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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10 views

Fourier coefficients of eigenforms

How does one prove that the fourier coefficients of a normalized eigenform for Hecke operators $T_p$ on $S_k(N)$ all lie in a fixed number field? If the proof is lengthy, a reference to a book that ...
3
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0answers
16 views

Terminology for the difference of real dimension and scheme-theoretic dimension

Consider the scheme $\mathrm{Spec} \left(\mathbb{R}[x_1,\cdots,x_n]/(x_1^2+\cdots+x_n^2-a)\right)$ where $a$ is a real number. Scheme-theoretically, this has dimension $n-1$. But the dimension of the ...
5
votes
1answer
54 views

Recommendation for books on topology (light reads) [on hold]

Are there any books on topology which can be read without having to do any exercises and look up definitions every second line? Something to read while relaxing, and not meant to replace a textbook ...
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0answers
22 views

Is there any PDE that applies specifically to Number Theory?

Given the advanced results obtained by analytic means in Number Theory, it puzzles me why I don’t recall ever seeing a partial differential equation used to good effect in Number Theory. Is there ...
5
votes
2answers
243 views

Are Vector Spaces over non-commutative fields ever studied?

Are Vector Spaces over non-commutative fields ever studied? If not, why? If they are, I'd like to learn a bit about them, could you guys recommend me a good book on the subject? Cheers.
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0answers
20 views

Books that introduce many subjects?

I am looking for a rigorous math textbook that introduces many subjects in university (undergraduate) math. One book that seems to meet this criteria is Alan Beardon's Algebra and Geometry but I ...
2
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0answers
26 views

Sources for simple probability brain teasers

I am searching for a book that can supply me with probability brain teasers, that can be solved using little arithmetic/mental math, paired with somewhat detailed solutions. Any suggestions? ...
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3answers
34 views

Book to learn Mathematical Probability theory? [on hold]

What are some good references to , good book to learn Mathematical Probability theory ? Please help .
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0answers
23 views

Book to learn Darboux integral

What are some good references to , good book to learn Darboux integral ( https://en.wikipedia.org/wiki/Darboux_integral ) ? Please help .
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0answers
13 views

Composition operators on fractional-order Sobolev spaces

Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication provided $s > 1/2$. This is proved for ...
2
votes
1answer
23 views

Multicategories with out-arities

Basically, my question is: Why the emphasis on domains in the notion of multicategory? I will now give the formal framework to state it correctly. Passing from categories to multicategories ...
2
votes
0answers
35 views

Estimating the “size” of the mathematical research literature

The other day I was telling one of my friends that mathematics, as a living science, possesses quite an extensive research literature. How extensive then, she asked. Unfortunately, I didn't have ...
3
votes
0answers
33 views

(Nonunique) Solvability of Sylvester Equation

I am interested in stating existence of solution of a Sylvester equation $$ AX - XB = C, $$ where $A$, $B$, $C$, and $X$ are $(n,n)$ matrices. Existence of a unique solution $X$ is given, if $A$ ...
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0answers
31 views

Book similar to Milnor's book [on hold]

I just finished to read "Topology from the differentiable viewpoint" of Milnor, and for me this was the perfect mathematic book : short, clear with really beautiful results proved at the end. (I'm ...
1
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0answers
21 views

largest distance between vertices on a polyhedron

I have a polyhedron defined by m inequations and n unknowns. I am interested in the largest distance between two vertices (the number of edges I have to follow from one vertex to another). I am ...
1
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0answers
31 views

What books you can recommend me for the following subjects? [on hold]

Really I am looking for rigorous and full content books (as Apostol, Spivak). Subjects: Mathematical Analysis, Theory of Rings and Statistics. Thanks all.
2
votes
1answer
32 views

Probability theory required for learning statistics rigorously

I would like to learn statistics rigorously. The only book that I can find that seems to do statistics rigorously is this book "Theory of statistics" by Schervish (which seems advanced): ...
1
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0answers
33 views

Anyone know a good standard reference for Lie group and Lie algebra facts?

I'm writing something and I need to refer to a mathematical fact; unfortunately I got it from Wikipedia, which does not source the specific piece of info! It relates to a choice of simple roots for ...
4
votes
0answers
40 views

Looking for reference on the appearance of $\pi$ in the central limit theorem

There are many books on $\pi$ out there, but I have not yet been able to find one that makes a serious attempt to make sense of the fact that $\pi$ (traditionally a geometric constant) appears in the ...
2
votes
1answer
42 views

literature on advanced calculus [on hold]

I need your opinions on this particular textbook: Advanced Calculus by Robert C. Buck. In my first year in college I finished two semesters of single-variable calculus and now I'm looking for a proper ...
0
votes
1answer
37 views

A road-map through “Combinatorial Set theory: With gentle intro to independence proofs”

I'm going to study independence proofs form Halbeisen's book. It seems that some material is not needed to study independence proofs, so it seems that the book contains more material than my needs. ...
1
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0answers
23 views

Any online videos on a course taught from Munkres?

Are there any vidoes available on the Internet --- for watching online or for download --- of any (general) topology course taught using the book Topology by James R. Munkres, 2nd ed? If so, please ...
0
votes
1answer
52 views

Mathematics in Chemistry [on hold]

What are the applications of mathematics in chemistry, biology ? Please give some references including text books if any.
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2answers
94 views

Ayn Rand and athematics

I am an honors undergraduate in mathematics. I have taken an interest in objectivism. I came across a discovery of Ms. Ayn Rand's in mathematics: In a triangle the inscribed circle touches the ...
0
votes
0answers
11 views

Eigenvalues of normalized adjacency matrix

Can anyone introduce some references on the eigenvalue estimation of normalized adjacency matrix, i.e., $W=D^{-1}A$ ($D$ is the degree matrix and $A$ is the adjacency matrix of the corresponding ...
0
votes
1answer
83 views

Books for Ordinals and Cardinals

I am looking for a nice introductory book to read to learn and master ordinals and cardinals. Please help me!
1
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1answer
35 views

Reference Request: Subgroup of free abelian group is free abelian

I have the following reference Question, meaning that I search for a reference for the following statement: Let $F$ be a free abelian group of finite rank and $U$ be a subgroup. Then there is a basis ...
0
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0answers
10 views

Reference on Malcev completion

I need a reference for learning Malcev completion, its associated group scheme and Lie algebra. Thanks!
0
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2answers
32 views

Prove (or derive) the de Polignac formula for the prime decomposition of $n!$

I can't seem to find any papers published dedicated to show that the de Polignac formula has a rigorous derivation. From Wikipedia's entry for the formula: Let $n \geq 1$ be an integer. The prime ...
0
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0answers
12 views

Inverse Laplace operator $\Delta^{-1}$ in $H^1_0, \ H^{−1}$.

Let open domain $\Omega \subset \mathbb R^n$, $u : \Omega \rightarrow \mathbb R$, $u \in H^1_0 (\Omega)$ and $f \in H^{−1}(\Omega)$. I'm looking for some known expressions of the inverse Laplace ...
3
votes
1answer
36 views

Jacobi Elliptic Functions built from Jacobi theta functions

I believe I understand the general theory of elliptic functions to an extent. What I can't seem to find is the distinct method which was used to show that a particular combination of Jacobi Theta ...
-1
votes
0answers
29 views

Partial Differential Equations Solver

Anyone knows a good website/program in which you enter your partial differential equation with the initial conditions and it simply solve it?
-2
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0answers
34 views

Asking for an article [on hold]

I tried to download this article: M. N. Huxley, Area, lattice points, and exponential sums, London Mathematical Society Monographs. New Series, vol. 13, The Clarendon Press Oxford University Press, ...
2
votes
5answers
125 views

Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”

If $C$ and $A$ are the circumference and area of a circle, then $$ \frac{C^2}{A}=4 \pi\; . $$ I'm looking for a reference to an elementary but rigorous proof of it. I'm particularly interested in ...
0
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0answers
29 views

Books for Real analysis similar to or having same essence like Charles Pinter Abstract algebra

I am looking for book which is similar to spirit of pinter's book mentioned in question that is which is suitable for self study and beginners. Can anyone recommend? Thanks
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2answers
110 views

Two questions on the Grothendieck ring of varieties

1) In the definition of the Grothendieck ring of varieties over a field $k$, which definition of the various notions of "variety" is chosen? Finite type and separated, or maybe more? 2) If ...
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0answers
11 views

An example of frame operator.

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that, for $A<B$ and for all $f\in ...
0
votes
0answers
24 views

Finding references about modular forms and symmetric power $L$-functions

I want to write an introduction to my thesis which is about modular forms and symmetric power $L$-functions. Could you give me good references to these two topics as simple as possible to get an ...
2
votes
1answer
35 views

Class field theory of imaginary quadratic fields

Can someone give me a good source that deals in detail with the Class Field Theory of imaginary quadratic fields? The sources on CFT that I have at hand only deal with CFT in general and then proceed ...
2
votes
2answers
33 views

Positiveness of partial sums of type $ \psi * D_N $

In his paper about Extremal Functions for the Fourier Transform (see, for example, here? https://projecteuclid.org/download/pdf_1/euclid.bams/1183552525), Jeffrey Vaaler, while trying to build ...
1
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2answers
92 views

Hardcore Abstract Algebra Book Request

I am going to start learning Abstract Algebra soon. I was originally going to start with Dummit and Foote, but I am starting to abandon that idea. I want to use a "hardcore" algebra book. I don't mind ...
0
votes
4answers
88 views

In need of a group theory textbook.

I am in need of a group theory textbook for a good summer review. I have already studied from various books (mostly "group theory" part from basic algebra books) and the lecture notes of my teacher, ...
8
votes
1answer
50 views

Mathematically rigorous books on programming and computer science

I would like a mathematical approach to programming languages and computer science, not just the theoretical aspects of computer science. Is there any such text out there? After all, the world has a ...
4
votes
2answers
45 views

About Mordell's Theorem (Elliptic Curves)

I've just finished the proof of Mordell's Theorem given in the book "Rational Points on Elliptic Curves " by Silverman. One of the key lemmas used in the proof of the theorem is: Let ...
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votes
0answers
41 views

Which harvard course should I take? [closed]

apologies if I'm on the wrong website or anything. I'll be starting math at Harvard next year but am at a loss as to which course I should start with. I have a good background in competition math (so ...
0
votes
2answers
45 views
+50

Reference request for this topics

I need a good reference to learn these topics Markov Chains in discrete time.    1.1. Classification of states, recurrence notions of transience.    1.2. Stationary measure.    1.3. ...
3
votes
1answer
85 views

Difference between proof-based calculus and analysis?

When I asked a teacher at school what the difference between calculus and analysis he said that calculus is essentially analysis without proofs. So where does proof-based calculus lie? Is it somewhere ...
6
votes
1answer
131 views

What proof uses both the Riemann Hypothesis and its negation?

Some time ago I happened to see a proof that was remarkable in that it used both the Riemann Hypothesis and its negation. That is, it considered the two cases: RH is true, and RH is false, obtaining, ...
1
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0answers
46 views

Proof of the formula for dimension of moduli of stable vector bundles on smooth curves

Let $C$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on $C$ of ...
3
votes
2answers
44 views

What are good references for the action of $\Gamma := \pi_1(S)$ on $S^1 = \partial \mathbb{H}^2$, where $S$ is a closed hyperbolic surface

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...