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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23 views

Brownian motion hitting probability and Martin capacity

Consider a Brownian motion $B_t$ in $\mathbb{R}^n, n\geq 3$ and the ball $B(0, r)$ of radius $r$ around the origin. Let $\overline{C}$ be a compact set inside $B(0, r)$ such that $C$ is open in $B(0, ...
-1
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0answers
12 views

Transient Brownian motion and stopping time

Let $B(t)$ be a Brownian motion in $\mathbb{R}^n$, or on a compact Riemannian manifold $M$ of dimension $n$, $n \geq 2$. Let us consider the stopped Brownian motion at a deterministic time $T$ (in ...
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1answer
15 views

Hopf fibration and exact sequence in homotopy

I have encountered the Hopf fibration $S^1\hookrightarrow S^3\twoheadrightarrow S^2$ when studying smooth principal $G$-bundles. Whenever I google the Hopf fibration, I encounter a remark which boils ...
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0answers
18 views

Reference request: how to check whether a set is invariant for a second order dynamical system?

I am looking for some examples where invariant set is proved for second order systems For a example, consider the Van Der Pol equation: $$\dot x_1 = x_2$$ $$\dot x_2 = -x_1 + 0.5(1-x_1^2)x_2$$ In ...
2
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2answers
26 views

References for the applications of probability in gambling

The intuition behind many theorems in probability comes from gamblers' games. I would like to know if there are any books or articles which cover some such connections between probability and its ...
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2answers
16 views

Books on Riemann Surfaces

I am starting a scholarship on geometry and the subject of research is going to be Riemann surfaces (we will focus on compact Riemann surfaces). I am finishing my undergraduate studies so my knowledge ...
1
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0answers
21 views

Software for solving first-order logic

Is there any class of software that can help me with the following problem in first order logic: given $\phi$ a formula with a "hole" in it (a subformula which is undetermined) and a particular set of ...
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0answers
25 views

Future learning for a math graduate in applied mathematics references

As a mathematics graduate with focus on programming we did a whole lot of coding of some mathematical statements (as well as proving them), but yet rarely giving real life examples and applications ...
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1answer
37 views

Real analysis reference for statistician

I'm a undergraduate statistics student, I think that learn Real Analysis can be useful to me in some points, can anyone suggest a introductory book for self-study ? I'm already multivariate calculus, ...
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0answers
19 views

Reference and Textbooks for the given topics [on hold]

This is my post in Maths Stack Exchange. I am a student of BS-MS course .I want to get some recommendations in books. I will be giving different competitive exams also. I need one textbook and one ...
1
vote
2answers
45 views

Reference request: When is a conic birationally equivalent to the projective line?

I am looking for a reference which contains the proof of the following theorem: "A conic $C$ defined over the field $\mathbb{F}$ is birationally equivalent to $\mathbb{P}^{1}(\mathbb{F})$ if and only ...
1
vote
1answer
23 views

Finite group representation on endomorphism ring

Let $\rho:G\to\mbox{GL}(V)$ be a finite dimensional representation of a finite group $G$. We can assume the base field is $\mathbb{Q}$, but it doesn't really matter. Then we also obtain a ...
2
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0answers
34 views

Proof of the Ribet's theorem

My question is very simple : My goal is to read a proof the proof of the epsilon conjecture proven by Ken Ribet (1986) which is an ingredient of the proof of the Fermat Last Theorem (I want the ...
1
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0answers
31 views

Congruence numbers

Having read about Stirling numbers of the second kind I am curious. The article says it shows the number of equivalence relations on a set $n$ with $k$ equivalence classes which makes sense to me from ...
-2
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0answers
35 views

Solving Math problems for entrance exams [on hold]

Could you please share what are the ways one goes about to solve the problems in various books. I will take an example but this would be true for any book. Arthur Engel's book on problems. I have seen ...
1
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0answers
15 views

Examples where ostragodsky's method is needed for integrating rational functions

I found out about Ostragodsky's method for integrating rational functions and thought it was pretty cool. However, I have never encountered any examples where it seemed needed (rather than just ...
2
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0answers
51 views

How does one read a formula with subscripts and superscripts?

An expression like $\Gamma_{ij}^k$ seems to be pronounced "gamma sub i, j upper k". Is this a generally accepted usage? Question. Is there a quotable source for such usage? Note that $k$ is not a ...
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0answers
18 views

Combination of certain linear-programming topics new?

I am writing a book on Linear Optimization. Its goal is to present material in a particular form which has not been encountered yet in the literature to the best of my knowledge. I am aiming at the ...
1
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0answers
20 views

Return probability of a SRW in an even number of steps

I am looking for some references for the following problem. Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)...
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0answers
44 views
+50

Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.

I have been scouring the internet for answers for some time and would therefore appreciate a reference or a proof since i'm not able to produce one myself. Let $(\mathcal{X},d)$ be a metric space, ...
0
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0answers
21 views

Fisher information for exponential family: Regularity conditions

for the Fisher-Information to be defined certain regularity conditions have to be fulfilled (like in Lemma 5.3. in Theory of Point Estimation by E.L. Lehmann or on slide 2 here: http://www.stat.nus....
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0answers
10 views

References request: two-queue, one-server model with pre-emptive queue priority and finite buffers

Sorry of the title is a mouthful. I'm developing a queue model with the following characteristics: Two queues: One contains an infinite number of people (Queue A) while the other (Queue B) is ...
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1answer
34 views

Good reference for partitions of unity?

I am reading about Sobolev Spaces and regularity theory of PDEs. The partition of unity lemma, as stated in Haim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations, is as ...
2
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0answers
26 views

Introduction to flag manifolds

What is a good self-contained introduction to the geometry of complex flag manifolds?
2
votes
2answers
69 views

Introductory Topology Book Recommendation for Economics

Would you please share your 2 cent on book recommendation for introductory topology book to graduate student in Economics. Have exposure to the first half of the yearlong analysis course in the ...
-3
votes
1answer
63 views

Help finding an article [closed]

Hello Recently I have been studying algebra and am in search of the following paper : Kac, V. G. Classification of simple $Z$-graded Lie superalgebras and simple Jordan superalgebras. Comm. Algebra 5 ...
1
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1answer
40 views

Uncertainty in a theorem about Sarrus numbers

From https://oeis.org/A001567 there is a theorem of Ray Chandler formulated: An odd composite number $2n + 1$ is in the sequence if and only if multiplicative order of $2\;(\text{mod}(2n+1))$ ...
2
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0answers
49 views

proof of Triangle Removal Lemma

Where can I find a proof of the following version of Triangle Removal Lemma (or any version equal to it)? Let $G(V,E)$ be a graph on $n$ vertices such that it contains $\varepsilon n^3$ triangles, ...
1
vote
1answer
53 views

Calculus of 1 variable [on hold]

What are some good web link or pdf link to understand(self study) calculus of variable intuitively?I am a 12 th grade student with some notions of maxima and minima and some other notions in one ...
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0answers
18 views

Books / lecture series on Homotopy theory

I want to read homotopy theory on my own so I want to know prerequisites, good books and if there is any lectures available which can help me. Links are welcome. Till now I have done point-set ...
2
votes
0answers
50 views

Problem sets on Abstract Algebra

Many times we ask about what books should we read to learn or know more about a math topic (Abstract Algebra, in this case). However, I would like to get a list of the exercises what should we solve ...
1
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0answers
47 views
+50

Hailstone collatz max sequence length upper bound of $260.5+x^{.43}$?

Let the Collatz function be defined as if $x$ even $c(x)=x/2$, if $x$ odd then $c(x)=3x+1$ over the naturals. Each operation is defined as a step. For example $3$ goes $(3,10,5,16,8,4,2,1)$ and takes ...
3
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1answer
55 views

Unsure on which sources to choose related to Calculus

I tried to get into Spivak's Calculus only to find that I've never been taught the type of Math presented there. First chapters talk about the properties of numbers, then mathematical induction, ...
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1answer
35 views

I've been working on Spivak and I'm on chapter 7. What are some good books to supplement Spivak for someone beginning to learn pure mathematics.

If I have too much difficulty with a concept/problem, then I'll just press on and solidify my understanding when the concept arises later by going back to it. This seems to be a lucrative method at ...
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0answers
13 views

Sheaves of Simplicial Rings?

Could someone provide me a reference for a treatment of sheaves of simplicial commutative rings? As in simplicial sheaves with a ring structure.
0
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0answers
16 views

Mapping cylinder of chain complexes via $-\otimes \Delta$

An instructor gave me a homework set where the mapping cylinder of a chain map $C_\bullet \xrightarrow{f} D_\bullet$ is defined as $(\Delta^1_\bullet \otimes C_\bullet) \oplus_{C_\bullet} D_\bullet$, ...
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0answers
59 views

Cardinallity increasing constructions

If we start with $\mathbb{Z}$ we can through localization get $\mathbb{Q}$, but that has the same cardinallity as $\mathbb{Z}$, so it doesn't increase cardinality for infinite sets, which is what I am ...
1
vote
1answer
41 views

Comparison of capacity of sets in $\mathbb{R}^n$

This is mainly in reference to this MSE post. Let $B_r \subset \mathbb{R}^n$ denote the ball of radius $r$ centered at the origin. Consider any set $F \subset B_1$. For all sets $\Omega \subset \...
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0answers
18 views

minimum distance of a linear codes

My question is about computing the minimum distance (weight) of a linear code. Assume that we have the generating matrix of the code. Then we can easily compute the weights of each row and of course ...
0
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0answers
28 views

Limit of the commutator of two elements?

Given a Lie group $G$ such the $\mathfrak{g}$ denoted its Lie algebra. Let $[g,g']_{G}$ the commutator of two elements $g,g' \in G$ and denoted by $[X,X']_{\mathfrak{g}}$ the Lie bracket of two ...
0
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2answers
39 views

Order Theory and Lattice Theory Synonymous?

Is Order Theory the same as Lattice Theory? Can anyone recommend good beginners text book on either?
1
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0answers
45 views

How to see that diagonal and tranvections matrices generate $GL_n(\mathbb{Z})$?

I'm trying to see how diagonal and transvection matrices generate $GL_n(\mathbb{Z})$. Is there any book that I can find a more detailed description of this problem? Thanks!
0
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0answers
48 views

classification of equilibrium points of 3d systems of ode's

I'm trying to find information about the classification of equilibrium points of 3d systems of differential equations, The qualitative analysis. I wonder if someone could refer me to some book or ...
2
votes
1answer
41 views

Capacity of a set in $\mathbb{R}^n$

The $2$-capacity of a set $\Omega$ sitting inside an open set $V \subset \mathbb{R}^n$ is given by $$\text{cap}_2(\Omega, V) = \inf_{u \in C^\infty_0(V), u|_\Omega \equiv 1} \int_V |\nabla u|^2 dx.$$ ...
2
votes
1answer
35 views

Expected number of duplicates

Suppose I have $m$ bins and throw $n\ll m$ balls into the bins uniformly at random. (In my application $n\sim m/\log m.$) What is the expected number of duplicates? In other words, if there are $k_i$ ...
0
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0answers
23 views

Implicitization Problem on Graphs?

I learnt the implicitization problem for varieties in introduction course on Algebraic Geometry. I am trying to understand how to formulate a similar implicitization problem on graphs where the ...
1
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0answers
37 views

Green's formulae, Stokes theorem, Gauss theorem, divergence theorem and Gauss-Green theorem?

I am getting really confused about the Green's formulae, the Divergence theorem and all those related equalities. For example, How is this formula exactly called? $$\int_\Omega \frac{\partial u}{\...
5
votes
1answer
53 views

Partitioning $\mathbb{P}^1(K)$ via the class group

Let $K\subset\mathbb{C}$ be a number field. There is a surjective map $\phi:\mathbb{P}^1(K)\to Cl(K)$ from the field to the class group, sending $[\alpha:\beta]$ to the class of the ideal $(\alpha,\...