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) $.

Can any one please help me prove or disprove this assertion?

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.

google sheets – Using IMPORTRANGE to juxtapose lists of varying lengths

Suppose I have two Google sheets, Doc1 and Doc2, both of which have only one sheet with two lists with the same schema, say two columns Name and Age. The number of entries in the lists is unknown, but if it makes the task easier, we can assume that they have at most 5 entries:

Two sheets of Google

I want to bring the two lists together and display them in a third "Summary" document, one after the other. I can show them using these formulas, in the Summary cells A1 and A6, respectively:

= IMPORTRANGE ("1cLqgopAWG ...", "Sheet1! A1: B5")
= IMPORTRANGE ("1564EXoW-s ...", "Sheet1! A1: B5")

However, this leaves a boring variable number of blank lines between the two:

Unwanted blank lines

while I would like to show them one directly above the other, as if (manually copied values ​​here to show what I want):

What I would like to achieve

If I try to use IMPORTRANGE in two contiguous rows (i.e. A1 and A2), I have a #REF! error ("The result of the array was not expanded because it would overwrite the data in A2."):

#REF! Fault

Is it possible to achieve what I want?
(Bonus points to remove the constraint on the maximum number of items.)

Telephoto – Selecting a tripod for use with very long focal lengths

I play with very long focal lengths with a Kenko 3x teleconverter. I had a hard time getting sharp pictures and I think camera shake is an important part of that.

Since I plan to buy a tripod to lock the objects.

What should I look for in a tripod in this context?
I'm currently using the 100-300 f4 sigma and a 5d mark ii canon, but is also planning to get a 150-600 mm zoom and the MTO-1000 mirror lens, so I wish work with a tripod.

I've heard that a heavy tripod is recommended in this context, is this correct and if so, how heavy is it?

Looking at the web, I found the Genesis A3 Tripod that looked promising. Am I on the right track with this?