Extending certain morphisms of sheaves of rings

Suppose I have three sheaves of rings $ A,B,C$ on a topological space $ X$ (or a site), with a map of sheaves of rings $ \gamma: A\otimes_{\mathbf{Z}} B\to C$ .

Suppose I have a continuous self-map of topological spaces $ f : X\to X$ (or a morphism of sites such that $ f^{-1}$ is an exact functor), that comes together with maps of sheaves of rings $ \alpha: f^{-1}A\to A$ and $ \beta: f^{-1}B\to B$ .

Call $ D$ the sub-sheaf of rings of $ C$ on $ X$ generated by the image of the map of sheaves of rings $ \gamma: A\otimes_{\mathbf{Z}} B\to C$ .

Does there exist a map of sheaves of rings $ \delta : f^{-1}D\to D$ that is compatible with $ \alpha$ and $ \beta$ ?

By compatible, I mean that $ \gamma\circ (\alpha\otimes\beta)$ should agree with $ \delta\circ (f^{-1}\gamma)$ , i.e. the obvious diagram of maps of sheaves of rings commutes.

Morally, I’m asking if, given a sheaf of rings $ C$ and two sheaves of subrings $ A,B$ of $ C$ such that $ \alpha$ and $ \beta$ exist, $ \alpha$ and $ \beta$ can be extended to the sheaf of subrings of $ C$ that $ A$ and $ B$ generate.

Attempt: For $ U$ open in $ X$ , we call $ D’(U)$ the sheaf of subrings in $ (f^{-1}C)(U)$ generated by the image of $ (f^{-1}\gamma)(U):(f^{-1}A)(U)\otimes_{f^{-1}\mathbf{Z}(U)}(f^{-1}B)(U)\to (f^{-1}C)(U)$ . A section in $ D’(U)$ in the image of $ (f^{-1}\gamma)(U)$ can be sent to a choice of preimage in $ (f^{-1}A)(U)\otimes_{f^{-1}\mathbf{Z}(U)}(f^{-1}B)(U)$ , then to its image in $ A(U)\otimes_{\mathbf{Z}(U)}B(U)$ via $ \alpha(U)\otimes\beta(U)$ , and then to $ D(U)$ . This assignment is well defined because $ f^{-1}$ commutes with equalizers, and then sends the kernel of $ (f^{-1}\gamma)(U)$ to the kernel of $ \gamma(U)$ . It is also additive and multiplicative, and hence uniquely extends to a ring homomorphism $ \delta(U) : D’(U)\to D(U)$ .

One should be able to check by hand that $ \delta(U)$ is natural in $ U$ , and gives a morphism of sheaves of rings $ D’\to D$ .

The question becomes whether $ f^{-1}D = D’$ as sheaves of rings. This sounds plausible but I’m not sure how to see it.

Spring Boot – Extending ‘Base’ or ‘Common’ JARs

Here is the situation at work:

We have a ‘framework/patchwork’ team that is responsible for setting up . However it is hardly consistent. Recently, we received implementation for work that did not include prior work that overlapped and everything broke until they fixed it. Not the first and I doubt the last.

My idea is that they should maintain one application that includes all this new application functionality as a smoke test (versions, properties, options) all in one app/ micro-service as proof it works before passing it down to us. Once that is tested (they do not), we would simply extend that JAR/ use as a parent for our own common library across our micro-services.

Is there an issue with this idea?

Any suggestions for going about this? (‘exclude’ configs, condition annotations, etc)

Extending Galois theory: Finding all additional operations requires to possibly express roots of all polynomials of a certain degree?

While doing an iota worth of digging into deceivingly basic introductions to Galois theory I had this wild thought, since one of the areas it’s helpful in is to prove that certain polynomials can’t have their roots expressed using a certain set of operations, then how about if we focused our attention on whether we can find all other operations for which that would be possible? Would such set of those operations form a group and behave nicely? More importantly has this been studied before?

(I assume that there exists other operations than “take 5th root” for e.g., otherwise this is a very non-interesting question)

Extending Apple Airport Extreme Network

Hi I’m trying to add a second router(a Netgear R7000) to extend my current network which uses a 5th generation Airport Extreme as the base. I am trying to set it up such that there is one continuous network with a single password. Right now it exists as three separate networks (Airport, Netgear and Netgear5G). I tried to set it up as an ethernet connected roaming network. The problem I have is the configuration pages shown on the Apple site (https://support.apple.com/en-us/HT204616) showing connection sharing options do not seem to be anywhere to be found on my current Airport utilities software. Does anyone know where these configuration pages can be found or how I can configure the routers into a single network? Thanks!

extending a partition of a number to get new partition

Let $ \lambda = (k_1^{m_1}\,k_2^{m_2})$ where $ 0<k_1<k_2$ be a partition of $ n$ in the power notation.

Let $ \mu = p_0^{r_0}\,p_1^{r_1}\,\cdots\, p_t^{r_t} \,(k_1^{m_1}\,k_2^{m_2})\,q_0^{s_0}\,q_1^{s_1}\,\cdots\, q_u^{s_u}$ where $ 0<p_0<p_1<\cdots<p_t<k_1<k_2<q_0<q_1<\cdots<q_u$ be a partition of some positive integer $ m>n$ .

Pictorially, I am adding new rows on top and bottom of the Ferrer diagram of $ \lambda$ .

My question is what is the relation between $ \lambda$ and $ \mu$ .

More precisely are they comparable in any natural partial order defined on partitions?

Kindly share your thoughts.

Thank you.

Extending the modulus of continuity to measurable functions

The “modulus of continuity” operator for continuous functions is defined here: Are continuous functions almost completely determined by their modulus of continuity?

I want to extend this operator to the space of measurable functions. Whatever the choice of extension, it should be invariant among functions in the same equivalence class, and assign the same moc as the basic version for those classes with a continuous representative.

The obvious approach seems to be to allow the moc to ignore null sets in the computation, but there is a problem as shown below:

The null set neglecting left moc (similarly define the right moc),

$ $ L(f)(x, e) := \sup \{d \ge 0 \,:\, f((x, x+d)-A) \subseteq (f(x) – e, f(x) + e)\ for some null set A\}.$ $

for fixed $ e$ is well defined a.e. for two functions in the same equivalence class. I.e. if $ f = g$ a.e. then simultaneously $ L(f)= L(g)$ and $ R(f) = R(g)$ a.e. as single variable functions of $ x$ . But because the null sets that they do not agree on can differ for each $ e$ , the two variable modulus of continuity (if I’m not mistaken) isn’t necessarily measurable.

There are some regularity properties of the function – it is right continuous in $ x$ , and monotone increasing in $ e$ . Is this enough to guarantee joint measurability? If not, what should be done to correct this?

How to set a default date when extending DateRangeItem FieldType plugin

I created a new field type plugin that extends DateRangeItem. My plugin only adds some more settings. I also created an own widget that extends DateRangeWidgetBase. My formElement function looks like this:

public function formElement(FieldItemListInterface $  items, $  delta, array $  element, array &$  form, FormStateInterface $  form_state) {   $  element = parent::formElement($  items, $  delta, $  element, $  form, $  form_state);   $  time_type = 'time';   $  time_format = $  this->dateStorage->load('html_time')->getPattern();    $  element['value'] += [     '#date_date_format' => 'none',     '#date_date_element' => 'none',     '#date_date_callbacks' => [],     '#date_time_format' => $  time_format,     '#date_time_element' => $  time_type,     '#date_time_callbacks' => [],     '#default_value' => $  this->createDefaultValue($  items[$  delta]->value, $  element['value']['#date_timezone'])   ];    ... }  protected function createDefaultValue($  date, $  timezone) {   $  date = parent::createDefaultValue($  date, $  timezone);   $  date->setDate(2018, 01, 01);   return $  date; } 

I only want the user to be able to enter a time value, but no date value. This works.

My problem is that when saving the field value it is always stored with the date of the current day – this seems to be the default value. (The time is set correctly according to what the user enters). I want to set a different fixed date instead.

So I’m overwriting createDefaultValue as shown above, but without any effect.

If I comment out the two lines

//'#date_date_format' => 'none', //'#date_date_element' => 'none', 

my new default value is shown and also saved correctly. But if set the two lines to not show the date, my default value is ignored.

How can I set a default value and not show the date (only time)?

Extending allocated primary partition with preceding unallocated volume

I tried to make a dual-boot system on my Razer Blade. When I allocated a seperate partition for my Linux installation, I suddenly got a new unallocated partition in front of my Windows partition. Now I can’t merge the C partition with the unallocated space in front of it, nor can I extend the partition to have enough space to be usable with my desired Linux installation. (TLDR) How do I make the unallocated space usable again?


Extending one pair of solutions to other pairs of solutions in a polynomial function

I have:

$ $ f(x,y)=yx^4+a(y)x^3+b(y)x^2+c(y)x+d(y)$ $ with:

$ $ a(y)=-(4y^2+6y+1)$ $ $ $ b(y)=6y^3+18y^2+11y+6$ $ $ $ c(y)=-(4y^4+18y^3+22y^2+6y+11)$ $ $ $ d(y)=y^5+6y^4+11y^3+6y^2+6$ $

I’m looking for $ x,y\in\Bbb Z^+ \text{ that solve } f(x,y)=0$

From how I formulated this function, I know that $ f(11,6)=0$ , and I’m wondering if there are any extension methods that could be used to find other solutions if they exist.