Show convergence of a sequence of resolvent operators

Let

  • $ E$ be a locally compact separable metric space
  • $ (\mathcal D(A),A)$ be the generator of a strongly continuous contraction semigroup on $ C_0(E)$
  • $ E_n$ be a metric space for $ n\in\mathbb N$
  • $ (\mathcal D(A_n),A_n)$ be the generator of a strongly continuous contraction semigroup on$ ^1$ $ B(E_n)$
  • $ \pi_n:E_n\to E$ be continuous and $ $ \iota_nf:=f\circ\pi_n\;\;\;\text{for }f\in C_0(E)$ $ for $ n\in\mathbb N$

Let $ \lambda>0$ and $ f\in C_0(E)$ . Assume$ ^2$ $ $ \left|\left(R_\lambda(A_n)\iota_nf\right)(x_n)-\left(R_\lambda(A)f\right)(x)\right|\xrightarrow{n\to\infty}0\tag1$ $ for all $ x_n\in E_n$ , $ n\in\mathbb N$ , and $ x\in E$ with $ \pi_n(x_n)\xrightarrow{n\to\infty}x$ . Are we able to conclude $ $ \left\|R_\lambda(A_n)\iota_nf-\iota_nR_\lambda(A)f\right\|_\infty\xrightarrow{n\to\infty}0;\tag2$ $ at least under suitable further assumptions (e.g. compactness of $ E$ )?

Note that the result holds if $ E_n=E$ for all $ n\in\mathbb N$ , $ E$ is compact and $ \iota_n$ is the identity for all $ n\in\mathbb N$ : https://math.stackexchange.com/q/3139957/47771.

We may note that by contractivity, $ (0,\infty)$ is contained in the resolvent sets of $ (\mathcal D(A_n),A_n)$ , $ n\in\mathbb N$ , and $ (\mathcal D(A),A)$ . Moreover, $ $ \left\|R_\lambda(A_n)\right\|,\left\|R_\lambda(A)\right\|\le\frac1\lambda\;\;\;\text{for all }n\in\mathbb N.\tag3$ $ This might be crucial.


$ ^1$ If $ S$ is a set, let $ B(S)$ denote the space of bounded functions from $ S$ to $ \mathbb R$ equipped with the supremum norm.

$ ^2$ If $ (\mathcal D(B),B)$ is a bounded linear operator on a Banach space and $ \lambda$ is a regular value of $ (\mathcal D(B),B)$ , let $ R_\lambda(B)$ denote the resolvent operator of $ (\mathcal D(B),B)$ .

How do I learn all the information from which I can ask a deleted question of mine properly? [on hold]

I never went to teacher’s college and never took a masters in teaching. I never worked for any research groups. I know almost nothing about what information math researchers are using in their research job.

My deleted question at https://mathoverflow.net/questions/327157/do-we-really-know-that-the-axiom-of-regularity-is-true had an answer which gave me some partial idea of which information I was missing. I clicked the check mark on it because I felt like it taught me that I was missing some information which felt like a satisfactory answer, but I’m still missing that huge pile of information. What type of information is it? How can I learn all that information? It could be useful for more than just asking that one question.

Is there a community of mathematicians that claim to have a feel for the real meaning of statements used in formal proofs in certain proof systems? If so, do they decide on the meanings of each of the statements used in a formal proof so whether one formal system is a subtheory or a supertheory of another formal system depends on the intended meaning of each string of characters that represents a statement?

Are there multiple communities of mathematicians each of which its people have a fundamentally different way of thinking? For example, one community of mathmaticians has an intuition for Zermelo-Fraenkel set theory and claims to have an understanding of the meanings of statements they use in the formal systems they use and another community of mathematicians does a similar thing except that they have an intuition for New Foundations which is totally incompatible with Zermel-Fraenkel set theory but they don’t collaborate with each other for research because they don’t understand the meaning of each other’s research results.

MongoDB query choosing the wrong index in winning plan, Though in executionTimeMillisEstimate as lower for the other index?

MongoDB Query chooses the wrong index in the winning plan. I have two indexes for the same field, one is a single field index and a Compound index with another field.

Eg. Field name: Field1, Contains Yes or No Field name: Field2, Contains 0 or 1 or 2 or 3

Index 1: {'Field1':1} Single Field Index Index 2: {'Field1':1,'Field2':1} Compound Index.

On Search Query {‘Field1′:’Yes’} for Field1 it uses the compound index, instead of single key index. Attached below is the query execution plan.

{     "queryPlanner" : {         "plannerVersion" : 1,         "namespace" : "xxxx",         "indexFilterSet" : false,         "parsedQuery" : {             "Field1" : {                 "$  eq" : "Yes"             }         },         "winningPlan" : {             "stage" : "FETCH",             "inputStage" : {                 "stage" : "IXSCAN",                 "keyPattern" : {                     "Field1" : 1,                     "Field2" : 1                 },                 "indexName" : "Field1_Field2_1",                 "isMultiKey" : false,                 "multiKeyPaths" : {                     "Field1" : [],                     "Field2" : []                 },                 "isUnique" : false,                 "isSparse" : false,                 "isPartial" : false,                 "indexVersion" : 2,                 "direction" : "forward",                 "indexBounds" : {                     "Field1" : [                          "[\"Yes\", \"Yes\"]"                     ],                     "Field2" : [                          "[MinKey, MaxKey]"                     ]                 }             }         },         "rejectedPlans" : [              {                 "stage" : "FETCH",                 "inputStage" : {                     "stage" : "IXSCAN",                     "keyPattern" : {                         "Field1" : 1                     },                     "indexName" : "Field1_1",                     "isMultiKey" : false,                     "multiKeyPaths" : {                         "Field1" : []                     },                     "isUnique" : false,                     "isSparse" : false,                     "isPartial" : false,                     "indexVersion" : 2,                     "direction" : "forward",                     "indexBounds" : {                         "Field1" : [                              "[\"Yes\", \"Yes\"]"                         ]                     }                 }             }         ]     },     "executionStats" : {         "executionSuccess" : true,         "nReturned" : 762490,         "executionTimeMillis" : 379131,         "totalKeysExamined" : 762490,         "totalDocsExamined" : 762490,         "executionStages" : {             "stage" : "FETCH",             "nReturned" : 762490,             "executionTimeMillisEstimate" : 377572,             "works" : 762491,             "advanced" : 762490,             "needTime" : 0,             "needYield" : 0,             "saveState" : 16915,             "restoreState" : 16915,             "isEOF" : 1,             "invalidates" : 0,             "docsExamined" : 762490,             "alreadyHasObj" : 0,             "inputStage" : {                 "stage" : "IXSCAN",                 "nReturned" : 762490,                 "executionTimeMillisEstimate" : 1250,                 "works" : 762491,                 "advanced" : 762490,                 "needTime" : 0,                 "needYield" : 0,                 "saveState" : 16915,                 "restoreState" : 16915,                 "isEOF" : 1,                 "invalidates" : 0,                 "keyPattern" : {                     "Field1" : 1,                     "Field2" : 1                 },                 "indexName" : "Field1_Field2_1",                 "isMultiKey" : false,                 "multiKeyPaths" : {                     "Field1" : [],                     "Field2" : []                 },                 "isUnique" : false,                 "isSparse" : false,                 "isPartial" : false,                 "indexVersion" : 2,                 "direction" : "forward",                 "indexBounds" : {                     "Field1" : [                          "[\"Yes\", \"Yes\"]"                     ],                     "Field2" : [                          "[MinKey, MaxKey]"                     ]                 },                 "keysExamined" : 762490,                 "seeks" : 1,                 "dupsTested" : 0,                 "dupsDropped" : 0,                 "seenInvalidated" : 0             }         },         "allPlansExecution" : [              {                 "nReturned" : 101,                 "executionTimeMillisEstimate" : 0,                 "totalKeysExamined" : 101,                 "totalDocsExamined" : 101,                 "executionStages" : {                     "stage" : "FETCH",                     "nReturned" : 101,                     "executionTimeMillisEstimate" : 0,                     "works" : 101,                     "advanced" : 101,                     "needTime" : 0,                     "needYield" : 0,                     "saveState" : 10,                     "restoreState" : 10,                     "isEOF" : 0,                     "invalidates" : 0,                     "docsExamined" : 101,                     "alreadyHasObj" : 0,                     "inputStage" : {                         "stage" : "IXSCAN",                         "nReturned" : 101,                         "executionTimeMillisEstimate" : 0,                         "works" : 101,                         "advanced" : 101,                         "needTime" : 0,                         "needYield" : 0,                         "saveState" : 10,                         "restoreState" : 10,                         "isEOF" : 0,                         "invalidates" : 0,                         "keyPattern" : {                             "Field1" : 1                         },                         "indexName" : "Field1_1",                         "isMultiKey" : false,                         "multiKeyPaths" : {                             "Field1" : []                         },                         "isUnique" : false,                         "isSparse" : false,                         "isPartial" : false,                         "indexVersion" : 2,                         "direction" : "forward",                         "indexBounds" : {                             "Field1" : [                                  "[\"Yes\", \"Yes\"]"                             ]                         },                         "keysExamined" : 101,                         "seeks" : 1,                         "dupsTested" : 0,                         "dupsDropped" : 0,                         "seenInvalidated" : 0                     }                 }             },              {                 "nReturned" : 101,                 "executionTimeMillisEstimate" : 260,                 "totalKeysExamined" : 101,                 "totalDocsExamined" : 101,                 "executionStages" : {                     "stage" : "FETCH",                     "nReturned" : 101,                     "executionTimeMillisEstimate" : 260,                     "works" : 101,                     "advanced" : 101,                     "needTime" : 0,                     "needYield" : 0,                     "saveState" : 10,                     "restoreState" : 10,                     "isEOF" : 0,                     "invalidates" : 0,                     "docsExamined" : 101,                     "alreadyHasObj" : 0,                     "inputStage" : {                         "stage" : "IXSCAN",                         "nReturned" : 101,                         "executionTimeMillisEstimate" : 0,                         "works" : 101,                         "advanced" : 101,                         "needTime" : 0,                         "needYield" : 0,                         "saveState" : 10,                         "restoreState" : 10,                         "isEOF" : 0,                         "invalidates" : 0,                         "keyPattern" : {                             "Field1" : 1,                             "Field2" : 1                         },                         "indexName" : "Field1_Field2_1",                         "isMultiKey" : false,                         "multiKeyPaths" : {                             "Field1" : [],                             "Field2" : []                         },                         "isUnique" : false,                         "isSparse" : false,                         "isPartial" : false,                         "indexVersion" : 2,                         "direction" : "forward",                         "indexBounds" : {                             "Field1" : [                                  "[\"Yes\", \"Yes\"]"                             ],                             "Field2" : [                                  "[MinKey, MaxKey]"                             ]                         },                         "keysExamined" : 101,                         "seeks" : 1,                         "dupsTested" : 0,                         "dupsDropped" : 0,                         "seenInvalidated" : 0                     }                 }             }         ]     },     "serverInfo" : {         "host" : "xxxxx",         "port" : 27017,         "version" : "3.6.0",         "gitVersion" : "xxxxx"     },     "ok" : 1.0 } 

The executionTimeMillisEstimate for single filed index is 0 where us executionTimeMillisEstimate for the compound index is 260, then why still it uses the compound index in winning plan. I am using a single field query for single field index why it uses compound index?

When a function space with compact-open topology has countable chain condition?

As in title,when a function space with compact-open topology has countable chain condition? Are there some sufficient and necessary conditions? Who give some references about this topic?

McCoy and Ntantu [Topological Properties of Spaces of Continuous Functions, Page 68] pointed out that Vidossich had prove that $ C_k(X)$ has ccc if $ X$ is submetrizable. Who can give a proof of this statement?

Thank you in advance.

Is $SL_n(\mathbb{Q}_p)$ virtually torsion free?

Recall that a group is virtually torsion free if it admits a finite index subgroup which is torsion free.

Question. Is it known whether $ SL_n(\mathbb{Q}_p)$ is virtually torsion free for n > 1?

Comments.

  1. Note that $ SL_1(\mathbb{Q}_p) = \mathbb{Q}_p^*$ is virtually torsion free, but this doesn’t really give any evidence that it should hold for $ n>1$ .
  2. We know by a theorem of Selberg that for a field $ K$ of characteristic 0, any finitely generated subgroup of $ GL_n(K)$ is virtually torsion free. However, this does not apply to $ SL_n(\mathbb{Q}_p)$ as it is not finitely generated; the diagonal matrices give a copy of $ \mathbb{Q}_p^*$ , which is uncountably infinite.
  3. A related question can be found here where it is shown that $ SL_n(\mathbb{Z}_p)$ is virtually torsion free as it is a compact $ p$ -adic analytic group.

Thanks in advance for the help!

Volume of $n$-sphere contained in $\ell_1$ ball

For a given $ r>1$ , what is the surface area of $ \mathbb S^{n-1}$ (the sphere of radius 1 in $ \mathbb R^n$ ) which is contained outside of the $ \ell_1$ ball of radius $ r$ ? Or equivalently, if $ X\sim U(\mathbb S^{n-1})$ , a point sampled uniformly from the sphere, what is the probability that $ \Vert X\Vert_1\geq r$ ?

This is easy to compute for $ r\geq \sqrt{n-1}$ , as the area is exactly $ 2^n$ spherical caps, and this has a clean, closed-form formula. For smaller values of $ r$ , however, these caps intersect, and the algebra gets worse.

The exact value of this probability matters less than approximate asymptotic bounds for $ n$ large.

A mistake in “Chtoucas de Drinfeld et correspondance de Langlands”

I have been told by some people that there is a mathematical mistake in the last section of “Chtoucas de Drinfeld et correspondance de Langlands” (L. Lafforgue) concerning the hyperplane section arguments which does not invalidate the main results. I was not able to identify the mistake myself. Could somebody equipped to answer this question tell me what the mistake was and how it could be fixed?

Is there a prime $q$ smaller than a given prime $p>5$ such that the inverse of $q$ modulo $p$ is an integer square?

Let $ p$ be a prime. For each $ k=1,\ldots,p-1$ there is a unique $ \bar k\in\{1,\ldots,p-1\}$ with $ k\bar k\equiv1\pmod p$ , and we call $ \bar k$ the inverse of $ k$ modulo $ p$ . In 2014 I investigated the set $ $ \{\bar q:\ q\ \text{is a prime smaller than}\ p\}$ $ and found that it contains an integer square if $ 5<p<2\times 10^8$ . (See http://oeis.org/A242425 and http://oeis.org/A242441.) This led me to formulate the following conjecture.

Conjecture. For any prime $ p>5$ , there is a prime $ q<p$ such that the inverse $ \bar q$ of $ q$ modulo $ p$ is an integer square.

For example, the inverse of $ 13$ modulo $ 23$ is $ 4^2<23$ , the inverse of $ 5$ modulo $ 61$ is $ 7^2<61$ , and the inverse of $ 11$ modulo the prime $ 509$ is $ 18^2<509$ .

QUESTION. What tools in number theory are helpful to prove the above conjecture?

When the action of the gauge group on the space of connections is free?

Let $ G$ be a compact Lie group. Let $ \mathcal{A}$ be the space of connections on the principal trivial $ G$ -bundle $ G\times \mathbb{R}^4$ possibly with some growth condition (to specify it is a part of the question). The gauge group $ \mathcal{G}:=Maps(\mathbb{R}^4\to G)$ acts on $ \mathcal{A}$ in the usual ways.

Can the action of $ \mathcal{G}$ on $ \mathcal{A}$ be free? E.g. for $ G=SU(2)$ ? If not, is it true that the set of connections with non-trivial stabilizers (or infinitesimal stabilizers) is ‘very small’ in some sense?

Remark. If $ G=U(1)$ then the action of $ \mathcal{G}$ on $ \mathcal{A}$ is free provided we impose a growth condition on connections such that they should vanish at infinity at least along a given direction.