## What are the minimum and maximum lengths of a Mainnet Bitcoin address?

The wiki is correct! The source you linked must have assumed that the address with the smallest encoding has version_byte=00, data=20*00, checksum=94A00911

``````\$ encodeBase58 00000000000000000000000000000000000000000094A00911
1111111111111111111114oLvT2
``````

Which has length 27. This address is valid and has been used on the blockchain! But it is not the shortest address.

I wrote this short bash script to find the minimum length address. It found that there were a total of 266 address of length 26. For anyone who is curious, these are the 266 addresses with length 26.

The maximum length address has version_byte=00, data=20*FF, checksum=FA06820B:

``````\$ encodeBase58 00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFA06820B
1QLbz7JHiBTspS962RLKV8GndWFwi5j6Qr
``````

Which has length 34. This has also been used to receive coins, as can be seen here.

## What are the focal lengths required for aerial photography in alt range 400-1200m?

first i am new in the field:

If i have a UAV fly in alt range between 400-1200m, and i want to capture photos like Google Satellite maps, and specifically i want the photos to have the same (or close to) scale of Google maps photos at the most zoomed level (zoom 23-24).

The UAV will have a full-frame mirror-less camera, so i want to know or help me to figure out the required focal lengths.

Thanks.

## focus – Are the internal focal focal lengths really uncompensated varifocal lenses?

If I shorten the focal length of a lens without taking further action, it will reduce the flange distance required for focusing to infinity – which means that if the actual flange distance is not changed, the focus will be closer than infinity.

Is this effect used to get an internal focus on the main lenses, and does this explain why the effective focal length decreases instead of increasing in such designs when it is focused more near?

In other words, do these lenses focus breathe BECAUSE you focus them, or do they focus breathe TO focus?

DO NOT ask questions about real cine lenses or ultra-large floating elements here.

(NB, although this may seem like a question of pure optical design not related to photography, it can become relevant when evaluating the best way to choose or modify lens adapters).

## geometric topology – Hyperbolic fillings of lengths between 6 and 2 \$ pi \$

What is the longest slope $$gamma$$ in addition to the Dehn surgery space of a cusped hyperbolic 3-collector $$M$$? Here the Dehn surgery space is the space of the fillings so that the hyperbolic structure on the filling $$M ( gamma)$$ can be achieved as a distortion of the original $$M$$.

This question is related to Ken Baker's question:

Exceptional hyperbolic obturations of 3 cusped hyperbolic collectors

However, Ken's question is concerned with the total number of tracks in this add-in. This question focuses on the longest slope of this type, where the length is measured by displacement at the limit of a horobille trim (as in the configuration for the $$6$$-the theorem or $$2 pi$$-Theorem).

Of course, it is possible that this question, as noted, has no workable answer, as there is no longest slope.

Here is a more carefully stated version:

What is the greatest $$L$$ so that there is a family of tracks $$gamma_i$$ in collectors $$M_i$$ such as $$lim_ {i to infty} length ( gamma_i) = L$$
and each $$M_i ( gamma_i)$$ is a hyperbolic variety such that the hyperbolic structure cannot be realized as a deformation of the hyperbolic structure of $$M_i$$?

Of course, the fact that the 6-theorem is clear implies that $$L geq 6$$. Also $$2 pi$$ Theorem says $$L leq 2 pi$$.

## Analytical geometry – a, b, c the sides, and l, m, n the lengths of the medians of a triangle. \$ k = frac {l + m + n} {a + b + c} \$. \$ k \$ assumes each value in \$ ( frac {3} {4}, 1) \$

Let a, b, c be the side lengths of a triangle and let l, m, n be the lengths of its medians.
So prove that $$k = frac {l + m + n} {a + b + c}$$can assume all the values ​​of the interval $$( frac {3} {4}, 1)$$ .

My attempt: I can prove it $$k in ( frac {3} {4}, 1)$$ by triangular inequality. But I can not prove that $$k$$ will assume each value in the interval $$( frac {3} {4}, 1)$$.

## How to transform two lists of unequal lengths into coordinates?

I have the following two lists;

``````a = {-1.8, -1.6, -1.4, -1.2, -1., -0.7, -0.6, -0.4, -0.2, 0., 0.2, 0.4,
0.6, 0.8, 1., 1.2, 1.4, 1.6, 1.8}
b = {.2, .4, .6, .8, 1., 1.2, 1.4, 1.6, 1.8}
``````

I want to turn this into coordinate points where each ai value is associated with each bi. Essentially (-1.8, .2), …., (- 1.8,1,8) would be the first 9 points and would create points for the 19 values ​​of a.

## graph theory – Random tree with constraint lengths

Let $$T$$ to be a tree labeled with nodes $$N = {1, dots, n }$$ and the edges $$E$$. Define the length from one edge $$e = {u, v }, u in N, v in N$$ to be $$l (e) = | u – v |$$. Leave him sequence of length $$L$$ of $$T$$ the sorted sequence of lengths of all the edges of $$T$$.

An example tree is presented in the link below. Each edge is marked by its length. The length sequence for this tree is $$(1, 1, 1, 1, 2, 2, 3)$$.

Tree example

My question is this: given a length sequence $$L$$, is there an efficient algorithm for generating a tagged tree? $$T$$ with sequence of length $$L$$, randomly to the uniform of all these trees? Or alternatively, list all trees with a sequence of given length $$L$$?

## Lens – Do mounting adapters change apparent focal lengths of lenses?

I have a Sony A6500 APS-C camera using a Sony 55-210mm lens.

I wanted more scope and so I rented a Sigma 18-300mm lens to use with a Sigma MC11 adapter.

By testing this configuration, I get an identical picture with the Sony lens at 200mm and the Sigma at 300mm.

The Sigma lens setting on 200mm gives a wider-angle picture compared to the Sony 200mm lens. It seems that the cropping factor of 1.5 is revised when using an adapter.

Why does the Sigma lens seem to have different focal lengths for the Sony lens? Is this caused by the adapter?

## List Handling – What is the best way to join tables that have different lengths on the same column values ​​that exist in both tables?

I think my problem is pretty simple, and in SQL it would be trivial. I have two tables

``````TableOne = {{a, x1}, {b, x2}, {c, x3}};
TableTwo = {{a, y1}, {c, y2} , {a, y3}, {a, y4}, {b, y5}, {c,y6}, {c, y7}}
``````

I want to be able to join these two tables where the values ​​in column 1 in both tables match such as:

``````DesiredResult =  {{a, x1, a, y1}, {c, x3 , c, y2} , {a, x1, a, y3}, {a, x1, a, y4}, {b, x2, b, y5}, {c, x3, c, y6}, {c, x3, c, y7}}
``````

I've tried with both `Select[]` statements inside a `Table[]` the structure and also examined `JoinAcross[]` but have not managed to get the desire effect. In SQL, it would be simple, something like:

``````SELECT Col1.Table1, Col2.Table1, Col1.Table2, Col2.Table2,  FROM table2 INNER JOIN table1 ON Col1.Table1  = Col1.Table2
``````

Or something similar.

## Geometry – A curious relationship between the angles and the lengths of a tetrahedron's edges

Consider a Euclidean tetrahedron with edge lengths edge lengths $$l_ {12}, l_ {13}, l_ {14}, l_ {23}, l_ {24}, l_ {34}$$ and dihedral angles $$alpha_ {12}, alpha_ {13}, alpha_ {14}, alpha_ {23}, alpha_ {24}, alpha_ {34}.$$
Consider solid angles
$$begin {split} & Omega_1 = alpha_ {12} + alpha_ {13} + alpha_ {14} – pi \ & Omega_2 = alpha_ {12} + alpha_ {23} + alpha_ {24} – pi \ & Omega_3 = alpha_ {13} + alpha_ {23} + alpha_ {34} – pi \ & Omega_4 = alpha_ {14} + alpha_ {24} + alpha_ {34} – pi \ end {split}$$
and perimeters of faces
$$begin {split} & P_1 = l_ {23} + l_ {34} + l_ {24} \ & P_2 = l_ {14} + l_ {24} + l_ {12} \ & P_3 = l_ {13} + l_ {34} + l_ {14} \ & P_4 = l_ {12} + l_ {23} + l_ {13}. \ end {split}$$
Then, the following cross ratios are equal:
$$[e^{iOmega_1}, e^{iOmega_2}, e^{iOmega_3}, e^{iOmega_4}]=[P_1, P_2, P_3, P_4].$$
Question: Is this known? I have found a proof of this affirmation (to be published soon), but it implies a rather delicate algebraic geometry. It will be very interesting for me to see a more basic approach.