What is it? How the solutions? There are no goals to publish (maybe blocked by the search engines, not eng …

What is it? How the solutions? My projects do not work and continuously tells me: There are no goals to publish (maybe blocked by the search engines, there are no engines that allow the extraction of chosen URLs, there are no scheduled publications)

You need WordPress Expert for quick solutions

Looking for a developer / wordpress expert, to fix some things on our few websites.
Waiting for your answer.

Thank you,

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nt.number theory – Question about parametric representations of solutions to $ x ^ 3 + y ^ 3 + z ^ 3 = n in mathbb N $

There are such representations for $ n = 1,2 $. However, according to the Wikipedia article, there seem to be no known (polynomial) parametric representations $ P, Q, R $ such that $ (P (m)) ^ 3+ (Q (m)) ^ 3+ (R (m)) ^ 3 = 3 $.

There are some $ n en mathbb N $ (apart from those excluded trivially with congenial conditions) for which it is known that there are no three polynomials $ A, B, C $ with integer coefficients such that $ (A (d)) ^ 3+ (B (d)) ^ 3+ (C (d)) ^ 3 = n $. And are there some general conjectures about the non-representability of some natural numbers? Does any heuristic suggest that it is rare to have such representation, as we have it for $ n = 1,2 $?

Solutions a $ y ^ 2 = x ^ 3 + x $ in $ mathbb {P} ^ 2 ( mathbb {F_p}) $.

Consider the elliptical curve $$ mathcal {C}: y ^ 2 = x ^ 3 + x. $$

Consider the reduction of $ mathcal {C} mod {p} $.

Some explicit calculations show that $ # tilde { mathcal {C}} ( mathbb {F} _3) = 4 $, $ # tilde { mathcal {C}} ( mathbb {F} _5) = 4 $, $ # tilde { mathcal {C}} ( mathbb {F} _7) = $ 8Y $ # tilde { mathcal {C}} ( mathbb {F} _11) = $ 12 (these totals include the point at infinity).

On page 138, Example 4.6, Silverman and Tate continue to say that "it is not difficult to verify that" $ 4 | # tilde { mathcal {C}} ( mathbb {F} _p) $ for all $ p geq 3 $.

However, I am struggling to verify this fact myself, and I would be very grateful for any help!

From working the smallest cases by hand, it seems that a natural division would be between $ p equiv 1 mod {4} $ Y $ p equiv 3 mod {4} $. However, if it seems that there must be some theorems about the solution in these two separate cases, I am losing myself at this time.

Differential equations – Different solutions with Mathematica and Wolfram Alpha.

I tried to solve this in a Mathematica notebook:


And the results are:

Sqrt[-1+Tanh[E^C[-1+Tanh[E^C[-1+Tanh[E^C[-1+Tanh[E^C[1] (t + C[2])]^ 2])},
Sqrt[-1+Tanh[E^C[-1+Tanh[E^C[-1+Tanh[E^C[-1+Tanh[E^C[1] (t + C[2])]^ 2]}}

But if I execute it in Wolfram Alpha the result is:


And the seconds are the correct results, I needed a real function, while the first ones are complex functions. Why does this happen? What assumptions do Wolfram Alpha make in its entry?

Thanks in advance

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Approximate solutions to non-polynomial equations.

Let's assume the following symbolic equations

$$ E = sqrt {p ^ 2 + m ^ 2} + frac {e d_1 (p)} {r} + frac {e ^ 2d_2 (p)} {r ^ 2} $$

being $ d_ {i} $ some generic function of $ p $.
I would like to find solutions to this equation in terms of $ p $ and for small values ​​of the constant. $ e $. How can I do this using Mathematica?

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pr.probability: definition of weak solutions for infinite SDEs in the same probability space

Suppose I have an SDE of the form
$$ dX_t = b (X_t) dt + sigma (X_t) dB_t + int _ { mathbb {R}} G_ {t -} (y) N (dtdy) $$
that I can solve weakly if I cut the last integral to the range on the set $ { mid {y} mid> epsilon } $.

To build a complete solution, I would like to define the sequence of weak solutions for the cut equations in the same probability space and show that both sides converge (like) to the complete solution (I can do this with particular coefficients that I have).

My idea was to take a weak solution for each equation, define it in the appropriate path space taking its distribution as the probability measures, take an infinite product of the resulting spaces and then define the corresponding processes through the composition of a canonical process and a projection.

Is something similar to this possible?