Geometría de ag.algebraic – Finite covers etale of products of curves

Probably this question can be formulated in a much greater generality, but I will simply expose it in the generality that I require. I work more $ mathbb {C} $.

Leave $ C_1, C_2 subset mathbb {P} ^ 1 $ be open subsets not empty and $ f: X to C_1 times C_2 $ A non-trivial finite etale cover. There exists $ i en {1,2 } $ such that the composition $ X to C_1 times C_2 to C_i $ Do you have unconnected fibers?

reference request – Finite covers of Boolean algebras for their subalgebras

It is a student exercise that no group can be represented as a union of the set theory of its two appropriate subgroups. The same can be shown for Boolean algebras. On the other hand, it is not difficult to prove that any infinite Boolean algebra $ mathcal {A} $ can be covered by your $ k $ appropriate subalgebras $ mathcal {A} _0, ldots, mathcal {A} _ {k-1} $, where $ k in mathbb {N} setminus {0,1,2,4 } $, such that $ mathcal {A} _i not subseteq bigcup_ {j neq i} mathcal {A} _j $ for each $ i <k $. However, I'm not sure if $ k = 4 $ I really should be excluded here. So, my question is this:

Q: Leave $ mathcal {A} $ Be an infinite Boolean algebra. Do your own subalgebras always exist? $ mathcal {A} _0, mathcal {A} _1, mathcal {A} _2, mathcal {A} _3 $ such that $ mathcal {A} = bigcup_ {i <4} mathcal {A} _i $ Y $ mathcal {A} _i not subseteq bigcup_ {j neq i} mathcal {A} _j $ for each $ i <4 $?

I suspect that a careful (and tedious) analysis of the cases could show that the answer is negative (equally for $ k = 2 $), but perhaps there is some intelligent proof (or refutation) of it. I am also asking you about possible references to documents or books in which such problems are studied.

(I asked the same question in the Math Stack Exchange, but the interest was literally zero).

Optics: Can lens covers become too tight to become accidental openings?

Vignetting is an opening effect, visible in the corners rather than in the center.

Since the border edge of the lens cover is a few centimeters in front of the lens, its effect is not evenly distributed throughout the image area, but is more prominent at the corners.

Then, before the hood has a real impact on the overall exposure, you will clearly see a vignetting.

A few years ago, when I was experimenting with several lenses and special hoods for my SLR camera, I built a card-box adapter that simulated my SLR camera body (correct focal length of the flange) with a cut-out that equals the film size of 24 by 36 mm. If looking from the "film layer" to the lens, could not see any part of the lens cover, vignetting could not be a problem.

Convex geometry – Volume of covers of a polytope.

Leave $ K $ be a politician in $ mathbb R ^ d $, exploit it by a factor $ lambda> 0 $. For a unitary vector. $ u in mathbb S ^ {d-1} $, $ lambda K $ has 2 support hyperplanes $ H_1 $ Y $ H_2 $ with its corresponding normal vectors towards the outside. $ u $ Y $ -u $. I only take into account $ u $It is such that $ H_1 $ Y $ H_2 $ they have a "good" position, which says they are not parallel to any face of $ K $, or equivalently, contain a single vertex of $ lambda K $.

Consider another parallel hyperplane. $ H_t $ such that the distance between him and $ H_1 $ is $ t $ (Of course $ t $ is smaller than the width of $ lambda K $ in the direction $ u $). So $ H_1 $ Y $ H_t $ define a limit of $ lambda K $.

My question is:
Is there a possibility that we can estimate the volume of that limit, in terms of $ u, lambda $ Y $ t $?

The easiest case is when $ t $ is smaller than the distance from the nearest vertex $ lambda K $ to $ H_1 $, then the volume is $ frac {1} {d} in ^ d $, where $ a $ is a constant depends on $ u $.

I mean, in general, it is very difficult to calculate the volume of capitals defined in a similar way to $ K $. But if we explode $ K $, some factors can be ignored when $ lambda $ It's big enough, so I hope there's a limit to the size of the lid volume.

insurance: if I pay a rental car with points and the credit card only covers taxes and deposits, is the CDW card valid?

I recently rented a car in the USA. UU That I paid with Enterprise points. I usually reject the CDW because my Chase card covers it when I pay. But when paying with points, the "rent" is not charged on the card, only the tax and the deposit. In the Benefits Guide, write that the condition for coverage is "Start and complete the full rental transaction using your card that is eligible for the benefit." Which is the case, except that the rental transaction does not include the cost of the rent, because it is covered by points. Does anyone know how credit card companies treat these types of cases?