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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.

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2answers
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Suggestions of books more complex than Spivak and Apostol for Calculus 1?

I've been using Apostol and reusing until got satisfied, I've done the same with Spivak, and I wan't to take another level now.
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0answers
33 views

Answer to this wierd 8 years old math question from school textbook [on hold]

2 taxies start at the beginning of their lane, in one of them is 4 passengers and in the other is 3 passengers, between the start point and end point a few passengers got in and a few got out of ...
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0answers
25 views

What is the base measure in measure theory?

I see the term "base measure" used frequently about measures. I do not completely get what that exactly means: Some examples are: Let $\cal F$ be the space of all probability density functions ...
4
votes
1answer
23 views

Functor preserving finite and filtered colimits

Is is true that a functor preserving both all finite colimits and all filtered colimits, preserves in fact all colimits? I read this somewhere and have tried to find a proof of it, but I can't find ...
0
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0answers
9 views

Wave cone of the curl operator

How can one compute the wave cone $\Lambda_{\mathcal A}$, defined as \begin{equation*} \Lambda_{\mathcal A}:=\bigcup_{|\xi|=1} \ker \mathbb A^k(\xi) \qquad\textrm{with}\qquad \mathbb A^k(\xi)= (2\pi ...
1
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1answer
33 views

Manifolds that admit Lorentzian metrics?

John Lee says in "Riemannian Manifolds: An Introduction to Curvature": With some more sophisticated tools from algebraic topology, it can be shown that every noncompact connected smooth manifold ...
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0answers
12 views

Reference for the multivariate Leibniz rule of many factors

I'm looking for a reference (a book/article) with a formula to $$ \frac{ \partial ^ k }{ \partial x_1^{k_1} ... \partial x_n^{k_n} } f_1(x) ... f_m(x) , $$ where $k=k_1+...+k_n$, $x=(...
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0answers
12 views

A reference for higher cohomologies of an $n$-dimensional manifold being trivial?

There is a theorem due to Grothendieck that states if $X$ is a complex manifold of complex dimension $n$ and $V$ any holomorphic vector bundle on $X$ then $H^i(X,V)$ is trivial for $i >n$. Can ...
1
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0answers
17 views

Representing a complex line as a directed ellipse

Consider nonzero $v = v_r + iv_i \in \mathbb{C}^n$, It can be thought of as an ordered 2-tuple of vectors $(v_r, v_i)\in \mathbb{R}^n\times\mathbb{R}^n$. The complex line generated by $v$ is $$\{r[(\...
1
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0answers
23 views

Name of Hardy Littlewood Lemma/Result

I am interested in finding the variation of argument of a certain analytic function $g(z)$ on a region of the form $\frac{1}{2}\leq \sigma \leq \sigma_1$, $t_0\leq t \leq T$ where $\sigma$ stands as ...
0
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1answer
22 views

Characterization of Strongly Regular Graphs

I am looking for a reference in which I can find a proof of the following result. A strongly regular graph is disconnected if and only if it is a disjoint union of complete graphs $K_n$ of the same ...
0
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1answer
31 views

An example from Contemporary Mathematics, Volume 18 that non-expansive maps may fail to have a fixed point.

Recently under the tutelage of a Mentor, we were told to complete this example where contractiveness plays a pivotal role in the Contraction mapping principle. I already have an example by taking $E=...
1
vote
1answer
36 views

Term for “inverse image” of element under set-valued map?

Say I have a function $f: A \to 2^B$. Given an element $b \in B$, I want to refer to the set $f_b := \{a \in A: b \in f(a)\}$. Is there a standard name for such sets? Notice it's not technically ...
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0answers
6 views

Reference request: general Turan density upper bounds

One problem I have encountered while doing research is that I find it difficult to find papers that were published decades ago. In particular, I am interested in "Extension of a theorem of Moon and ...
0
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0answers
32 views

Advice on Henri Cohen's Number Theory books

I am a graduate student willing to learn Number Theory. I have come across Henri Cohen's "Number Theory" (Vol. I and II) and I would like to hear from someone who has read these books before: are they ...
3
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0answers
21 views
+50

Examples of BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative? More precisely, I'd like to see an example of a function $$u_1 \in BV(\mathbb R^2; \mathbb R^2)$$ ...
1
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0answers
31 views

Partition a set into g groups, k different ways, such that no pair of elements is ever in the same group together more than M times

Over at Wolves and Sheep on puzzling.stackexchange.com, noedne's answer involves repeatedly partitioning a group of 99 sheep into a series of "test groups" such that All but one sheep are tested ...
1
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1answer
25 views

Examples of applying Dirichlet's approximation.

I've seen many examples of Dirichlet's approximation being proven , or other questions regarding to the theory of the approximation on this site and others but I would like to see a concrete example ...
1
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0answers
23 views

References for Lie algebra extensions, Poincaré

I will be posting this to the physics stack since this is a physics paper, but I figured the mathematicians would be able to suggest more comprehensive references for me. My future adviser just ...
-2
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0answers
65 views

Can we have complete closure for the known six arithmetical operators even with inclusion of the transfinites?

[EDIT] Lets define an implementation of the integer numbers as ordered pairs of cardinals with zero, so we stipulate that: $\langle 0,x \rangle = +x$ $\langle x,0 \rangle = -x$ Where $x=|X|$ for ...
2
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0answers
52 views

The automorphism group of the countable atomless Boolean algebra does not have ample generics

I was told that the automorphism group of the countable atomless Boolean algebra does not have ample generics. I assume that one would show this by using the Fraisse-theoretic characterizations of ...
2
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1answer
16 views

Coarea-like formula for BV function (not its derivative)

Let $f \in BV(\Omega)$. The coarea formula states that $$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$ Do we also have that $$f = \int_{\mathbb R} \chi_{\{f >h\}} \, dh$$ holds?
1
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1answer
24 views

Total variation and Lipschitz continuity

Let $f:B_R(0) \subset \mathbb{R}^N \to \mathbb{R}$ be a $L$-Lipschitz continuous function. Is it true that the total variation $|Df|(B_R(0))$ is controlled by the Lipschitz constant $L$? How?
2
votes
1answer
37 views

Reference Request for Small Cancellation Theory

I am looking for a self contained survey / paper / lecture notes on small cancellation theory and it's generalizations. I am aware of Lyndon and Schupp's textbook chapter and I have been recommended ...
2
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0answers
41 views
+50

Proving a property of a solution to a set of nonlinear polynomial equations

Consider the following system of equations for $R_{i}(\lambda)$ \begin{align} R_1 &= \frac{\lambda}{4}(1 + R_3 + 2 R_2 R_1) \tag{1.1}\\ R_2 &= \lambda \left[q + \left(\frac{1}{2} - q\right)...
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0answers
43 views

Proving limits using the $\epsilon-\delta$ definition [on hold]

Are there any guides on how to prove complicated limits by definition (either online or in a textbook) that provide both exercises and solutions?
4
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0answers
149 views
+100

Book recommendation for learning image processing as an application of Fourier analysis

I have searched around this website for some references of applications of Fourier transform in image processing, but did not find any satisfactory ones. I have a major in maths. In years I always ...
0
votes
1answer
31 views

Zermelo's Proof of the Well Ordering Theorem

I am trying to find the paper by Zermelo in the early 1900's in which he proved the Well Ordering Theorem (implicitly using the axiom of choice). Does anyone know where I can find this?
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0answers
13 views

C.J. van Duijn. An introduction to conservation laws: theory and applications to multi-phase flow. - PDF

I could not get the following reference and would like to know if someone can send me a PDF. I've researched at gen.lib, google and b-ok.org. If someone can help me with other suggesties of online ...
0
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0answers
11 views

Green's function for homogeneous PDE

I was looking for some Green's function method to solve a homogeneous PDE with nonhomogeneous boundary conditions (i.e., $Lu=0$ in $D$ with $u=f(\mathbf{x})$ in $\partial D$), but most of the ...
1
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1answer
21 views

(Existence part of) Neyman-Pearson via weak-* convergence

I would like a ask whether there is any statistical reference containing the following functional analytic argument for the existence part of Neyman-Pearson: Let $(R, \mathcal{F}, \mu)$ be a measure ...
0
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0answers
20 views

Divisor class group of vector bundle over an integral Noetherian scheme

Ler $X$ be an integral Noetherian scheme. Then one can show that taking the inverse image induces an isomorphism of Weil divisor class groups $\operatorname{Cl}(X)\to \operatorname{Cl}(\mathbb A^n_X)$....
3
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0answers
28 views

Regularity of harmonic functions

I have a question on a fundamental property of harmonic functions. Let $\Omega \subset \mathbb{R}^d$ be a domain. We define $H^{1}(\Omega)$ by \begin{align*} H^{1}(\Omega)=\{u \in L^{2}(\Omega,m) \...
10
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1answer
135 views

Research level algebraic topology

I became very interested in Algebraic Topology more or less recently, and learnt a lot of "classical algebraic topology", including : Hatchers' Book, and more categorical approaches here and there ...
1
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1answer
26 views

Book about summable families

after studying summable series, I know that there exists the concept of summable families. There is any book where I can find the proof of the basic properties of summable families? Thanks.
0
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0answers
26 views

List of extension theorems

As a post grad student, I have come across many results where a function with certain properties(eg-homomorphism) on a smaller algebraic structure is extended to a larger one. For example, extending ...
0
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0answers
14 views

Reference request for bigraded vector spaces

I'm currently reading up on Spectral Sequences in Algebraic Topology, and often times authors refer to graded vector spaces and bigraded vector spaces freely without defining them. I've found ...
0
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0answers
17 views

Reference for simple Markov chain construction

Let $M_n$ , $n\in\mathbb{N}_0$ be a Markov chain on a general state space $X$. Fix $m\in \mathbb{N}$. ¨ My question is if there's a name / reference for this trivial Markov chain on $X^m$ defined by ...
0
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2answers
30 views

Equivalent definitions for the differentiability in general

I'm looking for equivalent definitions for the differentiability in general . I'm really confused about the differentiation of multivariable functions . Also I want the proofs for equivalency of the ...
0
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0answers
28 views

How to prove $n$-to-$1$ mapping property for Blaschke products?

I have two questions regarding the Blaschke products. 1) I came across the following post Boundary behaviour of finite Blaschke products on the unit circle where it's mentioned that for any $\...
0
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1answer
23 views

Reference Request: Original $\Gamma$-convergence Paper

Does anyone know what the original paper/work of Ennio de Giorgi, where $\Gamma$-convergence first appeared is?
1
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0answers
50 views

Book about interpolation of functions by polynomials using linear algebra/projection

I've been asking a lot of questions about interpolation of functions using polynomials: Approximate $f(t) = 1-|2t-5|$ in $[2,3]$ by $p\in P_2$ by using the least squares method Approximate $f(x) = x^...
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0answers
30 views
+50

Heuristic on Sobolev and BV functions

Let $f: \Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a Sobolev or BV vector field. A heuristic that I've heard frequently is the following: $f$ is almost Lipschitz on a large "good" set but ...
0
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1answer
69 views

Underapproximating the exponential function from below [duplicate]

I think that for positive natural numbers t and n we have $$ \left(1+\frac{n}{t}\right)^t\ \le\ e^n\,. $$ Is this true? I have constructed a proof of it (which would probably take some lengthy ...
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0answers
41 views

Sparse Vandermonde matrix factorization.

How to factor a Vandermonde matrix? A Vandermonde matrix is a matrix $$V=\begin{bmatrix}1&x_1&{x_1}^2&\cdots&{x_1}^k\\1&x_2&{x_2}^2&\cdots&{x_2}^k\\&&\vdots\\...
2
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2answers
95 views

Looking for a different type of Linear Algebra book

Are there any good linear algebra books with lots of (mathematical, preferably algebraic or geometric-flavored) applications? E.g. I'm not so interested in the typical engineering-style applications ...
2
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2answers
84 views

Subgroups of $\operatorname{GL}_2(\mathbb Z/8\mathbb Z)$

Is there some program or a location which would allow me to work and calculate with the subgroups of the group $\operatorname{GL}_2(\mathbb Z/8\mathbb Z)$?
0
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1answer
36 views

Intuition of cocycles and there use in dynamical systems

I’ve come across several papers and lectures that use cocycles to talk about dynamics on a manifold. However, I haven’t come across an actual definition of what a cocycle is. Could someone give a ...
0
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2answers
29 views

Multi-deformed numbers

The following deformations of usual numbers are well-known: $$ [n]_q=\frac{q^n-q^{-n}}{q-q^{-1}}, $$ and $$ [n]_{pq}=\frac{p^n-q^{-n}}{p-q^{-1}}. $$ Question. Are there any meaningful further ...
10
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1answer
216 views

An expression for computing second order partial derivatives of an implicitely defined function

Let $\Phi(x,y)=0$ be an implicit function s.t. $\Phi:\mathbb{R}^n\times \mathbb{R}^k\rightarrow \mathbb{R}^n$ and $\det\left(\frac{\partial \Phi}{\partial x}(x_0,y_0)\right)\neq 0$. This means that ...