ap.analysis of pdes – Canonical forms on higher degree Jet bundles similar to the Liouville form

On a smooth manifold of dimension $n$, the application value of the canonical $1$-form, the Liouville form on $T^*(X)$, to the Hamiltonian mechanics is well known; $T^*(X)$ is a degree $1$-Jet bundle. My question is Do canonical forms similar to the Liouville form exist on higher degree Jet bundles?
I ask this because, beyond the invariant sub-principal symbol of a pseudodifferential operator, nothing much seems to be known to handle multiple characteristic problems, especially of the non-involutive
type. I am aware of Ivrii-type Fuchsian operators, already posing great difficulties.

graph theory – Minimizing the degree of outgoing edges in a digraph, does this problem have a name?

I have a problem which can be rephrased in this way.

Suppose $G = (V,E)$ is a digraph (directed graph) and for each $v in V$ we denote with $delta^+(v)$ the number of outgoing edges of the vertex $v$.

I’m looking for a way to swap the edges (so $(i,j) in E$ would become $(j,i)$) so that $max_{v,w in V} |delta^+(v) – delta^+(w)|$ is minimized.

Does this problem have a name in literature?

All real and complex roots of 10 degree polynomial

I am trying on
poly=x^10 + 7x^9 – 38x^8 – 192x^7 + 209x^6 – 1009x^5 + 5768x^4 – 19002x^3 – 2580x^2 – 99792*x^1 – 120960;

  1. I was trying "FindRoot & NRoots" to find real and complex roots, but this doesn’t work.

  2. Does Mathematica have its own implementation of double roots?

differential geometry – Computing degree of a map

Define $f:mathbb{CP}^1 rightarrow mathbb{CP}^1$ by
$$f((z_0,z_1))=(p(z_0,z_1),z_1^n),$$
where $p(z_0,z_1)=z_0^n+c_{n-1}z_0^{n-1}z_1+ dots c_1z_0 z_1^{n-1}+c_0 z_1^n$ is an arbitrary homogeneous polynomial. I would like to compute the degree of $f$. I think I need to use the following: for a regular point $x in mathbb{CP}^1$, the differential $df_x:T_xmathbb{CP}^1 rightarrow T_{f(x)}mathbb{CP}^1$ is orientation preserving.

I am not entirely sure how to show this. I know that $df_x$ is orientation preserving if its Jacobian is positive. However, can I compute it in charts? For instance, in the chart $U_0={(z_0,z_1) mid z_0 neq 0}={(1,z) mid zin mathbb{C}}$, we have
$$f((1,z))=(p(1,z),z^n),$$
whose Jacobian is $0$.

diophantine approximation – On the degree of irrationality of two irrational numbers and their rational (in)dependence

Let $x$ and $y$ be some irrational numbers. If the degree of irrationality of $x$ is the same as that of $y$, is it necessarily the case that $x$ and $y$ are rationally dependent ?

ADDENDUM: What if $x$ and $y$ are transcendental, specifically logarithms of some rational numbers ?

co.combinatorics – High degree differences in bipartite graphs

Consider a finite, simple and undirected graph $G=(V,E)$ with $V={v_1,dots, v_n}$. Let us define the quantity:

$$mathcal{I}_k(G) := sum_{1le i,j le n} mathbb{1}{Big{|mathrm{deg}(v_i)-mathrm{deg}(v_j)|ge k}Big},$$
i.e. the number of all those pairs $(v_i,v_j)$ with degree difference greater or equal to $k$.

In the answer to a previous question,

Pairs of vertices with high degree difference

I was able to prove, that:

For $k>frac{n}{2}$, we have

begin{equation} mathcal{I}_k(G) le 2k(n-k). end{equation}

Now, my current question is: does it also hold for bipartite graphs? More precisely, consider a bipartite graph with vertices $${v_1,v_2,dots, v_n} cup {w_1,w_2,dots, w_n}.$$
Is it also true that (still assuming $k>n/2$)
$$sum_{1le i,j le n} mathbb{1}{Big{|mathrm{deg}(v_i)-mathrm{deg}(w_j)|ge k}Big} le 2k(n-k)? $$
If we would think about those two problems in terms of adjacency matrices, then the first (solved) problem is about symmetric matrices and the current one is about general matrices without the symmetry condition. Computer verification for $nle 10$ confirms this conjecture. Unfortunately, the proof suggested in the previous case does not seem to work in this case (it is not clear how to generalize it).

What I tried doing was incorporating the Gale–Ryser and Ford-Fulkerson conditions (equivalent to the bipartite realization of double integer sequences as degree sequences). Unfortunately, I did not succeed.

As previously: I would be very grateful for any comment or insight. Maybe looking on this question, You will think of some other related problem or theorem that might be helpful. Maybe You are able to say what a general strategy might be appropriate to handle this problem. It may turn out that for some reason this problem is trivial but I have overlooked it. I will appreciate any help or advice.

reference request – Spline Interpolation error of higher degree

It is well-known that the interpolation error of a cubic spline has at best order $O(h^4)$, which results from polynomials of degree $3$.

Can I assume that, if one uses polynomials of degree $p$ and the
respective function to be interpolated $fin C^p((a,b))$, that the
interpolation error of this spline is $O(h^{p+1})$? Is something like this present in literature?

Generalization of real roots for a polynomial of $n$ degree.

Is there any specific condition on the coefficients of a polynomial of $n$ degree, so that all roots are real ?

I know that there is a condition on quadratic polynomials : $p(x) = ax^2+bx+c$, for both the roots to be real, $b^2-4acge0$.

programming practices – Is it unethical to call myself a software engineer without a degree

I am a software developer who has been programming for over 11 years, with a good understanding of computer science, high-level architecture design & etc.

In my company, I train other software engineering graduates as I have more experience developing software and applications than the software engineering graduates in my company.

Is it unethical to call myself a software engineer, considering my skill level is higher than an average software engineering graduate?

Context: my country does not regulate the usage of the word “Engineer” and anybody can call themselves an engineer.

nt.number theory – Formal degree of discrete series representations

Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $pi$ of $G$ is a discrete series if its matrix coefficients are in $L^2(G)$.

The, it is possible to define its formal degree $mathrm{deg}(pi)$ as the “norm” of the operator “matrix coefficient”, i.e. as the constant $d_pi$ such that for all $v, w in V_pi$ we have
$$| xi^pi_{v, w} |_2^2 := int_{G} |langle pi(g)v, w rangle|^2 dg = d_pi^{-1} |v|^2|w|^2.$$

It is known (e.g. Dixmier, C* Algebras, Prop. 18.8.5) that in this case the formal degree matches the Plancherel measure, i.e.
$$mathrm{deg}(pi) = mu^{rm Pl} (pi).$$

I would like to relate this to the dimension of the “cohomological class” of $pi$. As in this question, a discrete series $pi$ is $xi$-cohomological for a certain $lambda_pi$ (its infinitesimal character if I understand correctly). This $lambda_pi$ is an irreducible finite-dimensional representation of $G (mathbb C)$. Do we have
$$mathrm{deg}(pi) = mathrm{dim} (lambda_pi) quad ?$$

References for these matters are welcome.