Prove disprove an existence of mapping reduction between 2 sets

I am currently studying mapping reduction in computational theory and finding it hard to grasp the concept fully.

For reference, consider the following given WHILE-Prog sets:

A = { (p.d) | p doesn’t halt on input d } = Complementary HALT set. B = { p | p halts on exactly one input (the input is unknown) }

Is A < B. meaning, is there a mapping reduction from A to B?

Can someone suggest a hint?

knowing that A belongs to coRE did not help much. B doesn’t seem to belong to either RE nor coRE.


Open mapping theorem for complete non-metrizable spaces?

The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin’s Functional Analysis) occasionally state and prove more general versions, for instance for $ F$ -spaces (i.e. metrizable by a complete translation-invariant metric, but not necessarily locally convex). Again, these qualify as classical results. It seems that the Banach property can be relaxed to a much weaker completeness condition, as long as the space is metrizable.

Even in the absence of metric, we still have a notion of completeness in general topological vector spaces (every Cauchy filter/net converges). However, the proof of the open mapping theorem for $ F$ -spaces relies on the Baire category theorem, so this really uses the complete metric in a non-trivial way. This leads me to the following question:

Question 1. Does the open mapping theorem generalize to arbitrary (not necessarily metrizable) complete topological vector spaces?

As the open mapping theorem comes in various forms (see e.g. Rudin Theorem 2.11), I may have to be a bit more specific. I am particularly interested in the following question:

Question 2. Are there complete topological vector spaces $ X$ and $ Y$ and a continuous linear surjection $ T : X \to Y$ such that $ T$ is not open?

Compactness when mapping into a higher $L^p$ space and then back

Let $ q>p$ and let $ T:L^p[0,1] \to L^q[0,1]$ be a bounded linear operator. Let $ i: L^q[0,1] \to L^p[0,1]$ be the inclusion map (which is bounded). Is the composition $ i\circ T$ necessarily a compact operator on $ L^p$ ? We can assume $ p\in (1,\infty)$ if the uniform convexity of the underlying space helps in some way, though the boundary cases might be interesting themselves.

Context: This was inspired by this question, where the accepted answer covers the case $ q=\infty$ (in this case the composition is indeed always compact).

I’ve been at it for a couple days trying to prove and also to disprove it. It might be trivial with a silly oversight on my part, but on the other hand it could be inherently nontrivial as well, and I was hoping to learn some new fun fact about the geometry of Banach spaces in case of the latter.

Ideas: To prove it here are a couple ideas that I played around with. One suggestion is to somehow translate the problem to a statement about $ \ell^p$ spaces and then apply Pitt’s theorem. Unfortunately there seems to be no clear way to do this, for example any attempt at using a Fourier transform (which boundedly sends $ L^p \to \ell^{\frac{p}{p-1}}$ and $ \ell^p \to L^{\frac{p}{p-1}}$ for $ 1< p < 2$ ) will somehow “go the wrong way.” A second idea is to try to find a Banach space $ X$ which has the Dunford-Pettis property (e.g. a $ L^1(\mu)$ or $ C(K)$ space) and which nests in between $ L^p$ and $ L^q$ , i.e., $ L^q \subset X \subset L^p$ . This would easily show compactness of the inclusion, but finding such $ X$ seems not too easy. A third idea is to try to mimic the proof of Pitt’s theorem adapted to this $ L^p$ context. Basically suppose $ i\circ T$ was not compact. Then we can find $ f_n \in L^p$ with $ \|f_n\|_p=1$ and $ f_n \to 0$ weakly, and also $ \|Tf_n\|_p \ge \delta>0$ . Then (after perhaps passing to a subsequence) one may try to show that $ T$ is bounded below on the closed linear span of the $ f_n$ , which would imply that the image of $ T$ contains a closed infinite-dimensional subspace of $ L^p$ . And I’m farily certain that such a subspace cannot be contained in $ L^q$ . But formalizing these ideas might take some work.

On the other hand, here are some ideas for potential counterexamples if it turns out to be false. For one idea let us identify $ L^p[0,1] \simeq L^p(\Bbb R)$ and consider the Fourier transform $ \mathcal F:L^p(\Bbb R) \to L^{\frac{p}{p-1}}(\Bbb R)$ , where $ p<2$ . I’ve shown that $ \mathcal F$ is not a compact operator. Hence there is at least a noncompact operator from $ L^p[0,1] \to L^{\frac{p}{p-1}}[0,1]$ for $ p<2$ (which brings up another question of what if we restrict attention to $ q>p^*$ , then can we say that $ T$ itself is compact?… but perhaps that’s for another day). However, the difficult thing is to show that it remains noncompact when we compose it with the inclusion map, and I think that it actually becomes false. I played around with the Hermite functions $ \psi_n$ which orthonormally diagonalize $ \mathcal F$ on $ L^2(\Bbb R)$ , and I was able to compute that $ \|\psi_n\|_{L^p(\Bbb R)} \sim_n C_p n^{\frac1{3p}-\frac16}$ , so while these converge weakly to $ 0$ in $ L^2(\Bbb R)$ they fail to do so in $ L^p(\Bbb R)$ for $ p<2$ , hence they are useless for us. A second idea is to use some of the abstract mappings mentioned in this MO thread and the subsequent comments. Actually this might all be trivial from some proposition in Albiac-Kalton but I can’t access the book at the moment.

Is it possible to untangle two manifolds by mapping them to a higher space?

given two manifolds M_1 and M_2 embedded in some Euclidean space R^n, possibility linked in this sense :

Relation between Alexander duality and linking numbers

Can we find a PL-map f:R^n->R^m such that f(M_1) is not linked with f(M_2)? It seems to be intuitively but I cannot quit see it rigorously. Also, what is the condition on m ? how much larger than n it should be?

Mapping Cone and derived tensor product

This question is in some sense a continuation to this question: Derived Nakayama for complete modules

For the setting: Let $ A$ be a ring and let $ I$ be some finitely generated ideal in $ A$ . Let $ f\colon \mathcal C\rightarrow \mathcal D$ be a map of chain complexes of derived $ I$ -complete $ A$ -modules. I am trying to apply the “derived Nakayama” to the mapping cone of $ f$ to produce the following result:

Suppose that $ f’\colon \mathcal C\otimes^{\mathbf L} A/I\rightarrow \mathcal D\otimes^{\mathbf L} A/I$ is a quasi-isomorphism. Then $ f$ is a quasi-isomorphism.

To do this, I want to relate the mapping cone $ cone(f’)$ to the mapping cone of $ cone(f)\otimes^{\mathbf L}A/I$ but I am at a loss on how to proceed here. Any tipps or solutions are welcome.

chained conditional mapping – `ifDefined` method

Is there a more concise way of conditionally mapping over a value like in:

val userName: Option[String] = Some("Bob") val address: Option[String] = Some("Planet Earth")  val dbQuery = new Query()  val afterUserName = => dbQuery.copy(_.userName = u))     .getOrElse(dbQuery)  val modifiedQuery = => afterUserName.copy(_.address = a))     .getOrElse(afterUserName) 

I wish there was an ifDefined method available on all types like in the following block. This removes the .getOrElse(...) call.

dbQuery   .ifDefined(userName)((d, u) => d.copy(d.userName = u)   .ifDefined(address)((d, a) => d.copy(d.address = a) 

Stability for mapping class groups, spaces of sections, and polynomial coefficient systems

Let $ X$ be a simply connected space. Cohen and Madsen proved that the functor sending a surface with boundary M to $ H_i(Map(M,X))$ has polynomial degree $ \leq i$ . Here Map means the space of maps which agree with the base point on the boundary. Polynomial degree is defined in that paper and is originally due Ivanov I believe. Does this generalize to spaces of sections? That is, if $ E_M \to M$ is some bundle (say functorially built out of the tangent bundle of M) with preferred choice of section, is the functor sending $ M$ to $ H_i(\Gamma(E_M))$ polynomial of degree $ \leq i$ ? Here $ \Gamma$ means space of sections which agree with the basepoint section on the boundary. I am aware that Randal-Williams proved that $ \Gamma(E_M)//Diff(M)$ has homological stability. For simplicity, all I care about is surfaces with one boundary component.

A linear mapping from a finite dimensional space always attains its supremum over the basis of that space

Let $ f$ be a linear mapping from the finite dimensional normed space $ X$ to another normed space $ Y$ .

The proof for “A linear map from a finite-dimensional space is always continuous”, as given here, uses the fact that such a functional is continuous if and only if it is bounded.

Defining $ (e_1,\cdots, e_n)$ as a basis for $ X$ , somewhere in the proof it is taken for granted that $ M:= \sup_i\{||f(e_i)||\}$ does exist because $ X$ is finite dimensional.

Now maybe I am missing something here, but as far as I remember, theorems about a mapping attaining its sup and inf on f.d. spaces assume its continuity, which is what this theorem is trying to establish.

Any hints where to look for? How is it guaranteed that $ M$ as defined above is bounded?

8 – Drupal8 (JSON:API) to Drupal8 Migrating Taxonomy References not mapping

OK, for a number of reasons I’m doing a Drupal 8 – Drupal 8 migration using the JSON:API module on the source site. I’ve run into an issue attaching taxonomy terms to nodes. Here’s as far as I’ve gotten :

1) I created a vocabulary called “Photo Categories”

2) I’ve created and run import_drupal_taxonomy_photo_categories to import the terms. This works perfectly. I have manually checked to ensure they all get in. I use the UUID as the migration’s unique identifier.

3) I have a content type called Photos, it has a field called Photo Category which references the Photo Categories vocabulary.

4) From the source JSON in the relationships section I see this:

"field_photo_category": {    "data": [      {        "type": "taxonomy_term--photo_categories",        "id": "84c1e8bd-16f7-429f-b299-50fe43297d47"      },      {        "type": "taxonomy_term--photo_categories",        "id": "00931a31-d129-4c55-a962-f7ba11fed5a4"      }   ],   "links": {     "self": "https:\/\/\/jsonapi\/node\/photo\/1396b571-0ece-4a07-ab17-7327e030dfb5\/relationships\/field_photo_category",     "related": "https:\/\/\/jsonapi\/node\/photo\/1396b571-0ece-4a07-ab17-7327e030dfb5\/field_photo_category"   } } 

5) In the source > Fields section of import_drupal_type_photo I have:

-    name: field_photo_category   label: 'Field Photo Category'   selector: /relationships/field_photo_category/data 

6) In the process section I have:

  field_photo_category:     plugin: iterator     source: field_photo_category     process:        plugin: migration_lookup        migration: import_drupal_taxonomy_photo_categories        source: id 

As I understand it this should loop through the and do a migration lookup on each id. But nothing is getting mapped.

I have verified that the ids match the ids in the data section match the UUID from the taxonomy term JSON. I have checked the database and verfied that ids appear in migrate_map_import_drupal_taxonomy_photo_categories.soure1 and they map to the appropriate tids.

I have installed the migrate devel module and when I run the migration with –migrate-debug this is what the destination section look like:

  'field_photo_category' => array (2) [         array (1) [             'source' => string (36) "84c1e8bd-16f7-429f-b299-50fe43297d47"         ]         array (1) [             'source' => string (36) "00931a31-d129-4c55-a962-f7ba11fed5a4"         ]     ] 

So, my question, what am I doing wrong? I have the rest of this part of my migration laid out and this seems to be the final obstacle.

Thanks in Advance