Recent MathOverflow Questions

Estimating quality of projection

Math Overflow Recent Questions - Sat, 06/01/2019 - 11:30

I asked this question on math.se, but didn't receive an answer

Suppose we are given a vector $v$ and vectors $\mu_i$:

$v = \mu_1+\mu_2+...+\mu_m$, where $\mu_i \in R^n$, all $\mu_i$ are of unit length.

Oracle will give me $k$ vectors $\mu_{j_1}, \mu_{j_2},...\mu_{j_k}$ from the original set such that when I project $v$ onto subspace spanned by these vectors the length of the projection is highest possible. In other words, from the set of all combinations of $k$ vectors from $[\mu_1,...\mu_n]$ the $[\mu_{j_1}, \mu_{j_2},...\mu_{j_k}]$ give highest length of projection. Lets denote by $v_{\text{proj}}$ projection of $v$ onto $[\mu_{j_1}, \mu_{j_2},...\mu_{j_k}]$

I want to estimate quality of projection before oracle gives me this $k$ vectors. I want to give upper bound on $||v - v_{\text{proj}}|| $

As far as I understood it is very difficult to obtain these $k$ vectors by myself. However, I know that for any two vectors $\mu_i, \mu_j$, $||\mu_i-\mu_j|| \leq \alpha$, where $\alpha$ is a given positive number.

Small values of $\alpha$ will tell me that all $\mu_i$ are close to each other and heading towards same direction. I would suspect then that projection will be good, and its length will be close to the length of original vector. How can I use this to give an upper bound $||v - v_{\text{proj}}|| $?

My attempts:

Without loss of generality lets assume that $k$ optimal vectors are first $k$ vectors in the list, i.e $\mu_1,\mu_2,...\mu_k$. Lets denote by $P$ projection operator on the space spanned by $\mu_1,\mu_2,...\mu_k$.

$\|v - v_{\text{proj}}\| = \|v - P(v)\| = \|v - P(\mu_1+\mu_2+...+\mu_m)\| = $

$\|v - P(\mu_1) - P(\mu_2) - ... - P(\mu_m)\| = $

$ \| v - \mu_1 - \mu_2 - ... - \mu_k - P(\mu_{k+1}) - P(\mu_{k+2}) - ... - P(\mu_m)\| = $

$\|\mu_{k+1} - P(\mu_{k+1}) + \mu_{k+2} - P(\mu_{k+2}) + ... + \mu_{m} - P(\mu_{m})\|$

$\|v - v_{\text{proj}}\| \leq \|\mu_{k+1} - P(\mu_{k+1})\| + \|\mu_{k+2} - P(\mu_{k+2}) + ... + \|\mu_{m} - P(\mu_{m})\|$

$\|v - v_{\text{proj}}\| \leq (m-k)\alpha$

So in order to make $\|v - v_{\text{proj}}\| \leq \epsilon$, we need $k \geq \frac{m\alpha - \epsilon}{\alpha}$

I am not satisfied with this result because $k$ grows linearly with $m$. I want it to grow much slower, something like $\log(m)$. My goal is to show that under some constraints on $\mu_i$, we need only approximately $\log(m)$ vectors to approximate $v$.

I think the bound can be improved substantially. First Cauchy inequality isn't very tight and second, I used $|\mu_{k+1} - P(\mu_{k+1})\| \leq \alpha$ which is also very loose.

I am open for additional constraints on $\mu_1,...\mu_m$ to achieve logarithmic growth

Alex Ravsky on math.se has noted, that we also need a constraint on $\alpha$ in order to achieve logarithmic growth. Assume that $k$ $\leq n$, $\mu_i$ is th $i$-th standard ort of the space $\mathbb{R}^n$, and $\alpha = \sqrt{2}$. Then $\|v - v_{\text{proj}}\| = \sqrt{m-k}$

Small energy implies a lifting $\rho e^{i\theta}$

Math Overflow Recent Questions - Sat, 06/01/2019 - 11:20

Set $T^N$ the $N$-dimensional torus and $u\in H^1(T^N,\mathbb{C})$. Can I say that if the energy$$\int_{T^N}|\nabla u|^2 +\frac12\int_{T^N}[1-|u|^2]^2$$ is small enough (let say lower than some $\epsilon>0$), then $|u|$ is close to one, and therefore $u$ admits a lifting $u=\rho e^{i\theta}$ on the torus?. In that case, when can I assure that $\theta\in H^1(T^N,\mathbb{C})$. Any idea or comment is welcome. Thanks in advance!

$\mathrm{Sp}_n(q)$-conjugacy classes in $\mathrm{GL}_{2n}(q)$

Math Overflow Recent Questions - Sat, 06/01/2019 - 11:16

The symplectic group $\mathrm{Sp}_n(q)$ acts on $\mathrm{GL}_{2n}(q)$ by conjugation. All the literature I have found concerning the orbits of action of this kind is "Unipotent conjugacy classes in general linear groups" by Alperin, which is not really about what I am looking for.

Is there any work referring to this action and its orbits?

The best error term for the second moment

Math Overflow Recent Questions - Sat, 06/01/2019 - 11:06

Let $r_2(n)$ be the number of representations of a positive integer $n$ as a sum of two prime squares, i.e. $n=p^2+q^2$. Consider $S_1(x)= \sum_{n \le x} r_2(n)$ and $S_2(x) = \sum_{n \le x}r_2^2(n)$. For the first one by the prime number theorem we have $$S_1(x) = \frac{\pi x}{\log^2 x}+O\left(\frac{x \log \log x}{\log^3 x} \right).$$ In here

Rieger proved that $$S_2(x) = 2 \sum_{n \le x}r_2(n)+O\left(\frac{x}{\log^3 x} \right).$$ (this result improves in the error term on Erdos.)

Is there anything better known/provable about the error terms in the above formulas?

The square modulus of coordinates of a uniformly chosen point in complex projective space is uniform in the simplex

Math Overflow Recent Questions - Sat, 06/01/2019 - 10:36

I can't recall where I learned this (beautiful) fact, and I would like a reference (if possible, in a textbook):

Let $(z_0:\cdots:z_n) \in \mathbb{P}^n(\mathbb{C})$ be chosen uniformly at random w.r.t. the Fubini-Study metric, and normalized so that $|z_0|^2 + \cdots |z_n|^2 = 1$. Then the point $(|z_0|^2, \ldots, |z_n|^2)$ is uniformly distributed on the $n$-simplex (the set of $(p_0,\ldots,p_n)\in\mathbb{R}^{n+1}$ such that all $p_i\geq 0$ and $p_0+\cdots+p_n = 1$) w.r.t. its Euclidean metric.

This affords the quantum-mechanical interpretation that if we draw a random quantum entanglement of $n+1$ pure states, and we observe the pure state it is in, we get a uniform probability measure on the pure states. However, I'm not asking for a reference in relation to quantum mechanics.

This is, for example, proposition 1 in this paper but the only reference the authors give after calling the fact “known” is a 600-page book without any specific page number (shame!).

It is also mentioned in this blog post (and attributed to Bill Wootters); and the particular case $n=1$ is mentioned in this other blog post (in relation to the Box-Muller transformation); but I would like something more tangible than a blog post.

As a bonus question, if we take a uniformly random $(n+1)\times(n+1)$ unitary matrix (uniformly w.r.t. the Haar measure) and we look at the square norms of its columns, we get $n+1$ points on the $n$-simplex, each uniformly distributed by the above fact: does this distribution of $n+1$ points on the $n$-simplex have a standard name, and where might I learn more about it?

Why are there elementary equations that are not solvable in closed form?

Math Overflow Recent Questions - Sat, 06/01/2019 - 08:03

Elementary equations and closed-form solutions can be important in mathematics and in the natural sciences, e.g. for discovering and describing certain relationships.

$\log\colon x\mapsto\log(x)$; $x\neq 0$; $-\pi<\Im(\log(x))\leq\pi$ for all $x$

$\mathbb{L}$ denote the Liouvillian numbers (= Elementary numbers). $\mathbb{L}$ is the smallest field that contains $\mathbb{Q}$ and is closed under algebraic operations, $\exp$ and $\log$. The Elementary numbers are divided in the explicit elementary numbers and the implicit elementary numbers.
$i\in\mathbb{L}$

In [Chow 1999], Chow cites [Lin 1983] and writes:
"Let $\overline{\mathbb{Q}}$ denote the algebraic closure of $\mathbb{Q}$. Then Lin's result is the following.
Theorem 1. If Schanuel's conjecture is true and $f(x,y)\in\overline{\mathbb{Q}}[x,\exp(x)]$ is an irreducible polynomial involving both $x$ and $y$ and $f(\alpha,\exp(\alpha))=0$ for some nonzero $\alpha\in\mathbb{C}$, then $\alpha\notin\mathbb{L}$."

Chow continues: "The reader may check that our arguments generalize readily to other transcendental equations such as $x=\cos\ x$."
$ $

1. Can Lin's theorem be generalized to all irreducible polynomials $f$ with $f(x,y,z)\in\overline{\mathbb{Q}}[x,\exp(x),\exp(ix)]$ involving $x$, $y$ and $z$, or $x$ and ($y$ or $z$)?
$ $

2. Can Lin's theorem be generalized to all irreducible polynomials $f$ with $f(x,y,z)\in\overline{\mathbb{Q}}[x,\exp(A_1(x)),\exp(A_2(ix))]$ involving $x$, $y$ and $z$, or $x$ and ($y$ or $z$), wherein $A_1$ and $A_2$ are arbitrary algebraic functions?
$ $

3. To which kinds of Elementary equations can Lin's theorem further be generalized?
This question is necessary, because How to extend Ritt's theorem on elementary invertible bijective elementary functions is not the whole answer.
$\ $

[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448

[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50

Localification of a ringed space

Math Overflow Recent Questions - Sat, 06/01/2019 - 03:52

Call a ringed space local it if it lies in the image of the obvious faithful, non-full functor from locally ringed spaces to ringed spaces.

Given a ringed space, is there a map $f$ from it to some local ringed space such that any other map from it to a local ringed space much factor uniquely through $f$?

Intrinsicness of Hodge-theoretic properties of Galois representations in a general reductive group

Math Overflow Recent Questions - Sat, 06/01/2019 - 03:31

In the paper "The conjectural connections between automorphic representations and Galois representations" by Buzzard and Gee, it is said

"We say that $\rho$ is crystalline/de Rham/Hodge–Tate if for some (and hence any) faithful representation $H \rightarrow GL_N$ over $\mathbb{Q}_p$ , the resulting $N$-dimensional Galois representation is crystalline/de Rham/Hodge–Tate."

Here $\rho$ is a homomorphism from the absolute Galois group of a finite extension of $\mathbb{Q}_p$ to a reductive group $H$ (defined over some algebraic closure of $\mathbb{Q}_p$). How to prove the "hence any" claim?

Second question: given that the definition does not depend on this choice anyway, is there a way to give a definition not involving this choice (or is there a natural choice here)?

Hilbert class fields and transfer

Math Overflow Recent Questions - Sat, 06/01/2019 - 02:16

Let $K/k$ be an extension of number fields and $H_k$, $H_K$ their respective Hilbert class fields. Is there a transfer map from $\text{Gal}(H_k/k)$ to $\text{Gal}(H_K/K)$?

Orbit of an irreducible representation of a surface group under that action of the mapping class group

Math Overflow Recent Questions - Fri, 05/31/2019 - 20:10

Let $F$ be a closed oriented surface of negative Euler characteristic. Let $X^i(F)$ be the subset of the $SL_2\mathbb{C}$-character variety of the fundamental group of $F$ corresponding to irreducible representations. The mapping class group of $F$, $\mathcal{M}(F)$ acts on $X^i(F)$. Let $[\rho]\in X^i(F)$, is the orbit of the $[\rho]$ under the action of the mapping class group, $$\mathcal{M}(F).[\rho]$$ Zariski dense in $X(F)$?

Do coherent sheaves on rigid analytic spaces form an abelian category?

Math Overflow Recent Questions - Fri, 05/31/2019 - 13:01

It is said that the category of sheaves of abelian groups on a Grothendieck site(topology) is an abelian category. On the other hand, it is known that in usual algebraic geometry, given an variety (say, over $\mathbb C$), the category of coherent sheaves of $\mathcal O_X$-modules is abelian.

Now, a rigid analytic space $X$ inherits a Grothendieck topology and we can similarly define coherent sheaves of $\mathcal O_X$-modules with respect to this. So I would like to ask

Question: Do coherent sheaves on rigid analytic spaces form an abelian category?

In the first place, we know from BGR's book that taking kernels and images will preserve coherence in this setting. So the answer might be positive. But the axioms of abelian categories are too abstract for me, and I have no idea about how to prove or disprove this.

Sub-Gaussian decay of the measure of Euclidean balls

Math Overflow Recent Questions - Thu, 05/30/2019 - 15:22

Let $X$ be a random vector in $\mathbb{R}^d$ satisfying the following property: there exists $C_1,C_2>0$ such that $$\int_0^{+\infty}\mathbb{P}(\|X-\mu_0\|\leq\sqrt{t})\exp(-t)dt\leq C_1\exp(-C_2\|\mu_0\|^2)$$ for any $\mu_0\in\mathbb{R}^d$. Here $\|\|$ is the Euclidean norm in $\mathbb{R}^d$. If the above property holds, is the following statement true: there exists a sequence of vectors $\mu_n$ in $\mathbb{R}^d$ and a sequence of real numbers $t_n\to+\infty$ ($t_n$ may depend on $\mu_n$ for example $t_n=\|\mu_n\|^2/4$) such that: $$\lim_{n\to+\infty}\frac{\mathbb{P}(\|X-\mu_n\|\leq1)}{\mathbb{P}(\|X-\mu_n\|\leq \sqrt{t_n})\exp(-t_n)}=0$$

If this is not true, is there a counter example? Or is the the following result true? $$\lim_{n\to+\infty}\frac{\mathbb{P}(\|X-\mu_n\|\leq1)}{\int_0^{+\infty}\mathbb{P}(\|X-\mu_n\|\leq\sqrt{t})\exp(-t)dt}=0$$

I think this is intuitively true. The first display roughly specifies that the decaying speed of the density function of $X$ (if it exists) is sub-Gaussian. The convolution of the density function of $X$ (if it exists) and Gaussian function should decay slower than both functions resulting the limit to be 0. Is this intuition reasonable?

------------Update-------------------

A slightly more specific (assume existence of density) and mathematically elegant formulation of the problem to prove the third display is:

Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ and $g(x)=\exp(-\|x\|^2)$. Let $1_{B_1}$ be the indicator function of the unit ball centered at origin. Let $*$ be the convolution operation. Does the condition $$(f*g)(x)\leq C_1\exp(-C_2\|x\|^2)$$ for some $C_1,C_2>0$ imply $$\lim_{n\to+\infty}\frac{(f*1_{B_1})(\mu_n)}{(f*g)(\mu_n)}$$ for some sequence $\mu_n\in\mathbb{R}^d$? If this is not true, what additional regularity conditions on $f$ do we need?

finding function g with grad(f) x Lie(f,g)= 0 for given harmonic f

Math Overflow Recent Questions - Wed, 05/29/2019 - 17:18

I have a harmonic function f, Delta(f)=0. I need to find another scalar function g, such that: 1. grad(f).grad(g)=0 (gradients orthogonal) 2. grad(f) x Lie(f,g)= 0 (so that Lie derivative is either along grad(f), Lie(f,g) ~ grad(f), or Li(f,g)=0).

For example, in spherical coordinates {r,\theta,\phi}, if f=\phi then g=g(r,\theta) has Lie(f,g) ~ grad(f)

Is there a general procedure to find such g for a given f?

Cohomogy of local systems over CW-complexes

Math Overflow Recent Questions - Wed, 05/29/2019 - 16:40

Let $M$ be a finite CW-complex. Let $F$ be a finite rank local system over $M$ with coefficients in any field. Is it true that $dim(H^k(M,F))$ is at most the number of $k$-cells times $rank(F)$?

If $F$ is the trivial local system then this result is proven in almost any standard textbook in topology (or at least immediately follows from there). I believe that it should be true in the above generality and would be happy to have a reference.

When inverse of a real matrix has positive diagonal entries?

Math Overflow Recent Questions - Wed, 05/29/2019 - 16:17

This question originates from a numerical simulation where we observe that all diagonal entries of inverses of an ensemble of real matrices are positive. We expect there should be a reason for this but we don't know what to look for. Therefore any characterization of the kind of matrices described above could be very helpful.

Primes with given Hamming weight

Math Overflow Recent Questions - Wed, 05/29/2019 - 15:40

If I understand correctly, in the following MO-thread

Are There Primes of Every Hamming Weight?

two users of the site claim that it has been already proven that, for every sufficiently large $n \in \mathbb{N}$, there exist primes numbers with Hamming weight equal to $n$. Their claim is apparently supported by Theorem 1.2 of the paper "Primes with an average sum of digits" by M. Drmota, C. Mauduit, and J. Rivat.

Do you know if there is a text out there in which the deduction of the existence of primes with Hamming weight $n$ from the said theorem by Drmota, Mauduit, and Rivat is established in a thorough manner? Like other users of the site (see the sections of comments in the aforementioned thread), I am not totally sure of the veracity of such a claim. In case that you believe that there is no author out there that has dealt with this topic in detail but you consider that you've gotten the idea of the proof, would you be so kind as to explain it below as though I were a five-year old?

Thanks in advance for your help!

Semi-direct Product and Automorphisms of Weighted Projective space

Math Overflow Recent Questions - Wed, 05/29/2019 - 15:32

This is a follow up question stemming from the answer given by Sasha: Automorphisms of a weighted projective space

Let $X=\operatorname{Proj} k[x_1, x_2, x_3]$ denote the weighted projective space $\mathbb{P}_k^2(2,3,4)$ over an algebraically closed field $k$. Then any automorphism of $X$ is given by the mappings, $$ x_1 \mapsto a x_1 $$ $$ x_2 \mapsto b x_2 $$ $$ x_3 \mapsto c x_3 + d x_1^2$$ where $a,b,c \in k^*$and $d \in k$. The conclusion given is that the automorphism group of $X$ is isomorphic to $((k^*)^3 \ltimes k)/k^*$.

It seems that $(k^*)^3 \ltimes k) $ acts on $K[X]$ in two parts: First, $[a, b, c] \cdot [x_1, x_2, x_3]^T$ and then somehow add $d x_1 ^3$ to $x_3$.

How does one actually go about describing the action of $(k^*)^3 \ltimes k) $ on $X$?

Moreover, what exactly is the group structure defined on $(k^*)^3 \ltimes k) $? I I know that the multiplication should reflect composition of automorphism. So that if $\phi$ and $\psi$ are automorphism of $X$, given by $$ x_1 \mapsto a x_1, \ \ \ x_1 \mapsto a' x_1 $$ $$ x_2 \mapsto b x_2, \ \ \ x_2 \mapsto b' x_2 $$ $$ x_3 \mapsto c x_3 + d x_1^2, \ \ \ x_3 \mapsto c' x_3 + d' x_1^2, $$ respectively. Then the composition $\phi \circ \psi$ result in the automorphism given by the mappings, $$ x_1 \mapsto aa' x_1 $$ $$x_2 \mapsto bb' x_2 $$ $$ x_3 \mapsto c'c x_3 + c'd x_1^2 + d'a x_1. $$

Intuition behind the growth condition in the result of Griffin, Ono, Rolen and Zagier on Jensen Polynomials

Math Overflow Recent Questions - Wed, 05/29/2019 - 15:21

With great pleasure I read the recent paper of Griffin, Ono, Rolen and Zagier proving the surprising result that the Jensen polynomials $J^{d, n}_\alpha$ for a sequence $\alpha = \{\alpha(0), \alpha(1), \ldots \}$ of real numbers whose growth (?) is controlled in a certain way converges for fixed $d$ to a limiting polynomial of the same degree uniformly on compact subsets of $\mathbb{R}$.

Their main application is to some sequence of real numbers coming from the Riemann Xi function (see also this other MO question) but I already had lots of fun trying to see how this works out for much simpler sequences such as $\alpha(n) = 1$ or $\alpha(n) = 2^n$.

My question is however with the conditions in their main non-RH-related result: theorem 8 which reads:

Suppose that $\{E(n)\}$ and $\{\delta(n)\}$ are positive real sequences with $\delta(n)$ tending to $0$, and that $F(t) = \sum_{i =1}^\infty c_i t^i$ is a formal power series with complex coefficients. For a fixed $d \geq 1$, suppose that there are real sequences $\{C_0(n)\},\ldots,\{C_d(n)\}$, with $\lim_{n \to \infty} C_i(n) = c_i$ for $0 \leq i \leq d$, such that for $0 \leq j \leq d$, we have

$$\frac{\alpha(n+j)}{\alpha(n)} E(n)^{-j} = \sum_{i =0}^d C_i(n) \delta(n)^i j^i + o(\delta(n)^d) \qquad (*)$$ as $n \to \infty$. Then we have:

$$\lim_{n \to \infty} \frac{\delta(n)^{-d}}{\alpha(n)} J^{d, n}_\alpha \left(\frac{\delta(n)X - 1}{E(n)}\right) = H_{F, d}$$

uniformly on compact subsets of $\mathbb{R}$ where $H_{F, d}$ is defined by the generating function $F(−t) e^{Xt}=\sum_{m=0}^\infty H_{F,m}(X) \frac{t^m}{m!}$.

My question is about (*). Hopefully it is clear why I wrote above that I already had fun seeing what this theorem means even for really simple sequences $\alpha$: it is a priori not at all clear what $\delta, E$ or $C_i$ to take and one surprising thing I found is that (unlike their limits $c_i$) the sequences $C_i$ may depend non-trivially on the choice of the fixed value of $d$ even in cases we know a priori that the the limits exist for all $d$.

But managing to find sequence $E, \delta, C_i$ that 'work' is something quite different from understanding what is going on. My question is: what is, intuitively speaking, the set of conditions (*) trying to convey? Is it saying that the sequence $\alpha$ cannot grow too fast? Something else? Is it reasonable to think of finite sum on the right hand side as 'roughly a constant' so that the condition says that $\alpha$ grows more or less as $E^j$ where $E$ is the 'typical' value of $E(n)$. Ugh, as soon as I type it it stops making sense.

Any enlightenment is welcome here.

(Also on why the Latex is not parsing, btw. Update: never mind it didn't parse in the preview but does after posting)

Do we know what the impulse to "introduce" the Jordan canonical form was?

Math Overflow Recent Questions - Wed, 05/29/2019 - 15:13

Mo-ers,

Do you know how it was that the study of the Jordan canonical form began?

There are certain things that may be said once one has thought about the matter: for instance, one can say that the consideration of the Jordan canonical form of a matrix $A \in \mathbf{M}_{n \times n}(\mathbb{C})$ is valuable 'cause it facilitates certain commonplace computations related to $A$ (its powers, its exponential, etc.).

Do we know if the individual that first introduced the form in the mathematical realm (Camille Jordan? Wilhelm Jordan?) was somewhat motivated by this sort of things?Would you be so kind as to suggest a text in which one can find a very concrete, down-to-earth, and down to the nitty-gritty explanation of why the mathematician that "invented/discovered" the form did it?

On a lighter note, what's in your Weltanschauung the book on algebra (or rather linear algebra) that a person interested in mastering the Jordan canonical form should definitely peruse?

Minimum Volume of Intersection between two high-dim L1-norm balls

Math Overflow Recent Questions - Wed, 05/29/2019 - 14:51

Let B1 and B2 be two l1-norm balls in R^n with the same radius. The distance between the centers of B1 and B2 is d(B1, B2). Is there any deterministic method or probabilistic method to calculate the minimum intersected volume between the two balls B1 and B2?

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