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Questions tagged [reference-request]

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.

2
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1answer
19 views

Differential equation with “backwards product rule”.

If we have the following differential equation, ($h,f$ known, $y$ unknown): $$f'(x)y(x) + f(x)y'(x) = h(x)$$ it would be easy, since we could spot the derivative for a product: $$(f(x)y(x))' = h(x)$...
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0answers
6 views

Optimal transport with relaxed constraint on terminal distribution

I have read the topic on relaxing constraint on relaxing marginal constraints Optimal transport with relaxed constraint on marginals, where the constraint is expressed as the difference of initial and ...
-3
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0answers
27 views

Calculus and Physics for beginners. Suggestions for books

I would like to start brushing up on some math and physics. Mostly math that theoretical physicist use. Want to start with the basics so wanted some suggestions on books that'll help me with this. I ...
1
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0answers
12 views

Solving PDEs by adding small perturbation terms and taking limits

When trying to solve some PDE problem, sometimes we made approximation problem that is easier to solve than the original one. We do that by adding a new term(s) in the original problem. Then we try to ...
0
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1answer
25 views

What are some advanced books on metric space?

Metric space, with the additional notion of “distance between points”, has properties that are more “concrete” than a topological structure. After a basic study I saw a number of strange and ...
0
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1answer
39 views

Uniform Continuity in Topological Spaces?

Is it already in literature this generalized notion of uniform continuity in an arbitrary topological space (not necessarily in exactly the same form)? Let $(X, T_{1})$, $(Y, T_{2})$ be topological ...
3
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0answers
46 views

Looking for Errata, Gradshteyn and Ryzhik, 4th edition

This is the edition I have as a hardcover, and even though it's old, it has served me well. Except there's quite a few errors in there and I would like to know if there's a full list available ...
1
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0answers
36 views

Generalization of topologies with equivalence classes of sets

Is there a generalization of topological spaces which works on equivalence classes of subsets? To be a little bit more precise, I would think of something like the following: Let $X$ be a set and $P(...
1
vote
1answer
18 views

Function spaces from a geometrical viewpoint.

I'm wondering whether there exists some geometrical theories of functional spaces. I mean; function spaces ($L^p$ spaces for example) are called topological vector spaces (TVS). I'm interested in ...
-1
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0answers
14 views

The simplicity of the group PSU.

Can someone give a reference (with proof) to the fact that PSU (the projective special unitary group) is a simple group? Thanks Bill
0
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0answers
16 views

Is there free material for optimal design or do I need to buy books?

Is there free material for optimal design or do I need to buy books? While one may appreciate the effort of science writers, I believe this stuff should be public domain, since it's science. So can I ...
1
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0answers
17 views

Inverse problem : Finding a vector bundle (resp. with connection), given its characteristic class (resp. differential character)

Given a rank $n$vector bundle $\alpha :E \to M$, and an element $u \in H^k(BG, \mathbb{Z})$, $G=GL(n,\mathbb{R})$we can define its characteristic class $u(\alpha) \in H^k(M, \mathbb{Z})$ as $f_\alpha^*...
0
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1answer
32 views

(Soft Question) Largest known semiprimes with no known factors

Is there a list, similar to prime numbers and probable primes, of the largest semiprimes with unknown factors? Is there a list of numbers that are either semiprime or prime, with no known factors? Is ...
1
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0answers
15 views

When does the link of an algebraic singularity determines it algebraic type ?

Let $X \subset \Bbb C^n$ be an hypersurface with an isolated singularity $x$. I know that when $X$ is a curve, $\mathcal O_{X,x}$ is determined by its link (considered as a topological space ...
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0answers
20 views

Reference Markov martingale Harmonic function

I've just finished a course of stochastic process (discret martingale and markov chain). I would like to go further, I heard it exists a link between martingale markov process and harmonic functions. ...
-1
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0answers
43 views

advice for beginner in representation-theory

Thank you for you reading.I am in my second year of my undergraduate and I plan to study representation-theory in the future. I had courses in linear algebra, abstract algebra(Hungerford chapter I-V ...
0
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1answer
22 views

Necessary and sufficient conditions for $x'Ax = 0$

I came across the following problem and I am having a hard time thinking about it. Let $A$ be a $k\times k$ real matrix. Notice that I do not require that $A$ is symmetric, positive definite or ...
0
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0answers
19 views

Is it true that any Riemannian metric on cylinder can be deformed to product metric?

Is it true that any Riemannian metric on cylinder $\Bbb S^1\times \Bbb R$ can be deformed to standard product metric? Are there some standard references about such classifications?
9
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0answers
77 views

Peculiar pictures in advanced maths books

I have recently started reading Introduction to Symplectic Topology by McDuff and Salamon and I came across this picture: I find it very funny and really interesting. I read on Wikipedia that Ian ...
0
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0answers
11 views

Can WKL be proven in Monadic Second Order Logic with 2 Successors?

I have been reading the book Reverse Mathematics: Proofs from the Inside Out and learned of the Weak Kőnig's Lemma, which states that every finite subtree of the full binary tree has an infinite path. ...
0
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0answers
29 views

Quantization of a Lie algebra of symmetries

In quantum mechanics we have quantization map $Q$ that maps classical observables to quantum observables. If symplectic manifold $(M, \omega)$ is the phase space of a classical system then classical ...
1
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0answers
27 views

Is there a general approach for finding the Normalizer of a subgroup? [on hold]

I am currently facing a problem where I need to find the Normalizer of a subgroup. I have found examples where people compute the normalizer for specific subgroups, but no general approach. Does ...
0
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0answers
19 views

About the differences between definitions of “Capable Group”

I am looking for the properties of groups having "immediate Descendants", in other therm, "Capable Groups"; The problem that I fond is that "Capable Group" could have many meaning! So, could you ...
1
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1answer
40 views

Help me to find this magazine

Some years ago I've read a mathematical magazine about pigeonhole principle. I love the structure of the magazine, first it explains about the subject and after it has a lot of interesting exercises ...
0
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0answers
11 views

References of weighted norm distance relaxation?

In my practical work, I need to measure the distance between two vector $a$ and $b$ with dimensions weighted by vector $w$ in $\mathbb{R}^n$. I need to learn the weight for the distance, then use the ...
12
votes
2answers
111 views

The general proposition of Fermat

In his letter to Frenicle, dated 18th October, 1640, Fermat states the following (Point 8, translated) : If you subtract $2$ from a square, the remaining value cannot be divided by a prime which ...
0
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0answers
10 views

Means of powers of the zeta function

It is well known that the Lindel\"of Hypothesis is equivalent to the statement that $$\frac 1T\int_0^T|\zeta(1/2=it)|^{2k} =O(T^\epsilon)$$ for all positive integers $k$ and all positive real $\...
0
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0answers
23 views

What are some prerequisites for studying Elliptic Curves over $\mathbb Q$?

Suppose you had an undergraduate friend who's looking to take an introductory course on Elliptic Curves over $\mathbb Q$, in the context of Number Theory. The difficulty is around the level of ...
0
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0answers
19 views

Proof Ideas - Strong Induction, Pascal's Triangle and Fibonacci Numbers

I'm looking for attributes/characteristics/properties to prove about Pascal's Triangle or the Fibonacci numbers. Preferably something that requires a strong induction proof that is on the same level ...
1
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0answers
23 views

Another linear algebra textbook recommendation [duplicate]

Im looking for a linear algebra textbook that meets two criterion: The book should be proof-based. The book should try and motivate most, if not most of the ideas, of the geometry of linear maps and ...
1
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0answers
17 views

Proving inequality relation between two complex numbers and a positive real parameter. [duplicate]

Question: Prove that: $$|z_1+z_2|^2 \le (1+c)|z_1|^2+(1+{1\over c})|z_2|^2$$ where $z_1,z_2$ are complex numbers and $c$ is a positive real parameter. Solution: We can write $$|z|^2=z\...
1
vote
1answer
31 views

Homomorphism from $p$-adic to $l$-adic groups

I have seen and heard the statement that the $p$-adic and $l$-adic topologies are incompatible. I would appreciate a proof or references supporting this statement. More precisely, I am interested in ...
2
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0answers
23 views

Every holomorphic vector bundle on a stein manifold is Nakano positive?

I encounter this statement on page 53 of Takeo Ohsawa's L2 Approaches in Several Complex Variables but I don't know how to prove it. Could anyone explain it to me or give me a reference for it?
0
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1answer
17 views

The Hodge star operator and the wedge product: $\alpha \wedge (\star \beta)$

According to Wikipedia, The Hodge star operator on a vector space $V$ with an inner product is a linear operator on the exterior algebra of $V$, mapping $k$-vectors to $(n-k)$-vectors where $n=\...
0
votes
1answer
22 views

Identifying a wedge-to-metric formula

In this question, the original poster wrote: On every Riemannian manifold $M$, we can consider the Hodge $*$-operator, which is characterised by the following formula: $$a\wedge *b = (a,b)\nu.$$...
0
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0answers
10 views

Sinusoidal decomposition of signal

I have some data of periodic nature. The curve seems to be slightly irregular, and it makes sense to consider it as the sum of two or more different sinusoidals. I'm asking for a source or tools ...
2
votes
1answer
29 views

Reference Request - Developement of Geometry

I am looking for a reference that explains the developement of geometry that includes the developement from around the mid nineteenth century to modern geometry. By explain, I mean that it gives ...
1
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0answers
15 views

expressivity of graph directed IFS and L-systems

In my answer to Does there exist a L-system for this Pierced Diamond Fractal? I asserted that graph directed iterated function systems of similarities have equivalent expressive power to L-systems. ...
6
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0answers
78 views

Why do the Borwein integrals stop being $\frac{\pi}{2}$?

I just received the book "single digits - In praise of Small Numbers" by Marc Chamberland. In this book, he gives an interesting integral $$\displaystyle \int_0^\infty \dfrac{\sin x}{x} = \dfrac{\pi}...
0
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0answers
33 views

Any textbooks on applications of topology to natural and social sciences?

Here is a Mathematics Stack Exchange post on applications of topology. Now my question is, are there any easy-to-read textbooks that discuss (rather thoroughly) applications of elementary topology to ...
0
votes
1answer
19 views

Modular parametrization from equality of $L$-functions

In the literature, an elliptic curve $E/\mathbb{Q}$ is defined to be modular in two different ways 1) if there exists a nonconstant morphism $X_0(N) \to E$, 2) if there exists a modular form $f$ ...
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0answers
22 views

A geometric proof of Picard's little theorem

I'm preparing a presentation where I'd like to present a proof of Picard's little theorem using hyperbolic geometry. Picard's little theorem states that the range of an entire function can omit at ...
0
votes
1answer
29 views

Defining a cost function to find a rotation

Suppose I have pairs of vector $\left\{(v_i,w_i)\right\}_{1 \leq i \leq n}$, and I want to find an angle $\theta$ that describes an optimal rotation that aligns all the pairs. Now two possible cost ...
3
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0answers
48 views
+50

Minimizing the sum of KL divergences

Given a list of probability distributions $q_i$, what distribution $p$ minimizes the sum of KL divergences (if they exist) to and from each of them? That is, how do I determine $$\operatorname*{...
0
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0answers
21 views

Derivation of Sanov's theorem for continuous variables

Where can I find a derivation of Sanov's theorem for continuous variables? I am familiar with the derivation for discrete variables. I am looking hopefully for something similarly intuitive.
1
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0answers
23 views

Name for this property closely related to subadditivity

A subadditive function is a function $f : A \rightarrow B$ with the following property : $$ \forall x, y \in A,~~~~~f(x+y) \leq f(x) + f(y),$$ where both $A$ and $B$ are closed under addition. I'm ...
2
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2answers
273 views

Show that this ring is semi-simple

Let $\mathbb{F}_q$ be a field of order $q = p^m$, where $p$ is the characteristic of the field; a prime. Consider the ring $$R_n = \mathbb{F}_q[x]/\langle x^n - 1 \rangle $$ Now, I've read that this ...
0
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0answers
17 views

Statistics of excursions of 1D discrete random walks

Let $P(n)$ be a probability distribution on the integers. Suppose a random walker starts off at the origin and, at every positive integer time, takes a step of length $n$ with probability $P(n)$. ...
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0answers
21 views

Reference for the relation of the Casimir element to the Laplace Beltrami operator

Wikipedia says, "If $G$ is a Lie group with Lie algebra $\mathfrak {g}$, the choice of an invariant bilinear form on $\mathfrak {g}$ corresponds to a choice of bi-invariant Riemannian metric on $G$. ...
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0answers
16 views

How to compute a conditional probability related to Type II error?

I would like to know how to compute the following conditional probability. Let $X|\theta \sim \mathcal{U}(0,\theta)$ be a uniform random variable. Given a fixed $\theta = \theta_0 > 0$, one can ...