New answers tagged reference-request
6
votes
Books on complex dynamics that discuss polynomial mating?
I don't think that such a volume exists, but let me try and give some helpful references nonetheless. Use with caution: my knowledge of the literature is somewhat out-of-date.
There is a very brief ...
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9
votes
A basis-free formula for the determinant as a polynomial
Lang's observation is:
There is a unique linear functional $F\colon S^n(\operatorname{End}V)\to\mathbb K$ such that for any $A\in\operatorname{End}V$,
$$F(A^n)=\det(A).$$
There was another ...
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7
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Full expansion of $\det(I+\varepsilon A)$
From N. Bourbaki, Algebra I, p.529, Proposition 11:
$$\text{det}(I+\varepsilon A)=\sum_{k=0}^n\varepsilon^k\text{tr}\bigwedge^k(A)$$
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4
votes
What uniform Selberg–Delange estimate is needed to justify this contour shift?
I would be careful here. A purely logarithmic singularity typically suggests an x / log x scale for unweighted partial sums, whereas a singularity of the form (s - 1) log(1 / (s - 1)) carries an ...
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3
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Accepted
Does the mean value $\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^Tf(t)\,\mathrm dt$ exist if $f$ is given by a function in $L^1(\mathrm b\mathbb R)$?
The answer is almost yes.
Note that $(\mathrm b\mathbb R,\mathbb R)$ is an uniquely ergodic dynamical system. This can be seen directly by observing that any $\mathbb R$ invariant measure on $\mathrm ...
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1
vote
Reference request for uniform separation of closed sets
What Daniel Weber asks in the comments is exactly the right reduction: for a single pair $C,D\subseteq[0,1]$, this is just the standard lemma that disjoint compact sets in a metric space are ...
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8
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References on computability and intuitionism
You can look into realizability. There are current topics of research there that connect intuitionistic logic and computability, for instance:
Takayuki Kihara: Lawvere-Tierney topologies for ...
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2
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D-finiteness of Hilbert series of non-commutative invariant ring under reductive group
For finite groups $G \subset GL(V)$, Dicks and Formanek proved that the Hilbert series of $T(V)^G$ is of the form $$\frac{1}{|G|} \sum_{g \in G} \frac{1}{1-tr(g) t} $$ in Poincaré series and a problem ...
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7
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Orientation local system of a vector bundle
I am not aware of any references, but if we follow the usual references (Hatcher or Davis & Kirk), then a local coefficient system can be viewed as a fiber bundle $\mathcal{A} \to X$ with fiber an ...
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2
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Have the affine simplicial line arrangments been enumerated?
I have a construction of affine simplicial hyperplane arrangements showing that there exist a lot more than in the finite case.
The idea is to start with a known simplicial affine hyperplane ...
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3
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Name of a specific lattice
The latest version of gemini found an answer: It is called Post algebra. References:P. C. Rosenbloom, Post Algebras, I. Postulates and General Theory, American Journal of Mathematics 64 (1942), 167–...
8
votes
Accepted
F.g. graphs of groups with f.g. edge groups have f.g. vertex groups
Theorem 1.3 of this paper of Haglund and Wise should do it.
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6
votes
Accepted
Infinitesimal neighbourhoods for derived schemes
When $f: X \to Y$ is a map of prestacks locally almost of finite type admitting deformation theory, Gaitsgory and Rozenblyum gave a definition of
the (derived) $n$th infinitesimal neighborhood of $X$ ...
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1
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Does the Pósa rotation-closure lemma apply to a longest path on $k < n$ vertices?
Note that the assumption on your graph does not ensure the graph is Hamiltonian. An example is given by a complete bipartite graph $G$ with parts $A,B$ of size $3n/4$ and $n/4$ respectively. Then $|N(...
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2
votes
Accepted
Progress towards Selberg's conjecture A
For any nontrivial Dirichlet character $\chi$ let $L(s)=L(\chi,s)=\sum\frac{\chi(n)}{n^s}$ be the Dirichlet $L$-function. Recall it is an entire function of order $1$, as are its derivatives
$$L^{(k)}(...
- 35.6k
3
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Universal cover of a semisimple algebraic group as a functor
$\newcommand{\ssc}{{\rm sc}}$ It tunes out that the desired functoriality was proved by González-Avilés over an arbitrary non-empty base scheme.
Let $S$ be a non-empty scheme.
By an $S$-group we mean ...
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0
votes
Alternative proofs for the connectedness of the Mandelbrot set?
I have attempted a proof using Morse theory.
https://www.researchgate.net/publication/402825994_SOME_CONNECTEDNESS_THEOREMS_IN_REAL_POLYNOMIAL_DYNAMICS
I hope it is correct; it is the best I could do ...
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1
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Universally measurable function is $P$-a.e equal to a borel measurable function: Reference needed
Together with my remark a solution is given f.i. in Hewitt/Stromberg, Real and Abstract Analysis (1965), Exercise (12.63) (a) and the Hints in this exercise.
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4
votes
Accepted
Universally measurable function is $P$-a.e equal to a borel measurable function: Reference needed
Lemma. Let $(\Omega,\mathcal{B},P)$ be a probability space and $(\mathcal{Y},\mathcal{A})$ be a measurable spaces, where $\mathcal{A}$ is countably generated.
Let $f:\Omega \to \mathcal{Y}$ be a $\...
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2
votes
History of profinite groups, when was it first mentioned? What was the original definition?
In the paper On Galois groups of local fields by Kenkichi Iwasawa (Trans AMS 80, No. 2, 1955, 448-469), DOI:
p456:
Let $J$ be an arbitrary group. Let $\{N_\delta\}$ be the the family of all normal ...
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6
votes
Accepted
Alternative proofs for the connectedness of the Mandelbrot set?
Yes. Beyond the two proofs mentioned in the question, there is a third route through Teichmüller theory and quasiconformal deformations.
McMullen and Sullivan (Quasiconformal Homeomorphisms and ...
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2
votes
Accepted
A weighted sum over squarefree numbers involving Bernoulli numbers
Let's include $n=1$, and remove the negative sign, defining $$F(X)=\sum_{1\leq n\leq X} \mu^{2}(n)\frac{B_{\omega(n)}}{n},$$ noting that $\omega(1)=0$ and $B_0=1$. Then:
$$\boxed{F(X) = \frac{1}{\log ...
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20
votes
Accepted
What is $\Game \mathbf{\Delta}^1_1$?
The answer is no. Basically the strategies in Borel games appear too quickly.
For each $x$, let $M_x$ be the least transitive model $M$ of Zermelo set theory plus $\Sigma_1$-replacement with $x\in M$. ...
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1
vote
Accepted
Comprehensive research-level reference on martingale concentration inequalities (discrete time)
The book
Bercu, Bernard; Delyon, Bernard; Rio, Emmanuel
Concentration inequalities for sums and martingales. (English) Zbl 1337.60002
SpringerBriefs in Mathematics. Cham: Springer (ISBN 978-3-319-...
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4
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Pi Day: estimating pi using probability
Here's a fun one that I've heard called the Atlantic City method.
Choose any real number $x$ for an initial value and iterate the mapping
$$x\leftarrow\frac{2−x}{3−4x},$$
counting the number of times ...
3
votes
Pi Day: estimating pi using probability
Grover's algorithm works by
Setting up a uniform superposition state
Doing an iteration that amplifies the probability of the right answer, consisting of a phase flip and a "diffusion operator&...
3
votes
Reference for colimits in comma categories F ↓ G in which F and G are not assumed cocontinuous
I would guess you're well aware of this result, but Bird proved in his 1984 thesis Limits in 2-categories of locally presentable categories that locally presentable categories and right adjoints are ...
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6
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Pi Day: estimating pi using probability
Here's a few of various kinds.
The expected distance for 1D random walk after n steps is $\approx \sqrt{2n/\pi}$ for large number of steps.
The probability that $n$ integers are coprime $\approx \...
2
votes
Decay rate of measures on Cantor set
I have perused Kahane and Salem's book and, though I am willing to believe the book proves this result, I have not managed to find anything in there that obviously implies it, though it may well be ...
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2
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Asymptotics for partial sums of coefficients of $\frac{\zeta(s)^\alpha-1}{\alpha(\zeta(s)-1)}$
Just like in your previous question involving $\frac{\log\zeta}{\zeta-1}$, for most $\alpha$ there will be poles to the right of the $\operatorname{Re} s=1$ line which will prevent the Dirichlet ...
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5
votes
Accepted
Uniform asymptotics for counts of ordered factorizations $n_1\cdots n_t\le x$
As written, (H) is false. A correct version was established in a lucid paper of Hwang from 2000:
Hsien-Kuei Hwang, "Distribution of the number of factors in random ordered factorizations of ...
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4
votes
Sequence of $\pm 1$ defined by $i \cdot q$ mod $p$
One may translate between these and cutting sequences.
The relation is via the double branched cover, where the relator lifts to a geodesic on the Heegaard torus for the lens space of slope $q/p$.
In ...
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10
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Pi Day: estimating pi using probability
There are probably many that have to do with $1/\sqrt{2\pi}$ in the Gaussian pdf. In particular, let $Z \sim \text{Normal}(0,1)$ and let $X_n \to Z$ in probability, for example $X_n \sim \frac{\text{...
5
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History of profinite groups, when was it first mentioned? What was the original definition?
Although the term profinite was not used at the time, it is fair to give credit to Marshall Hall, Jr, who wrote the following paper:
Marshall Hall, Jr, A Topology for Free Groups and Related Groups, ...
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3
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Accepted
The ramifications of a general endomorphism of the projective space should be mild?
We can show that $e(f)$ is upper semicontinuous as a function from $\mathrm{Hom}_d(\mathbb{P}^n)$ to $\mathbb{Z}_{>0}$. Hence the upper bound of $e(f)$ for a general endomorphism $f\in \mathrm{Hom}...
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1
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Does Bohr's theorem indicate $1-\frac{\log\zeta(s)}{\zeta(s)-1}$ has a pole in $\Re(s)>1$?
Sharing my notes, here is a simple observation showing that the branch index must vanish for any $1$-point sufficiently far to the right.
Let
\begin{equation}
L(s):=\sum_{p}\sum_{r\ge1}\frac{1}{r\,p^{...
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19
votes
Accepted
Non-constructive proofs for the infinitude of prime knots?
There is an argument analogous to Euclid's proof of the infinitude of primes.
If $K_1, \dotsc, K_n$ are prime and distinct, then the Whitehead double of $K_1 \mathbin\# \dotsb \mathbin\# K_n$ is prime ...
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1
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Defining Euler's number via elementary euclidean geometry (and a dimension limit)
The Euler constant $e$ can be constructed by compass, straightedge and a catenary: take a rope and hang it fixing two ends of the rope. Under gravitational force, the rope creates a catenary described ...
- 46k
1
vote
Accepted
Removable and null-capacity sets in $W^{k,p}$ - not always the same?
You almost got the right idea, but you are using the wrong variant of spaces. Note that capacity is defined using sets of compact support. The closure of $C_c^\infty(\Omega)$ in $W^{k,p}(\Omega)$ is ...
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3
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Best practices for citing and documenting certified computations in a mathematics paper
There are established mathematicians who regularly publish about computation-heavy proofs. For example, people that come to mind are, in random order, Marijn Heule (SAT solvers and combinatorics), ...
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17
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Non-constructive proofs for the infinitude of prime knots?
I was going to say that one could use the infinitude of prime Alexander polynomials. However, proving that any Alexander polynomial is realized is most easily done by an explicit construction (see eg ...
- 72k
2
votes
Flat base change in the complex analytic setting
Theorem 1.3.2 of Fourier-Mukai transform on complex tori, revisited implies the following result. Let $f : X \to Y$ be a proper morphism of complex analytic spaces, and let $\mathcal{F}$ be a ...
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0
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Asymptotics for partial sums of coefficients of $1-\frac{\log\zeta(s)}{\zeta(s)-1}$
I wanted to add some numerical data and expansions that led to the original asymptotic's of $A(x) \sim \frac{x}{\log^2 x}$ and $1 - B(x) \sim \frac{1}{\log x}$ which are why Conrad's analytic result ...
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1
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Malcev's paper "On a class of homogeneous spaces" in English
It has been a long time, but in case someone is looking for an English translation of this article (as I was an hour ago) it can be found in the compilation of translation:
Mal’cev, A.I. American ...
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1
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Connected spaces with no nontrivial open connected subsets
That's very strange:
User munkres_fan (or similar) posted a highly appreciated answer essentially containing a link to a paper from A.E. Brouwer that claims to contain the solution, namely to ...
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0
votes
Multiplicity of Laplace eigenvalues
For (1), this may be counterintuitive. Yes, Uhlenbeck established properties for generic metrics. However, symmetry is not a generic property in measure. Rotational symmetry, that is having an ...
1
vote
Asymptotics for partial sums of coefficients of $1-\frac{\log\zeta(s)}{\zeta(s)-1}$
UPDATE: Thanks to a faster formula, the numerical tables now go up to $10^{10}$ but this is not enough to include big Bernouilli numbers and to show any sign of supra-linear regime.
Not an answer but ...
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6
votes
Accepted
Reference request for coherent complexes over a (co)-connective dg ring?
Let $A$ be a DG ring (with cohomological grading) and $D(A)$ its derived category. Assume $A$ connective, i.e. $A \in D^{\leq 0}(A)$.
In this case $A$ is compact object that generates $D(A)$. The ...
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3
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"Modern" references on trigonometric identities
I'll follow your "definitely yes".
In 1990 or 1991 I (re!)discovered simple trigonometric identities that I never saw anything similar withing the high school mathematics. I even called ...
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4
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Accepted
Asymptotics for partial sums of coefficients of $1-\frac{\log\zeta(s)}{\zeta(s)-1}$
Actually, a little thinking shows that $A(x)=\Omega(x^{1+\delta})$ for some $\delta>0$ (see edit: we can estimate $\delta >1/1000$ and likely to be around that size actually), hence the result ...
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