probability – Calculating the quotient $ frac { mu (E)} { nu (E)} $ of two measures with density w.r.t. the Lebesgue measure

Suppose that $ mu $ and $ nu $ are probability measures on $ mathbb {R} $ with density $ f $ and $ g $ w.r.t. the Lebesgue measure, i.e. $ mu = fdx $ and $ nu = g dx $. Is there an easy way to calculate the qutotient
$$
frac { mu (E)} { nu (E)}
$$

of a measurable set $ E $? This is a quotient of two integrals of $ f $ and $ g $. Can we calculate one single integral over the quotient of $ f $ and $ g $ instead?

Limits – Prove that $ frac {f (x) – (f * K_t) (x)} {t} bis – Delta f (t bis 0) $ for $ f in C_0 ^ { infty} ( mathbb {R} ^ n) $

Describe
$$ K_t (x) = frac {1} {(4 pi t) ^ { frac {n} {2}}} exp (- frac {| x | ^ 2} {4t}) $$
the heat core, $ f in C_0 ^ { infty} ( mathbb {R} ^ n) $. It is known that $ f * K_t to f (t to 0) $ point by point and $ L ^ p (p ge1) $. How can we prove it? $$ frac {f (x) – (f * K_t) (x)} {t} bis – Delta f (t bis 0) $$
holds ($ Delta $ denotes the Laplace operator)? I got a proof with the spectral resolution, but I wonder if there is a simpler proof with the basic analysis. Thanks a lot.

ordinary differential equations – what is the singular point of the ode $ frac {dy} {dx} = sqrt {y} + 1 $

Thank you for your reply to Mathematics Stack Exchange!

  • Please be sure answer the question. Provide details and share your research!

But avoid

  • Ask for help, clarify, or respond to other answers.
  • Make statements based on opinions; Support them with references or personal experiences.

Use MathJax to format equations. MathJax reference.

For more information, see our tips on writing great answers.

real analysis – why $ sum_ {n = 1} ^ infty frac {4 sin pi n} { pi pi n ^ 2} sin nx = 2 sin (x) $?

I used Wolfram Alpha to calculate this integral
$$ frac {1} { pi} int _ {- pi} ^ { pi} 2 sin x sin nx ; dx = frac {4 sin pi n} { pi pi n ^ 2} $$

So if $ n in mathbb {Z} $ That answer must be $ 0 $ Law?

Then I used Wolfram Alpha again for the calculation
$$ sum_ {n = 1} ^ infty frac {4 sin pi n} { pi pi n ^ 2} sin nx $$
and it is called
$$ sum_ {n = 1} ^ infty frac {4 sin pi n} { pi pi n ^ 2} sin nx = 2 sin (x) $$
How is that possible?

Algebra precalculation – Find $ n $ such that $ 365 left (1 – ( frac {364} {365}) ^ n – n frac {364 ^ {n-1}} {365 ^ n} right)> $ 1

i have to find $ n $ so that $ 365 left (1 – ( frac {364} {365}) ^ n – n frac {364 ^ {n-1}} {365 ^ n} right)> 1 $. The answer is $ n ge 28 $. I understand expanding the equation, rearranging it, taking the logarithm
$$ log (364)> log (365) + n log (364/365) + log (n) + (n-1) log (364/365). $$

I don't know what to do next. I would be happy if you could give me a hint.

nt.number theory – is there a non-negative sequence $ a_p $ so that $ sum_p frac {a_p} {p} $ converges but $ sum_p frac { sqrt {a_p}} {p} $ diverges?

Is there a real, non-negative sequence? $ a_p $ indexed on the prime numbers so that $ sum_p frac {a_p} {p} $ converges however $ sum_p frac { sqrt {a_p}} {p} $ diverged? If so, what is an example of such a sequence, and if not, how can this be proven?

(This was the result of examining the presumption distance for multiplicative functions in analytical number theory. A sequence that meets these conditions is required to find a multiplicative function $ f $ so that $ sum_p frac {1 – Re (f (p))} {p} $ but converges $ sum_p frac {| 1 – f (p) |} {p} $ diverges.)