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.

learn more… | top users | synonyms (3)

5
votes
0answers
68 views

when a space is a product of eilenberg mclane spaces for $\pi_1(X)$ is not abelian

In When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$. , I discuss in my answer when an Abelian cw complex $X$ is a product of Eilenberg-Maclane spaces, and show that it ...
0
votes
2answers
30 views

For a family of sets $\mathbb{U}$, $\cup_{arbitrary}(\cap_{finite} U)$ $\forall U \in \mathbb{U}$ is stable under $\cap_{finite}$.

The weak topologies of a Banach Space are constructed by taking a family $\tilde{B}$ consisting on all finite intersections of $\mathbb{U}$ and then taking arbitrary unions of sets of $\tilde{B}$. I ...
1
vote
0answers
23 views

Machine Learning: are there other functions similar to the softmax?

Recall in probability and machine learning softmax is defined as: $\sigma(\mathbf{z})_j = \dfrac{e^{z_j}}{\sum_{k=1}^K e^{z_k}}$ for $j = 1, ..., K.$ where $\sigma: \mathbb{R}^k \to (0,1)$ ...
3
votes
1answer
38 views

References on the moduli space of flat connections as a symplectic reduction

In their Yang Mills equations over Riemann surfaces paper, Atiyah & Bott famously remark that the moduli space of flat connections on a principal bundle over a compact orientable surface may be ...
1
vote
0answers
47 views

Distributions of components to distribution of vector

Suppose that I have independent variables $x_1,\ldots,x_n$ with tractable (not necessarily identical) distributions. I'm interested in the distribution of $\boldsymbol{x}=(x_1,\ldots,x_n)'$ and, if ...
2
votes
1answer
58 views

Counting the numnber of (labelled and unlabelled) rooted trees on $n$ vertices with height $h$

As far as I know, the number of labelled rooted trees on $n$ vertices is $n^{n-1}$. Is there a known result for counting the number of (labelled and unlabelled) rooted trees on $n$ vertices having ...
1
vote
1answer
40 views

Reference for the $\Bbb A^1_k$-rigidity of abelian $k$-varieties

Is there a reference that shows, for a field $k$, that abelian $k$-varieties are $\Bbb A^1_k$-rigid? A smooth variety $X$ over $k$ is $\Bbb A^1_k$-rigid if and only if the canonical map $$ {Hom}_{...
0
votes
0answers
61 views

Is there any linear algebra textbook presented using logical symbols?

I'm currently going through a book called Linear Algebra Done Right by Axler, and to be honest, his book seems to be very loose with what things he defines. For instance , the symbol 0 could be mean a ...
1
vote
1answer
24 views

Universal UHF algebra

I am reading on quasidiagonal $C^*$-algebras. There the phrase "the universal UHF algebra" appears. I know what UHF algebras are, but I don't know what the universal UHF algebra is. I would be glad ...
2
votes
1answer
36 views

How to make a probabilistic sense of the semigroup of a positive operator

Consider the operator $\mathcal{L}$ acting on the function $f:\{0,1\}\mapsto \mathbb{R}$ defined as following: $$\mathcal{L}f(x)=f(1-x)-f(x)$$ This is the infinitesimal generator of a continuous time ...
0
votes
0answers
15 views

Ask for reference convergence of implicit euler method for initial value problem with dissipative source term

I am considering the convergence of implicit euler method for solving the following initial value problem: \begin{cases} u'(t)=f(t,u(t)),t\in[0,T]\\ u(0)=u_0\in \mathbb{R}, \end{cases} where $u:[0,T]\...
0
votes
1answer
45 views

Reference for: simple closed curves generate the fundamental group

In a 2-complex $X$, it is "obvious" that the simple closed curves through the $0$-cell $v$ generate the fundamental group $\pi_1(X, v)$. (By a "simple closed curve" I mean a loop which does not self-...
2
votes
0answers
36 views

Name of dominated convergence for sums

Having a sequence $(a_n(j))_{n}$ where every element of the sequence also depends on $j\in\mathbb{N}$. If $\sum_{n=1}^\infty \sup_{j\in\mathbb{N}} |a_n(j)| < \infty$, then the following (assuming ...
0
votes
0answers
26 views

Reference Request: Monologues on Lie Groups/Algebras and Differential Geometry

I find that before really diving into a subject, I prefer to get a general idea of it. For instance, before studying ergodic theory through a standard textbook I enjoyed Paul Halmos' lecture notes on ...
0
votes
0answers
23 views

Name or reference about a inequality with integrals?

I have wrote down some class notes and I think I copied something wrong. It is an integral inequality; $$\iiint_{B^n}|\nabla\psi|^2\frac{1}{|x|^{n-2}}dV\leq C\iint_{\partial B^n}|\psi|^2dA$$ where $C$...
2
votes
1answer
30 views

What are the good references on tame hereditary algebras?

I have Thomas Brustle's Typical Examples of Tame Algebras, but I still do not have a systemic understanding of what tubes are and what regular roots of a tame hereditary algebra are. I'm also looking ...
3
votes
1answer
29 views

One to one map $f$ equal to its power series

Across a difficult exercise sheet I encountered this exercise : Let $f$ be a continuous map from $\bar D$ the closed unit disk (in $\mathbb{C}$) to $\mathbb{C}$. We suppose that $f$ is one to one ...
3
votes
1answer
32 views

“Shape” of solutions of 2nd order homogeneous ODEs

Consider a second order homogeneous ODE: $$P(x)y''+Q(x)y'+R(x)y=0.$$ If $P,Q,R$ are constant functions, then we know that the general solution has the form $$y=c_1e^{r_1x}+c_2e^{r_2x},$$ $$y=c_1e^{...
2
votes
0answers
27 views

How do linear operators acting on paths of Gaussian processes influence the covariance function?

It is well-known that applying a linear transformation $A$ on an $n$-dimensional centered Gaussian distribution with covariance matrix $\Sigma$ results in another centered Gaussian distribution with ...
4
votes
1answer
47 views

Finite Almost Simple Groups

I want to study finite almost simple groups but I am not sure which would be the best texts to look at. Can someone please refer me to some books that teach the theory of finite almost simple groups?
1
vote
0answers
68 views

The free cocompletion of a complete locally small category is complete

$\DeclareMathOperator{\colim}{colim}\newcommand{\cat}{\mathbf}\DeclareMathOperator{\Nat}{Nat}$I'm looking for a reference that talks about the free cocompletion $\hat{\cat C}$ of a (large) locally ...
4
votes
0answers
69 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 ...
0
votes
0answers
55 views

Find a polynomial such that this proposed root finding algorithm fails.

Is this polynomial root finding algorithm below known, and under what conditions for the choice of polynomial coefficients does it find at least one root? Description of the algorithm: Consider the ...
2
votes
1answer
42 views

Is there a projective morphism from the quadric surface to the projective plane with degree 1?

Is there a projective morphism from the quadric surface $\mathbb{P}^1\times\mathbb{P}^1$ to the projective plane $\mathbb{P}^2$, with degree $1$?
2
votes
1answer
60 views

Cauchy's real line and math philosophy till XIX

I have to write an essay concerning philosophy of mathematics until the end of $XIX$ century. I've heard that the reason why the Cauchy's theorem (if continuous functions $f_n \rightarrow f$ then $f$ ...
2
votes
1answer
68 views

$λ=log(2)$ for the tent map – which basis for the logarithm?

If $\lambda$ is the largest positive Lyapunov exponent of a piecewise linear dynamical chaotic discrete in time map, then is there a relationship between the entropy $h$ and its $\lambda$. According ...
5
votes
0answers
44 views

When is a mapping the proximity operator of some convex function?

Sorry for cross-posting from MO. It's been a few days and the question hasn't received any attention there. So, is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which ...
0
votes
1answer
49 views

Is there a list of recommended problems to do in each chapter of Spivak's Calculus anywhere?

I've recently been self-studying Spivak's Calculus, and since I don't have the time to do every problem from every chapter at a and finish at reasonable rate, I've looked for a course syllabus or ...
0
votes
0answers
57 views

Intuitive, short explanation of differential forms and exterior calculus

Are there any introductory lecture notes on differential forms and exterior calculus, preferably aimed at physics students studying General Relativity and Black holes? I have some familiarity with GR ...
2
votes
0answers
140 views

Isothermal coordinates

Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E*(du^2+dv^2)$ and where $E>0$ is an harmonic function? I know that this metric is a special kind ...
0
votes
0answers
49 views

Book reference for theory of differential equations (not Coddington's book)

I'm looking for references to study theory of ordinary differential equations. I'm looking for a similar book to Coddington's book, theory of ordinary differential equations but not this one, because ...
0
votes
0answers
13 views

References to: If $C\subset\mathbb{R}^n$ is convex and $0\notin C$ then there exists $v\in C$ such that $C$ is in the closed halfspace $H_v$.

For each $v\in\mathbb{R}^n$, we define the notation $H_v=\{u\in\mathbb{R}^n:\langle u,v\rangle\geq0\}$, where $\langle\cdot,\cdot\rangle$ denotes the usual inner product in $\mathbb{R}^n$. Recently, ...
1
vote
2answers
57 views

Proving Euler's spiral is an isometric embedding with bounded image

$\newcommand{\al}{\alpha}$ I am trying to prove Euler's spiral is an isometric embedding of $\mathbb{R}$ into $\mathbb{R}^2$ with bounded image. Here is the definition of the spiral: $(*) \, \,\al(t)...
2
votes
0answers
42 views

Field extensions that decompose into towers of degree$\leq n$ extensions

Let $F$ be a field and let $n$ be a natural number. Consider the class of field extensions $E/F$ that decompose into towers $E=E_k/E_{k-1}/\cdots/E_1/E_0=F$ such that $[E_{i+1}:E_i]\leq n$ for $i=0,1,...
5
votes
1answer
117 views

Coordinate-free notation for tensor contraction?

I am not sure if I can prevent this question from being too vague or with too large an overlap with other similar math.SE questions, but I will do my best... A standard linear operation in tensor ...
0
votes
0answers
43 views

Groupoid objects in the category of algebras

Can anyone give me some references where I could read about groupoid objects in the category of algebras? References about groupoid objects in other categories would also be welcome.
0
votes
1answer
46 views

Best source to study partial differential equations (PDE) [duplicate]

Want to understand partial differential equations (linear and non-linear) more deeply. I am not a mathematician and I am more interessted in a more practical source that is teaching this topic from a ...
0
votes
0answers
52 views

Order Magic riddle

I have to solve this problem for some course and no one in my class has an idea how to solve it... There are 4 objects on a table: spoon, fork, knife and a spatula, which are positioned in an order ...
0
votes
0answers
31 views

Lebesgue-Stieltjes integral and related topics

The theory of stochastic integration relies on the concept of the Lebesgue-Stieltjes integral. However, it is hard to find a textbook that handles this concept in detail. Take, for instance, Chung ...
1
vote
0answers
57 views

$\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$

Conjecture For $n \ge 1 $ , $m \ge 1$ $\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$ where $\pi\left(n\right)$ is the prime counting function . Does this conjecture ...
1
vote
1answer
49 views

Reference: Mahlo cardinals remain Mahlo in L

The following is stated on Wikipedia for Mahlo cardinals. Unfortunately, it's not sourced. Where can I find details? I wasn't able to google any articles dealing with Mahlo cardinals in L. Since ...
13
votes
2answers
140 views

Egg vs. chicken: trig functions, exponential, real and complex

This is something I was shaky about when I took calculus, real analysis, and then complex analysis. Specifically, is the following chain of definitions circular in any way? Define the set $\mathbb{N}...
0
votes
0answers
40 views

A request of a journal theorem

I am reading the paper Groups whose proper subgroups are finite-by-nilpotent by Maoqian Xu and he has some references that I can't find. Does somebody have (B. Bruno, On p-groups with nilpotent-by-...
5
votes
0answers
76 views

Irrationality of $\min$ and $\arg\,\min$ of $\Gamma|_{[1, 2]}$

The Gamma function achieves a local minimum at $x^* \approx 1.46163$, where $\Gamma(x^*) \approx 0.88560$. Can $x^*$ and $\Gamma(x^*)$ easily be proven irrational? Are they transcendental?
1
vote
1answer
29 views

Degenerate zeroes, fundamental theorem of algebra.

The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two ...
0
votes
3answers
54 views

Reference request: Analytic study of the trigonometric functions

I'm looking for a source (or sources) which develop a complete theory of the trigonometric functions with no reference to circle geometry. That is, it is purely analytic. The starting point could be (...
0
votes
0answers
11 views

K-wise identical marginal distributions

Suppose I have two joint distributions described by the two sequences of random variables,$X_1, \ldots, X_n$; $Y_1, \ldots, Y_n$. Is there a name/theory/reference for when these two distributions ...
1
vote
0answers
31 views

Continuity of PDE solutions with respect to coefficients

Suppose I have a PDE, for example the Fokker-Planck one: $$ \frac{\partial}{\partial t}u(x,t)=-\frac{\partial}{\partial x}(\mu(x,t)u(x,t))+\frac{1}{2}\frac{\partial^2}{\partial x^2}(D(x,t)u(x,t)). $$ ...
1
vote
1answer
21 views

Where should I begin the study of fixed point theory, especially of multi-valued maps?

How should one begin one's study of fixed point theory, especially of multi-valued maps? What background --- in topology, analysis, functional analysis, algebra, and set theory --- should one have? ...