What is the difference between a collection of Turing Machines and a family of Circuits?

Given a Collection of Turing Machines $ T_1, T_2, T_3,…T_n$ where $ T_1$ denotes that the Turing machine can only take in an input of size 1. Is there any difference in computational power to a family of Circuits $ C_1, C_2, C_3,…C_n$ ?

What if we assumed that each Turing machine, encoded special information for that specific input size to make each instance efficient?

If there is no difference in computational power, then maybe we could use this to define non-uniform algorithms instead of circuits?

efficiently calculate nearest common ancestor in a family tree (each person has two parents)

I’m well aware of ways to efficiently calculate the lowest common ancestor in a tree of nodes which converge to a single root (ie, each node has only one parent). Just iterate back to root for each person, then walk back from root tossing off anything common.

In a matriarchal society, for example, this could be used to quickly calculate how any two people are related as long as only the mothers are considered.

But if both parents are considered, eg mother AND father, then the algorithm just described breaks down.

So I wondered is there an algorithm to tell two people how they are related in a family tree where both parents are considered? For example, see the Icelandic genealogy app (https://www.islendingabok.is/) which does precisely that. How’s it done, algorithmically speaking?

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The underlying curve of a family of genus zero $n$ punctured curves

Let $ X$ be a curve of genus zero over an algebraically closed field $ k$ so that $ X \cong \mathbb{P}_k^1$ . Let $ (C, s_1, \cdots, s_n)$ a $ n$ punctured genus zero curve over $ k$ where $ s_i: k \to C$ are sections. Sine $ C$ is a genus zero over $ k$ , we have that $ C \cong \mathbb{P}_k^1$ .

Now let $ B$ be an affine scheme over $ k$ and suppose consider the family of $ n$ -punctured genus zero curves $ (C \to B, s_1, \cdots, s_n)$ where $ s_i:B \to C$ . Let us always assume that $ n \ge 4$ .

At this point, I tend to come across statements inferring that since the fibers of the family $ (C \to B, s_i)$ have non non-trivial automorphism $ C \cong \mathbb{P}_B^1$ . Here I by no means am claiming that the family $ (C \to B, s_i)$ of $ n$ -punctured curves is trivial but rather that the (total space) of the family of genus zero curves over $ B$ “underlying” the family $ (C \to B, s_i)$ is trivial.

Why does the automorphism group of the fibers of a family $ (C \to B, s_i)$ being trivial imply that the underlying family of genus zero curves over $ B$ is trivial? Again I mean this in the sense that the total space $ C \cong \mathbb{P}_B^1$ .

Can 2 members of a family use the same APC kiosk when they are entering the US under different visa programs?

Regarding APC kiosks, what happens when one member of the family is entering the US with an ESTA form and the other one with a B1/B2 Visa? Can they use the APC kiosk together? At certain moment, the system asks whether you are travelling under visa waiver program or B1/B2 visa or if you are a US or Canadian passport holder or if you have a green card. In my case, I will select “travelling with ESTA” but I’m afraid that when choosing “yes” to the question “are you travelling with other member of your family?” the system will only require to repeat passport scanning and fingerprints for the second person and will not ask the form of entry which is different from mine. Hope the question is clear.

Gare du Nord to Gare de Lyon transfer time for a family

When I book with the Eurostar site to a destination in the south of France, they only offer a short Paris transfer time (54 minutes) between Gare du Nord Eurostar and Gare de Lyon TGV. Is this easily achievable with 2 adults and 2 walking toddlers (no buggy)? How would you travel it to not be in a rush, and are there any tips to avoid surprises and delays?

Non-liner optimization of a family of functions

I’d like some advice about the following problem (resolution methods, problem category, etc.). The context is about the coregistration of several images all-in-once ($ n$ images, $ f_{i,j}$ the function that maps coordinates $ (x_i, y_i)$ to $ (x_j, y_j)$ .


Let $ f_{i,j}^k(x, y)$ a polynomial function of order $ k$ from $ \mathbb{R}^2$ to $ \mathbb{R}^2$ , with $ i, j \in [1, n]$ .

For example, if $ k=0$ , such functions can be expressed as $ f_{i,j}^0(x, y) = (a_{ij}^0, a_{ij}^1)$ . Or if $ k = 1$ : $ f_{i,j}^1(x, y) = (a_{ij}^0 + b_{ij}^0 x + c_{ij}^0 y, a_{ij}^1 + b_{ij}^1 y + c_{ij}^1 y)$ .

Let’s assume $ k$ is fixed, it will be omitted in the following.

This family of functions should respect as much as possible following the property: $ f_{i,j} = f_{i, k}(f_{k, j})$ .

I want to use this property to decrease the number of unknowns and obtain robust estimates of coefficients. Moreover, I have a large set of matches that can be expressed as such: $ v = f_{i,j}(u)$ .

My problem is the following: how can I compute robustly the functions $ f_{1,i}$ ?


If $ k = 0$ , the problem is quite simple: $ f_{i,j} = f_{i, k}(f_{k, j}) = f_{i, k} + f_{k, j}$ so i can write an overdetermined linear system and solve it with a Moore-Penrose inverse or an algorithm such as RANSAC.

If k = 1 or 2, I don’t really known how to proceed. I think I could try to design a custom convergence scheme with a predetermined equation resolution order and some iterations to converge.

As an example, if I solve for $ f_{1,2}$ , then to get $ f_{1,3}$ I can use my matches of the form $ v = f_{1,3}(u)$ but also the matches such as $ v = f_{2,3}(u) => f_{1, 2}(v) = f_{1,3}(u)$

Thanks, Thomas

CSS to change font family in modern communication site

I have a requirement to change the font family in SharePoint modern communication site. I have found a following extension: https://github.com/SharePoint/sp-dev-fx-extensions/tree/master/samples/react-application-injectcss which can be used to inject css.

Question is, what css class do i need to use to change the font throughout the site? Can someone please help, thanks in advance.

Visa Family Permit

Good Morning. I am writing to you to ask you about entry to UK to visit my wife sister. I am from Yemen and I am married to Polish citizen. At the moment I live in Germany and I have Family Member resident card in the paper Under Law No. 10 form. Can I in to UK on this card with out visa or permit member? If not when I will apply for permit member can I send some of my documents in Germany language.