Math Overflow Recent Questions


Etale cohomology of localizations of henselian rings

Fri, 03/09/2018 - 20:41

Let $R$ be a ring (say noetherian of finite Krull dimension, possibly with additional hypotheses) henselian along the ideal $(p)$, and let $\hat{R}$ be the $p$-adic completion. Is it true that the étale cohomology of $R[1/p]$ and $\hat{R}[1/p]$ with mod $p$ coefficients coincide? I believe that this should be true (for $K$-theoretic reasons), but I was wondering if there is a direct argument.

Variation of constant formula for SPDEs

Fri, 03/09/2018 - 20:06

Given a separable real Hilbert space $H$.

Assume that $A$ is the generator of a pseudocontractive semigroup and $Q$ be a bounded operator.

Then, let $(W(t))$ be a Wiener process taking values in a Hilbert space $H$ with positive trace-class covariance matrix.

It is known that for such systems stochastic evolution equation has a unique solution.

$$dX(t)=AX(t) dt + Q dW(t)$$

$$X(0)=x \in H$$

That is, there is a flow $\varphi$ such that $X(t)=\varphi(t)x.$

Now one might think about looking at

$$dY(t)=AY(t) dt + Q dW(t)+ y f(t)$$

$$Y(0)=x \in H$$

where $y \in H$ and $f$ is a sufficiently regular scalar function.

Classical ODE theory suggests that the solution to the second equation in this case can be written as

$$Y(t)= \varphi(t)x + \int_0^t \varphi(t-s)y f(s) ds.$$

Does anybody know a proof of this fact? It has to be true, but I would like to know a bit more about assumptions that one has to make etc.

I know that SPDEs are a very delicate field.

Reference for flatness in complex-analytic geometry

Fri, 03/09/2018 - 18:25

What is a good reference for flat morphisms of complex-analytic spaces? (The book by Grauert and Remmert doesn't treat them).

Topics I'm interested in: openness of flat maps, descent for coherent analytic sheaves.

What is against having distinct membership relations on sets in the Platonic realm?

Fri, 03/09/2018 - 15:31

This question is in connection with the question that I've asked at:

Where do models of false theories exist?

The answer to that question was that any consistent theory can have its primitives be re-interpreted in such a manner as to come true. So the difference between a false theory and a true theory is one of reference, a true theory is one whose sentences are satisfiable in the part of the Platonic realm that it refers to.

Now according to that answer, I'll pose the following possibility and the question is what is against that possibility:

Now let's assume that there exists a Platonic world $P^{sets}$ of all sets, and two Platonic worlds $P^{\in_1}$ , $P^{\in_2}$ of primitive ordered pairs of sets, these are taken to represent distinct membership relations between sets. So the ordered pairs in realms $P^{\in_1}$, $P^{\in_2}$ only have sets as their projections, so $P^{sets}$ is their domain, so they represent two kinds of membership relations between sets $\in_1$ and $\in_2$ relations defined on the same domain, as:

$y \in_1 x \iff \exists p \in^* P^{\in_1} (p=\langle y,x \rangle)$

$y \in_2 x \iff \exists p \in^* P^{\in_2} (p=\langle y,x \rangle)$

So for example the sentence $\exists x \forall y (y \not \in x)$ would be:

$\exists x \in^* P^{sets} \forall y \in^* P^{sets} (\not \exists p \in^* P^{\in_1} (p=\langle y,x \rangle))$

$\exists x \in^* P^{sets} \forall y \in^* P^{sets} (\not \exists p \in^* P^{\in_2} (p=\langle y,x \rangle))$

Where $\in^*$ is the membership relation between sets and $P^{sets}$ and between ordered pairs of sets and the realms $P^{\in_1}, P^{\in_2}$.

Now since we are having two membership relations on sets defined after two Platonic realms, then we can have two theories each referring to one of these membership relations, so no confusion of reference would raise (as far as each theory is speaking correctly about the part of the Platonic realm that it refers to), and so both theories would be TRUE in that Platonic world. Accordingly we can have both membership relations obeying all rules of $\text{ZF}$ and yet one of them obeying $\text{Choice}$ while the other negating it.

So this would mean that the answer to as whether choice or negation of choice is true about membership in sets, is to say that there are two kinds of membership in sets, one fulfills choice and the other negates it.

I don't see anything in the definition of a true theory [from a Platonic perspective] that can go against that possibility. Why should there be just one kind of membership in sets? there is no rule to the effect that no two distinct relations in the Platonic world can have the same domain, actually, this is not the case with the standard model of arithmetic, for it does have distinct relations having the same arity defined on the same domain of standard naturals (exp: the binary relations $Successor$ ,$<$; the ternary relations $+ ,\times $) so why not have the same situation with sets?

Along the same lines of this argument, we may have two membership relations one obeying $\text{CH}$ and the other negating it, on the SAME domain of all sets.

This question is intended to be answered from a Platonistic perspective.

Number of solutions for the inequality with square roots

Fri, 03/09/2018 - 13:57

Let $M$ be some large real number and $\delta>0$. I would like to estimate the number of solutions for the inequality


where $n_i$ are positive integers with $M<n_i\leq 2M$ for all $1\leq i\leq 4$. Let us denote this quantity by $N(M,\delta)$.

There is a paper by O.Robert and P.Sargos which provides the bound for a bit more general quantity. In our case their work gives

$$N(M,\delta)\ll M^{2+o(1)}+\delta M^{4+o(1)}.$$

I think, some heuristics suggest that when $\delta\gg 1$ this bound is more or less optimal. But I'm interested in the case when $\delta=o\left(\frac{1}{M}\right)$. So, my question is: can the bound of Robert and Sargos be improved for small $\delta$ and when should we expect that $N(M,\delta)\ll M^{2+o(1)}$? For example, we always have $N(M,M^{-2})\ll M^{2+o(1)}$ but can we do better (i.e. prove this bound for larger $\delta$) or at least are there any conjectures about this?

Regular partial tilting modules in wild hereditary algebras

Fri, 03/09/2018 - 13:32

Let $k$ be an algebraically closed field. Let $Q$ be a connected wild quiver. Let $\mathcal{R}_1$, $\mathcal{R}_2$ be two regular components of the Auslander-Reiten quiver of $kQ$ that contain stones. When is it true that there exists quasi-simple stones $M\in\mathcal{R}_1$ and $N\in\mathcal{R}_2$ such that $Ext^1(M,N)=Ext^1(N,M)=0$ ?

Using Auslander-Reiten formula and the fact that the modules we care about are regular this problem is essentially a problem about $Hom$ between regular components. We know that there are $card(k)$ many regular components due to (XVIII.1.8) in Elements of the Representation Theory of Associative Algebras. We further know that maps between regular components always exist in the wild hereditary case due to (XVIII.2.6) of the same book. Due to Lemma 1.4 in Kerner, 1991 we can further reduce this problem to a problem about quasi-simples alone. However when do such maps occasionally not exist between some quasi-simples seem to either be an open question or be some existing result I'm currently unaware of. May I ask whether anyone has done any work on this problem?

Can I check the accessibility of a functor on directed colimits of presentable objects?

Fri, 03/09/2018 - 07:45

Let $F: \mathcal{K} \to \mathcal{C}$ be a functor between $\lambda$-accessible categories, you can assume $\mathcal{C}$ to be Set if needed.

Is it true that $F$ is $\lambda$-accessible if and only if it preserves $\lambda$-directed colimits of $\lambda$-presentable objects?

Semigroups admitting field actions

Fri, 03/09/2018 - 06:12

Let $(S,*)$ be a semigroup, and $\mathbb{K}$ a field. Consider an action $$ \mathbb{K} \times S \to S, ~~~~~~~~~~~ (k,s) \mapsto k.s, $$ satisfying, for all $s,t \in S$, and $k,l \in \mathbb{K}$,

1) $~~~ k.(l.s) = (kl).s$

2) $~~~ k.(s*t) = (k.s)*t = s*(k.t)$,

3) $~~~~ 1_{\mathbb{K}}.s = s$.

Does such an object have a name, or is is easily seen to be equivalent to a standard structure? If such things are studied, what can one say about them?

Such a commutative semigroup admits an equivalence relation $$ s \simeq t, \text{ if there exists a } \lambda \in \mathbb{C}, \text{ such that } s = \lambda t. $$ Does the quotient have $S/\simeq$ have a name. For example, might one call it the projectivization of $S$?

If $S$ is a monoid does anything extra interesting happen?

Constructive proof of existence of non-separable normed space

Fri, 03/09/2018 - 05:48

I am looking for a constructive proof of one of the following two statements. If they are not constructively provable, I would be very thankful for an explanation as to why that is so (i.e., at which point in a proof must non-constructive means be employed?).

  1. There exists a normed space X such that for all Y $\subset$ X, if Y is denumerable, then Y is not dense in X.

  2. There exists a normed space X such that for all Y $\subset$ X, if Y is dense in X, then Y is not denumerable.

I'd consider a proof constructive if it includes no applications of the:

  • Law of Excluded Middle: $\phi$ $\lor$ $\neg$$\phi$
  • Law of Double Negation Elimination: $\neg$$\neg$$\phi$ $\rightarrow$ $\phi$
  • Axiom of Choice or any of its equivalents (Zorn's Lemma, etc.).

Suggestions would be much appreciated.

partially commutative monoid

Thu, 03/08/2018 - 10:55

Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ whenever $\{a,b\} \notin E(G)$ and $c \in I$ is arbitrary.

I have the following questions about the monoid $M(G)$.

  1. Is this monoid $M(G)$ well studied in the literature?

  2. What are some algebraic combinatorics or general combinatorial significance of this monoid?

Thanks for your time and have a good day.

Spectrum of orthogonality graph (2)

Thu, 03/08/2018 - 03:05

The orthogonality graph, $\Omega(n)$, has vertex set the set of $\pm 1$ vectors of length $n$, with orthogonal vectors being adjacent.

I am only interested when $4|n$, since otherwise $\Omega(n)$ is empty or bipartite. I am keen to know the spectrum of $\Omega(n)$ - the eigenvalues and their multiplicities. In particular I am seeking the inertia of $\Omega(n)$ - that is the numbers of positive, zero and negative eigenvalues. Many thanks Clive

Number of zeroes of derivatives of the polynomial

Wed, 03/07/2018 - 04:15

What are the maximal number of zeroes the polynomial of degree d and all of its derivatives can have at $k$ points? Say polynomial is over $\mathbb(C)$. I am interested in this question for a constant $k$, say 3. If $k\rightarrow \infty$ clearly polynomial can have $d$ zeroes and and its $i$'th derivative can have $d-i$ zeroes. The problem is that ussually polynomial and its derivatives have different zeroes.

Does there exist an example of the polynomial with more than k+d zeroes for him and its derivatives?

Relationship between decay constant (T) and Rt60

Wed, 03/07/2018 - 04:09

In acoustics, concerning the decay of energy in a sound field or of a resonance in, for example, a musical instument or a loudspeaker, two time periods are related; the 'relaxation time' for the signal to decay to 36% (e-1) of intitial value, and Rt60, deemed the point at which the energy has, for all practical purposed, diminished to zero.

One confusion is that acoustic energy is difficult to measure, so sound pressure is used as this is easy to measure, with time and frequency. Hence, the theoretical approach is based on energy dissipating to one millionth of original, where as working in sound pressure (analogous to volts), it is a diminution to one thousandth of original pressure level.

I find in published literature two seemingly inconsistent terms:

T (secs) (diminution to 36%) x 6.9 = Rt60 (secs) and T (secs) (diminution to 36%) x 13.8 = Rt60 (secs)

Only one can be correct.

Note that 6.9 is half of 13.8

I suspect that this is somehow related to the power/amplitude 10log or 20log element, but despite days on this, I'm just not bright enough to work it out!

Your thoughts would be most appreciated.


Source 1:

Source 2:

Many thanks!

On minimum of a function?

Wed, 03/07/2018 - 03:49

Given $A,B\in\Bbb R^{n\times n}$ is there a technique find $\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_2$?

Do maximally almost periodic groups embed homeomorphically into their Bohr compactifications?

Wed, 03/07/2018 - 03:37

If $G$ is a Hausdorff topological group and $bG$ is its Bohr compactification, Wikipedia says that $G$ is called maximally almost periodic (MAP) if and only if the natural map $i : G \to bG$ is injective. Equivalently, MAP groups are those groups the points of which are separated by the finite-dimensional unitary representations. My question is:

if $G$ is MAP, is $i$ a homeomorphism onto $i(G)$, or is it only injective?

(I would like to know if topological properties and regular measures on $i(G)$ can be pulled back on $G$.)

Integer factorization

Wed, 03/07/2018 - 03:26

Is there a way to find out the first few digits of the factors of the RSA numbers.(RSA-1024 or RSA-2048)

I do not want to get all the digits but only first 4-5 digits.

Existence of equation about the product of the divisor sum function

Wed, 03/07/2018 - 02:55

Let $\sigma_k(n)$ be the sum of the $k$-th powers of the positive divisors of $n$ and $\mu(n)$ be the Möbius function.

As Arithmetic function - Wikipedia mentioned, there is an equation that $$\sigma_k(u) \sigma_k(v) = \sum_{d~| \gcd(u, v)}{d^k \sigma_k\left(\frac{u v}{d^2}\right)}$$ which can also be represented as $$\sigma_k(u v) = \sum_{d~| \gcd(u, v)}{\mu(d) d^k \sigma_k\left(\frac{u}{d}\right) \sigma_k\left(\frac{v}{d}\right)}.$$

I wonder to know if there exists an equation between $\sigma_k(u v w)$ and $\sigma_k(u) \sigma_k(v) \sigma_k(w)$?

The Chow monoid is an algebraic set

Wed, 03/07/2018 - 01:55

Let $X\subset \mathbb{P}^N(\mathbb{C})$ be an irreducible algebraic variety. The ($p$th) Chow monoid of $X$ is given by $$C_p(X)=\bigsqcup_{d\geq 0}C_{p,d}(X),$$ where $C_{p,d}(X)$ are the $p$-cycles having degree $d\geq 0$. It's very well known that every $C_{p,d}(X)$ is a complex algebraic set, but however, I'm not able to find a reference in which this fact is proved. Can you help me?

On the total number of hamiltonian cycle in I-graph [migrated]

Wed, 03/07/2018 - 00:53

prove that the total number of hamiltonian cycle in I-graph where n is odd and j=k is $(2n)^2$

Definition of I-graph

Which continuous function is optimal for sieving?

Tue, 03/06/2018 - 23:25

In 1968, Barban and Vehov considered [1] the problem of determining for which continuous functions $\rho:\mathbb{R}^+\to [0,1]$ satisfying certain properties ($\rho(t)=1$ for $t\leq U_0$, $\rho(t)=0$ for $t>U_1$) the sum $$S(x) = \sum_{n\leq x} \left(\mathop{\sum_{d|n}}_{d\leq U_0} \rho(d) \mu(d) \right)^2$$ was minimal. (Assume from now on that $x>U_1>U_0$. They showed that the choice $\rho(t) = \rho_0(t) = \log(t/U_0)/\log(U_1/U_0)$ for $U_0<t\leq U_1$ was grosso modo optimal, in the sense of giving a sum $S(x)$ whose main term is no more than a constant times the minimal value that such a sum $S(x)$ could take. Later, Graham showed that, for $\rho(t) = \rho_0(t)$ as above, $S(x)$ actually asymptotes to $x/\log(U_1/U_0)$.

(a) What is the easiest way to show that this is the least possible leading term in an asymptotic for $S(x)$ for any function $\rho$ satisfying the above conditions? (That the order of magnitude is correct is clear from estimates on rough numbers, but such estimates have Buchstab's function in front.)

(b) What work, if any, has been done to date on the second-order term in the asymptotics? Graham gave a bound of the form $x/\log(U_1/U_0) + O(x/\log^2(U_1/U_0))$. I recently finished showing that, for $\rho$ as above and $U_0\ggg \sqrt{X}$, $S(x)$ asymptotes to $x/\log(U_1/U_0) - c x/\log^2(U_1/U_0)$, where $c$ is an explicit constant that is positive under some broad conditions ($U_1/U_0\geq 8$). I have no idea as to whether $c$ is the best value one could obtain for any $\rho$ satisfying the conditions.

In the range $U_1\lll X$, one could of course compare the corresponding constant $c'$ to the constant coming from Selberg's sieve. At the same time, one could change the conditions slightly (requiring $\rho$ to be the rescaling of a continuous function $\eta$ independent of $U_0$, $U_1$ and $X$, say) so as to exclude Selberg's sieve from consideration -- and then we would only have an upper bound on $c$. Is anything better known?

[1] M. B. BARBAN AND P. P. VEHOV, On an extremal problem, Trudy Moscov. Obsch. 18 (1968), 83-90.