Why my Ubuntu22.04 cannot link to the Internet after Softwares updates &install aptotude?

Could anyone please clarify to me what active particles, in particular active deformable particles are? I have never heard of them, and I am quite curious

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic zero.

Is there a description of $operatorname{Aut}(operatorname{Coh}(X))$, i.e. the autoequivalences of the category $operatorname{Coh}(X)$?

Clearly it contains $operatorname{Aut}(X)ltimesoperatorname{Pic}(X)$ as a subgroup.

$$int_{n+1}^{infty}f(x)dx le R_n le int_{n}^{infty}f(x)dx$$

What does this actually mean?

Let’s use n=5.

The $R_5$ is the error of the partial sum $S_5$

That error is less than the sum of the remaining terms from 5 to $infty$ ?

Why is that?

Also, why is it greater than sum of the remaining terms from 6 to $infty$ ?

Let $n$ be a positive integer and $p$ be a prime. Let $v_p(n)$ be the $p$-adic order of $n$, i.e., the exponent of the highest power of $p$ that divides $n$. I would like to know if there is a quick and simple proof for the following congruence relation.
$$
sum_{j=1}^{lfloor log_{p} n rfloor} {n-1 choose p^j-1} equiv v_p(n) ;mbox{(mod }pmbox{)}.
$$

Key ideas involved in a ‘not so simple proof’ can be found in http://math.colgate.edu/~integers/w61/w61.pdf

Best wishes.

Considering $(-Delta)^s: H^s(Omega) to L^2(Omega)$, is it possible to show that this operator is closed (continuous)?
For instance, taking a sequence $(u_n) subset H^s(Omega)$ with $u_n to u$ in $H^s(Omega)$, we need to show that $(-Delta)^su_n to (-Delta)^su$ in $L^2(Omega)$.

Attempt:

Consider
$$lim_{n to +infty}int_Omega |(-Delta^s)u_n(x)|^2 dx = lim_{n to +infty} int_Omega Big(int_Omega frac{u_n(x) – u_n(y)}{|x – y|^{N + 2s}} dyBig)^2 dx.$$

So my idea would be to use some inequality (perhaps something similar to the Poincaré inequality) to obtain the norm of $u_n$ in $H^s(Omega)$ on the right side. But, I don’t know how to get rid of this one on the right side of the expression above.

I am studying ribbon graphs on surfaces. I am reading this paper (page 5).

But I have confused about how to make surfaces using the ribbon graphs. Actually, we call this surfaces as associated ribbon surfaces.

The definition of ribbon graph is given in definition 1.5 of the above mentioned paper.

Consider the following ribbon graph

To get its associated ribbon surface, first we do its edge refinement by adding each edge a degree two vertex. Then consider the half edges incident to each vertices of the graph.

Then replace the half edges with thin strips with the orientation on boundaries (this orientation of the boundaries follows the orientation of each vertex), which is as follows:

My question is how we connect the strips corresponding to the two half edges following the orientation of their boundaries to form ribbons?

Please help.

Let $Z$ be a $[0, infty)$-valued random variable satisfying $lambda E[g(Z+1)] = E[Zg(Z)]$ for all indicator functions $g$ of Borel subsets of $[0, infty)$.

Prove that $mathcal{L}(Z) = Poisson(lambda)$.

Hint: Consider $1_{(n,n+1)}$ for $n in mathbb{N}_0$.

Proof:

Let $n in mathbb{N}_0$ and g = 1_{(n,n+1)}, then holds:
begin{align*}
lambda E[g(Z+1)] = lambda P[Z+1 in (n,n+1)] = lambda (P[Z+1 = n+1] – P[Z+1 = n]) = lambda P[Z = n] – lambda P[Z = n-1]).
end{align*}
But what is
begin{align*}
E[Zg(Z)] = E[Z cdot 1_{(n,n+1)}(Z)] = ?
end{align*}

Have I got everything right so far? If so, how should I proceed now?

Thanks for the help!

Bézout’s theorem states that a linear diophantine equation $ax+by=c$ has integer solutions if and only if $gcd(a,b)|c$. Then is there a theorem regarding the existence of a unique integer solution for a system of two linear equations $ax+by=c$ and $px+qy=m$, where $a,b,c,p,q,m$ are positive integers?

Additionally, can the theorem be generalized to the condition where there are more than two unknowns?

I’m learning about normal modes right now and I have a question pertaining to the way the equations of motion are written vis-a-vis summation notation. The book defines the equations of motion of a system of $n$ masses attached to springs, whose displacements are $q_j$ as the following: $$underset{j}{sum} (A_{jk}q_j + m_{jk}ddot{q_{j}}) = 0$$My quandary is caused by the subscript $k$ that appears in the undetermined coefficients $A_{jk}$ as well as the masses $m_{jk}$. I just want to know what values of $k$ to use here in this equation in the general case. Since it’s not explicitly stated, I’m sure there’s an implicit understanding there.

Let’s take for instance $n = 2$. So, we have $2$ masses attached to springs and $2$ solutions for the displacements (from equilibrium) $q_j$ of each mass. Then we must have $2$ equations of motion. I’m assuming $A$ and $m$ are $2 times 2$ matrices and then is it the case – in this particular situation – that we have the relation $delta_{jk}(A_{jk}q_j + m_{jk}ddot{q_j})$? Such that our equations of motion become $$A_{11}q_1 + m_{11}ddot{q_{1}} = 0\ A_{22}q_2 + m_{22}ddot{q_{2}} = 0$$

If this is true, then for the case $n gt 2$ how do we treat the matrices $A$ and $m$? Do they just become $n times n$ matrices? Where the elements on the diagonal are the nonzero ones?