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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13 views

About inverse system definition

In the definition of an inverse system in the category of groups some authors ask for the index set to be a partially ordered set, others a directed set (and others both conditions). I am dealing ...
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1answer
50 views

What are some examples of unconventional fields?

We started talking about fields in my foundations of mathematics class, and since the symbols we are using are + and •, I keep catching myself giving them properties of multiplication and addition. ...
0
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0answers
48 views

Fact on Iwasawa module

The following fact falls under the category of Iwasawa modules. Let $M$ be a torsion free finitely generated module over the non commutative noetherian ring $\Bbb{Z}_p[[G]]$, (where $G$ is a p ...
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2answers
130 views

An analogous definition of Fourier transform $\hat f(u) = \int_{-\infty}^{+\infty} f(t) \exp(- i u t) dt$ for sinc-function.

We know the definition of Fourier transform $$\hat f(u) = \int_{-\infty}^{+\infty} f(t) \exp(- i u t) dt \ \ \ (*)$$ It is widely used in the analysis in the frequency of dynamical systems, in the ...
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1answer
18 views

Smallest integer $N(\epsilon)$ such that $K\subset \bigcup_{n=1}^{N(\epsilon)}B(x_i,\epsilon)$

In a metric space, a set $K$ is said to be totally bounded if for each $\epsilon>0$ there exist a finite number of balls $B_1,B_2\dots B_{N(\epsilon)}$ with radius $\epsilon$ which covers $K$. ...
1
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1answer
46 views

Lower bound on convexity radius in terms of injectivity radius (without using curvature)

Let $M$ be a complete Riemannian manifold, and let $C$ be a subset of $M$. We will say $C$ is convex if for any points $p,q \in C$, there exists a unique normal minimal geodesic $\gamma$ joining $p$ ...
3
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5answers
86 views

Can anyone suggest a book on Fourier Analysis containing many good problems

I am taking a basic course in Fourier Analysis in my undergrad Analysis class and I know the theory and related theorems. However, this is a relatively new zone for me and I would like a book that ...
2
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1answer
47 views

Does the Riemann tensor encode all information about the second derivatives of the metric?

In answer to this question I suggested the following as a motivation for the definition of the Riemann tensor: Let two $\mathcal M$ and $\mathcal N$ be two dim-$n$ Lorentzian manifolds with ...
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0answers
26 views

Angle condition for $a^2+c^2=nb^2$

Find a necessary and sufficient angle condition (independent of $a,b,c$ -- see under "what I have got so far" for examples) such that $a^2+c^2=nb^2$ where $n$ is a positive integer. Note: As usual ...
1
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1answer
59 views

Analogue of splitting field in several variables

Let $k$ be a field, and $P \in k[X]$. Consider the extensions $k \subset L \subset K$, where $L$ is a splitting field for $P$ over $k$ and $K$ is the algebraic closure of $k$. Then (by definition) all ...
3
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2answers
66 views

In triangle $ABC$, $a^2+c^2=3b^2$

In triangle $ABC$, we have $a=BC$, $b=CA$ and $c=AB$ as usual. What is a necessary and sufficient condition for $a^2+c^2=3b^2$ to hold? I created this problem as a generalization of $a^2+c^2=2b^2$ ...
3
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1answer
45 views

Triangle geometry: $BC^2+AC^2=n\cdot AB^2$.

I am looking for information regarding which triangles $ABC$ satisfy $BC^2+AC^2=n\cdot AB^2$ for $n=1,2,3,...$. I'm sure that work has already been done in this area since it is a fairly simple ...
0
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0answers
19 views

Manifolds from a “fiber bundle” viewpoint

I am starting to study fiber bundles and its relations with field theories in physics, and I would like to know some books in manifolds/differential geometry that are oriented for a complete study in ...
0
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1answer
22 views

Consequence of linear combination in matrix .

If a column of a matrix is linear combination of another column, what are the consequences ? Several terminology coming into my mind to relate with this such as Rank of the matrix ; Determinant ...
0
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0answers
21 views

Precalculus book recommendation (list provided)

Looking for a precalc book to brush up my knowledge before I start calculus. I know a bit about functions, trig identities/equations etc, but just a few bits here and there and would like a ...
2
votes
1answer
59 views

Chain rule in several complex variables: Wirtinger derivatives

Let $\Omega,\Omega'\subseteq\Bbb C^n$ open, $F:\Omega\to\Omega'$ holomorphic invertible function; it's a variable change, so let's call $F(z)=\tilde z$. Let $r:\Omega\to\Bbb R$ twice differentiable, ...
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0answers
14 views

Mathematical foundations of formal semantics in linguistic

I am looking for information on the mathematical foundations of formal semantics in linguistic. After some time, I found this book (Mathematical methods in linguistics / by Barbara H. Partee, Alice ...
1
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1answer
24 views

Rigorous pre-calc book with answers

I'm looking for a rigorous pre-calculus book so I can start learning Calc and beyond. I have taken some precalc topic, but have a 40%ish comprehension rate. I've done a few bits on limits, continuity, ...
1
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0answers
106 views

Topology and Planetary Nebulae

I apologize ahead of time if this receives any down-votes, but I was just reading a text on topology when the idea struck me: has any mathematician or, for that matter, any topologist, analyzed the ...
1
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1answer
53 views

Definition of a smooth complete integral pointed algebraic curve

Can anybody give me a reference to understand the definition of "a smooth complete integral pointed algebraic curve"? I'm beginning to study the paper "upper bounds for the dimension of moduli spaces ...
7
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1answer
81 views

(Co)homology theory and electrical circuit

I have read that one of the origins of the theory of (co)homology is the study of electrical circuits by Poincare. I'd like to know more about that. Could someone sugest any reference on this subject? ...
1
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0answers
30 views

Explanations for the Hilbert symbols

Are there some elementary sources which help me to understand the Hilbert symbol and the proof of the Hasse-Minkowski-theorem? If you know anything which explains it well (except J.-P. Serre's book!) ...
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2answers
26 views

Ceil function equation

I'm interested whether there is some standard way or any heuristic tricks to solve equations of the following type (N is unknown, $\alpha$ is positive irrational number, m is positive integer): ...
0
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2answers
52 views

Good book on Lebesgue Theory [duplicate]

I am a graduate student and I need a suggestion for a good book in Lebesgue Measure Theory with good exercises and if its possibly with hints or solutions. Thank you.
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2answers
54 views

Learning about convex optimisation

I'm interested in learning a bit about convex optimisation. The wikipedia article contains the following paragraph: The convexity of $f$ makes the powerful tools of convex analysis applicable. ...
1
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0answers
18 views

Expected value of Brownian Bridge evaluated at a stopping time

Denote by $B$ a Brownian bridge process, by $B(\omega)$ a realization of it and by $B_t$ the projection to the time point $t \in [0,1]$. Now let $c < 0$ and $$t^*(\omega) = \sup\{t \in [0,1]: ...
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1answer
22 views

Concerning ideals of $\mathbb Z[\sqrt m]$ and $\mathbb Z[\sqrt m] [x] $

For a given integer $m<-1$ or non-square integer $m>1$ , how do we calculate the quotient ring $\mathbb Z[\sqrt m]/I$ , for example its order or whether it is a field or has zero divisors or not ...
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0answers
67 views

Do we have this type of integral expression of Bessel function of the first kind?

Let $z=\lambda+i\mu$ with $\mu>0$. Then for any $r>0$, $k=1,2,3, \cdots$. Do we have the following identity $$ \int_{r}^{\infty}{\frac{t}{\sqrt{t^2-r^2}}(\frac{1}{t}\frac{d}{dt})^k ...
1
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0answers
29 views

References about algebraic/differential geometry in French

Aside from learning mathematics, I am learning French, so I would like to practise both at the same time if possible. Do you know of any good references about complex/algebraic geometry or ...
0
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0answers
36 views

Seeking guide for project.

I have to submit a project within 2 months for 4th semester(M.Sc). I wish to do it on knot theory, although I know little about it. My plan is to make it an introduction to the subject and to ...
1
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0answers
21 views

Reference for derived functor

I'm following a course in algebraic geometry and in 2-3 month we will see the cohomology of schemes using derived functors. I don't know anything about it, (and about category theory in general), ...
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0answers
27 views

Chromatic number for sets of five or more elements

Definition. Given the set $D$ of positive integer numbers, we construct the distance graph for the integers, which vertices are $\Bbb{Z}$ and two numbers $x$ and $y$ are connected if the ...
3
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1answer
117 views

A textbook for a rigorous introduction to Stochastic Analysis with emphasis on stochastic differential equations

I'm looking for a good textbook for an introduction to Stochastic Analysis, preferably one that focuses on rigour. I am familiar with measure theory and basic probability theory. The direction I am ...
1
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0answers
22 views

maximum principle on compact manifolds with boundary

Let us consider the equation $Lu + f(u) = 0$ on a compact manifold $\overline{M} = M \cup \partial M$ with boundary, with Dirichlet boundary conditions. $L$ is a linear elliptic operator, and $f$ ...
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0answers
14 views

Solving a linear functional equation

Working with Green functions, I have found to solve the following equation $$ -\omega^2G(\omega)-m^2G(\omega)+\kappa\sum_{n=-\infty}^\infty b_nG(\omega-n\omega_0)=1 $$ where $m$, $\kappa$ and ...
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0answers
25 views

Looking for reference giving exposition on Pell's Equation

I know the continued fraction convergence method of finding a solution of Pell's equation . I am looking for other methods of finding the smallest or at least one solution of Pell's equation which is ...
2
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0answers
24 views

Terminology for subsequences?

Note: I'll index all my sequences by $\mathbb N$, so I drop the indices in the sequence notation. The notion of a subsequence of a sequence $\{a_n\}$ is a sequence obtained by deleting some terms in ...
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2answers
121 views

Evaluating $\sum_{\gcd\left(m,n\right)=1}\frac{1}{m^2n^2}$

I was wondering how one would evaluate the sum $$\sum_{\gcd\left(m,n\right)=1}\frac{1}{m^2n^2}.$$ The first thought that came to mind to to try something like this: ...
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0answers
27 views

Polynomials satisfying Similar Appell Sequence Properties

Appell sequences, $\{P_k(x)\}$ are sequences of polynomials that have the following characteristics. $1. P_0(x)=1$ $2. P'_n(x)=nP_{n-1}(x)$ Now I have found a few sequences of polynomials that ...
2
votes
2answers
132 views

Further Reading on Stochastic Calculus/Analysis

I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal. So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on ...
3
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0answers
31 views

Complete intersections with respect to different sections

Let us consider the Whitney sum $E$ of holomorphic line bundles $L_1,\dots, L_k$ on a smooth (projective) variety $X$. For a generic global section $s$ of $E$, the zero locus $Z_s := s^{-1}(0_E)$ ...
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2answers
132 views

approximate vanishing in Pontryagin dual

Let $\{n_k\}\subseteq \mathbb{Z}$ to be any given sequence of integers, and suppose it satisfies the following property: (*) For any $\lambda\in A\subseteq \mathbb{T}$(the unit circle), ...
0
votes
0answers
16 views

Estimate of roots of polynomial with positive, decreasing coefficients

I am looking for guidance about the size of roots of a polynomial $\sum a_kx^k$ where the coefficients are positive and decreasing, $0<a_{k+1}<a_k$ for each $k$. My hope is that the roots (real ...
1
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1answer
20 views

Positive Integral Solutions to Matrix Equation

Given a matrix, $M$, and a vector ${b}$, with positive integral entries, how many solutions are there to the equation $$Mx = b$$ where $x$ is a positive integral vector? Any solution or reference to ...
0
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0answers
60 views

An English translation of the paper “Expansion in Fourier series and integrals with Bessel functions” by Levitan

In my research, I've come across some results which can be found in the aforementioned paper by Levitan (see here). Several authors simply paraphrase the work in the paper, particularly the results ...
0
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0answers
23 views

Exponential and regular Diophantines?

I am looking for a reference on connections between exponential and "regular" (polynomial) Diophantine equations. For example, I was wondering about the Catalan-Mihailescu problem and I thought of the ...
1
vote
1answer
41 views

Logic built on “strings”

Are there (studied) logics or fragments of logic in which there are two unary functions Z, O that are used to build (binary) strings, for example: $\epsilon$ is the empty string, if $x$ is a string ...
1
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1answer
37 views

Chebyshev's theorem on the distribution of primes

I a lecture V. Arnold says that Chebyshev had proved that the limit $$\lim_{n\to \infty}\frac{\pi(n)}{n/\mathrm{log}(n)}$$ if exists is equal to one. Where I can find the proof? Thanks!
3
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1answer
79 views

What is the prerequisite knowledge for Navier–Stokes Existence and Smoothness problem?

I am highly interested in the Millennium Problem of Navier–Stokes Existence and Smoothness (also here) and my aim is to reach some level of knowledge to do research on it. The problem seems simple to ...
0
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0answers
17 views

Can matrix representation be extended to free modules?

Let $V$ be a finite-dimensional $F$-vector space and $T$ be an $F$-endomorphism on $V$. Let $\alpha$ be an $F$-basis for $V$. Then, there is a natural $F$-linear isomorphism ...