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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2
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0answers
31 views

Literature about Ultrafilters

I am in the early stages of planning my senior project and was wondering if anybody had some recommendations of literature about the applications of ultrafilters in social choice theory, along with ...
1
vote
1answer
23 views

Fractional Sobolev spaces on closed manifolds

Let $M$ be a closed manifold and $0<s<1$. How is the fractional Sobolev space , $H^s(M)$ defined? In particular if $M$ is a closed smooth simple curve in the place how is $H^{1/2}(M)$ ...
0
votes
0answers
22 views

Reference Request for Complex Analysis (with some specificity regarding Ahlfors and Cartan)

I am a self-studier and am making my second pass through Complex Analysis. I have read the reference request posts many times. Yet perhaps I could get some advice as to the relevant benefits of ...
0
votes
0answers
31 views

Deformation theory in a paper of Bogomolov and Tschinkel

I am trying to read this paper http://www.math.nyu.edu/~tschinke/papers/yuri/00ajm/ajm.pdf, by Bogomolov and Tschinkel. I had 0 preparation in the theory of deformation of complex structures, but in ...
0
votes
1answer
26 views

linear algebra texts suggestions

I am looking for a textbook about linear algebra. I want one with a pure math/algebraic approach and not one with a geometric or a applied/numerical approach. Do you have any suggestions? Thank you
3
votes
1answer
39 views

Conjectured diagonal Ramsey numbers

While doing some reading on the Wikipedia page for Ramsey numbers, I stumbled upon OEIS sequence A120414, which lists the allegedly conjectured values of the diagonal Ramsey numbers $R(n,n)$. I ...
3
votes
1answer
57 views
+100

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ ...
1
vote
1answer
25 views

Reference to line parametrization

Defining two lines in space, $\mathbb{R}^3$, as: $l_1: \textbf{a}_1+\lambda_1\textbf{b}_1$ $l_2: \textbf{a}_2+\lambda_2\textbf{b}_2$ The line to line intersection condition is: $\textbf{b}_1\cdot ...
0
votes
1answer
93 views

Is it necessary to read point set topology to read differential geometry?

I want a quick insight in differential geometry but it is hard to start directly although i have done courses in calculus and basic algebra .is it necessary to get through point set topology and ...
1
vote
2answers
53 views

Is there a metric proof for the Caristi theorem?

I'm writing a paper about hyperconvexity in metric spaces and came across Caristi's theorem: Let $(X, d)$ be a complete metric space and $\phi\colon X \to \mathbb{R}^+$ a continuous function. An ...
0
votes
1answer
30 views

Book recommendations for reading A. Okounkov and A. Vershik's approach for complex irreducible representations of symmetric groups?

Does anybody have book recommendations for reading A. Okounkov and A. Vershik's approach for complex irreducible representations of symmetric groups? Preferably, I am looking for a book that is ...
4
votes
0answers
84 views

Textbook on infinite loop spaces

I'm looking for a good update reference covering the material in first three chapters of "Adams, Infinite loop spaces" (specially construction of delooping functors and group completion) with exact ...
2
votes
0answers
30 views

Reference for proof of Hochschild-Kostant-Rosenberg for Hochschild cohomology

Is there a place where there is a full proof of the Hochschild-Kostant-Rosenberg Theorem for Hochschild cohomology? I am aware of many places where the result is proven for Hochschild homology i.e. ...
1
vote
1answer
74 views

Learning mathematical concepts

Our teacher loves to test us on pure concept based questions and test if we really know what we are doing when learning a particular lesson. For example, when we first started learning about ...
-1
votes
0answers
26 views

Info on Mean Value Inequality

I understand the mean value theorem and I can understand the proofs I've seen using the mean value inequality, but often they use steps that are really unintuitive to me. Does anyone know of a ...
1
vote
1answer
42 views

What are some good books on vector analysis in higher dimenesion

What is some good books specifically on vector analysis in higher dimension? Standard vector calculus book usually only introduced double and triple integral method
1
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0answers
16 views

Domain of $A^{1/2}$ on $L^2(\mathbb{T}^2)$

Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, ...
3
votes
1answer
56 views

Proof of Fermat's last theorem for $n=5$ using primitive roots of unity?

I've been reading "An introduction to the theory of numbers" by Hardy and Wright and they gave a nice proof of Fermat's last theorem for $n=3$ by proving that there are no solutions to ...
0
votes
0answers
26 views

Does Thompson's Calculus hit all the essentials needed?

I found out in my preliminary research for my undergraduate degree that I need a bit of undergraduate-level mathematics for my topic at the very minimum to start with, up to ODEs. So I found a ...
5
votes
1answer
81 views

What book or website has nice, colorful diagrams illustrating real quadratic integer rings?

I'm sure you all have seen diagrams, colorful or not, illustrating prime numbers in $\mathbb{Z}[i]$ and $\mathbb{Z}[\omega]$, with some of them helpfully pointing out the inert and splitting primes ...
3
votes
1answer
25 views

Self-adjointness under relatively bounded perturbation

Let $T$ be a densely defined linear operator on a Banach space $X$. Another operator $A$ satisfying $\mathcal{D}(T) \subset \mathcal{D}(A)$ is called a relatively bounded perturbation of $T$ if ...
1
vote
2answers
32 views

Reference request for a special kind of numbers.

Let $q$ be an element of a field $k$ (possibly $\mathbb{C}$), different from $-1$ and $1$. We have $$[n]=\frac{q^n-q^{-n}}{q-q^{-1}}=q^{n-1}+q^{n-2}+\dots+q^{-n+1}$$ Where $n$ is a natural number. ...
3
votes
2answers
66 views

Reference to complete proof that integral closure of $\mathbb{Z}$ in $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$?

Where can I find a complete proof to the fact that the integral closure of $\mathbb{Z}$ in $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$ (the Gaussian integers are the integral closure of $\mathbb{Z}$ in the ...
0
votes
1answer
44 views

Poincaré type inequality

Consider a sequence of functions $f_n \in H^1(M)$, the first Sobolev space of a complete (possibly noncompact) Riemannian manifold. If we have the normalization $\Vert \nabla f_n\Vert_{L^2} = 1$, ...
0
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0answers
27 views

Krein Milman Property

If a closed bounded (not compact) set $X$ in a Banach space $B$ (like $L^1$) has extreme point(s), must the max of a linear functional defined on $X$ occur at one of them? I suppose it depends on $B$. ...
1
vote
1answer
42 views

Seeking Recommendation on Theoretical Multivariable Calculus textbooks

I am a college sophomore with double majors in mathematics and microbiology. I wrote this email to seek your advice on selecting a theoretical, proof-based textbook on the multivariable calculus. I ...
3
votes
1answer
34 views

Further examples of Principal Ideal Domain that are not Euclidean Domains

In several courses of algebra, I've heard that not all PIDs are EDs, and the canonical example is $\mathbb{Z}\left[\dfrac{1+\sqrt{-19}}{2}\right]$ which I've heard over and over. Some cursory research ...
45
votes
4answers
3k views

Is “A New Kind of Science” a new kind of science?

A couple of years ago I was reading "New Kind of Science" (NKS) by S. Wolfram, and it presented lot of interesting ideas for a young Physics undergraduate. Now that I am studying Mathematics however, ...
3
votes
1answer
104 views

Mathematics books that tell you what is really happening? [closed]

Many book I've read teach you symblobic manipulations instead of pointing out what's really happening. So my question would be: what mathematics textbooks don't do that? Books that rather than listing ...
3
votes
2answers
63 views

A reference for a combinatorial identity

I have come across this identity from study of species. I am not posting my method but I am interested in knowing whether it arises in some other contexts as well. The identity is: $$\sum ...
4
votes
1answer
124 views
+50

Why are logarithms of trigonometric functions useful?

I have noticed that in many trigonometric tables the logarithm of the trigonometric values are given. Why this is given and not the actual values of the trigonometric functions? For example, instead ...
1
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0answers
37 views

Is there a companion to the book 'A Synopsis of Elementary Results in Pure and Applied Mathematics' by George S. Carr?

A Synopsis of Elementary Results in Pure and Applied Mathematics by George S. Carr is as most of you probably know a book that was famously used by the great mathematician Ramanujan. It is said he ...
0
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0answers
29 views

Open conjectures in number theory that is easy to do some programming for

I have a to do a project in number theory that we are assigned that we should do some programming for that is not the collatz conjecture, so any suggestion would be really great.
1
vote
1answer
17 views

Let $V$ be a finite dimensional vector space over a field $F$. Prove the dependence of a set of vectors $\in V$ and reference request

Let $V$ be a finite dimensional vector space over a field $F$. Show that if $u_{1}, \dots, u_{n}, v_{1}, \dots, v_{n} \in V$ and $u_{1}, \dots, u_{n}$ are linearly independent, then there are only ...
3
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0answers
14 views

Which subject deals with questions about texture mapping in computer graphics? [migrated]

When we do texture mapping in computer graphics, we care about following questions: How are texture coordinates(also called UV coordinates) generated for a specific geometry? How can we measure the ...
3
votes
2answers
112 views

Proper axiomatization of Euclidean Geometry that Euclid would approve of

It is relatively common knowledge that Euclid's axiomatization is not sufficient to prove all the things that Euclid wants to, and that there are other axiomatizations out there that strengthen ...
1
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0answers
23 views

Learning about Markov Chains

I am trying to learn about how to use markov chains for complicated probability problems. I have been looking for different materials to learn these but haven't had much luck. Does anyone have any ...
2
votes
0answers
51 views

A question on (odd) perfect numbers

(Note: This has been cross-posted to MO.) Let $\sigma(x)$ be the (classical) sum of the divisors of $x$. A number $N \in \mathbb{N}$ is called perfect if $\sigma(N)=2N$. An even perfect number $U$ ...
17
votes
2answers
223 views

To whom do we owe this construction of angles and trigonometry?

I've come across what is, to me, the most precise, beautiful and thorough definition of what we know of as the angle between two vectors. I say this because most literature either skims over things ...
2
votes
0answers
49 views

20th century books on geometry

I've heard something about the fact of some old geometry textbooks, dated to the beginning of the 20th century approximately, have a structure composed by a problem, the solution and then something ...
0
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0answers
23 views

Table of Fourier series

I found that there are very good references on Fourier integral transform but none on Fourier series. Do you happen to know one?
0
votes
0answers
26 views

Identifying a sequence of numbers from an optimization problem in $L^1$

Question Does there exist general closed form solutions (or some sort of recurrence relation) to the system of equations: $$\begin{align} x_0 &= -1\\ x_{k+1} &= 1\\ \sum_{j = 0}^k (-1)^j ...
6
votes
1answer
170 views

Conjectured Primality Test for $N=8\cdot 3^n-1$

Definition Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ , where $m$ and $x$ are nonnegative integers . Conjecture Let $N=8\cdot 3^n-1$ ...
1
vote
1answer
20 views

Reference request: Right pseudo inverse

Suppose, I have a matrix $P\in\mathbb{R}^{m\times n}$ with $\text{rank}(P)=m$ and I'm searching for a right pseudo inverse $P^{+R}$. Since I'm working with symbolic matrices in a computer algebra ...
4
votes
1answer
90 views

Existence of bijection that reorders elements?

Suppose I have some function $f:\mathbb{R}\to[0,1]$. Does there necessarily exist a bijective mapping $g:\mathbb{R}\to\mathbb{R}$ such that $g(x)\leq g(y)$ implies $f(x)≤f(y)$? If not, does it help if ...
1
vote
2answers
84 views

How can we detect if a topological space has “holes”.

I realize that this question might seem ambiguous, is there a topological notion for what a Hole is? I think it has something to do with the fundamental groups of the topological space but I don't ...
0
votes
0answers
17 views

Domains whose Green functions is explicit or can be approximated explicitly?

The only examples I keep finding are upper half plane (and tilted) and sphere (eg. Evans). Can you suggest some other domains? If not, how about any good books or papers documenting the progress ...
0
votes
1answer
78 views

Help with an inequality in Cazenave's book “Semilinear Schrodinger equations”

I'm reading Cazenave's book "Semilinear Schrodinger equations" and I found this inequality at page 84 $$\vert\vert u_1\vert^\alpha u_1-\vert u_2\vert^\alpha u_2\vert\vert\leq C (\vert ...
3
votes
1answer
78 views

Affine variety and dimension

I'm working on a paper about representation of quivers and Gabriel's theorems. See this .pdf if you're interested ; but I guess you can answer my question without knowing anything about quivers, or at ...
1
vote
4answers
127 views

Calculus books recommendation (intermediate level)

:) I would like to ask for some intermediate level textbook for calculus (single variable), or, at least, some supplement to Spivak's Calculus for better understanding on how to approach and solve ...