known that $PD$ implies projective uniformization. Assuming $AD$, is there an analogous theorem that holds for all subsets of the plane (where the uniformizing functions are reasonably definable ...
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52 views

Is there a dictionary for math notation?

As a non-mathematician who loves the elegance of mathematics, I'm often confused about certain syntax I see. For example, $2+2$ is "2 plus 2" obviously. But things get more complicated when, for ...
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24 views

Computing the de Rham Cohomology of $\mathbb{S}^2$ using Cech Cohomology

This might be a naïve question but I have decided to give it a try as I cannot figure out a way to give it a satisfactory answer. I've learned about de Rham's theorem relating de Rham and Cech ...
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61 views

Open problems in Lie theory

I been studying lie theory for some time. Beside classification related problems what are some examples of open problems in the lie world? Especifically in the topological/differentiable structure of ...
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59 views

Harmonic map into $S^n \times \mathbb{R}$

Consider a harmonic map $\Phi : \Sigma \to S^n \times \mathbb{R}$, where $\Sigma$ is a surface, and the metric on $S^n \times \mathbb{R}$ is given by the product metric. Choose local spherical (polar) ...
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128 views

Properties and geometrical interpretations of a specific planar vector law.

Genesis of the following thoughts. One usually define the following internal composition law on $\mathbb{R}^2$: $$+:\left\{\begin{array}{ccc} ...
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80 views

Mathematics Wallpapers

I know that this sounds very silly. But I don't know where else to ask. Is there a good free site for mathematics wallpapers , pictures etc ? Most of the time it is very difficult to find exact ...
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41 views

What are some concrete examples of how Graph Theory can be applied to Finite State Machines?

Conceptually, Graph Theory is not very hard to understand, even its terminology is very intuitive, however, it appears to have a very wide range of applications, one of which is the field of Finite ...
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47 views

Signal processing and algebraic geometry

Signal processing is a pretty huge branch of what I would (maybe wrongly) call electrical engineering. I have heard here and there whispers of interesting connections between signal processing - in ...
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69 views

Book on Algebraic Spaces?

I'm actually studying "the geometry of scheme" of Eisenbud Harris, after that I want to begin with a generalization of scheme that is algebraic spaces, I know only the book of Knutson "Algebraic ...
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108 views

Two formulae for $\pi$, probably known?

I stumbled upon (in the literature) two identities for $\pi$, but they were not referenced as they are probably well-known. Hoping someone could point out who found them first. Basically, the ...
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61 views

Original proof of Ljunngren's equation

The equation $$x^2=2y^4-1$$ was studied and solved by Ljunngren, who showed that 1,1 and 293,13 are the only integer solutions.However, his proof was very difficult and L.J.Mordell thought there must ...
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Category theory without sets

I have been reading Mac Lane's Categories for the Working Mathematician, and the prospect of developing category theory without any use of set theory is mentioned more than once in the book, but never ...
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107 views

Reference request: are levelsets/levelspaces (as defined here) used anywhere in mathematics?

Thinking about unimodal distributions in multiple dimensions, I came up with the following. Given a function $f : Y \leftarrow X$ where $X$ is a topological space, define that a levelset of $f$ is a ...
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110 views

P.A. Grillet, “Abstract Algebra” linear algebra prerequisites

The author says in the introduction that he only assumes knowledge of linear algebra. I wonder how much linear algebra is actually assumed there? A question for those who have read the book. ...
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70 views

Is the Hilbert Scheme $\operatorname{Hilb}_P(X)$ independent of the choice of ample line bundle?

The construction of the Hilbert scheme of a projective scheme $X$ requires us to fix an ample line bundle $L$ on $X$ in order to define the Hilbert polynomial $P$. Suppose that $S \subset X$ is a ...
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73 views

Best sources on complete transforms (classic orthonormal transforms) and overcomplete transforms in signal processing

In the introduction section of a thesis I read a little about classic orthonormal transforms such as Fourier, discrete cosine and wavelet transforms and their application in signal processing. Then ...
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60 views

Seeking more information regarding the function $\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right].$

Define a function $\varphi : \mathbb{N} \rightarrow \mathbb{N}$ as follows. $$\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right]$$ The motivation is that according to ...
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105 views

Baker Campbell Hausdorff formula for unbounded operators

Baker Campbell Hausdorff formula says that for elements $X,Y$ of a Lie algebra we have $$e^Xe^Y=\exp(X+Y+\frac12[X,Y]+...),$$ which for $[X,Y]$ being central reduces to ...
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122 views

Given $g$ find an $f$ which is solution for $L f = g$. How do I do this?

I am learning about Stochastic processes. To characterize uniqueness of solutions to a given Stochastic differential equation, I need to find for each continuous function $g :\Bbb{R}^2_+ \to \Bbb{R}$ ...
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102 views

How to derive Clopper-Pearson interval's F and beta approximation?

It is well-known that there is an approximation of the Clopper-Pearson exact Confident Interval for binomial test. Wiki It just simply claimed, without any reference that: Because of a ...
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130 views

Pointless Topology Text

What's a good textbook I can use to learn more about pointless topology? Will I need more than a course in regular, old point-set topology and an algebra course which included some category theory to ...
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55 views

Can anyone identify the function that represents this infinite product?

$$\lim_{\omega \to \infty} \prod_{N=1}^{\omega} {{1+e^{b \cdot c^{-N}}} \over 2}$$ For instance, the Lerch Transcendent is a analogous example of a special function that defines the sum of a useful ...
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29 views

Properties of spectra

I've been trying to understand spectra better, and I was wondering what would be a good source to learn from. My issue with the classical treatments (e.g., Adams, Switzer) is that people keep telling ...
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Are the ring of integers of the constructible numbers a Euclidean domain?

I suspect that since Euclid uses the Euclidean Algorithm to perform division on constructible numbers in Elements, the ring of integers of the constructible numbers are a Euclidean Domain, but I have ...
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45 views

Intuition for homotopy (co)limits in triangulated categories

The following definition is taken from Daniel Murfet's Triangulated Categories Part I notes. Let $\mathcal T$ be a triangulated category with countable coproducts. Suppose we are given a ...
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27 views

terminology for a “forward flow” type of random digraph

I am trying to find a characterization of the probability that vertex $1$ is connected to an arbitrary large vertex $N$ in a random digraph. The difference from typical random digraphs is that if ...
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106 views

Normal curvature of geodesic spheres

I would like to ask the community for a reference on the following property of geodesic spheres. Let $(M,g)$ be a compact Riemannian manifold without conjugate points and $\tilde{M}$ its universal ...
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45 views

Shaking a box of rocks (Optimal Packing)

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when ...
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186 views

Has this difference equation :$x_{n+1}=Ax^2_{n}-Bx^2_{n-1}$ been studied before?

I posted this problem before in MO but only I would like to know if this difference equation :$$x_{n+1}=Ax^2_{n}-Bx^2_{n-1}$$ where $A(\theta)= \cos(\theta)$ and $B(\theta) =\sin(\theta)$ are ...