Results for query "le du"

Mongoose findOne is not working for a particular query, yet working for others

I am trying to work with a mongoDB using mongoose, here is an example element from the db:

{
"_id": {
"$oid": "59bc026679f3a71ed02e4812"
},
"name": "Le Ruban blanc",
"customID": 80074,
"description": "Un village protestant de l'Allemagne du Nord à la veille de la Première Guerre mondiale (1913/1914).L'histoire d'enfants et d'adolescents d'une chorale dirigée par l'instituteur du village et celle de leurs familles : le baron, le régisseur du domaine, le pasteur, le médecin, la sage-femme, les paysans.

Confusing rhythms/tuplets in Le Sacre du Printemps?

I've been studying Igor Stravinsky's Le Sacre du Printemps, a wonderful piece known for being quite strange.For example, here's a small excerpt from the first part when the flute comes in:



The rhythms here make absolutely no sense.

Direct proof of fact $u \in C(U)$ satisfies $|Du| \ge 1$ in sense of viscosity if and only if property holds

Is it possible anybody could sketch me a direct proof of the fact that $u \in C(U)$ satisfies $|Du| \ge 1$ in the sense of viscosity if and only if the following property holds?If $V \subseteq U$ is open and bounded and $\varphi \in C^\infty(U)$ satisfies$$\begin{cases} |D\varphi| \le 1 & \text{in }V \\ \varphi \le u & \text{on }\partial V,\end{cases}$$then $\varphi \le u$ on $\overline{V}$.

estimation of a vector-function

Then $x(t)=\int_0^t\dot x(u)du$, whence
$$|x(t)|\le c_1\int_0^t du_1|x(u_1)|$$
$$\le c_1^2\int_0^t du_1\int_0^{u_1} du_2|x(u_2)|$$
$$\le\dots
\le c_1^k\int_0^t du_1\int_0^{u_1} du_2\dots\int_0^{u_{k-1}}du_k |x(u_k)|
$$
$$\le
c_1^k\int_0^t du_1\int_0^{u_1} du_2\dots\int_0^{u_{k-1}}du_k M_t
=M_tc_1^k t^k/k!\to0$$
as $k\to\infty$
for each real $t>0$, where $M_t:=\max_{0\le u\le t}|x(u)|<\infty$.

Poincaré lemma for distributions

For simplicity, let us check the case of a $1$-form
$$
u=\sum_{1\le j\le n} u_j dx_j,\quad u_j\in \mathscr D'(\mathbb R^n),
$$
and assume that $du=0$, i.I want to prove that there exists $a\in\mathscr D'(\mathbb R^n)$
such that
$da=u$.

Non-algebraic Hecke characters

But it is an observation of Weil (Sur le theorie du corps de classes, 1951) that not all Hecke characters (that is, automorphic forms for $\mathrm{GL}_1(\mathbb{A})$) correspond to Galois representations.The Weil group was later defined as an extension of the absolute Galois group that contains the non-algebraic Hecke characters too.

Find $\lim_{x\to 1}\int_{x}^{x^2}\frac{1}{\ln {t}}\mathrm dt$.

$$
For $x>1$ we then have $1\le e^u\le x^2$, hence
$$ \int_{\ln x}^{2\ln x}\frac{\mathrm du}{u}\le f(x)\le x^2\int_{\ln x}^{2\ln x}\frac{\mathrm du}{u}$$
(and similar for $x<1$)
with a now well-known integral: $\int \frac{\mathrm du}u=\ln |u|+C$.$$


HINT:

Write $\frac{1}{\log(t)}=\left(\frac{1}{\log(t)}-\frac{1}{t-1}\right)+\frac{1}{t-1}$.

Help with $H^2$ regularity proof in Evans.

82 implies
\begin{align}
\int_u |v|^2 \, dx &\le C \int_U |D(\zeta^2 D_k^h u)|^2 \, dx \\
&\le C \int_W |D_k^h U|^2 + \zeta^2 |D_k^h Du|^2 \, dx \\
&\le C \int_U |Du|^2 + \zeta^2 |D_k^h Du|^2 \, dx
\end{align}


Theorem 3(i): assume $1\leq p<\infty$ and $u \in W^{1,p}(U)$ then for each $V \subset \subset U$

\begin{equation} \|D^h u\|_{L^p(V)} \leq C\|Du\|_{L^p(U)} \end{equation} for some constant $C$ and all $0<|h|<\text{dist}(V, \partial U)$.By the Leibnitz rule:
$$\int_U |D(\xi^2 D^h_ku)|^2\;dx \le C \int_U |(D\xi^2)D^h_ku|^2 + \xi^2|DD^h_k u|^2\;dx$$
since $(a+b)^2 \le 2(a^2 + b^2)$.

Probability Density Function to Cumulative Distribution Function (Integration Problem)

This thing is begging loudly for a particular substitution:
$$
F_X(x)=2\beta\int_0^x ue^{-\beta u^2} \, du = \int_0^x e^{-\beta u^2} \Big( 2\beta u \, du\Big) = \int_0^{\beta x^2} e^{-w} \, dw = \cdots\cdots
$$

You wrote $F_x(x)$ where you needed $F_X(x).Note that $F_X(x) = \Pr(X\le x)$ and the expression $X\le x$ is incomprehensible if you don't know the difference between $X$ and $x$.