Math Overflow Recent Questions


Why c.p.c order zero maps induce morphism between cuntz semigroups

Tue, 03/06/2018 - 22:48

Maybe a naive question:

Right now I am reading the paper ``Completely positive maps of order zero'' written by Wilhelm Winter and Joachim Zacharias. I do not quite understand the last corollary when they proved the induced map between Cuntz semigroups of a c.p.c order zero map between algebras preserves addition. Why we need the map is order zero?

To be simple, suppose that $A, B$ are $C^*$-algebras. Let $a,b\in A_+$ and $\varphi: A\rightarrow B$ be a c.p.c order zero map. They claim that if $a,b\in A$ are orthogonal, then so are $\varphi(a), \varphi(b)\in B$, whence $$\varphi(a\oplus b)=\varphi(a)\oplus \varphi(b)$$.

The first $\varphi$ is interpreted to be the amplification of $\varphi$, i.e. $\varphi\otimes id_{M_2}$ I believe. However, I guess that the equation above hold for every c.p.c map. Because $a\oplus b= a\otimes e_{11}+b\otimes e_{22}$ and $\varphi\otimes id_{M_2}(a\otimes e_{11}+b\otimes e_{22})= \varphi(a)\otimes e_{11}+ \varphi(b)\otimes e_{22}=\varphi(a)\oplus \varphi(b)$.

I believe I must miss something. But I cannot find where I was wrong.

Thank you for all helps!

Do commutative rings with "interesting" Jacobson radicals turn up "in nature"?

Tue, 03/06/2018 - 21:09

Let $R$ be a commutative ring. Let's say that the Jacobson radical $J(R)$ of $R$ is uninteresting if

  1. $J(R)$ coincides with the nilradical, or

  2. $J(R)$ is the intersection of a finite number of maximal ideals.

It seems as if most rings used in algebraic geometry have uninteresting Jacbson radical:

  • Every finitely-generated commutative algebra over a field or over a Dedekind domain is Jacobson, so its Jacobson radical coincides with its nilradical, and so is uninteresting by (1).

  • Every local or semilocal commutative ring has finitely many maximal ideals, and so has uninteresting Jacobson radical by (2).

For a nonuninteresting example, take the localization $R = \mathbb Z[x]_S$ where $S = \{f(x) \in \mathbb Z[x] \mid f(0) = 1\}$. The maximal ideals of $R$ are of the form $(p,x)$ where $p \in \operatorname{Spec} \mathbb Z$. So the Jacobson radical is $(x)$, which is not uninteresting.

But this example seems rather artificial to me; for example I don't know anywhere a ring like this would show up in algebraic geometry.

Showing an alternating integer series is never $0$

Tue, 03/06/2018 - 20:50

The following series arose in some work related to Hilbert Functions of ideals of points

$$\sum_{k = 0}^m (-1)^k {2m+2 \choose k}[2m(m+1)-k(2m+1)]^{2m-1}.$$

Experimentally, this series is always negative for $m > 0$ and decreases incredibly quickly. We only need that this is never $0$. We have tried rational roots, taking first differences, but have been unable to make much progress. Any help or suggestions are appreciated.

What is the difference between exactness and optimality of an algorithm?

Tue, 03/06/2018 - 19:40

I'm studying some papers related to graph partitioning (GP). It is well-known that the GP problem is NP-Complete. Based on my understanding, it means that there is no polynomial time solution to solve this problem, or there is no optimal solution for that. The following paper mentioned this fact in its introduction: "An exact algorithm for graph partitioning" However, they provided an exact solution for GP using branch-and-bound algorithm. Isn't it a paradox?, I mean I assume that if a problem is NP-Complete, there is also no exact solution for that, right?

Almost-disjoint sequence of sets at singular cardinals and stationary reflection

Tue, 03/06/2018 - 18:04

Let $\mu$ be a singular cardinal of countable cofinality. Let $ADS_\mu$ be the assertion that there exists $\langle A_\alpha\subset \mu: \alpha<\mu^+\rangle$ such that for all $\beta<\mu^+$, there exists $F: \beta \to \mu$ such that $\langle A_\alpha\backslash F(\alpha): \alpha<\beta\rangle$ is pairwise disjoint. It is a well-known fact that $ADS_\mu$ implies the failure of $Refl^*([\mu^+]^{\omega})$, which says for any stationary subset $S$ of $[\mu^+]^\omega$, there exists $W\in [\mu^+]^{\aleph_1}$ with $\omega_1\subset W$ and $cf(\sup W)=\omega_1$ such that $S\cap [W]^{\omega}$ is stationary.

$Refl([\mu]^{\omega})$ is almost the same assertion as $Refl^*([\mu^+]^\omega)$ except we do not require $cf(\sup W)=\omega_1$ (so it's weaker). It is also known that these two principles in general are not equivalent.

My question is: does $ADS_\mu$ imply the failure of $Refl([\mu^+]^\omega)$?

Where do models of false theories exist?

Tue, 03/06/2018 - 17:53

I have some difficulties in understanding the [uni]verse platonic view. How are we to understand the existence of a model of a false theory? what is the relationship of this model to the platonic world of sets?

My personal try is the following, let's assume the existence of a platonic universe $P^{sets}$ of sets, and also another universe $P^\in$ that serves as a relational universe, i.e., that can stand as a representative of the relation epsilon $\in$ in the real world. For example $P^\in$ might be a universe of some object representation of "ordered pairs" of sets, these ordered pairs are primitive ones, they stand for really existing objects exemplifying ordered pairs in a platonic realm. So $P^\in$ would be a really existing platonic membership relation between objects of $P^{sets}$.

Now when we say for example the rules of $ZFC$ are true, in a platonic sense, would that be taken to mean that the sets in $P^{sets}$ would be related to each other by the platonic membership relation in a manner that abides $ZFC$ rules.

So now when we say that $ZFC$ can interpret $ZF\neg C$ then this mean that the later is consistent (should $ZFC$ be consistent) so there would exist a model of $ZF\neg C$, but how I'm to understand the existence of this model in the platonic world? is it the case that the interpretation would be a pair of subsets of $P^{sets}$ one representing a domain and the other is a set of set-ordered pairs (non primitive ones, i.e. interpreted ordered pairs as special kinds of sets), and so the rerlation object interpreting the membership relation is not a real relational one, i.e. not composed of primitive membership ordered pairs??? as it is the case with ZFC? so this model is deemed as not real?

Is that a good analogy? I mean I want to conceal two matters: that there EXISTs a model of a false but consistent theory, if we are to hold platonistic views, then this existence must be in some reality? otherwise the assertion "there exists" wold have no meaning, now this reality must be taking part in the true platonic world of sets? where else it would be? On the other hand we must differentiate this kind of existence of a model of a false theory from the existence of a model of a true theory! that's why I used the "primitive ordered pairs" to represent the relations in the theory, a true theory would have its relation sets being real in the platonic sense, i.e. composed of primitive ordered pairs abiding the rules of that theory, while a false one would not have them real, i.e. the relation on its domain is not composed of primitive ordered pairs.

I always had the following impression:

for every effectively generated theory $T$ we have: $$ Con(T) \to \exists T^* (True(T^*) \wedge T^* inteprets T)$$

the idea can be seen in the above anaology, if a set theory $T$ is consistent but false, still the model of $T$ must be a part of $P^{sets}$ and so there must be a true theory $T^*$ (i.e. $T^*$ satisfied in $P^{sets}$) that can itnerpret $T$, otherwise how can we explain that a model of $T$ must exist? exist where?

Is that impression correct?

I visualize the whole platonic world of mathematics $P^{math}$ as the union of the platonic worlds of the individual disciplines i.e. the union of $P^{arith}$ , $P^{Geom}$ , etc..., I think standard models of those disciplines are the real ones represented by the Platonic realms of those disciplines, i.e. they have real relation sets, as opposed to the fake or false theories which don't have real grounded relations in the platonic world.

I know that this question is in some sense philosophical, but "platonism" especially the [uni]-versed one, is an almost ongoing working assumption of most innovative work in mathematics and set theory, so it is important to at least shed some light on that aspect.

Homomorphic Images of C^*-Algebras

Tue, 03/06/2018 - 17:42

If A and B are C^*-algebras that are algebraically isomorphic to each other, does this imply that they are *-isomorphic to each other?

Naturality of Moore-Postnikov systems

Tue, 03/06/2018 - 14:16

Where in the literature can I find a naturality statement for Moore-Postnikov towers of maps? Something like the following:

Let $f:X\to A$ and $g:Y\to B$ be maps of connected CW-complexes which both admit a Moore-Postnikov tower of principal fibrations. Then a commuting diagram $\require{AMScd}$ \begin{CD} X @>f>> A\\ @V \Phi V V @VV \phi V\\ Y @>>g> B \end{CD} (possibly with some extra conditions) induces maps $\Phi_n:X_n\to Y_n$ between the $n$-th stages of the towers of $f$ and $g$, for all $n\ge1$.

Time Complexity of the Word Problem for Finite Permutation Groups

Tue, 03/06/2018 - 11:15

Given a finite permutation group, i.e. a subgroup of the symmetric group on $n$ symbols in terms of generators, what is the complexity of the word problem? That is, computing if two words in the generators represent the same group element?

Generalizing the Fourier isomorphism between Sobolev spaces and weighted $L^2$ spaces to (locally) compact groups?

Tue, 03/06/2018 - 09:59

Motivating examples:

  1. Let $V$ be a real vector space with Haar measure $dv$. The fourier transform induces the following topological isomorphism: $$H^s(V,dv) \cong L^2(V^*,(1+|v^*|^2)^sdv^*)$$ The LHS being the $W^{s,2}$-type Sobolev space and the RHS a weighted $L^2$ space on the dual.

  2. Consider $S^1$ a circle with haar measure $d\theta$. Fourier transform Induces a topological isomorphism: $$H^s(S^1,d\theta) \cong l^2(\mathbb{Z},(1+|n|^2)^s dn)$$ Here the LHS is again the sobolev space and on the RHS $dn$ is the counting measure on $\mathbb{Z}$ (This example can be generalized to any finite dimensional torus by suitably modifying both sides).

Notice we get the usual Fourier transform isomorphism by putting $s=0$.

Question: Is there a generalization of the above to non-abelian Fourier transforms on (locally?) compact lie groups? (assuming of course the existence of a Plancherel measure).


Suppose $G$ is a compact (connected) lie group (for simplicity), let $dg$ denote the Haar measure and $d\mu$ the Plancherel measure on $\hat{G}$. Does there exist a function $q: \hat{G} \to \mathbb{R}$ (a weight, hopefully $q$ will be just the eigenvalue of the Caisimir) s.t. the non-abelian Fourier transform induces a topological isomorphism

$$H^s(G,dg) \cong \{ T \in\hat \oplus_{\pi \in \hat{G}} End(V_{\pi}): [ \pi \mapsto ||T_{\pi}||_{HS}] \in L^2(\hat{G},q^sd\mu) \}$$

What are the possible eigenvalues of these matrices?

Mon, 03/05/2018 - 07:50

First, here is a baby version of the question, that I already know the answer to. Consider complex Hermitian $4\times 4$ matrices of the form $$\left[\begin{matrix}a I_2&A\cr A^*&b I_2\end{matrix}\right]$$ where $A \in M_2(\mathbb{C})$ and $a,b \in \mathbb{R}$ are arbitrary. Can any four real numbers $\lambda_1 \leq \lambda_2 \leq \lambda_3\leq \lambda_4$ be the eigenvalues of such a matrix, or is there some restriction? Answer: there is a restriction, we must have $\lambda_1 + \lambda_4 = \lambda_2 + \lambda_3$.

The real question is: what are the possible eigenvalues of Hermitian $8\times 8$ matrices of the form $$\left[\begin{array}{c|c}aI_4&A\cr \hline A^*&\begin{matrix}bI_2& B\cr B^*&cI_2\end{matrix}\end{array}\right]$$ with $a,b,c\in\mathbb{R}$, $A \in M_4(\mathbb{C})$, and $B \in M_2(\mathbb{C})$? Can any eight real numbers be the eigenvalues of such a matrix? (I suspect not. If they could, that would tell you that any Hermitian $8\times 8$ matrix is unitarily equivalent to one of this form.)

Deformation of "Hecke modification"

Sun, 03/04/2018 - 14:56

Let $X$ be a smooth curve over $\mathbb{C}$. I wish to compute the deformation of the following data $(E,F,x)$. $E$ and $F$ are locally free sheaves over $X$ and $x$ is a point on $X$. They satisfy:

1) $E\longrightarrow F$ is a injective map of sheaves over $X$.

2) $F/E$ has length 1 and is supported at $x$.

How to compute the first order deformation of $(E,F,x)$ and describe it as a subspace of $H^{1}(End(E))\oplus H^{1}(End(F)) \oplus T_{x}X?$

Homogeneity of the space of semicontinuous functions

Sun, 03/04/2018 - 03:13

I am interested in the topological homogeneity of function spaces.

Question. Let $X$ be a Tychonoff space, let $USC(X)$ be a space of upper semicontinuous functions on $X$ and let $USC(X)^+$ be a space of non-negative upper semicontinuous functions on $X$.

  1. Is the space $USC(X)$ topologically homogeneous ?
  2. Is the space $USC(X)^+$ topologically homogeneous ?
  3. What about $USC(X,[0,1])$ ?

The canonical form of the first Painlevé equation

Sun, 03/04/2018 - 02:40

The first Painlevé equation is traditionally written as $$y''=6y^2+x. $$ Using scaling in both the dependent and independent variables, one can transform this equation into $$Y''=aY^2+bx $$ for arbitrary complex constants $a,b$. My question is: is there any special reason for choosing the coefficients $a=6,b=1$ in the traditional form? If so, what is it?

My guess regarding the $a=6$ part is that this is the only choice which makes the principal part of the Laurent series expansion around any pole $x_0$ have the form $$\frac{1}{(x-x_0)^2},$$ shared by the solutions of the Weierstrasß differential equation $$\wp''=6\wp^2-\frac{1}{2}g_2. $$ However, I can't see any reason for preferring $b=1$ over any other value.

How to kill a $\Sigma_{n+1}$-correct cardinal softly(n>1)?

Sun, 03/04/2018 - 02:11

A cardinal $\kappa$ is $\Sigma_n$-correct iff $V_\kappa \prec_n V$. For n>1, how to force a $\Sigma_{n+1}$-correct cardinal to be $\Sigma_{n}$-correct but not $\Sigma_{n+1}$-correct?

For $n=1$, we can force GCH below $\kappa$ and then violate GCH at $\kappa$.

If we assume some large cardinals, there are more partial answers, but the general situation is not clear to me.

relation between weakly convergence and pointwise convergence?

Sun, 03/04/2018 - 02:02

Suppose $f_k$ converges weakly to $g$ in $L^p$ space for $1<p<\infty$, does there exist a subsequence of $f_k$ converges to $g$ almost everywhere?

Assassins in zero-dimensional local rings

Sun, 03/04/2018 - 01:20

During a study of the behaviour of assassins and torsion functors (cf. this paper), I met the following problem about assassins in $0$-dimensional local rings.

Let $R$ be a commutative ring, and let $\mathfrak{a}\subseteq R$ be a nil ideal (i.e., an ideal consisting only of nilpotent elements). We consider the following statements:

(1) $\Gamma_{\mathfrak{a}}(M/\Gamma_{\mathfrak{a}}(M))=0$ for every $R$-module $M$;

(2) ${\rm Ass}_R(M/\Gamma_{\mathfrak{a}}(M))=\emptyset$ for every $R$-module $M$.

(Clearly, in both statements it suffices to consider only monogeneous $R$-modules $M$.)

Here, $\Gamma_{\mathfrak{a}}(M)=\bigcup_{n\in\mathbb{N}}(0:_M\mathfrak{a}^n)$ denotes the $\mathfrak{a}$-torsion submodule of $M$, and ${\rm Ass}_R(N)$ denotes the assassin of an $R$-module $N$, i.e., the set of prime ideals of $R$ of the form $(0:_Rx)$ for some $x\in N$.

One can show that (1) implies (2). A bunch of examples I checked indicates that also the converse might hold, at least if $\mathfrak{a}$ is the maximal ideal in a $0$-dimensional local ring. So:

Does (2) imply (1) if $\mathfrak{a}$ is the maximal ideal in a $0$-dimensional local ring?

Is Det-Stoch a factorization of the Giry Monad?

Sun, 03/04/2018 - 01:04

Stoch is the category of Measurable spaces and stochastic maps. It is the Klesli category of the Giry monad. Deterministic theories form a subcategory of Stoch. Specifically, the objects are just objects in Stoch and the morphisms are all isomorphisms. Call this category Det-Stoch. Is this category a factorization of the Giry Monad? Does it restrict the Giry monad to additive Gaussian noise? The reason I ask this, is that the probability distributions we should expect to see when taking measurements of systems that obey deterministic theories should just be additive noise. This is in contrast to quantum theory, where the theory itself has a delicate interplay between the physics and information and randomness.

On distribution of a number theoretic quantity associated with a subspace

Sun, 03/04/2018 - 00:14

Take $A,B,C,D$ pairwise coprime with $$n<A,B,C,D<2n$$ $$ n/8<|A−B|,|C−D|,|A−C|,|A−D|,|B−C|,|B−D|$$ and consider the space spanned by $3\times 4$ matrix $$N=\begin{bmatrix} -D&C&0&0\\ -B&0&A&0\\ 0&0&-D&C \end{bmatrix}.$$

Denote the $\Bbb Q$-linear space spanned by rows of $N$ by $T_{A,B,C,D}\subseteq\Bbb Q^4$ and denote the set of non-zero $\Bbb Z$ vectors in $T_{A,B,C,D}$ by $T_{A,B,C,D}[\Bbb Z]^\star$.

  1. Is there a name for the quantity $\min_{v\in T_{A,B,C,D}[\Bbb Z]^\star}\|v\|_\infty$ where $\|v\|_\infty$ is largest coordinate by magnitude of vector $v$?

  2. How is $\min_{v\in T_{A,B,C,D}[\Bbb Z]^\star}\|v\|_\infty$ distributed as a function of $A,B,C,D$ chosen with the constraints above (at least consider $A,B$ and $C,D$ each a coprime pair and $n<A,B,C,D<2n$) and what is its average value?

Simulations and heuristics suggest a value between $\Omega(n^{1/2})$ and $\Omega(n^{2/3})$ with $\Omega(n^{2/3})$ being the most likely possibility of lower bound for expected value.

Measuring the non-commutativity of the codifferential and pullbacks

Sun, 03/04/2018 - 00:08

$\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\operatorname{T\N}}$ $\newcommand{\TstarM}{T^*\M}$

This is a cross-post.

Let $\M,\N$ be $d$-dimensional oriented Riemannian manifolds, and let $f:\M \to \N$ be smooth; Let $\delta=d^*$ be the adjoint of the exterior derivative.

Let $\omega \in \Omega^1(\N)$. Is it true that $$ \delta(f^*\omega)-f^* \delta \omega=\omega(\delta df),$$ where $\delta df=\tr(\nabla df)$ is the tension field (or laplacian) of $f$.

If not, what is the right expression for $\delta(f^*\omega)-f^* \delta \omega$?

Here is a generalization for $k$-forms:

$df \in \Omega^1(\M,f^*{\TN})$, and for $1 < k \le d$, let $\bigwedge^k df\in \Omega^k\big(\M,\Lambda_k(f^*{\TN})\big)$ be the induced map on exterior powers.

Denote by $\nabla^{\Lambda_k(f^*{\TN})}$ the induced connection on $\Lambda_k(f^*{\TN})$ (by the Levi-Civita connection on $\N$), and by $\delta_{\nabla^{\Lambda_k(f^*{\TN})}} $ the adjoint of $d_{\nabla^{\Lambda_k(f^*{\TN})}}$.

Conjecture: Let $\omega \in \Omega^k(\N)$. Then

$$ \delta(f^*\omega)=w_{f(p)} \big(\delta_{\nabla^{\Lambda_k(f^*{\TN})}} (\bigwedge^k df)\big)+f^* \delta \omega? $$

I verified this conjecture for conformal maps, but I am interested in the general case.

This question can be thought of as "measuring the non-commutativity of the codifferential and pullbacks".

Alternatively, it can be viewed as a "Leibnitz rule" for the codifferential: To differentiate $f^*\omega$, we first need to differentiate $f$, and then differentiate $\omega$, both derivatives add additively their parts.

If we can solve the case of one degree forms, we might be able to proceed using induction.