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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4 views

Reference needed: Chi-square goodness of fit test, independence test, …

I want to really understand what is a Chi-square test and how it works: when is it needed, what motivates its use, etc. The same thing is needed for "Independence tests" and analysis of variances. Is ...
1
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1answer
23 views

What are those variations of norms called?

Let $V$ be a vector space with a function $\|\cdot\|$ on it that satisfies all the axioms of norms except for scalability condition $\|\alpha \mathbf{x}\| = |\alpha| \|\mathbf{x}\|$ replaced with ...
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1answer
16 views

Reference about conformal map

I am here because I want to know if someone knows of some good e fast books or references about conformal map . More precisely I need of the propeties of the conformal maps on manifolds with boundary. ...
0
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0answers
10 views

“Asymptotic” $\mathbb{R}$-algebras

Definition. By an asymptotic $\mathbb{R}$-algebra, I mean an $\mathbb{R}$-algebra $F$ of functions $\mathbb{R} \rightarrow \mathbb{R}$ satisfying: $$\mathop{\forall}_{f:F}\left[\left(\lim_{x ...
2
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1answer
43 views

Error in Billingsley?

Problem 8.25 in the third edition of Probability and Measure by Billingsley (1995, p. 142) is as follows: Suppose that an irreducible [Markov] chain of period $t>1$ has a stationary ...
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0answers
16 views

What do we call the number that measures how good of an asymptote $g$ is to $f$, and what are the basic results about this number?

Suppose we have a (potentially very complicated) smooth function $f : \mathbb{R} \rightarrow \mathbb{R},$ and we're trying to approximate it (in the limit as the input value goes to $+\infty$) by a ...
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1answer
13 views

Zeitz's ACoPS vs Larson's PSTP

Which of the following books is better to prepare for a mathematical competition at the undergraduate level? The art and craft of problem solving (ACoPS) or Problem solving through problems (PSTP). ...
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1answer
31 views

Density of spaces $C_0^{\infty}(\mathbb{R})$, $W_2^2(\mathbb{R})$ and $L^2(\mathbb{R})$ in each other

Let's consider following spaces: $L^2(\mathbb{R}) = L^2(\mathbb{R}, \mathbb{C}, \mu_L)$ --- space of $\mathbb{C}$-valued functions defined on $\mathbb{R}$ for which the square of the absolute value ...
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2answers
172 views

Works of Kurt Gödel

I'd like to know how to get started with Gödel's work and theorems. I have a decent knowledge of tensors, Einstein field equations. When it comes to logic and set theory, I'm a beginner. Can anyone ...
0
votes
1answer
18 views

Conditions for convergence of derivatives from pointwise convergence

Let $\{f_n\}_n$ be a sequence of functions $\mathbb R \rightarrow \mathbb R$ which converges pointwise to $f$, ie: $$f_n(x) \rightarrow f(x) \hspace{10pt}\hbox{for all $x$}.$$ What additional ...
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0answers
7 views

Geodesics of Sasaki metric

I would like to ask the community for a reference on the following question: Let $(M,g)$ be a Riemannian manifold and $(T^1M,g_S)$ be the unit tangent bundle with the Sasaki metric. Is it true that ...
2
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1answer
28 views

Error estimation for the Wallis product

From the Wallis product we know $$\prod_{k=1}^{\infty} \left(\frac{2k}{2k-1} \cdot \frac{2k}{2k+1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot ...
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0answers
26 views

Kaplansky characterization of principal Artin ring

I would like to learn the proof of this paper of Kaplansky where it is proven that for a commutative ring every module split as sum of cyclic module iff the ring is an Artin principal ideal ring (well ...
1
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1answer
34 views

Gradient flow of Dirichlet energy

I have heard that the gradient flow of the Dirichlet energy gives a solution of the heat equation, i.e. if $u(t,x) \in C^1( [0,\infty) \times \mathbb R^d)$ solves $$ u_t(t,x) = - dE(u(t,x)), $$ where ...
2
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0answers
44 views

QFT and topology

I have had a course in topology, I have heard of homotopy quantum field theory and topological field theory, but I dont know anything about QFT, what would be a good starting point to learn about the ...
2
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0answers
20 views

Properties of spectra

I've been trying to understand spectra better, and I was wondering what would be a good source to learn from. My issue with the classical treatments (e.g., Adams, Switzer) is that people keep telling ...
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0answers
7 views

Nakayama automorphism $\sigma$ of Hecke Algebra $^0H^f_n$ is not inner for $n\geq 3$?

With $R=\mathbb{Z}[q_1,q_2]$, the Hecke algebra $H^f_n$ of $S_n$ is defined to be the $R$-algebra generated by $T_1,\dots,T_{n-1}$ satisfying $T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1}$, $T_iT_j=T_jT_i$ if ...
2
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0answers
29 views

Besides the $3x + 1$ problem, for which similar problems are still unresolved regarding trayectory?

Generalize the $3x + 1$ problem as $cx \pm 1$, where $c$ is a positive odd integer and $x$ is a positive integer iterated through the function as far as possible to discover a cycle. If $x$ is even, ...
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0answers
60 views

Lebesgue Premeasure via Transfinite Induction

If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of ...
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1answer
26 views

Reference on manifolds with boundary

I am here because I want to know if someone knows of some good e fast books or references about manifolds with boundary. Help me please.
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0answers
25 views

Reference request about Thm which use Transversality to compute Homotopy Groups [duplicate]

I'm following the following notes, and my attention was caught by Theorem $1.1.4$. I am unable to find any reference of the proof. Could you suggest me some books in which there is a proof of this ...
3
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1answer
43 views

graph partition, second smallest eigenvalue.

In spectral graph partition theory, the eigenvector corresponding to the second smallest eigenvalue of the laplacian matrix of a graph, in general, is used to partition the graph. What is the ...
0
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1answer
15 views

Lattice points in simplices - reference request

I found this paper http://homepages.math.uic.edu/~yau/35%20publications/An%20upper.pdf which, in formulas (1.2) and (1.3), relates the number of non-negative and positive integer values that are ...
4
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1answer
81 views

A Guide to the Proof of Fermat's Last Theorem from the Modularity Theorem

A few years ago, a friend of mine told me that he had taken an advanced undergraduate number theory class (so something that assumed only a knowledge of algebra and mathematical maturity) which ended ...
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0answers
19 views

reference for unsolved problems in ETCS [on hold]

I am looking for unsolved problems in the theory of "elementary theory of category of sets" are there references for (foundational) problems in the category of sets as foundation?
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1answer
72 views

The greatest book on calculus ever written, if Spivak didn't exist [on hold]

If Spivak, supposedly, did not exist....who would replace him as the author of the greatest book on calculus ever? And please, don't just list a name of books like an absolute imbecile. Give us your ...
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0answers
93 views

Advanced Differential Geometry Textbook

In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses. They are Switzer Algebraic Topology: Homology and Homotopy and Whitehead ...
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0answers
21 views

Finitely generated abelian groups and finite index subgroups

I want a proof or reference for the following fact: "Let $G$ be a finitely generated abelian group and let $\phi:G\to G$ be an injective homomorphism. Then the index $[G:\phi(G)]$ is finite." I ...
2
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0answers
36 views

Global sections of sheafification of Cohen-Macaulay module

Let $S=k[x_0,\ldots,x_n]$ be the polynomial ring over a field $k$ with the standard grading. Let $M$ be a finitely generated graded Cohen-Macaulay $S$-module of dimension at least two. Let ...
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1answer
24 views

Recommend a software for solving math.

I am working with long geometry equation having long chains of other equations in each equation. So recommend a math solving software that may help me solve those long equations and save my time.
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0answers
27 views

Recommend resources of Abstract Algebra which contain more historical development

I read the Wikipedia page about Abstract Algebra. There is a sentence that says Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and then ...
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0answers
20 views

Reference for a couple of terms, $\underline{\operatorname{Hom}}_X(-,-)$ and $\boxtimes$

I have a couple of questions on symbols. What are the names for $\underline{\operatorname{Hom}}_X( \mathscr{F},\mathscr{G})$ for sheaves on a scheme $X$, and $\boxtimes$? And what would be a ...
2
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0answers
32 views

What calculus material can prepare me for MCMC?

I am looking to revise calculus from scratch to move on to Monte Carlo Markov Chain Methods and Quasi Monte Carlo Methods. I studied calculus properly a couple of years ago however it was back in high ...
2
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0answers
13 views

Solutions to Dirichlet problem on manifolds with boundary

I am looking for a reference for the following assertion: Let $M$ be a Riemannian manifold with boundary, and $f:\partial M \rightarrow \mathbb{R}$ be smooth. Then there exists a unique smooth ...
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0answers
24 views

Recommend a book for complex analysis. [duplicate]

I am going to have an introductory course in complex analysis. So please recommend me a book in complex analysis.
3
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1answer
39 views

What does the symbol $H^1_0(\Omega)$ mean?

Here $\Omega \subset \mathbb{R}^n$ is a closed disc centred at $0$ with radius $r$.The book I am reading is assuming the Dirichlet boundary condition on $\Omega$ and claiming that the dual of ...
4
votes
1answer
51 views

Is there an English version of Johann Bernoulli's integral calculus lectures?

The name of lectures of integral calculus written by Johann or Jeans Bernoulli (he is called by both names as far as I know) might be " lecciones mathematicæ de calculo integral"; I must mention that, ...
0
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1answer
33 views

Cross-disciplinary math papers [closed]

Can anyone tell me of any papers in math that belong to more than one field, such as a paper in topology and analysis, or graph theory and boolean algebra? Any such paper would be welcome, but I am ...
0
votes
1answer
24 views

Continuum hypothesis and non measurable set

This is from Chap 8 of Real and Complex analysis of Rudin. The author does not present a proof (using the continuum hypothesis) for the existence of the function $j$. Where can I find such a ...
3
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0answers
52 views

The form of the zeta function of an elliptic curve over a finite field

I seek a (very) elementary proof that the zeta function of an elliptic curve $E$ over $\mathbb{F}_q$ has the form $$Z(T)=\frac{1-aT+qT^2}{(1-T)(1-qT)}.$$ Something tedious and computational making use ...
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0answers
45 views

Vector bundles on Hirzebruch surface $\mathbb{F}_2$

I would like to know a classification for all holomorphic vector bundles on the second Hirzebruch surface $\mathbb{F}_2$. Is this known? What is known? In particular, I'm looking for holomorphic ...
1
vote
1answer
32 views

References about moduli space of abelian varieties with level structure

In the course of one of my research project, I have been advised to try to have a look to "Moduli Space of Abelian Varieties with Level Structure". I am interested in references where this topic is ...
2
votes
2answers
179 views

Why Study Homological Algebra?

I'm very interested in learning Homological Algebra. But I'm not sure about the prerequisites for learning this. My current knowledge in algebra consists of Abstract Algebra (Group,Rings,Fields), ...
2
votes
0answers
45 views

Constructing $\text{Hilb}_P(X/S)$ as a locally closed subscheme of $\text{Hilb}_P(\mathbb{P}^n/S)$?

For a projective scheme $X/S$, how do I construct $\text{Hilb}_P(X/S)$ as a locally closed subscheme of $\text{Hilb}_P(\mathbb{P}^n/S)$? ($P$ is Hilbert polynomial.) Can I get a reference to this ...
4
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1answer
72 views

2011 USAMO Problem 3, Hexagons.

In hexagon $ABCDEF$, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A = 3\angle D$, $\angle C = 3 \angle F$, and $\angle E ...
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1answer
23 views

Fourier Series Relation Time - Frequency

I want to study and understand the relation between time and frequency with the help of Fourier Series. Can you indicate me some papers, or some example?
0
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0answers
11 views

Metropolis Markov Chain and Mixing Time

I have a statistical mechanical system, which I would like to sample with the Gibbs distribution using a Metropolis-Markov chain. I think the following are standard questions, but I am not sure what ...
1
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1answer
21 views

Lusin property (N) for functions of several variables

I just read in a paper by Martio and Zeimer$^1$ that smooth functions ($C^1$) of several real variables have the have the Lusin property (N). I have two questions. First, could someone give me a ...
0
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0answers
37 views

Does any one have a link to Jorm Steuding's Probablistic Number Theory? [closed]

I was reading it through the browser and the link on CiteSeer died. :( I will download it this time. Thanks. Regards, -EM
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0answers
14 views

Reductions of structure groups and sections of coset bundles

I'm looking for a reference for the following proposition: Let $G$ be a Lie group and $H$ a (closed) Lie subgroup of $G$. Let $E \to B$ be a principal $G$-bundle. Then reductions of the ...