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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8
votes
1answer
87 views

Diophantine equations for polynomials

I know that there has been work on diophanitine equations with solutions in poynomials ( rather than integers ) of the Fermat and Catalan type $x(t)^n+y(t)^n=z(t)^n$ ; $x(t)^m-y(t)^n=1$ and these have ...
1
vote
0answers
56 views

Books with similar coverage to Linear Algebra Done Wrong

Axler's book is great, but for my immediate purposes, it isn't suitable. I've been looking at the Table of Contents of Linear Algebra Done Wrong by Treil starting at p. 5 of this document but there's ...
2
votes
1answer
42 views

Question on syzygies

It is hard to formulate a question, but I want to ask about a reference/recipe for computing syzygies in general. For example, on $\mathbb{P}^1_{(x:y)}$ there is an exact sequence $0\longrightarrow ...
1
vote
2answers
43 views

Line in projective space is an example of a curve of genus $0$?

Let $L$ be a line in the projective space $\mathbb{P}^n$ over a field $k$. Is a line $L$ an example of a curve of genus $0$ in $\mathbb{P}^n$. I was wondering if I could verify this with someone, ...
2
votes
2answers
43 views

Reference/Literature on Cantor Sets

I recently came across Cantor sets in my analysis course while doing a hw problem and find them extremely fascinating. I am trying to find other notes/literature on cantor sets with no success. I only ...
2
votes
0answers
21 views

Website for sharing solutions/proof verification?

Is there a website for sharing solutions to exercises in math books? I'm self-studying math and I find solution manuals like this very helpful. When I do an exercise, I usually scribble down a few ...
0
votes
2answers
52 views

Is $x_1^d + x_2^d + x_3^d + x_4^d + x_5^d= 0$ a geometrically integral hypersurface in $\mathbb{P}^4$?

Let $d>2. $Let $X$ be a surface defined by $x_1^d + x_2^d + x_3^d + x_4^d + x_5^d= 0$ in $\mathbb{P}^4_{\mathbb{Q}}$? I am interested in finding out if this is a geometrically integral hypersurface ...
1
vote
0answers
26 views

Hilbert's inequality for $\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{\left(n-m\right)^\lambda}\right|$.

We know that, the Hilbert's inequality for double series states $$\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{n-m}\right|\leq\pi \sum_n |a_n|^2$$ for $a_n\in\mathbb C$. I'd like to know if inequalities ...
2
votes
0answers
39 views

Construction of Tate curve and formal schemes

In the notes websites.math.leidenuniv.nl/geom/tate.ps (and probably in other places), there is a construction of the Tate curve, where the steps are summarized below. 1) Take ...
3
votes
0answers
49 views

Short exact sequence on $\mathbb{P}^1$

Let F be a torsion free sheaf of rank $n+4$ over $\mathbb{P}^1$ which fits in the SES $0\longrightarrow\mathcal{O}_\mathbb{P^1}(-3)^{n+2}\longrightarrow ...
3
votes
0answers
81 views

Symmetry and trivial solutions to Pell equations

Below is a representation of the solutions to the equation $x^2-Dy^2=1$ for $6(6-1)\leq D \leq 6(6+1)$: \begin{array}{c} & 30 & 31 & 32 & 33 & 34 & 35 & 36 & 37 & ...
1
vote
0answers
24 views

Reference request: correlation and spectral analysis of stochastic processes

I'm wondering if anyone knows of a reasonably rigorous text on stochastic processes that discusses specifically things like the autocorrelation, spectral density, and other "correlation and spectral" ...
0
votes
0answers
44 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 ...
2
votes
1answer
59 views

Representation of regular languages by monoids [closed]

I'm interested in representation of regular languages by monoids, and in particular of how to use this kind of representation to get a recognizer. I have found some references on the web, but does ...
1
vote
2answers
65 views

No Galois Theory in Godement's Cours d'Algebre?

I just procured an English translation of Godement's Cours d'Algebre and was interested in reading the treatment of Galois Theory. I started to look for the relevant chapter in the ToC, but to my ...
0
votes
0answers
17 views

Decidability of the theory $(\mathbb{R},+,*,\max,\leq)$

I know that the theory $(\mathbb{R},+,*,\leq)$ is decidable (Tarski 51) and I was wondering wether the decidability status is known when extending this theory with the $\max$ operator. Do you have ...
23
votes
2answers
349 views

A diophantine equation with only “titanic” solutions

I made a note some time ago that I had read in some book that the equation $$313(x^3+y^3)=t^3$$ has positive integer solutions, but that these are so large that it would be absolutely hopeless to ...
1
vote
1answer
44 views

Does this function have a name? What other properties does it have?

Let $x < y$ be real numbers and let $ a$ satisfy $0<a<1$, Does the function $ z = a y +(1-a)x$ have a name? What properties does this function have? This particular function has come up a ...
1
vote
2answers
54 views

how to find null space basis directly by matrix calculation

The problem of finding the basis for the null space of an $m \times n$ matrix $A$ is a well-known problem of linear algebra. We solve $Ax=0$ by Gaussian elimination. Either the solution is unique and ...
2
votes
1answer
35 views

Geodesics are minimizing in a simply connected manifold without conjugate points?

Let $\tilde M$ be a compact Riemannian manifold, without conjugate points. Denote by $M$ its universal cover. Then in this paper, it is claimed that every geodesic is globally length minimizing. Why ...
0
votes
0answers
40 views

Quick question about number of positive summands in a sum of $p$-adic integers

I've started reading recently on $p$-adic numbers online. Forgive me if the question is silly. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers and let $a_1, \ldots, a_k \in \mathbb{Z}_p$. If ...
0
votes
2answers
57 views

Reference request on complex projective algebraic geometry

I am looking for a reference on complex algebraic projective geometry. Specifically, I would like to become more acquainted with notions like the dimension and the degree of a projective algebraic ...
0
votes
0answers
33 views

Prove that The set of Sentences over a theory $T$ is a Cartesian Closed Category

i am sorry to bother but, I have doubts with this problem: In some elementary theory $\;$ $T$ $\;$ consider the set $S=\{p,q,\ldots \}$ of sentences of $T$ as a preorder, with $p\leq q$ meaning "$p$ ...
0
votes
0answers
14 views

Exercise 2.1.5 in An Introduction to Random Matrices by Zeitouni et al.

I have a question regarding exercise 2.1.5 on page 19 in this book: http://www.wisdom.weizmann.ac.il/~zeitouni/cupbook.pdf I would like a reference or help on this exercise. The exercise asks the ...
4
votes
3answers
963 views

Lack of rigour in Spivak's Calculus book?

I logged on today with this exact question: Ellipse definition I found it disconcerting for him to say that it was clear that $a > c$ when $a$ could be equal to $c$ (a straight line) or maybe even ...
0
votes
0answers
76 views

What variant of exponential smoothing is used in the VEGAS numerical integration algorithm?

The VEGAS numerical integration algorithm uses the following procedure to update a vector x of length n: ...
3
votes
2answers
530 views

How many times should I roll a die to get 4 different results?

What is the expected value of the number $X$ of rolling a die until we obtain 4 different results (for example, $X=6$ in case of the event $(1,4,4,1,5,2)$)? I'm not only interested in technical ...
1
vote
0answers
11 views

Reference for the “geometry” or “arrangements” of subspaces of a vector space?

Inspired by Section $5$ of Chapter $1$ in Kostrikin & Manin's famous "Linear Algebra and Geometry", I am searching for a book or paper on the geometry or arrangement of subspaces in a ...
0
votes
0answers
16 views

Reference request: Inverse problem with stochastic error term

In many inverse problems there is an an error term resp. disturbance like $\|{y_\delta} - y \| \le \delta$ with noise level $\delta$, because only noisy data $y_\delta$ are known. Now I'm interested ...
3
votes
1answer
62 views

Is there a book on **Topology** with the desired properties

I have completed courses in Real Analysis,Abstract Algebra,Linear Algebra. My next topic is Topology and Metric Spaces. I am searching for books in this book..Until today I have used the following ...
1
vote
1answer
31 views

Galois group of a polynomial over $\mathbb{C}[t]$

To find the Galois group of the polynomial $X^3-X-t\in\mathbb{C}[t]$, an approach is to compute the discriminant (equal $(2-\sqrt{27}t)(2+\sqrt{27}t)$) which is not a square in $\mathbb{C}[t]$ so the ...
0
votes
1answer
25 views

Normalization of ring of polynomials

Let $x_1(t),...,x_n(t)\in\mathbb{C}[t]$ be such that $\mathbb{C}[t]$ is finite as a $\mathbb{C}[x_1(t),...,x_n(t)]$-module and that $\mathbb{C}(x_1(t),...,x_n(t))=\mathbb{C}(t)$. How to show that ...
0
votes
1answer
28 views

Invariant subfields and Galois group

Let $f$ be an irreducible polynomial of degree $n$ over $\mathbb{Q}$. Let $K$ be the splitting field of $f$ over $\mathbb{Q}$. Let $\alpha\in K$ be a root of $f$. Let $H$ be the subgroup of $G$ that ...
1
vote
1answer
45 views

Intersection of Kummer extension [closed]

Let $p$ and $q$ be two prime numbers and $\omega$ be the primitive 3rd root of unity. The splitting field of $X^3-p$ over $\mathbb{Q}$ is $K_p=\mathbb{Q}(p^{\frac{1}{3}},\omega)$ and we have a similar ...
0
votes
1answer
20 views

Convergence of the Ratio of Consecutive Terms of Positive Recurrence Sequences (Reference Request)

I would like to locate a source that shows (or gets pretty close to) the following. Let $S$ be a positive (integer) recurrence sequence $S_n=S_{n-1}+S_{n-k}$ for some $k>1$. Then, ...
0
votes
0answers
8 views

$\left(n+1\right)\times \left(n+1\right)$ algebra isomorphic to Bose-Mesner algebra?

The Wikipedia article on association schemes claims regarding Bose-Mesner algebras: There is another algebra of $\left(n+1\right)\times \left(n+1\right)$ matrices which is isomorphic to ${\mathcal ...
0
votes
0answers
9 views

Locally nilpotent derivations on noncommutative rings

I am interested on LNDs on noncommutative rings, specially noncommutative polynomial rings. It is being hard to find via Google. Anyone know any reference?
2
votes
0answers
36 views

How are weakly universal Turing machines actually defined?

For what I know, the definition of a universal Turing machine is something along the lines of the following (of course, details might vary from source to source): A Turing machine $M$ is called ...
3
votes
1answer
70 views

Computational Topology and Lie Group Theory [closed]

I study Machine Learning and my limited background in math is enough to understand all the popular algorithms and methods. However, recently, Topology has been successfully applied to Data Analysis ...
1
vote
0answers
18 views

Approximation theory in Lp spaces (Reference Needed)

I am looking for some reference on approximation theory in Lp spaces. I have found a number of papers like: paper1 , paper2 etc. I was wondering if there is a book or a monograph that will contain ...
0
votes
0answers
29 views

Did Ackermann produce a finitary consistency proof of second-order $PRA$?

In Wilhelm Ackermann's Doctoral Thesis (it is claimed, by Richard Zach, for one, in his paper "The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program", arXiv: ...
0
votes
0answers
26 views

Any good Linear Algebra textbook that gives a good geometrical intuition?

I want a book that maybe has pictures in it or just the author trying to explain what things mean geometrically.
5
votes
0answers
59 views

Geometry of the zeros of a power series.

This is probably a basic question that is easily googlable, but it seems that I dont have the right keywords. So my question is, having some power series $$ f(z)=\sum_{k=0}^{\infty}C_{k}z^{k}, ...
0
votes
0answers
14 views

Book filled with proofs pre-Decartes and post-Decartes?

I've been reading some books that argued about the difference of thinking in the resolution of problems with the Greeks methods and with Decartes methods. I wonder if there is some book that presents ...
0
votes
1answer
23 views

Precise analysis of physical interpretations of elliptic

When I read Evans' PDE, I find the below content. I am curious about it. Where I can get the precise analysis ?
2
votes
1answer
35 views

Discrete systems with complicated basin boundaries?

I am trying to come up with the strategy to write my Master's thesis in mathematics. At the moment it is as follows: Finding a (preferably) discrete dynamical system that possesses at least 3 ...
0
votes
0answers
18 views

A recommendation for introductory to spherical harmonics

My Background: So I'm familiar with multivariable calculus, Fourier series, ODE, a little bit of PDE, and a bit of linear algebra. Yesterday I read about Legendre polynomials and their ...
1
vote
0answers
32 views

ODE methods with one evaluation per time step and more state

Predictor-corrector method gives a "Predict–Evaluate–Correct" method for stepping $y' = F(y)$ (autonomous) with one evaluation of $F()$ per time step, and state $[y, step]$: ...
35
votes
4answers
3k views

What's between the finite and the infinite?

I'm wondering if there are any non-standard theories (built upon ZFC with some axioms weakened or replaced) that make formal sense of hypothetical set-like objects whose "cardinality" is "in between" ...
0
votes
0answers
51 views

Reference request on a trigonometric identity (third incarnation)

This is the third incarnation of this question because I've again made this identity simpler, this time by doing what by hindsight seems like the natural thing to do: expressing it in terms of complex ...