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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?

## Calculate \$ int_ {0} ^ { infty} frac {e ^ {- alpha x} beta ^ x gamma ^ {x -1}} { gamma (x)} dx \$

To let $$alpha, beta, gamma> 0$$ and let $$Gamma$$ denote the gamma function. How can the following integral be calculated?
$$int_ {0} ^ { infty} frac {e ^ {- alpha x} beta ^ x gamma ^ {x -1}} { gamma (x)} dx$$
Is there a useful transformation?

## 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 $$0 and the constants $$0 < alpha, beta, gamma leq 1$$ are very close to our ceiling $$1$$for all real numbers $$x With $$L for a suitable choice of a constant $$L$$,

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