ct.category theory – What does a family of conservative fibre functors on $n$-topoi see?

Let $mathcal{C}$ be a site and $f:mathcal{F}to mathcal{G}$ a morphism of $n$-sheaves. If we assume that $mathcal{C}$ is nice, then there exists a conservative family of fibre functors ${phi_i}_{iin I}$ such that $f$ is an isomorphism if and only if for all $iin I$, the induced morphism $phi_i(f)$ is an isomorphism. However, I feel I’ve always been told that isomorphisms of categories are the “wrong” thing to look out for, and that I’d rather should consider equivalence of categories. Hence I wonder if a conservative family of fibre functor can tell us something about this, that is to say:

If for all $iin I$ the $phi_i(f)$ are natural equivalences, does it follow that for any $Uin text{Ob}(mathcal{C})$ the induced morphism $f(U):mathcal{F}(U)to mathcal{G}(U)$ is a natural equivalence?

Also, does a conservative family of fibre functors detect essential surjectivity, that is to say if for all $iin I$, the $phi_i(f)$ are essentially surjective, does it follow that for any $Uin text{Ob}(mathcal{C})$ the induced morphism $f(U):text{Im}(f)(U)to mathcal{G}(U)$ is essentially surjective, where $text{Im}(f)$ denotes the image $2$-sheaf?

optimization – Why are the payoffs of this family of maximization problems convex?

This is an exercise for the reader from my optimization course. A simple profit-model is
$$max_x pcdot q(x) – c^Txquadtext{ s.t. }quad xge 0$$
for some price $p$, costs $c$ and a continuous function $q:mathbb R^nrightarrow mathbb R$. Now for fixed $c>0$ and an arbitrary continuous function $q$, denote the solution of the problem by $x(p)$ (we may assume it exists). Show that the function
$$g(p)=pcdot q(x(p))-c^Tx(p)$$
of the optimum payoff for given price $p$ is convex on $(0,infty)$.

Since $q$ is not assumed to be differentiable, I don’t think calculus can be used.
On the other hand, if I work with $g(lambda p_1+(1-lambda)p_2)$, I don’t know what to do with$x(lambda p_1+(1-lambda)p_2)$.

Is there some relationship/property of $x(p)$ that I’m missing here that would allow me to do a direct convexity proof? Or is there an indirect way to show that $g$ is convex?

color – What’s the point of scanning family photos at a higher resolution for later enlargement other than restoration?

I have a question about an answer on Genealogy & Family History Stack Exchange. According to this blog post found at the Library of Congress:

You can scan [photos] at a higher resolution [I’m assuming higher than
600 ppi]. However, in most cases, all you will see are the defects [on
those same photos].

The blog further states that:

If the original you have to work with is a 4 x 6 inch print, and you
scan it at 600 or 1200 pixels per inch, [one] could then make the
equivalent of an 8 x 12 inch print, but it’s not likely to give you
better quality.

However, the answer to the G&FH question suggests scanning an extremely valuable family photograph at 2400 ppi to do the necessary color separations before using the same photo for the dust jacket of a hypothetical family book. Is this a useful suggestion, or does the Library of Congress advice about not scanning at higher resolutions still apply?

Is this something which makes sense only if you are going to do color separation in an image editing tool, or would there be other reasons to use a higher resolution?

Would the same advice apply to black-and-white and/or color slides? The LoC blog suggests scanning those at extremely high resolutions.

statistics – Canonical form and exponential family

Suppose you have a random variable X, who’s distribution depends on $theta$.
If X is a part of the exponential family of distributions, X can be written in a certain form, namely:
$$f_theta(x)=h(x)*c(theta)*e^{T(x)*zeta(theta)}$$

Is the above expression the ‘Canonical’ form of the distribution? If not, what is the relationship between the Canonical form of a distribution which is part of the exponential family (or any distribution if I am mistaken).

visas – EUSS Family Permit to U.K

I am an Indian and my spouse is an EU citizen with a presettled status so I applied for the EUSS family permit from Dubai. My biometrics appointment was on 17/11/2020. I received an email on 15/12/2020 from UKVI saying:

“With regards to your enquiry about the status of your application, I can confirm that we have made a decision on your application. We cannot discuss the outcome of your application on-line or by telephone. However the visa application centre will be in contact with you in regards to collection or delivery of your passport.”

When I contacted the VAC, they said they have no update yet and I still need to wait. Anyone with a similar case please advise what do I need to do. I’m thinking if my visa is approved, then what date will the vignette sticker be stamped on my passport.

ag.algebraic geometry – Conditions for a family of functions to have generic zeros

Let $emptysetneq Fsubset C((0,1)^n,mathbb{R})$ be a family of functions. Are there known conditoins for the zeros of $F$ to be generic; in the sense that:

  • For every $g in C((0,1)^n,mathbb{R})$ there exist some $f^g in F$, such that for every $x in (0,1)^n$ the following holds:
    $$
    g(x)=0 Leftrightarrow f^g(x)=0
    .
    $$

More generally, given such a family $F$, what can we say about the set of functions having common zeros with some element thereof; i.e.:
$$
mathcal{Z}(F)triangleq left{
gin C((0,1)^n,mathbb{R}):, (exists f^g in F),(forall x in (0,1)^n), g(x)=0 Leftrightarrow f^g(x)=0
right}?
$$

Are these types of problems studied?

General Comment: I wasn’t sure how to classify this question; excuse me if I don’t have the optimal tags.

visas – EEA family permit refused

I applied for an EEA family permit on the basis that my brother is Swedish national. I have got the refusal although its stated that if EEA national is planning to move to UK within six months of date of application he can apply for his extended family member. They also didn’t accept the birth certificate which I had provided. I am attaching a copy of decision. Should i appeal this decision?

You state that your brother is a Swedish national. You have provided
evidence that your sponsor holds a Swedish passport.

Only those family members referred to under Article 2 of the Directive
2004/38/EC have an automatic right to join or accompany an EEA family
member to another member state when that EEA national is exercising a
Treaty right.  Article 3 of Directive 2004/38/EC provides the basis
for a member state to consider other relatives, such as ‘extended
family members’ and determine the terms of entry and residence to
such ‘beneficiaries’ in accordance with their own domestic
legislation. (Article 3(2)).  The United Kingdom has transposed the
terms of Article 3 into Regulation 8 of the Immigration (European
Economic Area) Regulations 2016. As Regulation 8(4) makes clear, the
United Kingdom is allowed to set terms on when it will accept
extended family members and allow them to reside in the United Kingdom
as family members of an EEA national.  To apply for an EEA permit as
the extended family member of an EEA national in accordance with
regulation 8 of the Immigration (European Economic Area) regulations
2016, you must satisfy that you are financially dependent on your
sponsor.  Guidance states that financial dependence should be
interpreted as meaning that the family member needs the financial
support of the EEA national or his or her spouse/civil partner in
order to meet the family members essential needs in the country where
they are present – not in order to have a certain level of income.
The applicant must also provide evidence to show their EEA national
sponsor has enough money to support them and the applicant is reliant
on them for this.  I note that on your application you state that you
are financially supported by your sponsor. As evidence of this you
have provided money transfer remittance receipts. However, I am not
satisfied that this sufficiently demonstrates that you are dependent
on your sponsor to meet your essential needs. Unfortunately, this
limited amount of evidence in isolation does not prove that you are
financially dependent on your sponsor. I would expect to see
substantial evidence of this over a prolonged period.  Furthermore,
the fact of transferring money is not evidence that it is needed by
the recipient. You have not provided any evidence regarding your own
financial situation such as bank statements or other documents
indicating financial ingoing and outgoings. You have provided no
evidence to demonstrate yours and your family’s circumstances
including your income, expenditure and evidence of your financial
position which would prove that without the financial support of your
sponsor your essential living needs could not be met.  As evidence
of your relationship with your sponsor you have provided yours and
your sponsors birth certificate that was registered on 26th September
2020 confirming that your date of births are 1996 and 1991, these
registrations took place 24 and 29 years after your births. Due to the
length of time between birth event and registration this certificate
cannot be accepted as reliable evidence in the absence of other
relevant birth documentation issued at the time of the event or other
credible documentation evidencing your parentage.  Every application
received by this office is assessed on its own merits using the
evidence that has been supplied with the application and all other
evidence available to me. However, the submission of repeat
applications within a short space of time is unlikely, without
significant, detailed additional evidence, to satisfy the Entry
Clearance Officer and alter the decisions that have previously been
made. I therefore refuse your EEA Family Permit application because I
am not satisfied that you meet all of the requirements of regulation
12 (see ECGs EUN2.23) of the Immigration (European Economic Area)
Regulations 2016.

How to protect a vulnerable Family Member?

I apologize if this isn’t the best exchange to be asking in, but I’m a little new at this.

Problem: A close relation outside my local area with declining cognitive function is demonstrating increasingly poor infosec practices. I have a plan to try preserver their online freedom while keeping them reasonably safe, but have no idea if it’s any good.

Details: Relation has issue with poor impulse control, and frankly a long unaddressed history of terrible infosec. They have multiple email accounts, some dating back decades, and little to no hesitation about opening extremely questionable attachments. I am aware of at least one successful remote access phishing attack. They are prone to visiting sites that pose security threats.

Devices: The user works mostly on a windows machine,and has an iOS phone. The residence also has an OSx (x86) machine, other iOS devices, and a small number of networked devices. (printer/smart tv/etc.)

My Skill Level: Technically competent w/ limited coding skills, but decent hardware skills. Little to no experience with network management or Linux. I do have an embarrassing abundance of free time at the moment.

Current Plan:

  • Toss the users current system after an expert retrieves critical files from it
  • Hardware authenticator & password manager
  • Purchase a subscription-based anti-virus program
  • Implement DNS filtering via PiHole to block malicous IPs
  • Setup remote access behind a VPN. (Preferable with a simple hardware switch to start & stop the service that also shoots an sms to me)
  • Harden the Pi with something like Tripwire IDS & Rkhunter
  • Wipe/factory reset all networked devices to the best of my ability
  • Find/implement a method to segregate the users computer from other networked devices
  • Attempt* to migrate the user to a new set of email address, or at least retire the most dangerous ones
  • Attempt* to revoke the user’s administrative access
  • Attempt* to migrate the user’s from windows to OSx or even Chrome OS**

*I say attempt, not due to a lack of technical knowledge, but with regards to user buy-in.

**The user has previously resisted a proposed migration to a chromebook, on the basis they will lose access to excel.

I would greatly appreciate any and all feedback this community could offer. As a novice, I’m well aware I may be missing low hanging fruit or pursuing wildly impractical solutions.