Recent MathOverflow Questions

Prime counting. Meissel, Lehmer: is there a general formula?

Math Overflow Recent Questions - Fri, 04/13/2018 - 11:17

I am looking for a general forumla to count prime numbers on which the Meissel and Lehmer formula are based:
$$\pi(x)=\phi(x,a)+a-1-\sum\limits_{k=2}^{\lfloor log_2(x) \rfloor}{P_k(x,a)}$$

Wiki - prime counting - Meissel Lehmer

More precisely, I am looking for the detailed description of the $P_k$ for $k>3$.

$P_k(x,a)$ counts the numbers$\leqslant x$ with exactly $k$ prime factors all greater than $p_a$ ($a^{th}$ prime), but in the full general formula, this last condition is not necessary.

The Meissel formula stops at $P_2$ (and still uses some $\phi$/Legendre parts)
Wolfram - Meissel

The Lehmer formula stops at $P_3$ (and still uses some $\phi$/Legendre parts)
Wolfram - Lehmer

I don't find anything about the general formula (using all the $P_k$ terms). Is there any paper on it? Why stop at $P_3$? is it a performance issue?

Lehmer vaguely talk about it in his 1959 paper
On the exact number of primes less than a given limit

Deleglise talks about performances here
Prime counting Meissel, Lehmer, ...

Thanks

Edit: by "a general formula on which the Meissel and Lehmer formula are based", I meant the one they tend to (with all $P_k$), not the one they started from (Legendre, with no $P_k$). Sorry if it was not clear.

Stochastic differential equation under transposition

Math Overflow Recent Questions - Fri, 04/13/2018 - 11:01

The setting:

Let $B_t$ be standard scalar Brownian motion.

I am considering the follow SDE

$$dY(t) = TY(t) \ dt + S Y(t) \ dB_t, \text{ and }Y(0)=Y_0 \in \mathbb R^n.$$

for square (time-independent) matrices $T \in \mathbb R^{n \times n}$ and $S\in \mathbb R^{n \times n}.$

The solution is given in terms of a stochastic flow $\Phi_1: \mathbb R^n \rightarrow \mathbb R^n$ (I completely subpress the randomness in the notation, here) as

$$Y(t):=\Phi_1(t,0)Y_0.$$

Now, consider the following equation where we conjugate $T$ and $S$ and study

$$dZ(t) = T^*Z(t) \ dt + S^* Z(t) \ dB_t, \text{ and }Y(0)=Y_0 \in \mathbb R^n.$$

This one has also a solution $$Z(t):=\Phi_2(t,0)Y_0.$$

My question: Is it true that $Z(t)$ has the same law as $\Phi_1(t,0)^* Y_0$? It almost looks like nothing, but I fail to see why this holds.

Any comment/remark/idea is highly appreciated.

Do two dimensional representations with the same adjoint representation differ by a character?

Math Overflow Recent Questions - Fri, 04/13/2018 - 05:56

Let $K$ be a field of characteristic not equal to $2$. Let $\text{ad} : \text{GL}_2(K) \to \text{GL}_3(K)$ be the adjoint representation, obtained by $\text{GL}_2(K)$ acting on $2 \times 2$ matrices with trace $0$ by conjugation. Suppose $\rho_1, \rho_2 : G \to \text{GL}_2(K)$ are representations of a group $G$ such that $\text{ad}\rho_1 \cong \text{ad}\rho_2$ over $K$. Is there a character $\eta : G \to K^\times$ such that $\rho_1 \cong \rho_2 \otimes \eta$ over $K$? That is, if $x \in \text{GL}_3(K)$ such that $\text{ad}\rho_1 = x(\text{ad}\rho_2) x^{-1}$, can one produce $y \in \text{GL}_2(K)$ and $\eta$ as above such that $\rho_1 = y \eta (\otimes \rho_2) y^{-1}$?

I am most interested in the case when $K$ is a finite field. In that case, I believe one can treat the case when the projective image of $\rho_i$ is isomorphic to $\text{PSL}_2(K)$ or $\text{PGL}_2(K)$ fairly easily by using that those groups have automorphism group $\text{P}\Gamma\text{L}_2(K)$. But I'd like a proof that does not break into cases based on the subgroups of $\text{PGL}_2(K)$.

Change of variables Levy process

Math Overflow Recent Questions - Thu, 04/12/2018 - 11:36

Let $L$ be a Lévy process and define $M_t:=L_t-t\mathbb E(L(1)),$ then $M$ is a centred martingale.

Now consider the stochastic integral for $f$ a continuous process

$$\int_0^t f(t-s) \ dM_s,$$

is it then true that this is equal $\int_0^t f(s)\ dM_s$ almost surely?

I believe the answer should be yes, because of a linear change of variables is compatible with the iid increments.

Find the equation of the cone whose generating curve is X2 + Y2 + Z2 = a2 and X + Y + Z = 1, whose vertex is (0, 0, 0)

Math Overflow Recent Questions - Thu, 04/12/2018 - 11:34

Find the equation of the cone whose generating curve is X^2 + Y^2 + Z^2 = a^2 and X + Y + Z = 1, whose vertex is (0, 0, 0).

Approximation for ideals

Math Overflow Recent Questions - Thu, 04/12/2018 - 11:27

Let $A$ be a ring, $A_0$ a subring of $A$ equipped with an injective endomorphism $f : A_0\to A_0$, and such that $A$ is the direct limit:

$$A = \varinjlim_{f^n : A_0\to A_0, n\ge 0} A_0.$$

Let $I\subset A$ be an ideal. Does there exist an ideal $I_0\subset A_0$ such that $I = \varinjlim I_0A_0$?

My guess is $I_0 := I\cap A_0$, where the intersection takes place in $A$, and $A_0$ is the copy of $A_0$ in $A$ given by the injective map $A_0\to\varinjlim A_0$.

Tensor product and quotients of it

Math Overflow Recent Questions - Thu, 04/12/2018 - 10:55

Let $A,B$ be Banach algebras, and $I$ be a closed two sided ideal of $A$ and $J$ be a closed two sided ideal of $B$. Is the relation $A\hat{\otimes}B/I\hat{\otimes}J\cong A/I\hat{\otimes}B/J$ true?(I mean topological isomorphism and projective tensor product). If it isn't true in general, is there conditions that make it true?

Converting non-homogenous linear equations to homogenous form

Math Overflow Recent Questions - Thu, 04/12/2018 - 10:25

I have a set of 14 linear equations with 14 unknowns, but they are in non-homogenous form.

Can I convert the system to homogenous form simply by rearranging such that each of the 14 equations is equal to 0?

e.g. x + y = z becomes x + y - z = 0

Or will this change the solution?

Argument for differentiability of a certain quotient of smooth functions

Math Overflow Recent Questions - Thu, 04/12/2018 - 10:03

I have what is in essence a basic analysis question.

To make working out a certain example a bit easier I found that I need to find existence of a function $f\in C^\infty(\mathbb{R})$ with the following properties:

  1. $f$ is increasing
  2. $f(x)=0$ for all $x\leq 0$
  3. $f(x)=1$ for all $x\geq 5$
  4. $\frac{f(x)}{x}<1$ for all $x>0$
  5. For any $y>-1$ the function $F(x,y)=\frac{f(x)}{f(x+f(x)y)}$ which is immediately well-defined and smooth for $x>0$ can be extended to a smooth function on $\{(x,y)| y>-1\}$.

To find a function $f$ that satisfies the first 4 conditions is relatively simple by tweaking one that looks like $e^{-\frac{1}{x^2}}$. So the main question is whether $f$ exists such that 5. is satisfied. After some discussion it seems like it should be possible even setting $F(x,y)=1$ for $x\leq 0$, but I have not been able to convince myself fully yet. The idea is to rewrite the denominator as $f(x(1+\frac{f(x)}{x}y))$ and use the fact that $\frac{f(x)}{x}$ vanishes to order $n$ at $x=0$ to deduce $n$ times differentiability of $F(x,y)$. Then, since $f^{(n)}(0)=0$ for arbitrary $n$ we find that smoothness.

Any insights would be appreciated of course!

Chain condition in iterated forcing

Math Overflow Recent Questions - Thu, 04/12/2018 - 09:52

(I apologize for the long question, which has no mathematical content. Just looking for the right reference.)

In their celebrated paper [ST1971] introducing iterated forcing, Solovay and Tennenbaum showed the following theorem:

Theorem A: If $(B_\alpha: \alpha< \omega_1)$ is an increasing sequence of complete subalgebras of the Boolean algebra $B_{\omega_1}$, and $\bigcup_{\alpha<\lambda} B_\alpha$ is dense in $B_\lambda$ for each limit ordinal $\lambda\le \omega_1$, and if all $B_\alpha $ ($\alpha<\omega_1$) satisfy the ccc (countable chain condition), then also $B_{\omega_1} $ will satisfy the ccc.

This theorem is used to show that the finite support iteration of ccc forcings is again ccc. (Theorem 6.3 in [ST1971])

An essential fragment of the proof given in the paper is due to Silver. (The authors write that Silver's version is "quite a bit simpler than [our] original proof".)

Essentially the same proof shows the following theorem:

Theorem B: Let $\kappa $ be regular uncountable. Let $(P_\alpha:\alpha <\kappa)$ be an iteration of forcing notions with direct limit $P_\kappa$, and assume that the set of stages $\delta<\kappa$ where $P_\delta$ is the direct limit of the previous forcings is stationary in $\kappa$. If all $P_\alpha$ satisfy the $\kappa$-cc, then so does $P_\kappa$.

(In Jech's book, Theorem A is 16.9/16.10, and Theorem B is 16.30. The latter theorem has a 3-line proof, starting with "Exactly as the proof of 16.9.")

Question: To whom should Theorem B be credited?

  • To Solovay-Tennenbaum, whose unpublished original proof of Theorem A most likely also showed Theorem B?
  • Or to Silver, whose proof of Theorem A definitely can be easily generalized to a proof of Theorem B?
  • Or to "Silver/Solovay-Tennenbaum", or "Silver's proof in [ST]"?
  • Or just to Solovay? Solovay's remarks in [K2011] indicate that the proof of the iteration theorem is his. But I think it is customary (at least in mathematics) not to divide credit between the coauthors of a paper, unless such a division is explicitly mentioned in the paper.
  • Or to somebody else, who first explicitly formulated Theorem B?

(I am asking because I want to add a remark to a proof of a variant of Theorem B; the proof will be a variant of the S/S-T proof.)

[K2011]: Akihiro Kanamori: Historical remarks on Suslin's Problem, in: Juliette Kennedy and Roman Kossak, editors, Set Theory, Arithmetic and Foundations of Mathematics: Theorems, Philosophies, Lecture Notes in Logic, volume 36, 1-12. Association for Symbolic Logic, 2011. (MR2882649. http://math.bu.edu/people/aki/18.pdf )

[ST1971]: Solovay, R. M.; Tennenbaum, S. Iterated Cohen extensions and Souslin's problem. Ann. of Math. (2) 94 (1971), 201–245. (MR0294139, DOI:10.2307/1970860)

Description of Koszul dual of Sklyanin algebras

Math Overflow Recent Questions - Thu, 04/12/2018 - 09:46

It is well-known that Sklyanin algebras are Koszul, but, is it known an explicit description of the dual algebra Ext_A(k,k)? (I mean in terms of generators and relations)

Uniform Asymptotic Approximation of the Whittaker function

Math Overflow Recent Questions - Thu, 04/12/2018 - 09:28

I would like to know if there exist a uniform asymptotic approximation of the Whittaker function $W_{\kappa,i\mu}(x)$ for $\kappa<0$, $x >0$, and with $\mu \to +\infty$. The case of $\kappa \ge 0$ is treated in [1], but I was not able to find the asymptotics for negative $\kappa$ in the literature.

[1] T. M. DUNSTER, Anal. Appl., 01, 199 (2003).

Inner radius of a random convex hull

Math Overflow Recent Questions - Thu, 04/12/2018 - 09:15

Let $\sigma_1,\ldots,\sigma_M$ i.i.d. random vectors in $\mathbb{R}^d$, and for notational convenience, let $\Sigma=(\sigma_1,\ldots,\sigma_M)$. I am interested in understanding $$ \gamma(\Sigma) = \min_{\lambda\in\Delta_M} \Big\|\sum_{i=1}^M \lambda_i \sigma_i\Big\|_2, $$ where $\Delta_M=\{\lambda \in\mathbb{R}_+^M: \sum_i \lambda_i=1\}$, is the $M$-dimensional simplex. I am primarily interested in the cases of the distribution being the standard Gaussian and the uniform probability on the hypercube $\{-1,+1\}^d$.

Here is what I know:

  1. If we consider continuous distributions, the sigmas are linearly independent with probability 1 when $M\leq d$, thus this quantity should be strictly positive in this regime. In the discrete case, the latter claim should still hold with high probability.
  2. For the discrete case, the function $\gamma(\cdot)$ is Lipschitz for the Hamming distance, so it concentrates around its mean
  3. Similarly, for the Gaussian case one can prove $\gamma(\cdot)$ is Lipschitz for the Euclidean norm (more precisely, the Frobenius norm of $\Sigma$ as a matrix), so it concentrates around its mean.

By the last two observations, I am now mostly interested in understanding $\mathbb{E}_{\Sigma}[\gamma(\Sigma)]$, as a function of $M$. Clearly, for $M=1$, and for my distributions of interest, $\mathbb{E}[\gamma]=\sqrt{d}$, and I believe that for $M>d$, $\mathbb{E}[\gamma]\approx0$ (although I don't have a proof).

My question is how to compute (or lower bound) this expectation as a function of $M$. Connections with the literature are also welcome. As a final comment, I tried to lower bound the expectation using the Khintchine inequality, but the minimum in between seems to ruin the approach.

PS: $\gamma(\Sigma)$ represents the largest possible (origin centered) ball not touching the simplex generated by the vectors $\sigma_1,\ldots,\sigma_M$; which is similar, but not equivalent to the inner radius of the (symmetrized) convex hull. So better suggestions for a title are also welcome.

A difficult determinant

Math Overflow Recent Questions - Thu, 04/12/2018 - 08:57

The $N\times N$ determinant $$D(a,\vec{b})=\det\left( \frac{(2N+a+b_j-i-j)!}{(N-j)!(N+a-i)!}\right)$$ has the nice form $$D(a,\vec{b})=\prod_{j=1}^N\frac{(N+a+b_j-j)!}{(N+a-j)!}\prod_{i=j+1}^N\frac{(b_i+b_j-j+i)}{(i-j)}.$$

I would like to know if the generalization where $a$ is allowed to vary with $i$ has a nice expression as well, $$D(\vec{a},\vec{b})=\det\left( \frac{(2N+a_i+b_j-i-j)!}{(N-j)!(N+a_i-i)!}\right)=?$$

I know Krattenthaler has this great paper about determinants, but I was not able to find help there.

Section of a holomorphic line bundle with given differential at a zero

Math Overflow Recent Questions - Thu, 04/12/2018 - 08:56

Let $X$ be a compact Kähler manifold of dimension $n$ with a given Kähler metric $\omega$. Let $L$ be a hermitian holomorphic line bundle on $X$ whose metric is positive. Let $x_0\in X$.

I would like to construct a section $s\in H^0(X,L)$, so that $s(x_0)=0$ and $ds(x_0)=\alpha\neq 0$ for given $\alpha$. Moreover, I want to control the $L^2$ norm of $s$ by some function of $|\alpha|$.

Here $ds$ is the differential of $s$, since $x_0$ is a zero of $s$, $ds(x_0):T_{x_0}X \rightarrow L_{x_0}$ is intrinsically defined.

The question is similar to a special case of Ohsawa-Takegoshi theorem, where the prescribed information is only $s(x_0)$. If there is a good solution to my question, then I would like to know if it is possible to replace the single point $x_0$ by a more general analytic subset of $X$?

Functional Taylor expansion for differential entropy

Math Overflow Recent Questions - Thu, 04/12/2018 - 08:50

Consider an continuous distribution $F$ with density $f$. The (differential) Shannon entropy of $f$ is

$h(f)=-\int f(x)\log f(x) dx$.

In the literature of differential entropy estimation, oftentimes analyzing the performance of an estimator relies on a functional Taylor expansion (sometimes termed `Von Mises Expansion'). For two densities $f$ and $g$, the expansion of $h(f)$ about $g$ reads as (see, e.g., https://www.cs.cmu.edu/~aarti/Class/10704_Spring15/lecs/lec5.pdf)

$h(f)=h(g)+\int \big(\log g(x) +1\big)\big(g(x)-f(x)\big)dx + O\big(||f-g||_2^2\big)$

However, I couldn't fully figure out under what assumptions on $f$ is this expansion valid. Would very much appreciate a clarification on what are the minimal assumptions on $f$ for the above to hold true.

In particular, does the expansion apply when $f=p\ast\gamma$, where $p$ is a compactly supported density and $\gamma$ is a Gaussian (i.e., $f$ is a convolution with Gaussian), and $g$ is some density estimator of $f$ (say, a kernel density estimator)?

Proof Verification: Application of Perron's Theorem [on hold]

Math Overflow Recent Questions - Thu, 04/12/2018 - 08:19

My homework: For $P \in \mathbb{R}^{n \times n}$ with $P > 0$. Show that the dominant eigenvalue $\lambda(P)= \min \{\lambda \in \mathbb{R}_{\geq 0} \mid Px \leq \lambda x \text{ for some nonzero } x \geq 0\}=\min \Gamma(P)$.

My proof: Perron's Theorem says if $P \in \mathbb{R}^{n \times n}$ with $P > 0$, then $P$ has a dominant eigenvalue $\lambda(P)>0$ with an corresponding eigenvector $v>0$. Therefore, $\lambda(P)\in \Gamma(P)$. Furthermore, $Pv\leq \lambda v$ for any $\lambda\in \Gamma(P)$. So, $\lambda(P) v\leq \lambda v$. Therefore, $\lambda(P)\leq\lambda$. Hence, $\lambda(P) = \min_{\lambda\in\Gamma(P)} \lambda$.

My main concern is this sentence: Furthermore, $Pv\leq \lambda v$ for any $\lambda\in \Gamma(P)$.

In the question, we define $\Gamma(P)$ by $Px \leq \lambda x \text{ for some nonzero } x \geq 0$. Picking an arbitrary $\lambda\in \Gamma(P)$, I don't know whether I can replace $x$ by $v$, where $v$ is a positive dominant eigenvector. In other words, does "for some nonzero $x\geq 0$" means any nonnegative $x$ works?

closed decomposition of locally compact Hausdorff space

Math Overflow Recent Questions - Thu, 04/12/2018 - 08:10

Let $X$ be a locally compact Hausdorff space and suppose that $X$ can be written as the disjoint union of countably many non-empty closed subsets. Is at least one of the subsets clopen?

On proof of Sauer's lemma

Math Overflow Recent Questions - Thu, 04/12/2018 - 00:25

Let $X$ be a (possibly infinite) set. We consider a subset $H$ of the set $\{0,1\}^X$ of functions $X\to\{0,1\}$. Given a finite subset $B\subset X$, we denote by $H_B$ the set of restrictions to $B$ of elements of $H$, that is, the image of $H$ by the restriction map $\{0,1\}^X\to\{0,1\}\to B$. If $H_B=\{0,1\}$ (or equivalently if $|H_B| = 2^{|B|}$), we say that $H$ shatters $B$.

(definition provided here on page 69)

Consider $X,B,H$ as above. Fix an element $c_1\in B$. Given $h\in\{0,1\}^X$, define its "brother" $\hat{h}\in\{0,1\}^X$ as equal to $1-h(c_1)$ on $c_1$ and equal to $h$ elsewhere. Define $H'$ as the set of $h\in H$ such that there exists $g\in H$ such that $h$ and $\hat{g}$ coincide on $B$.

(definition is on page 75)

I am considering the proof of Sauer's lemma here from page 74. It is proven by induction, and the proof consists mostly in saying that

$$|H_A|\le|\{B \subseteq A : H \text{ shatters }B \}| \ \ \ \ \ (1)$$

I want to add additional requirement, which leads to

$$|H_A| = |\{B \subseteq A : H \text{ shatters }B \}| \ \ \ \ \ (2)$$

all the time when this requirement is satisfied and when it is not satisfied, I want to prove that

$$|H_A| < |\{B \subseteq A : H \text{ shatters }B \}| \ \ \ \ \ (3)$$

Am I correct in the assumption that the only requirement for it to be true is $H' = H$ for all $H'$ defined on connected with them subsets $B$? If it is true, then is it correct to say, that strict equality '<' as in (3) is achieved if and only if there exist some element in A, which either classified as 0 or as 1 by all functions in H? And when for every element $x$ from $A$ there exist $h_1$, $h_2$ from $H$ so that $h_1(x) = 0$ and $h_2(x) = 1$, we have equality '=' as in (2)?

I am asking this as part of finding an answer to that question of mine.

What are good articles/books on the psychology of mathematical research?

Math Overflow Recent Questions - Wed, 04/11/2018 - 22:11

I am thinking about about advanced texts similar to Polya's 'How to solve it?'. Quite a few good articles of such a kind are published under Philosophy of Mathematics, but that dwells on a very different domain generally.

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