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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0
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0answers
11 views

Does anyone know of a no-nonsense intro to basic logic that I can give to a Year 11 student?

I'm looking for some material on propositional and first-order logic to give to a Year 11 student. I want to be able to write the null-factor law as: $$\forall (a \in \mathbb{R},b \in \mathbb{R}) \,ab ...
3
votes
2answers
48 views
+50

Reference Request - Statistics Book with exercises

I'm looking for an as complete as possible statistics book with exercises, including the following topics: Probability Review Random Variables and Samples Descriptive Statistics Estimation (...
0
votes
3answers
111 views

Survey articles in Commutative/Homological algebra

I am a graduate student interested in Commutative algebra/Homological algebra. I am comfortable with first eight chapters of Atiyah. I am familiar with some algebraic geometry, first two chapters of ...
1
vote
0answers
19 views

Questions on color theory, mostly linear algebra related

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The human ...
0
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0answers
7 views

Proving $\text{disc}(A) = [\mathcal O_K:A]^2 \text{disc}(\mathcal O_K)$ for a subalgebra $A$ of the ring of integers of a number field.

I would like to have a reference of a proof for the following result (where $K$ is a number field), that I found in this answer: If $A$ is a $\mathbb Z$-subalgebra of $\mathcal O_K$ which spans $K$...
0
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0answers
14 views

Scaled gradient descent

Consider the unconstrained minimization \begin{align*} \min_{x\in\mathbb{R}^n}f(x) \end{align*} One iterative approach to obtaining a solution is to use the gradient descent algorithm. This algorithm ...
0
votes
2answers
64 views

Alternatives to Chapters 8-10 of Rudin's PMA

S.E advisers, I have been hearing that the chapters on multi-varaible analysis in Rudin's PMA are almost nothing like previous insightful chapters in the single-variable analysis, and I verified ...
0
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2answers
320 views

Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
2
votes
0answers
25 views

Basic question: Curvature transforms under Complexified Gauge Transformation

Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ is a self adjoint complexified gauge ...
2
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0answers
36 views

Homeomorphisms of the circle

I know that there is a vaste litterature about the group of the homeomorphisms of the circle. I would a good reference to start the study of this topic. Thanks in advance.
0
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1answer
50 views

Olympiad Books for Primary Students

I am a teacher of gifted program in primary school and currently I am developing Olympiad Curriculum (topic-wise) for my students. I have those topics that could need some help in terms of questions: ...
0
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0answers
14 views

Homotopy type of some lattices with top and bottom removed

There was an interesting question on MO which OP removed by some reason. Here is a (more or less) equivalent form. Take a finite cartesian product of finite linear orders, and remove top and bottom. ...
14
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1answer
712 views

How many digits of the googol-th prime can we calculate (or were calculated)?

Here, a lower and upper bound for the $n$-th prime are given. Applying the given bounds $$n(\ln(n\cdot\ln(n))-1)<p_n<n\cdot\ln(n\cdot\ln(n))$$ and the approximation $$p_n\approx n(\ln(n\...
0
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0answers
39 views

translation of “Der kanonische Modul …”

Do you know a note that is the translation of the following in English? J. HERZOG et al., "Der kanonische Modul eines Cohen-Macaulay-Rings," Lecture Notes in Mathematics No. 238, Springer-...
1
vote
1answer
32 views
+50

References on Hauptmoduln

Here it is said that a Hauptmodul (a generator of a modular function field) is unique up to a Möbius transformation. My impression is that it is really hard to find references on Hauptmoduln and ...
0
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0answers
10 views

Program to deal with Grobner basis for ideals in universal enveloping algebras.

Is there any way to deal with calculations in ideals of the universal enveloping algebra of the current algebra $\mathfrak{sl}_2\otimes \mathbb C[t]$? Particularly, I am interested in Grobner basis ...
2
votes
1answer
25 views

Characterization of a square matrix.

I would like to see a proof to this fact. For a square matrix the following are equivalent: $A$ has a right inverse. $A$ has rank $n$, where $A$ is $n \times n$. $A$ is invertible.
0
votes
1answer
95 views

Text book on solid geometry/stereometry, without involving analytic geometry

As the title says I'm searching for a textbook, about solid geometry, without involving analytic geometry. The material which the book should cover is the stereometry learned in the eastern bloc. An ...
0
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0answers
42 views

(Reference Request) Proofs for basic facts about regular functions on algebraic sets.

I am writing an assignment about algebraic and analytic sets in $\mathbb{C}^n$ and, when searching for references, came across the book Algebraic Geometry III. The book is a bit out of my depth, yet ...
45
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25answers
4k views

What books should I get to self study beyond Calculus for someone about to start undergrad mathematics?

I am struggling to pick out books when it comes to self studying math beyond Calculus. My situation is as follows. I have taken all math courses at my school (up to Calc BC and AP Stats) and I have ...
0
votes
0answers
24 views

Russian journals (similar to Monthly) publishing English papers

I am looking for journals comparable to the American Mathematical Monthly that are published in Russia but that they include English articles too. However I guess for each field the same question ...
1
vote
0answers
24 views

What is a good introductory book on Rational Choice Theory for a mathematician?

I'm interested in Rational Choice Theory as an approach to political science. Amongst other, related subjects, I'd like to know a thing or two about Arrow's impossibility theorem (and other aspects of ...
1
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2answers
60 views

Books on Group Theory.

I am looking for a book/note that has a good collection of advance results,theorems on finite group theory. By advance, I mean theorems after Sylow,Lagranges(I am considering theorems of Sylow,...
1
vote
2answers
54 views

Cyclically reduced words

This is just a reference request. I'm trying to find out whether there are some well developed notes/theory out there (books and the like) focusing on cyclically reduced words in groups. Quickly ...
0
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0answers
32 views

Books recommendation

Which books and topics should I study to solve following problem, book should start from beginning. pls help. Let $\mathbb{N}$ denote the set of natural numbers. Define a function $T:\mathbb{N}\...
2
votes
1answer
67 views

How does a complex algebraic variety know about its analytic topology?

This question has two parts. The first is a reference request regarding a result I assume is standard, and the second is a soft question asking for philosophy and intuition about an issue the first ...
2
votes
1answer
71 views

Discrete logarithm modulo powers of a small prime

Is there an efficient way to compute $x$ in $2^x \equiv b \pmod {p^m}$, where $p$ is a small odd prime and $m$ could be a large integer? I know the solution is of the form $x=\phi(p^m) k + y$ for ...
4
votes
1answer
207 views

Every holomorphic map between Kähler manifolds is harmonic

I was reading the Wikipedia article on harmonic maps and saw the following statement in the 'examples' section: Every holomorphic map between Kähler manifolds is harmonic. I am not that familiar ...
-3
votes
1answer
39 views

Formal definition of “proexample”. [on hold]

Where in the literature do we find the preferred formal definition of “proexample” as in: the number zero is a proexample for the existential sentence "some integer is neither positive nor negative"? ...
64
votes
16answers
2k views

Unconventional mathematics books

I've recently purchased Oliver Byrne's reproduction of Euclid's Elements. It's a beautiful tome, that's rather unique in its presentation of the material as it represents many of Euclid's proof as ...
4
votes
1answer
54 views

What is an injection in a topos?

I am reading the paper of Andrej Bauer that proves there is a realizability topos in which there is an injection from the internal Baire Space $\mathbb{N}^\mathbb{N}$ to the natural numbers $\mathbb{N}...
6
votes
1answer
74 views

Perfect circles in the Mandelbrot set?

It is known that the boundary of the period 2 hyperbolic component of the Mandelbrot set is a perfect circle of radius $\frac{1}{4}$ centered at $-1$. Moreover it is known that the boundaries of the ...
1
vote
2answers
1k views

What is a good book on basic high school math (algebra, geometry, trig etc.)?

What is a good book on basic high school math (algebra, geometry, trig etc.)? I want a book that presents mathematics in a rigorous manner and with emphasis on creativity rather than memorization. If ...
0
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0answers
38 views

what is the asymptotic expansion of this $_2F_2$ function?

We need to expand the function $_2F_2(a+b x,1; 1+a+b x, b x; x)$ near $x=+\infty$. Where $a$ is complex, $b>1$. When $x\to+\infty$, both the parameters and the variable goes to infinity, we can ...
1
vote
1answer
17 views

Connection between focal points and singularities of the normal exponential map

I am looking for nice references on focal points of Riemannian submanifolds. In particular, I would like to see a proof for the connection between focal points and singularities of the normal ...
1
vote
1answer
25 views

What does a function of matrices do to the eigenvalues of matrices in its domain? Two examples and request for generalization if possible

I think, for example, that if $\lambda$ is an eigenvalue of a matrix $A$, then $\lambda^2$ is an eigenvalue for $A^2$ and that $\frac{1}{\lambda}$ is an eigenvalue for $A^{-1}$ provided $A$ is ...
1
vote
1answer
29 views

Convergence of finite metric spaces to an infinite one

Let $\{(M_i, d_i)\}$ be an infinite sequence of finite metric spaces, where $|M_i|$ is strictly increasing with $i$. Is there a standard definition of what it means for the sequence $\{(M_i, d_i)\}$ ...
0
votes
1answer
32 views

How to show this equality for operator norm?

Let $(X,\Sigma_\mu,d\mu)$ and $(Y,\Sigma_\nu,d\nu)$ be two positive $\sigma$-finite measure space and let $M(d\mu)$ and $M(d\nu)$ be spaces of complex-valued $d\mu$-measurable and $d\nu$-measurable ...
1
vote
1answer
26 views

What mathematical background is preliminary for reading and understanding books/papers on wavelets?

Please excuse my english. I have had the following math courses for mechatronics engineering education: Calculus (single and multivariable) Linear algebra (introductory) Differential equations (ode'...
7
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3answers
432 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
0
votes
0answers
2 views

Reference request for "isomorphism upto compact kernel /cokernel “

Let $A$, $B$ be abelian topological groups with a map $f :A \to B$. Assume also that the kernel and cokernel of this map are compact. Then we call f an isomorphism upto compactness. Now let $A, B, C$...
0
votes
1answer
15 views

Riemann-Stieltjes integral and unbounded variation

I was studying the Riemann-Stieltjes integral and I was looking for the proof that function with unbounded variation can't be integrated. Thanks
1
vote
0answers
109 views

The number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit $d$-...
16
votes
1answer
172 views
+100

Origins of Differential Geometry and the Notion of Manifold

The title can potentially lend itself to a very broad discussion, so I'll try to narrow this post down to a few specific questions. I've been studying differential geometry and manifold theory a ...
6
votes
1answer
63 views

A topology on the natural numbers: a set is closed if it contains the divisors of all its elements

Is there a name of or a reference to the following topology on $X=\mathbb N$: $A\subseteq X$ is closed if and only if $n\in A\wedge m|n\implies m\in A$?
-2
votes
1answer
97 views

What are some good books and materials for studying rings and fields theory? [closed]

I will very soon be introduced to the subject. I have heard this is one of the most important part of undergraduate algebra. I want to develop clear understanding in it from the beginning. I have ...
0
votes
0answers
18 views

Original statement of Wiener's $1/f$ theorem

I'm studying Wiener's 1/f theorem, and I got curious about which was its original statement.I've been looking online but found nothing. I want to know if Wiener also proved the $n-$dimensional ...
3
votes
0answers
64 views
+50

Continuity of PDE solutions with respect to coefficients

Suppose I have a PDE, for example the Fokker-Planck one, in which I am mostly interested: $$ \frac{\partial}{\partial t}u(x,t)=-\frac{\partial}{\partial x}(\mu(x,t)u(x,t))+\frac{1}{2}\frac{\partial^2}{...
6
votes
5answers
904 views

Formal definition of “counterexample”.

What is the preferred formal definition of “counterexample” as in: zero is a counterexample for "every integer is either positive or negative". Where in the literature is the notion of “counterexample”...
2
votes
0answers
47 views

Provable Hamiltonian Subclass of Barnette Graphs

Given a bicubic planar graph consisting of faces with degree $4$ and $6$, so called Barnette graphs. We can show that there are exactly six squares. Kundor and I found six types of arrangements of the ...