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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3
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0answers
108 views

Complex Root of Unity Analogue of Forward Difference Operator

In my studies I have come across a couple of operators; in particular; $$\Delta[f(x)]=f(x+1)-f(x)$$ $$\Delta^*[f(x)]=f(x+1)+f(x)$$ $\Delta$ has been called the Forward Difference Operator. I was ...
2
votes
0answers
19 views

A sufficient condition for factorization in a complete local ring

I think something like the following statement is true, but I don't recall a reference. Suppose $f(x,y)\in k[[x,y]]$ is power series with no constant or linear terms. Then, if the quadratic terms ...
1
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1answer
125 views

Clarification if a disconnected function has a derivative at defined points.

I know so far for a derivative to exist. -The point should not exist as a discontinuity -It should not have a vertical tangent -There should be no sharp corner/ cusp at the point ...
1
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0answers
46 views

Estimates for the Dedekind number $M(9)$

The Dedekind number $M(n)$ is the number of antichains in the partial order of subsets of $\{1,\dotsc,n\}$. It is only known for $0 \leq n \leq 8$. Question. What are some known upper and lower ...
1
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3answers
121 views

Book on foundational reasoning of standard arithmetics “curriculum”

I am interested in a book that is about arithmetics but the presentation is not just the known to all formulas but the foundational logic behind it. The closest example I can think about is the way ...
0
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1answer
47 views

Using derivatives to get some trigonometric identities

Is there a way of using derivatives to get some trigonometric identities in a straight-forward fashion? I use to forget them, so that would help me a lot... For example, since when we get the ...
0
votes
1answer
47 views

Reference for theorem? Inequality of integrals of increasing function over two distributions

I have a monotone increasing function $H(x)$ and two distributions with CDFs $F_1$ and $F_2$, where $F_1(x) \leq F_2(x)$ everywhere. The domain is $[0,\infty)$. This seems like it must be true: $$ ...
1
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2answers
48 views

DKW-style $\ell_{\infty}$ bounds for sum of i.i.d. random functions: $\to [0,1]$

Let $\mathbf{G}$ be the set of (edit: convex) functions $g: X \to [0,1]$, where $X$ is a compact subset of $\mathbb{R}^d$ or something like that. Suppose I have a distribution $D$ on $\mathbf{G}$. ...
2
votes
1answer
56 views

Why is even codimension necessary to apply excision for the Euler characteristic?

In this answer on MathOverflow, it is claimed that $$\chi(X/Z)=\chi(X)-\chi(Z)$$ holds for complex subvarieties $Z$ only because $Z$ has even codimension. It is implied that for $Z$ with odd ...
3
votes
1answer
72 views

Lie groups pre-requisites and reference

What are the minimum pre-requisites in analysis (differential geometry) required to study Lie-groups? And for that material, what are some good references? I have done basic courses in Metric spaces, ...
10
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0answers
93 views

Category Theory Zoo

There are a few very useful websites when it comes to either finding a specific object with certain properties (and maybe lacking other properties) or finding out which properties a certain object ...
20
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4answers
2k views

What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed ...
0
votes
0answers
30 views

“Peak lemma” and explicit monotone subsequence

Looking at the proof of Bolzano–Weierstrass theorem, it found an interesting lemma (called the peak lemma here) : Every sequence $(x_n)_{n\in \mathbb{N}}\in \mathbb{R}^\mathbb{N}$ has a monotone ...
0
votes
1answer
18 views

Asymptotic bounds on sum of primes

Let $p_i$ denote the $i$th prime number, and let $p_k\#$ denote the $k$th primorial, $p_k\# \overset{\textrm{def}}= \prod_{i \le k} p_i$. I am interested in asymptotic upper bounds for the ...
1
vote
1answer
30 views

Is the braid category biclosed and bicomplete?

Let $\mathcal{B}$ be the braid category, as in Categories for the Working Mathematician §XI.4 p.262 (objects are natural numbers and morphisms are the braids $n\to n$). Then this can be given a ...
0
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0answers
22 views

Best resources for learning about regular and context free languages

I would like to train myself when it comes to finding out if a language is regular or context free. I would be grateful for pointing what are the best places/books for training.
1
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1answer
683 views

video lectures on Lie algebra

Is there any video lecture on first course on Lie algebra available online? , by the first course I mean, The complete book of Introduction of Lie algebra and its representation theory by James ...
6
votes
1answer
56 views

Survey of varieties of non-standard analysis?

Is there a reliable, reasonably up-to-date, survey article doing a "compare and contrast" on varieties of non-standard analysis?
6
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1answer
52 views

Classification of homomorphisms $\mathbb Q \to \mathbb C^\times$

Are there any textbooks which discuss/classify the injective group homomorphisms from $\mathbb Q$ (under addition) into $\mathbb C \setminus \{0\}$ (under multiplication)?
0
votes
0answers
14 views

Hamioltonian Circuit of Planar Graph of Order $2^n$

$G$ is a planar graph of order(= number of vertices) $2^n$. Questions: When $G$ has a Hamiltonian Circuit? Is there a polynomial or quasi polynomial time algorithm to decide whether $G$ has a ...
1
vote
1answer
701 views

Expected state of a Markov chain

Let's start with a slightly trivial Markov chain defined as follows: the beginning state is called $1$ and the set of states is $\mathbb{N}$. At each step, when the current state is $n$, the ...
2
votes
0answers
44 views

Cartesian product of two graph's sets of edges

If $G=(U,V,E)$ is an undirected bipartite graph, that means that there are no edges in $E$ between vertices from a set in $U$ to/from a vertex in $V$. There are only edges between the sets $U,V$. I ...
1
vote
1answer
61 views

Where does the group $\mathbb Z/(a)\oplus \mathbb Z/(a^2)\oplus \cdots $ arise?

Let $a>1$ be an integer, and consider the infinite abelian group $$ V_a=\bigoplus_{j=1}^{\infty}\mathbb Z/{a^j\mathbb Z}. $$ Can anyone provide references to places where this (or related) groups ...
4
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0answers
45 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 ...
1
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4answers
43 views

Introductory text on nets

I have learned point-set topology using filters. Now I do functional analysis where we are using nets to do topological stuff. Therefore I search an introductory text on nets that is suitable for this ...
0
votes
1answer
15 views

Formula for choosing $x$ elements from a set containing $n$ elements, with repetition allowed

I've been searching around for a formula for the number of cmbinations for choosing $x$ elements from a set containing $n$ elements. For instance, for the set $(1,2,3)$ we have $10$ different ways of ...
1
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0answers
38 views

Is there a formula for the $i$th Chow group $CH_i(X\times\mathbb{P}^n)$?

Is there a formula for the $i$th Chow group $CH_i(X\times\mathbb{P}^n)$ for a variety $X$? I heard there was a formula in one of Fulton's texts, but I've been scouring them and can't find one.
0
votes
0answers
24 views

How do I prove shoenflies theorem for $\mathbb{R}^2$?

I studied the contents in Munkres-Topology. In this text, the author uses basic algebraic topology to prove Jordan curve theorem. Then, he wrote that "If $C$ is a simple closed curve in $S^2$, the ...
2
votes
1answer
52 views

Set of the vertex sets to make connected graph into disjoint sets of vertices?

Suppose a non-directed graph G with vertices V and paths P. What is the name for the vertex sets to make break the graph by removal of some vertices?
0
votes
1answer
58 views

Can one use the Hilbert-Ackermann Consistency Theorem to prove the consistency of $PRA$?

In his textbook Mathematical Logic, Shoenfield states the Hilbert-Ackermann Consistency Theorem as follows: "Consistency Theorem (Hilbert-Ackermann): An open theory $T$ is inconsistent iff there is ...
7
votes
2answers
373 views

Stochastic geometry, point processes online lecture

Does any of you know where to find online lecture/podcast introducing stochastic geometry and/or point processes? Thank you! Riccardo
8
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3answers
7k views

Functional analysis textbook (or course) with complete solutions to exercises

I am a Ph.D. student in economics and I plan to study functional analysis by myself either this winter or the next summer. I am currently looking for a textbook, and since I am studying it by myself, ...
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5answers
2k views

The definition of metric space,topological space

I have read some books in analysis. All of them define metric space, topological space or vector space directly, without any reason. Therefore, I want to know the background of the definition - the ...
0
votes
1answer
17 views

Existence of hypergraphs with large parameter values

By a result of Erdös, proved using the probabilistic method, there exist graphs of arbitrarily large chromatic number and girth. What are the corresponding results for hypergraphs (given some ...
2
votes
1answer
50 views

Complex Elliptic Surfaces without sections

Is there a description of smooth complex projective surfaces without sections? While working on a problem a surface $X$ showed up with the following property: it is a non-ruled surface that has an ...
3
votes
2answers
234 views

Resources to understand real world usage of linear algebra

I've learned linear algebra basics at university and really liked it, so I decided to learn it more deeply. Secondly, I want to work in computer science and I think linear algebra knowledge could be ...
5
votes
0answers
120 views

How does commutative and/or differential algebra think about total derivatives?

If we apply the "operator" $\frac{d}{dx}$ to the polynomial $xy$, we get the expression $y+x\frac{dy}{dx}.$ (Source: high school.) Thinking of $xy$ as an element of the polynomial ring ...
2
votes
0answers
103 views

Are “$S$-monoids” known to be good for anything?

I came up with the following... ...Definition. Let $(S,\wedge,1_S)$ denote a fixed but arbitrary monoid. (In the examples I have in mind, $S$ is always commutative and idempotent. But ...
0
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1answer
40 views

Translate this theorem from Endliche Gruppen

In a paper I was doing a reference is given from "Endliche gruppen" by Huppert. I do not understand german and google translator was also not much helpful. Can some translate this theorem or much ...
1
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0answers
41 views

Some questions about S.Roman, “Advanced Linear Algebra”

Question for those who have studied Roman's book "Advanced Linear Algebra". How self-contained is this book. Can I study determinants directly from this in context of exterior algebra and tensor ...
1
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0answers
63 views

Assigning values to a divergent integral?

Question If I can assign the series of the zeta function to: $$ \zeta(-1) \to 1+2+3+\dots$$ why can't we assign the integral $$ \int_{0}^{\infty} x dx \to 0$$ and it still have some physical ...
2
votes
2answers
162 views

The singular homology and cohomology of manifolds vanishes in high dimensions

Let $M$ be an $n$-manifold. It seems that there are two results that the $p$-th singular homology and cohomology of $M$ are zero if $p>n$. But I can not find them in my books of algebraic ...
4
votes
1answer
64 views

Name of a category constructed from the action of a group on a category

Let $G$ be a group acting on a category $C$. That is we have a morphism of groups $G \to Aut(C)$. We can now form a new category as follows: Its objects are tuples consisting of an object $x$ of ...
1
vote
1answer
25 views

Need information on the following multi-group homomorphic structure

For the sake of simplicity I will describe the problem with a three group structure. Suppose there are three groups $G_1$, $G_2$ and $G_3$. Suppose also there is a binary map $M:G_1\times G_2\to G_3$ ...
2
votes
0answers
42 views

Relationship between intuitionistic logic and infinite dimensional vector spaces.

Some time ago, I've heard that there was a relationship between intuitionistic logic and infinite dimensional vector spaces. More precisely, the fact that $\neg \neg \phi \to \phi$ may not be "true" ...
0
votes
1answer
24 views

Christoffel symbols of $S^n$ in polar coordinates

Consider the usual local polar coordinates $\theta_1, \theta_2,..., \theta_n$ on $S^n$. We were taught about Christoffel symbols today and I am trying to see what the Christoffel symbols of $S^n$ ...
0
votes
1answer
29 views

What is a complexed-valued harmonic function?

In Gameline's Complex Analysis, there is an exercise: Show that if $h(z)$ is a complex-valued harmonic function (solution of Laplace's equation) such that $zh(z)$ is also harmonic, then $h(z)$ is ...
5
votes
3answers
1k views

What's a good book for a beginner in high school math competitions?

Also, I want to make it clear: Beginner. I'm getting really frustrated trying to study for math competitions: On the one hand, there are books teaching the high school curriculum, but that's it. I ...
1
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0answers
23 views

Hessian of the Stereographic projection

Consider the stereographic projection from the sphere $S^n$ onto $\mathbb{R}^n$, and take the usual local spherical (polar) coordinates $\omega_1,..,\omega_n$ on $S^n$ (coming from its embedding in ...
1
vote
1answer
28 views

Connection between quivers and representations of Lie algebras

Can anyone recommend a reference to study the connection between quiver theory and representation theory of Lie algebras? Supposedly those two things have something to do with each other, with the ...