Fa.functional analysis – Reference request on the Min-Max theorem

Consider the following min-max problem

$$ inf_ {x in M} sup_ {y in N} F (x, y), $$

or $ F: M times N to mathbb R $ is Lipschitz and $ y mapsto F (x, y) $ is concave for all $ x in $ M. Could we drift $ inf_ {x in M} sup_ {y in N} F (x, y) = sup_ {y in N} inf_ {x in M} F (x, y) $ if $ M subset mathbb R ^ m $ and $ N subset mathbb R ^ n $ are the two compact?

PS: To the best of my knowledge, the reference to the Min-Max theorem comes from Mr. Sion: https://msp.org/pjm/1958/8-1/pjm-v8-n1-p14-p.pdf , the convexity of $ x mapsto F (x, y) $ is missing in my case. All comments or references are very appreciated!

The entity reference view fails after the drupal 8.7.6 update

Recently, I updated my Drupal site from 8.6.14 to 8.7.6.

A view of the reference of the previously functional entity connected to a Form AutoComplete field no longer works. The circle of rotation continues to rotate and does not match the user's input to an entity.
(The ER View View concatenates a user's email address and company name to facilitate user selection: "joe@smith.com – Smith Plumbing, Inc.")

The error in the log is:

Error: Class 'DrupalviewsPluginEntityReferenceSelectionXss' not found in DrupalviewsPluginEntityReferenceSelectionViewsSelection->stripAdminAndAnchorTagsFromResults() (line 272 of /Applications/MAMP/htdocs/chapter6/core/modules/views/src/Plugin/EntityReferenceSelection/ViewsSelection.php)

I've added "Use Drupal Components Utility XSS" to ViewSelection.php and now the error is

Recoverable fatal error: Object of class DrupalviewsViewExecutable could not be converted to string in DrupalCoreEntityEntityAutocompleteMatcher->getMatches() (line 69 of /Applications/MAMP/htdocs/chapter6/core/lib/Drupal/Core/Entity/EntityAutocompleteMatcher.php) 

Has anyone else ever experienced it in the last drupal 8?

reference query – Which ring spectra are the simplest homotopy boundaries?

I will surely mark this request by reference: I am sure that many things are known about this issue, I am too ignorant to even guess where to look. What makes me particularly stupid is the suspicion of actually seeing the answers and not being able to remember where.

I only know two examples of what I want to know, but only one conscious: these are complex cases and I would like to see simpler cases.

First of all, I heard that tmf (or TMF?) Is the inverse limit of the homotopy of all elliptical ring spectra. (As an aside: do we really need all d & # 39; them? Is there a small elliptical spectrum diagram that is sufficient to obtain it?)

Second, I've heard that orthogonal K theory is the spectrum of homotopic fixed points for an involution on K complex theory. And this (perhaps) so-called Galois theory of ring spectra represents many of them as fixed points of homotopy under the actions of finite groups on better understood groups. And this really algebraic K theory of any ring is in itself a homotopic inverse limit.

Hope that simpler instances that I would like to read somewhere:

Are there any interesting explicit diagrams of Eilenberg-MacLane ring spectra whose inverse homotopy limits produce something interesting, like the same complex K theory? Here, I am aware of the construction of the Snaith Producing Unit from a K (Z, 2) -s chain, but this seems to be a direct limit rather than the opposite limit, and I think it does not say nothing on the ring structure. Or does it?

In general, can one go up the chromatic levels by forming limits of homotopy? Is what Lubin-Tate's theory is called in this context can be formulated in these terms? Can complex cobordism be obtained as the homotopic limit of some "smaller" ring spectra? And what about the spectrum of the sphere?

Gt.geometric Topology – Reference Request for Knotted Trivalent Graphs

After coming across "The Algebra of Knotted Trivalent Graphs and the Phantom World of Turaev" by Dylan Thurston, I wonder if there are more readings in the generalization of plane-node diagrams.

The main application concerns materials including many detailed explanations of worked examples of KTG algebra operations.

References likely to answer the following questions would be particularly interesting.

Is there a Seifert algorithm for trivalent graphs knotted in $ mathbb {R} ^ $ 3 who hopes to produce a minimal genre seifert at least as often as planar diagrams can?

Let's say you start with a planar diagram for a node k $ and apply Seifert's algorithm and end up with a minimal surface of the genus Seifert, $ S $. Is there a trivalent graph integrated on $ S $, of the form to be described, which captures k $ as KTG? The shape of the trivalent graph on $ S $ being where at each vertex, exactly two of the incident edges come from arcs on the boundary of $ S $, so arcs in the node, and the third incident edge is a bow properly embedded in the surface? Or a more general form where we can consider this steerable surface as having two sides a $ + $ and one $ – $ and two properly integrated bows on opposite sides are allowed to cross each other without counting for 4-valent.

reference request – A list of evidence of "Coherent topoi have enough points"

For my research, I would like to read all the known texts evidence very classic result "Coherent topoi have enough points", by Deligne.

Ref 1: D3.3.13 in the sketches of an elephant

provides a very logical proof of the statement, I would like to see more geometric or more theoretical category evidence.

reference request – ODE almost linear

Let $ A, B $ be $ n times n $ matrices. I am interested in the following ODE in $ mathbb {R} ^ n $

$$ frac {dx_t} {dt} = Ax_t + Bx ^ + _ t $$

or $ x ^ + = (x ^ + _ {1, t}, …, x ^ + _ {n, t}) $ and $ ( cdot) ^ + $ is the rectifier: $ r ^ + = max {0, r }. $

Does this type of EDE have a name? And are there any known stability criteria? Has it been studied by anyone in general?

The closest I've found is the "linear threshold networks" studied here for example. I appreciate any reference similar to this system.

oracle sqldeveloper – how can enable, disable constraints if the table has partitions by reference?

j & # 39; uses oracle12C as a database and using get_ddl to get ddl of database objects. Now, I have two tables Table 1 and Table 2 and Table1 has partition, Table2 uses the main constraint of Table1 as a reference constraint and also uses PARTITIONED BY REFERENCE.For example: –

TABLE1 is: –

create table parent_emp(
empno      number  primary key,
job        varchar2(20),
sal        number(7,2),
deptno     number(2)
partition by list(job)
( partition p_job_dba values ('DBA'),
  partition p_job_mgr values ('MGR'),
 partition p_job_vp  values ('VP')


TABLE2 is: –

    "ENAME" VARCHAR2(10), 

My problem is that I want to disable the activation constraints of table1 and table2, but when I run the script, I get a tracking error.

  1. alter table parent_emp disable the constraint SYS_C0010720 cascade;

it is used to disable the primary key of Table1 but generating the following error: –

02297. 00000 - "cannot disable constraint (%s.%s) - dependencies exist"
*Cause:    an alter table disable constraint failed becuase the table has
           foriegn keys that are dpendent on this constraint.
*Action:   Either disable the foreign key constraints or use disable cascade

I understand this, so I tried to disable the constraint of table2 and execute the following query.

  1. alter table reference_emp disable FK_EMPNO constraint cascading;

but that gives me the following error: –

alter table reference_emp disable constraint FK_EMPNO cascade
Error report:
SQL Error: ORA-14650: operation not supported for reference-partitioned tables

Please suggest me how can I disable, activate the constraints.

reference request – What happens when the head is in the leftmost cell and that an infinite time asks the Turing machine to move the head to the left?

I'm looking for articles with an explicit answer to the following question:

What happens when the head is placed in the leftmost cell and that an infinite time asks the Turing machine to move the head to the left?

The article "Theory of Infinite Time Calculable Models" contains the following succinct note:

The calculation stops only when the stop the state is explicitly obtained and, in this case, the output is the one that is written on the output tape. (If the head falls off the tape, no output is given.)

Does this mean that we can introduce a special program? Fault that occurs if the head falls off the tape, stops the calculation in exactly the same way as stop done but we have to ignore all the programs that reach the Fault State?

The article "The computing forces of Turing machines in infinite time on α-tape" contains the following information:

When the device is prompted to move the head to an undefined location, its default position is $ C_0 $. This occurs when a left move instruction occurs while the head is on a boundary ordinal cell because there is no "next cell left" determined in such cases.

I'm not sure that the notion of limit ordinal cell here includes the leftmost ($ 0-th) cell, but in all cases, the model of $ alpha $The Turing bands described in this document differ from the original model. This document is therefore not entirely relevant.

The first part of my question is: does anyone know of any other documents dealing with the problem? (I remember very well reading a detailed view of the problem in a document a few years ago, but for some strange reason, I can not find this document now.)

The second part of my question is: can we simply replace the instruction "move your head to the left" with the instruction "do not move your head" in such situations? That is, if the head is in the leftmost cell and the current instruction is "move the head to the left and go to the state $ s_i $", The machine does not move the head at all, but goes into state $ s_i $. Does such a solution in any way affect the computing power of Turing machines in infinite time?

reference request – From Laplacian case to general divergence form

What are some tips that can be used to generalize results for elliptic / parabolic equations in a related domain $ Omega $ with lapacian $ Delta u $ in the form of general divergence $ text {div} (A (x) nabla u) $.

Case 1: $ A $ is a constant matrix. In this case, we can use the change of variable $ v (x) = u (Ax) $ ($ A $ symmetrical and uniformly elliptical) to obtain the case of Laplacian. But this change sends $ Omega $ at $ A Omega $ and we can miss the regularity properties of $ Omega $.

Case 2: $ A $ is not constant. In this case, the previous variable change does not work here.

For Case 1, how can we reserve the regularity of $ Omega $ under $ A $?

In case 2, is there any trick to directly obtain the general result? Sometimes we have to impose restrictions on the ellipticity constant of $ A $ (eg higher or lower than a given number). Is there a trick to remove these restrictions in this case?

Thank you for any suggestions or references.

reference query – How to find literature (research)?

I've noticed that if we want to get into a new branch of mathematics, it's really hard to find good literature to begin with. Of course, if we look for the name of the subject in Google, we will find a book of 1000 pages containing all the theorems and proofs established. But for starters, an informal survey article is probably more useful. How to find such articles? In each particular case, should each person go to see his teacher or is there a uniform place where one can look for such literature?

I've also noticed that one of the main problems encountered if one wishes to start a research is that all the people involved in the current literature (who write all the new research papers) know all the usual tools used to solve problems of a specific type, and if one wants to start participating, one has the feeling that it lacks pre-knowledge. But this knowledge may only be communicated verbally by professors to doctoral students and, as a foreigner, it is difficult to find appropriate literature. But if there is one, where can I get it?