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Calculation and Analysis – Tell MMA that $ int_0 ^ 1 alpha beta (x) dx = alpha int_0 ^ 1 beta (x) dx $

Why does MMA exclude? alpha from the integral in the first case but not in the second?

Integrate(alpha*beta(x),x) 
(* alpha Integrate(beta(x), x) *)

Integrate(alpha*beta(x), {x, 0, 1})
(* Integrate(alpha*beta(x), {x, 0, 1}) *)

and how to have Integrate(alpha*beta(x), {x, 0, 1}) return alpha*Integrate(beta(x), {x, 0, 1})?

st.statistics – uniqueness of the reverse information projection with $ beta $ divergences

There is a lot of literature on this problem
$$ min_ {p in mathcal {P}} D (p | q), $$ namely the information projection problem, e.g. $ D $ the Kullback-Leibler divergence. To the $ mathcal {P} $ convex there is a unique such projection.
There are results for the reverse information projection $ min_ {q in mathcal {P}} D (p | q) $ also.

The question of uniqueness for the (direct) projection was tackled when $ D $ is the $ beta $-Divergence defined as $ D_ beta (p | q) = sum_ {i = 1} ^ nd (p_i | q_i) $
Where $$ d_ beta (x | y) = frac {1} { beta ( beta-1)} left (x ^ beta + ( beta – 1) y ^ { beta} – beta xy ^ { beta-1} right), $$
which interpolates between Kullback-Leibler ($ beta = 1 $) and some other important distances (Euclidean for $ beta = 2 $, Itakura-Saito for $ beta = 1 $).

However, I couldn't find anything for the rear projection $ min_ {q in mathcal {P}} D_ beta (p | q), $ With $ mathcal {P} $ convex. The strict convexity of $ y mapsto d_ beta (x | y) $ when $ 1 leq beta leq 2 $ makes the uniqueness of the reverse projection obvious, but the convexity does not apply to other values ​​of $ beta $, I would be surprised if this problem was not addressed anywhere in the literature where my question came from.

Type theory – in the lambda calculation with products and sums $ f is: [n] to [n]$ $ beta eta $ corresponds to $ f ^ {n!} $?

$ eta $Reduction is often described as a consequence of the desire for functions that are syntactically the same. When it comes to products and functions, it is enough, but when it comes to sums, I don't understand how one can reduce the same functions to a common term.

For example, it is easy to check a function $ f: (1 + 1) to (1 + 1) $ is the same at certain points $ lambda x.f (fx) $or more generally $ f $ is the same at certain points $ f ^ {n!} $ when $ f: A to A $ and $ A $ has exactly $ n $ Residents. Is it possible to reduce $ f ^ {n!} $ to $ f $? If not, what is the weakest extension of the simply typed calculation that enables this reduction?

analytical number theory – inequalities $ pi (x ^ a + y ^ b) ^ alpha leq pi (x ^ c) ^ beta + pi (y ^ d) ^ gamma $ including the prime function, where the Constants are very close to $ 1 $

To let $ pi (x) $ Be the prime count function, I'm curious whether a suitable variant of the second Hardy-Littlewood conjecture (this corresponding Wikipedia)
$$ pi (x ^ a + y ^ b) ^ alpha leq pi (x ^ c) ^ beta + pi (y ^ d) ^ gamma tag {1} $$
can be proven where the constants $ 0 <a, b, c, d leq 1 $ and the constants $ 0 < alpha, beta, gamma leq 1 $ are very close to our ceiling $ 1 $for all real numbers $ x <y $ With $ L <x $ for a suitable choice of a constant $ L $,

Question. Is it possible to prove any statement of the type $ (1) $ under the mentioned conditions for constants $ 0 <a, b, c, d leq 1 $ and constants $ 0 < alpha, beta, gamma leq 1 $ all of these (all together /
at the same time) very close to $ 1 $for all real numbers $ x <y $ for a suitable one $ L <x $? Many thanks.

I don't know if that kind of suggestions $ (1) $ are in the literature or are essentially the same original second Hardy-Littlewood conjecture when we request that these constants be very close $ 1 $,

If there is relevant literature, answer my question as a reference request and I will try to find and read these statements from the literature.

references:

(1) G.H. Hardy and J.E. Littlewood, Some problems of “Partitio numerorum” III: About the expression of a number as a sum of prime numbersActa Math. (44): 1-70 (1923).

Sequences and Series – Prove that for every $ phi in [alpha, beta]$, there is $ {x_ {n} in (0,1]| n = 1,2 … } $, so that $ lim_ {n rightarrow infty} f (x_ {n}) = phi $.

To let $ f (x) $ a continuous function on (0,1) and $ lim inf_ {x rightarrow 0 ^ {+}} f (x) = alpha $ and $ lim sup_ {x rightarrow 0 ^ {+}} f (x) = beta $, Prove that to everyone $ phi in ( alpha, beta) $, there are $ {x_ {n} in (0,1) | n = 1.2 … } $ so that $ lim_ {n rightarrow infty} f (x_ {n}) = phi $,

I try to think about this question with IVT. Can anyone suggest a clue to this question?

Abstract algebra – General method for determining $ mathbb {Q} ( gamma) = mathbb {Q} ( alpha, beta) $ by specifying $ alpha $ and $ beta $

I am currently reading S. Langs "Undergraduate algebra". According to the primitive root element theorem (field theory chapter), there are a number of exercises to find a primitive element of extensions and then their degrees. However, I don't even know how to start. They are as below:

  1. Find one item at a time $ gamma $ so that $ mathbb {Q} ( alpha, beta) = mathbb {Q} ( gamma) $, Prove every statement you make.

on) $ alpha = sqrt {-5}, beta = sqrt {2} $

b) $ alpha = sqrt (3) {2}, beta = sqrt {2} $

c) $ alpha = $ Root of $ t ^ 3 -t + 1 $ . $ beta = $ Root of $ t ^ 2-t-1 $

d) $ alpha = $ Root of $ t ^ 3 -2t + 3 $. $ beta = $ Root of $ t ^ 2 + t + 2 $

$ quad $2. Find the degrees of the fields $ mathbb {Q} ( alpha, beta) $ over $ mathbb {Q} $ in any case from exercise 1.

I think exercises a) and b) are pretty much the same, but I'm not sure about c) and d).

$ tan frac { alpha} {2} tan frac {beta} {2} = frac {1-e} {1 + e} $

If P is a point of the ellipse $ frac {x ^ 2} {a ^ 2} + frac {y ^ 2} {b ^ 2} = 1 $ Whose focal points are S and S ’. Let the angle PSS ’=$ alpha $ and $ PS’S = beta $ then prove it $ tan frac { alpha} {2} tan frac {beta} {2} = frac {1-e} {1 + e} $

I know that S (ae, 0) and S ’(- ae, 0) and $ b ^ 2 = a ^ 2 (1-e ^ 2) $ But I don't know how to solve it further

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