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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1
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0answers
22 views

Number of squarefree numbers and the Basel problem

Who discovered/proved that there are about $$ \frac{x}{\zeta(2)} $$ squarefree numbers up to $x$, or (roughly) when was this first known? Today I think this is considered 'obvious', but I don't know ...
0
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1answer
12 views

Runge's Theorem for meromrophic functions

Is there a name for this extension of Runge's theorem? Theorem: Let $K\subset\mathbb{C}$ be compact, and let $A\subset K^c$ be a set which intersects each component of $K^c$. Let $f$ be meromorphic ...
1
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0answers
10 views

A Mertens-like product over primes

MathWorld's page Prime Products gives the 'related result' (7) to Mertens' theorem: $$ \lim_{n\to\infty}\log p_n\prod_{k=1}^n\frac{1}{1+1/p_k}=\frac{\pi^2}{6e^\gamma}. $$ Does this identity have a ...
1
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0answers
22 views

Multivalued Functions for Dummies

I have been studying complex analysis for a while, but I still cannot "get" how multivalued functions work. Despite having it explained to me many times, my brain cannot process it. For example, I ...
0
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0answers
20 views

Does a $K_n$ with $n$ pendants have a name?

Consider the graph we get by taking the complete graph on $n$ vertices, and then attaching a pendant vertex to each of the $n$ vertices by an edge. Does such a graph have a name, i.e. do such graphs ...
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0answers
10 views

Reducing uncertainty of a mathematical model with data (Process control)

I know this is a very broad question, but need suggestions, link to good reference papers etc. So here is the question: I have an uncertain model whose parameters are static (not changing with time) ...
2
votes
0answers
44 views

Real form and real structure on a complex Lie group

E.B.Vinberg and A.L.Onishchik in their book give the following two definitions. For a complex Lie group $G$ its real Lie subgroup $H$ is called a real form of $G$, if a) the Lie algebra $L(H)$ of ...
0
votes
1answer
23 views

Ratio of degrees of nodes in Graph

I have a question regarding to graph and ratio of degrees of nodes in graphs. See the following image: I'm going to find a relation between $A$ and $C$. So, I count all links from $C$ to $B$s $= ...
2
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0answers
12 views

Good Problem Book for Convergence Concepts in Probability

I need a good book containing many challenging exercises (or problems) on Convergence Concepts in Probability. The topics I have covered are: Borel-Cantelli Lemmas Modes of Convergence and ...
2
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0answers
21 views

Multiplication operators are sectorial

Consider the multiplication operator $M_a$ on $L^p(M)$, where $M$ is a Riemannian manifold, and $a$ is a non-negative function. An operator $A$ is said to be sectorial if there exists $\theta \in (0, ...
2
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0answers
54 views

Challenging problems in algebra (book recommendation)

Could you suggest me a book/web page where I can find challenging/hard problems in algebra (possibly with solutions) for an undergraduate student (groups, rings, fields, Galois theory)? Thanks in ...
0
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1answer
9 views

any two uncountable Borel subsets of a Polish space are Borel isomorphic

I'm trying to find a proof that any two uncountable Borel subsets of a Polish space are Borel isomorphic. I've been trying to find it in Kechris' "Classical Descriptive Set Theory" but I've been ...
0
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0answers
23 views

Beyond Schwarz Lemma

Let $f(z)=a_1z+a_2z^2+a_3z^3... $ be a Schwarz function then by lemma $|a_1|\leq 1 $.But what is known about the higher coefficients? For example ; what can be said about $ max [a_1+a_2]$ ? Is there ...
4
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1answer
44 views

Good sources to learn about Geometric Analysis

So far the topics that have captured my interest the most are analysis and geometry. I want to learn a bit more about geometric and global analysis. There seems to be no shortage of geometry books but ...
2
votes
0answers
18 views

Resolvent estimate of hyperbolic Laplacian

Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form $$\Vert (-\Delta - \lambda ...
-3
votes
1answer
91 views

“Notes on Set Theory” - Badly Written? [on hold]

I hope I'm not breaking any rules by asking this particular question, but I honestly can't think of a better place to inquire about this. A few days ago I managed to get my hands on "Notes on Set ...
1
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0answers
2 views

Consistency/range conditions for (integral) transform mapping into higher-dimensional space

I am interested in learning more about what the (formal) implications are when transforming a $n$-dimensional function space (e.g., the space of all $\mathbb{R}^n\to\mathbb{R}$) to a higherdimensional ...
1
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0answers
18 views

Continuity of Translation and Dilation on $L^p$ spaces

Let us consider any $f \in L^p(U)$, where $U \subset \mathbb R^n$ is open, and $1 < p < \infty$. We know the translation operator $f(x) \mapsto f(x+a)$ and the dilation operator $f(x) \mapsto ...
0
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0answers
25 views

Bertini's papers in english

I'd very interested by finding translation in english of some papers of Bertini. I don't want necessarly a paper from himself, for exampe I was very happy to find this paper : ...
1
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1answer
20 views

Why can a function in $L^1(\partial \mathbb{D})$ be represented by a Fourier series?

I am looking for a reference to the claim that for any $f\in L^1(\partial \mathbb{D})$, where $\partial \mathbb{D}$ is the unit circle in $\mathbb{C}$, ...
2
votes
0answers
9 views

Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if $A$ and $B$ are two $n \times n$ Hermitian matrices, and $[A,B]=C$. I'd like a ...
0
votes
1answer
23 views

Is this a metric on $\mathbf P\mathcal H$?

Let $\mathcal H$ be a real or complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. On the projective space $\mathbf P\mathcal H = \left(\mathcal H\setminus\{0\}\right)\big/{\sim}$ ...
0
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0answers
25 views

An introduction to monodromy in topology and algebra

I'm looking for some basic references on monodromy and monodromy groups, in particular I'd like something which describes the interplay between the topological definition (in the theory of covering ...
0
votes
0answers
54 views

How we get the topics of our textbooks in math? [on hold]

My point is which processes have happened till we get such kind of undergraduate Algebra, Analysis or Topology topics? Sometimes I think about the vagueness that these materials carry since I don't ...
1
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0answers
24 views
+100

Mean value theorem for random variables

In a proof I am trying to understand a mean value theorem for random variables is used. It is stated that $$f(X+Y)=f(X)+f^\prime(X+\theta Y)(Y-X)$$ for real valued random variables $X$ and $Y$ and ...
3
votes
0answers
11 views
+50

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - ...
0
votes
2answers
56 views

Seeking advice from all [on hold]

I've come back to education after 4 years and I feel very out of practice, currently I am studying a-levels and need to pass with excellent grades for my ill fathers sake as it is his last wish. I am ...
3
votes
1answer
37 views

Book which covers these contents of the same level of Fulton's book.

My question is very specific. I'm studying Chapter 8 of Fulton's algebraic curves book and I would like to find another book (or online sources) which covers these contents: Divisors, the Vector ...
0
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0answers
42 views

Moscow Institute of Physics and Technology Mathematics Competition Question

As mentioned in the title, does anyone know where can I obtain past question papers from Moscow Institute of Physics and Technology? Any link will be greatly appreciated. The following is one of ...
1
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3answers
36 views

What book is good in studying beginning optimization?

Recently, I heard some talks about Optimization. And I am beginning to love that field. I want to study beginning optimization, what book can you recommend for me? Also what tips can you give to a ...
2
votes
0answers
26 views

Determination of quartic Gauss sums

Typically, the Gauss sum over $\mathbb{F}_p$ of order $k$ means the quantity $$\sum_{n=0}^{p-1} e^{2\pi i n^k/p}.$$ In the book Gauss and Jacobi Sums of Berndt, Evans and Williams, a more general sum ...
0
votes
2answers
39 views

A-Level/GCSE Geometry textbook? Geometry for STEP and MAT?

everyone. I have been looking for a book that covers the most elementary parts of Geometry, such as similar triangles, circles(arc, sector and others), Pythagorean theorem, Sine and Cosine Laws, so ...
0
votes
1answer
59 views

Prerequisites for learning general topology

I want to learn general topology in order to apply it in electromagnetism. I am an undergraduate student and I have a background in linear algebra (not at an advanced level), linear differential ...
0
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0answers
41 views

Feynman Integration

Could somebody please recommend some places to learn Differentiating under the Integral sign? I need to learn this technique as a lot of Integrals can be solved using this. Many thanks!
0
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1answer
26 views

Interpreting d as an operator in differential calculus.

I am enjoying this mathematics book for the general public called Measurement by Paul Lockhart. For the most part, I am happy with his metaphors and intuitive explanations of the different concepts, ...
4
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0answers
56 views

Does a space with peoperty A have a topological name?

As we know, If $X$ is a Tychonoff pseudocompact space, then for every decreasing sequence $\cdots\subset W_2\subset W_1$ of nonempty open subsets of $X$ the intersection $\bigcap_{i=1}^{\infty} ...
1
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0answers
48 views

Suggest a reading list to start TQFT

What would be books that would give the necessary prerequisities to study TQFT? I want to read something like Kock's Frobenius algebras and 2d TQFTs, I only know enough math that got me through a ...
6
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1answer
55 views

Books that start with questions? [on hold]

Does anyone know of any books that start with a relevant question, study it from different perspectives and then show some mathematics? I would appreciate it if you posted any that you know of, ...
1
vote
1answer
17 views

Slight generalisation of the distribution of Brownian integral

I think I have seen once that if the processes $\sigma$ and $W$, a Brownian motion, are independent then one has that $$ E \left[\exp \left(iu\int_t^T \sigma_s \, dW_s\right) \mid \mathcal{F} \right] ...
1
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0answers
22 views

Classification of $\mathbb{G}_m$-torsors?

Is there a nice proof or reference to one of the theorem that any $\mathbb{G}_m$-torsor is isomorphic to $\mathcal{L}\setminus z(S)\to S$? Here I am denoting by $\mathcal{L}\to S$ to be a line bundle ...
2
votes
0answers
42 views

Hilbert's reduction of second order logic to first order logic

I have read on the internet a theorem of Hilbert that says that we can reduce every second order theory to a first order theory. So there exists only one logic: first order logic. I cannot find it ...
2
votes
0answers
31 views

How should I learn the Mathematical Proofs?

S.E advisers, What is the most efficient way to learn the basic proof methodologies, which are essential for studying the mathematical analysis and number theory? I am very interested in studying ...
1
vote
1answer
21 views

Tutorials on LDPC error correction codes

Please consider this as soft question. Recently, I have been studying channel coding and in particular error correction codes. I am looking for best tutorial (easy to understand) on LDPC error ...
1
vote
1answer
38 views

Inverse Laplace Transform of $1/(s+1)$ without table

The pole is on the left half plane, so $\gamma =0$ $$\frac{1}{2i\pi}\int ^{i\infty}_{-i\infty}\frac {e^{st}}{s+1}ds$$ substituting $iu=s$ $$\frac{1}{2i\pi}\int ^{\infty}_{-\infty}\frac ...
1
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0answers
16 views

Arc length in curvilinear coordinates - reference request [on hold]

Can someone recommend a website or a book with solved problems regarding (advanced mathematics): Transformation from cartesian to curvilinear coordinates Parametrization of the spatial curve Arc ...
2
votes
2answers
26 views

explain the solution and/or suggest a different one

I have come across the following problem, in my calculus II course, about improper integrals: problem: Find the following limit, if it exists. $\displaystyle\lim_{x\to 1} \int\limits_{x}^{x^2} \! ...
0
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0answers
16 views

Is there an explicit formula for the Fourier transform of $(z-|\xi|^{\alpha})^{-1}$ on $\mathbb{R}$?

Let $0<\alpha< 2$, and $z=\lambda+i\mu$, where $\mu\ne 0$. Consider the following Fourier transform on $\mathbb{R}$ $$g(z,x)=\int_{\mathbb{R}}\frac{e^{ix\cdot\xi}}{(z-|\xi|^{\alpha})}d\xi$$ ...
0
votes
0answers
19 views

Looking for a paper on Weakly uniform bases

I want to find an old paper: R.W. Heath, R.W. Lindgren, Weakly uniform bases, HOUSTON JOURNAL OF MATHEMATICS. 2(1) (1976) 85–90 Could someone help me? A link is also welcome. Thanks!
1
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0answers
26 views

Reference request: Measure theory books using $\omega(\alpha) = |\{f>\alpha\}|$

I am working from Wheeden and Zygmund's Measure and Integral, and they prove theorems such as $\int_E f = -\int_{-\infty}^{+\infty} \alpha d\omega(\alpha)$ where $\omega(a) = |\{x: f(x)>\alpha\}|$ ...
3
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1answer
57 views

Is it known whether any positive integer can be written as the sum of $n$ different squares?

Is it known whether any sufficiently large positive integer can be written as the sum of four different squares? I know that every positive integer can be written as the sum of four not necessarily ...