Recent MathOverflow Questions

Twin primes conjecture and extrapolation method

Math Overflow Recent Questions - Sun, 11/05/2017 - 15:57

Let $(p_1, p_2)$ be a twin prime pair, where we include $(2, 3)$. If $p_1 \equiv 3$ mod $4$ then we let $t_{(p_1, p_2)} := p_2 ^ 2 / p_1 ^ 2$ otherwise, we let $t_{(p_1, p_2)} := p_1 ^ 2 / p_2 ^ 2$. Also define $t_{(2,3)}:=3 ^ 2/2 ^ 2$.

I conjecture that the product $$ \prod_{(p_1, p_2): \text{twin primes}}t_{(p_1, p_2)} =\tfrac{3 ^ 2}{2 ^ 2} \cdot \tfrac{5 ^ 2}{3 ^ 2}\cdot \tfrac{5 ^ 2}{7 ^ 2}\cdot\tfrac{13 ^ 2}{11 ^ 2} \cdot\tfrac{17 ^ 2}{19 ^ 2} \cdot\tfrac{29 ^ 2}{31 ^ 2} \cdot\tfrac{41 ^ 2}{43 ^ 2} \cdot \tfrac{61 ^ 2 }{59 ^ 2} \cdot \tfrac{73 ^ 2}{ 71 ^ 2}\cdot \tfrac{101 ^ 2 }{ 103 ^ 2}\cdots $$

is equal to $\pi$. (If this is true then twin prime numbers are infinity many.)

Some numerical values of partial products:

3.1887755102040816321 to $10^1$,
3.2055606708805624550 to $10^2$,
3.1290622219773513145 to $10^3$,
3.1364540609918890779 to $10^4$,
3.1384537326021492746 to $10^5$,
3.1417076006640026373 to $10^6$,
3.1417823471756806475 to $10^7$,
3.1415377533170544536 to $10^8$,
3.1415215264211035597 to $10^9$,
3.1415248453830039795 to $10^{10}$,
3.1415126339547108140 to $10^{11}$,
3.1415144504088659201 to $10^{12}$,
3.1415142045284687040 to $10^{13}$,
3.1415144719058962626 to $10^{14}$,
3.1415384423175311229 to $10^{15}$

Can we find a few more decimal places using the extrapolation method?

Calculation of logarithmic capacity?

Math Overflow Recent Questions - Sun, 11/05/2017 - 11:31

I am reading this paper about "Numerical approximation of the logarithmic capacity of domains", and there (on the third page) I found simple formulas for logarithmic capacity of simple figures like squares and equilateral triangles.

For a better understanding, I tried to recalculate those by myself several times, but I could not succeed. Is there a way that I can find those calculations or something similar that I can use as a guide?

Irrational number with known probability distribution on digits

Math Overflow Recent Questions - Sun, 11/05/2017 - 10:21

Is there any irrational number that is known the probability distribution of digits?

Something like 0 appears 10% of time, 1 appears 10% of time, etc.

Probably irrational numbers that are defined by a construction on digits like:

1234567891011121314....

You can prove digits distribution but I am interested in numbers that is not defined this way like pi, e, or square root of prime.

Is there any advance on this topic?

Linear equations with absolute values

Math Overflow Recent Questions - Sun, 11/05/2017 - 08:19

Assume we have a set of equations in $x \in \mathbb{R}^n$

$$|a_i\cdot x|=b_i$$

where $a_i \in \mathbb{R}^n$ and $b_i>0$ are given. Could such a system be solved efficiently?

  1. In a theoretical machine storing reals with perfect accuracy.
  2. An approximate solution taking rounding errors into account.

This is equivalent to having $2$ possible values for each linear combination, which is a valid question over any field. Does this equivalent problem over a fixed finite field have an efficient solution? Or is it perhaps known to be NP-complete?

So far, I concluded that squaring both sides of the equation we get linear equations in $y_{ij} = x_i x_j$. However, this helps only if the system is quadratically overdetermined.

Question: How to find the smallest value $x$ satisfying the equation: $x^2 = a \pmod c$ (known is $a$ and $c$, $c$ is not the prime)?

Math Overflow Recent Questions - Sun, 11/05/2017 - 02:09

Question: How to find the smallest value $x$ satisfying the equation: $x^2 = a \pmod c$ (known is $a$ and $c$, $c$ is not the prime)?

Using the Tonelli-Shanks algorithm and the Chinese remainder theorem does not always give me the smallest $x$ satisfying condition.

Is there any solution for calculating the smallest $x$?

Does anyone have an idea?

---- Edit:

I will describe more accurately my problem:

Let's assume that we are looking for a solution: $x^2 \equiv 1024 \pmod{1302}$.

We need to know the distribution of the factors $1302$, so $1302 = 2 \cdot 3 \cdot 7 \cdot 31$.

Now, using the Tonelli-Shanks algorithm we calculate for all divisors:

For 2:

$1024 \equiv k_1 \pmod 2$

$k_1 = 0$

$x_1 ^ 2 \equiv k_1 \pmod{2}$

$x_1^2 \equiv 0 \pmod{2}$

$x_1 = 0$

For 3:

$1024 \equiv k_2 \pmod 3$

$k_2 = 1$

$x_2^2 \equiv k_1 \pmod{3}$

$x_2^2 \equiv 1 \pmod{3}$

$x_2 = 1$

For 7:

$1024 \equiv k_3 \pmod 7$

$k_3 = 2$

$x_3 ^ 2 \equiv k_3 \pmod{7}$

$x_3^2 \equiv 2 \pmod {7}$

$x_3 = 4$

For 31:

$1024 \equiv k_4 \pmod 31$

$k_4 = 1$

$x_4 ^ 2 \equiv k_4 \pmod{31}$

$x_4 ^ 2 \equiv 1 \pmod{31}$

$x_4 = 1$

Then we solve the system of equations from the Chinese remainder theorem. We know the factors and also the values of $x$ from the formula: $x ^ 2 \equiv c \pmod {p}$ where $p$ and $c$ are known.

We solve the system of equations.

$x \equiv 0 \pmod{2}$

$x \equiv 1 \pmod{3}$

$x \equiv 4 \pmod{7}$

$x \equiv 1 \pmod{31}$

The Chinese remainder theorem comes out $x = 32$, and this is the good, smallest solution: $32 ^ 2 \equiv 1024 \pmod {1302}$ - quite trivial case.

The problem, however, is that it does not always agree. And so I write why.

In the above case, for example, for the first factor $31$ I assumed that I found a result equal to $1$. I do not necessarily have to find exactly $1$ as well:

$(31-1)^2 \equiv k_4 \pmod{31}$

$(31-1)^2 \equiv 1 \pmod{31}$

$30^2 \equiv 1 \pmod{31}$

The above is that for $30$ will also be $1$.

For such a system of equations (new value at $31$):

$x \equiv 0 \pmod{2}$

$x \equiv 1 \pmod{3}$

$x \equiv 4 \pmod{7}$

$x \equiv 30 \pmod{31}$

From the Chinese remainder theorem we get $x = 2944$. This also agrees, because $2944 ^ 2 \equiv 1024 \pmod {1302}$ but this is no longer the samllest possible value (smallest possible is $x = 32$).

Knowing the first value ($1$ for factor = $31$), the second one ($30$ for factor = $31$) that fits is easy to calculate as I did here.

However, since all combinations of values will be $2 ^ k$ (where $k$ is the number of prime factors (in different example). I have not found a way to do some search for these combinations in a better way than brutal-force.

Any ideas for that so I'm looking for.

Can someone suggest something?

Are all infinite graphs $3$-weak-edge colorable?

Math Overflow Recent Questions - Sun, 11/05/2017 - 01:39

Let $G=(V,E)$ be a simple, undirected graph such that every vertex has degree at least $2$. Given $n\in\mathbb{N}$, a map $c:E \to \{1,\ldots, n\}$ is said to be a weak coloring if for every $v\in V$ the edges adjacent to $v$ do not all have the same color. (More formally, we want the restriction $c|_{E(v)}$ to be non-constant, where $E(v) = \{e\in E: v\in e\}$.)

These two nice posts by Mikail Tikhomirov and Brendan McKay respectively show that for every finite graph there is a weak edge coloring with $3$ colors. I tried to carry through their arguments with transfinite induction to infinite graphs - without success.

Question. If $G=(V,E)$ is an infinite simple undirected graph, is there a weak edge coloring $c:E \to \{1,2,3\}$?

Gauge group of tangent bundle and diffeomorphism group

Math Overflow Recent Questions - Sun, 11/05/2017 - 00:28

I'm not exactly a differential geometer, so I hope this isn't too elementary a question.

From a naive point of view, it seems as if there are two natural group actions on the space of connections on the tangent bundle of a manifold $X$: the group of diffeomorphisms, which acts by pull-backs, and the group of gauge transformations, which acts by, well, gauge transformations.

Now, there isn't any obvious relationship between the two groups and, as far as I can tell, the gauge group is a much more natural and much more convenient object to study.

However, on the other hand, it is certainly possible that for given diffeomorphism $\phi: X \to X$ and a connection $A$, we have a gauge equivalence between $\phi^{*}(A)$ and $A$. I'm interested in understand when this occurs in general. In particular, for which diffeomorphisms is it true that $\phi^*(A)$ is gauge equivalent to $A$, for every connection $A$?

I tried attacking this, at least for diffeomorphisms isotopic to the identity, by viewing the isotopy as coming from a flow on $X$ and writing down a differential equation satisfied by a one-parameter family of gauge transformations inducing the same map on some fixed connection, but it didn't seem to lead anywhere.

Thanks.

Compute characteristic classes of principal bundle over closed surfaces

Math Overflow Recent Questions - Sat, 11/04/2017 - 10:07

Let $G$ be a connected Lie group and $\Sigma$ a closed oriented surface. We know that principal $G$-bundles $P$ can be topologically classified by a characteristic class $c(P)\in H^2(\Sigma,\pi_1G)\cong\pi_1G$.

The following is my question:

Let $G$ be a semisimple connected (or even compact) Lie group, $\beta\colon\pi_1(\Sigma)\to G$ a group homomorphism, and $\Sigma$ a closed oriented surface. Consider the universal cover $\tilde{\Sigma}\to\Sigma$ which is a principal $\pi_1(\Sigma)$-bundle. Form the associated bundle $\tilde{\Sigma}\times_\beta G$ which is necessarily a principal $G$-bundle over $\Sigma$. Hence, it should have a characteristic class $c(\tilde{\Sigma}\times_\beta G)\in H^2(\Sigma,\pi_1G)\cong\pi_1G$. Is there a way to compute this characteristic class in terms of $\beta$?

The difficulty here is that $\pi_1(\Sigma)$ is not connected, and hence I cannot use the functoriality of the characteristic class. Another one is that I am not sure how to compute the induced map $B\beta\colon B\pi_1(\Sigma)\to BG$ concretely.

I've asked this question on MSE and haven't got any answers yet.

Does $x, y\in \mathcal{R}$, $z\in (\mathcal{E}^\dagger)^*$ with $x\cdot y= z$ imply $x,y\in (\mathcal{E}^\dagger)^*$

Math Overflow Recent Questions - Sat, 11/04/2017 - 04:21

Let $\mathcal{R}$ be the Robba ring and $\mathcal{E}^{\dagger}$ the elements of $\mathcal{R}$ that are bounded at 0 (so the coefficients of the powerseries are bounded. Is it true that $x, y\in \mathcal{R}$, $z\in (\mathcal{E}^\dagger)^*$ with $x\cdot y= z$ implies $x,y\in (\mathcal{E}^\dagger)^*$?

If not, is it then true that $x\in \mathcal{R}$, $ y, z\in (\mathcal{E}^\dagger)^*$ with $x\cdot y= z$ implies $x\in (\mathcal{E}^\dagger)^*$?

This kind of seems to be implied in some of the papers of Colmez about $(\varphi, \Gamma)$-modules, for example here https://webusers.imj-prg.fr/~pierre.colmez/triangulines in Proposition 4.2, but I might be mistaken and some other argument is implicitly used.

Is every continuous microlocal operator a pseudo-differential operator?

Math Overflow Recent Questions - Fri, 11/03/2017 - 16:38

Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space. Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal. By being microlocal I mean that the wave front sets satisfy $WF(Af)\subset WF(f)$ for all $f$. (For another version, one could consider the singular supports instead of wave fronts, but I assume the answer wouldn't be different.) Does it follow that $A$ is a pseudo-differential operator?

This is a variation on this question about continuous endomorphisms on the Schwartz space. There one only assumed linearity and continuity, and the answer was negative. The other question was about $\mathcal S$ instead of $\mathcal S'$, but I need distributions to allow singularities.

I realize that there are many classes of pseudo-differential operators. The question is whether such a microlocal operator is always a ΨDO of some kind. Any details on what kind would be very interesting, of course.

Compactification of open manifolds in the form of a manifold( with zero Euler characteristic)

Math Overflow Recent Questions - Fri, 11/03/2017 - 14:25

Edit: According to the interesting comments of Michael Albanese and Nick L we revise the question as follows:

By manifold compactification of a manifold $M$ we mean a compact manifold $\tilde{M}$ which contains $M$ as an open dense subset.

Assume that $M$ is an open connected manifold which admits a manifold compactification. Does $M$ necessarily admit a manifold compactification with zero Euler characteristic?

Retractions for completely positive unital maps, with particularly nice norms

Math Overflow Recent Questions - Fri, 11/03/2017 - 12:19

Consider a non-singular, completely positive, unital map $\Psi: \mathbf M_k(\mathbb C) \to \mathbf M_h(\mathbb C)$. This map will have one or more retractions. Does $\Psi$ admit a retraction $\Phi: \mathbf M_h(\mathbb C) \to \mathbf M_k(\mathbb C)$, such that $\lVert \Phi \rVert = \lVert \Phi \rVert_{\mathrm{cb}}$?

(This question is a follow-up to a previous question, in which it was established that $\Psi$ may have retractions which do not have this property.)

Modular forms for different groups than $SL(2,\mathbb Z)$

Math Overflow Recent Questions - Fri, 11/03/2017 - 12:18

I know some theory of "classical" modular forms, that is functions in the complex upper-half plane satisfying

$f(\frac {az+b} {cz+d})=(cz+d)^kf(z)$

I know one can study modular forms on finite-index subgroups of $SL(2,\mathbb Z)$. But I have not seen much theory of modular forms on arbitrary Fuchsian groups. Which are the most interesting cases of such groups? Can somebody recommend a good reference?
I have also come across Hilbert and Siegel modular forms, but I don't have these in mind as an answer to this question. I wonder whether one can use arbitrary Lie group instead of $SL(2,\mathbb R)$, which is what the Wikipedia page about automorphic forms suggests, but I am not on a level to tackle the theory.

Upper bounds for lattice points in orbits, and representations of binary quadratic forms

Math Overflow Recent Questions - Fri, 11/03/2017 - 11:25

Write $\mathbb{Z}^{a\times b}$ for the $a\times b$ integer matrices. Let $Q\in\mathbb{Z}^{n\times n}$. Let $G=O(Q)$ be the orthogonal group of $Q$. For $X_0\in \mathbb{Z}^{n\times 2}$, set $$ R'(T,Q,X_0) = \{ X\in X_0G(\mathbb{Z}) : |X_{ij}|\leq T\}. $$

Dubious conjecture: If $Q$ is indefinite then $$ \#R'(T,Q,X_0) \ll_{Q,\epsilon} T^{\max\{0,2n-6\}+\epsilon} \tag{$\star$} $$ for all $\epsilon>0$, with an implicit constant depending only on $Q$ and $\epsilon$.

Question: Is this true? How about simple (?) examples like $Q=I_3$ or $Q=I_4$?

Background

Let $B\in\mathbb{Z}^{2\times 2}$ be nonsingular and symmetric. I am interested in the number of solutions $$ R(T,Q,B) = \{ X\in \mathbb{Z}^{n\times 2} : XQX^T =B, \, |X_{ij}|\leq T\} $$ with height up to $T$, when $Q$ is indefinite. A naive guess based on the circle method would be that $$ \#R(T,Q,B) \ll_Q 1+T^{2n-6} \tag{$\ast$} $$ with an implicit constant depending at most on $Q$. The set $R(T,Q,B)$ is the union of finitely many $G(\mathbb{Z})$-orbits $R'(T,Q,X_0)$, so the conjecture above is a weak version of this.

Note that ($\ast$) is true for large $n$. For instance when $Q$ is indefinite and $n\geq 12$ this (and much more) follows from the asymptotic result of Brandes. (The condition $n\geq 12$ follows from the comment at the end of section 1 in the subsequent paper.) On the other hand ($\ast$) is certainly false for small $n$, though I don't know of a good example in the indefinite case.

If $B$ is positive definite and $Q$ has signature $(p,q)$ then it is a special case of Theorem 1.4 of Eskin, Mozes and Shah that $$ \# R(T,Q,B) \sim C T^{\min\{2,q\}(n-1-\min\{2,q\})} $$ for some $C= C(Q,B)$, unless a certain volume is infinite. In the latter case, the comments after formula (1.4) of Duke, Rudnick and Sarnak suggest that the right-hand side of ($\ast$) just changes by a factor of $\log T$. Perhaps a proof of ($\star$) in this case is implicit in their work, but I have not been able to verify this.

where should start for learning numerical analysis? (it's URGENT)

Math Overflow Recent Questions - Fri, 11/03/2017 - 06:14

can anybody here help me, please?I really fed up. I am a master student and I have a numerical analysis course and it's very difficult for me cause I have not had experience in this subject. whatever I read I just mix-up more. Thank you.

Do averaged binary symmetric channels maximize mutual information?

Math Overflow Recent Questions - Fri, 11/03/2017 - 05:39

This is a refined version of Do binary symmetric channels maximize mutual information?, which was answered negatively.

Let the random variables $(X, Y)$ be a doubly symmetric binary source with parameter $0 \le p \le 1/2$, i.e., $X,Y \sim \text{Bernoulli}(1/2)$ and $P(X \neq Y) = p$.

Define the two regions $\mathcal{A}, \mathcal{B} \subseteq \mathbb{R}^3$:

  • $\mathcal{A}$ consists of all points $(R_0, R_1, R_2)$ such that there exist binary random variables $U,V$, satisfying the Markov chain $U - X - Y - V$ and \begin{align} R_1 &\ge \mathrm{I}(U;X) , \\ R_2 &\ge \mathrm{I}(Y;V) , \\ R_0 &\le \mathrm{I}(U;V) , \end{align} where $\mathrm{I}(\cdot;\cdot)$ is mutual information.

  • $\mathcal{B}$ consists of all points $(R_0, R_1, R_2)$ such that there exist probabilities $a,b \in [0,1]$ with \begin{align} R_1 &\ge 1 - \mathrm{H}(a * p) , \\ R_2 &\ge 1 - \mathrm{H}(b * p) , \\ R_0 &\le 1 - \mathrm{H}(a * p *b) , \end{align} where $a*b := a(1-b)+(1-a)b$ is binary convolution and $\mathrm{H}()$ is the binary entropy function.

My Question: Is $\mathrm{conv}(\mathcal A) = \mathrm{conv}(\mathcal B)$, where $\mathrm{conv}()$ denotes the convex hull?

Some comments:
The definitions of $\mathcal{A}$ and $\mathcal{B}$ are almost the same, except that the channels $X \to U$ and $Y \to V$ are required to be symmetric for $\mathcal{B}$ and hence $\mathcal B \subseteq \mathcal A$.
The question Do binary symmetric channels maximize mutual information? asked whether $\mathcal A = \mathcal B$ and a counterexample was provided for $p=0$, which does not seem to apply here.

Enquiry on an integral involving the logarithmic integral

Math Overflow Recent Questions - Fri, 11/03/2017 - 05:12

Is it true that there exists some constant $A$ such that

$$\int_{2}^x \frac{1}{t}{\Big[\frac{d}{dt}Li(t)\Big]} \mathrm{d}t=\log\log x + A + O(x^{-1/2+\epsilon})$$ for any $\epsilon>0$, where $Li(z)=\int_{2}^z \frac{dz}{\log z} $ ?

About 2-homogeneous polynomial on finite dimensional spaces

Math Overflow Recent Questions - Fri, 11/03/2017 - 05:00

For a Banach space $X$, the map $p:X\rightarrow\mathbb{C}$ is called a 2-homogeneous polynomial if there exists a bilinear form $A$ on $X$ such that $p(x)=A(x,x)$, the norm of this polynomial is defined by $sup_{\Vert x\Vert=1}|p(x)|$.

Take $X=l_\infty^n$, a finite dimensional space with maximum norm. I wanna show any 2-homogenous polynomial $p:l_\infty^n\rightarrow\mathbb{C}$ has a norm-preserving extension to $l_\infty$, a complex sequence space with supremum norm, i.e. there is an extension $\hat{p}:l_\infty\rightarrow\mathbb{C}$ so that $\Vert p\Vert=\Vert\hat{p}\Vert$ and $\hat{p}$ is a polynomial on $l_\infty$.

Any related articles are appreciated!

Noetherian rings as homomorphic image of finite direct product of Noetherian domains?

Math Overflow Recent Questions - Fri, 11/03/2017 - 05:00

A theorem of Hungerford says that : Every PIR (principal ideal ring , obviously commutative ) is a homomorphic image of a finite direct product of PID s . My question is , is there a similar criteria for Noetherian rings, i.e. : Is every Noetherian ring a homomorphic image of a finite direct product of Noetherian domains ?

This Is every Noetherian Commutative Ring a quotient of a Noetherian Domain? shows that we cannot expect every Noetherian ring to be a homomorphic image of a Noetherian domain.

What is the meaning of with "multiplicity two" in the following:

Math Overflow Recent Questions - Fri, 11/03/2017 - 04:58

Recall that the spectrum of an invertible harmonic oscillator is continuous and covers the real axis with multiplicity two.

Thanks a lot

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