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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1answer
42 views

Examples of how abstract algebra is used to find concrete solutions to a mathematical model?

All references I've seen to abstract algebra show how it helps in the representation of mathematical models...are there any examples of using abstract algebra to calculate actual solutions to a ...
2
votes
1answer
78 views

Example of function not Fourier Invertible in $L^1$

It is well known that, if $1 < p \le 2$, then, for every $f \in L^p$, $$ \int_{[-R,R]^n} e^{-2\pi i x \cdot y} \hat{f}(y) dy \rightarrow f(x) $$ As $R \rightarrow \infty$, in the $L^p$ sense. ...
0
votes
0answers
22 views

Looking for resources on these topics from Linear Algebra

I am looking for Characteristic roots and characteristic vectors of a linear transformation or of a matrix, Algebraic and Geometric multiplicity of a characteristic value, Cayley-Hamilton theorem, ...
0
votes
0answers
28 views

Need recommendation for following topics in combinatorics

I have to do following topics for my exam .I have 2 months time .However i have never done any combinatorics except that of high school (Permutations ,Combinations etc ) .I want a book which covers ...
3
votes
2answers
124 views

Functional Analysis Question?

I have a question about functional analysis. I know in finite dimensional space $\mathbb{C}^n$, all bases have the same cardinality. However let us consider $L^{2}[-\pi,\pi]$ which has TWO bases ...
0
votes
0answers
20 views

References for microlocal versions of some theorems

I am trying to introduce myself into Microlocal Analysis. In particular, motivated by some results in Inverse Problems, I would like to find good references for the microlocal versions of Helgason and ...
0
votes
0answers
12 views

Sub gaussian concentration for Lipschitz functions

It is well know that: if $f:\mathbb{R}^m\to\mathbb{R}$ is a Lipschitz function with Lipschitz constant $L$, and $X_1,\dots X_m$ are i.i.d random variables s.t. $X_i\sim N(0,1)$, then for any $t>0$ ...
5
votes
0answers
67 views

Infinite Sums which turn out to be Riemann Integrals

I'm looking for examples of infinite series which look hard to evaluate at first, but become very simple when viewed as a Riemann integral. An example would be $$\frac{1}{n+1}+\frac{1}{n+2}+ \ldots ...
1
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2answers
523 views

Where can I find a copy of the Project Euler questions?

The Project Euler page is currently offline, and I would very much like to do the problems still. Does anyone know where I can find a transcript of the problems?
0
votes
1answer
19 views

An inductive limit of amenable groups is amenable

It is a Theorem that an inductive Limit of amenable Groups is amenable. Could someone sketch me a proof of this, or give me a reference? I couldn't find one. Thanks in advance. Edit: I wanted it for ...
6
votes
2answers
144 views

Cancellation problem: $R\not\cong S$ but $R[t]\cong S[t]$ (Danielewski surfaces)

I would like to understand why the two rings $$ R={\mathbb{C}[x,y,z]}/{(xy - (1 - z^2))} \\ S=\mathbb{C}[x,y,z]/{(x^2y - (1 - z^2))} $$ are not isomorphic, but $R[t]\cong S[t]$. This example is ...
1
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0answers
55 views

Recommendations for a thorough logic textbook

I'm looking for a (possibly introductory) textbook on logic that covers the motivation behind conventions in logic, like the definition of the implication. Prof. J. Lau has an excellent webpage, ...
0
votes
0answers
9 views

A reference to study Boundary conditions of diffusion processes

I am trying to learn about Wentzell Boundary condition and (A,L) diffusion in the sense of Watanabe's paper (On the existence and uniqueness of diffusion processes with Wentzell's boundary condition ...
23
votes
3answers
717 views

Making some standard theoretical physics argument rigorous

In theoretical physics one often encounters the following rationale: if $f$ and $g$ are functions on $\mathbf{R}^n$, satisfying some technical conditions, and $\displaystyle\int_\Omega f=\int_\Omega ...
1
vote
2answers
110 views

Reference Request Scott's Trick

Does anyone know of a reference for Scott's Trick. I can't find it in Set Theory-Jech?
5
votes
0answers
54 views

a new(?) operation using products of multiplicities

Does the operation $$n \odot m := \prod_{p \text{ prime}} p^{v_p(n) \cdot v_p(m)}$$ on positive integers have a common name? Has this operation been studied somewhere? Notice that $\odot$ is ...
5
votes
1answer
149 views

Algebraic geometry papers for beginners

What are some papers/books suitable for a beginning graduate student interested in algebraic geometry? I have taken commutative algebra and a classical algebraic geometry class, but I have no other ...
1
vote
2answers
146 views

Good ODE Books That Explain How Solution Methods Came To Be and Their Justifications

As part of the mathematics program offered at my college, I took an introductory ODE course a few semesters back. This was the one math course in my entire college career that I was totally lost in. ...
4
votes
1answer
3k views

Probability of finding at least k consecutive heads in N coin tosses?

There are quite a few topics on this question already but I couldn't find a well-explained solution. Please point me towards some relevant literature or theory to analyze this problem. $K$ ...
3
votes
1answer
59 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 $A$ ...
2
votes
0answers
15 views

Understanding $SL_3(D)$ where D is a central division algebra

Suppose that $K$ is a non-archimedean local field of positive characteristic and $D$ is a four-dimensional central division algebra over $K$. The group $SL_{3}(D)$ can be embedded as a $K$-form of ...
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votes
0answers
20 views

Topological entropy and degree of smooth mappings

Where can I find the literature "Topological entropy and degree of smooth mappings" by Misiurewicz. Thanks for any help.
18
votes
1answer
285 views

Review on Riemannian Geometry

I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian ...
3
votes
2answers
42 views

Is there a theory of induced representations for semigroups?

Given a semigroup $G$, a subgroup $H\subseteq G$ (not merely a subsemigroup) and a representation $\rho: H\rightarrow GL(V)$ for some vector space $V$, is there a canonical definition of an induced ...
4
votes
1answer
59 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
vote
1answer
75 views

Modeling, Measuring, and Maximizing “Mixedness”

Possible key-terms: combinatorial optimization techniques; simulated annealing; genetic algorithms; Kirkman's schoolgirl problem; Steiner triple systems; orthogonal regrouping. Background: My class ...
3
votes
1answer
73 views

Explanation of a joke on abelian groups (grapes).

Q: What's purple and commutes? A: An Abelian grape. Q: What is lavender and commutes? A: An Abelian semigrape. Q: What's purple, commutes, and is worshipped by a limited number of people? A: A ...
0
votes
0answers
15 views

Does the Laplacian commutes with elements of the basis of the Lie algebra?

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. I know that if $g$ is semi-simple then the Laplace-Beltrami operator on $G$ agrees with the Casimir element and therefore commutes with ...
0
votes
2answers
74 views

quadratic Gauss sum over a power of 2

Is there a general formula for the generalized quadratic Gauss sum defined by $$ G(a,b,c)=\frac{1}{c}\sum_{n=0}^{c-1}e\left(\frac{an^2+bn}{c}\right) $$ where $e(x)=\exp(2\pi ix)$ and $c$ is a power of ...
1
vote
2answers
92 views

Reference request for this topics

I'm a second year undergraduate statistic student, I need a good reference to learn these topics Markov Chains in discrete time.    1.1. Classification of states, recurrence notions of transience. ...
7
votes
0answers
324 views

Ramanujan and sum of four cubes

This is more a question on History than proof itself. About a decade ago, a college professor and a Math coach told us about this beautiful theorem: Every multiple of 6 can be written as a sum of ...
5
votes
1answer
55 views

A product version of Riemann integral

Motivated by Riemann sum in Riemann integral and motivated by relations between infinite series and infinite products we ask: Assume that $f:[0, 1]\to \mathbb{R}$ is a positive function. Assume ...
6
votes
2answers
537 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.
1
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0answers
27 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 ...
2
votes
1answer
41 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
0answers
22 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 ...
1
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0answers
25 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
votes
0answers
29 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? ...
8
votes
1answer
54 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 ...
1
vote
1answer
119 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!
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3answers
45 views

Book to learn Mathematical Probability theory? [closed]

What are some good references to , good book to learn Mathematical Probability theory ? Please help .
1
vote
0answers
30 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 .
0
votes
1answer
36 views

Superelliptic curves

I'm trying to find information on superelliptic curves and how to solve them over the integers. The equation is $$y^k = f(x)$$ where $k=3$ and $f$ has degree $d=3$. Does anyone know any ...
3
votes
1answer
43 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 ...
4
votes
2answers
213 views

Dynamical systems and differential equations reviews/surveys?

I would be very glad if someone could point me to modern reviews/surveys on these topics. To be concrete, I'll provide some examples: S. Smale, Differentiable dynamical systems D. V. Anosov, On the ...
5
votes
1answer
115 views

How can I calculate the formula of this fractal-like structure?

I did the following fractal-like structure manually, and I was trying to convert it to a formula (or an algorithm including formulas) to compute some parts of the drawing, but I get lost due to the ...
1
vote
0answers
28 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 ...
2
votes
1answer
49 views

literature on advanced calculus [closed]

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
45 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. ...
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votes
2answers
114 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 ...