What is the relationship between the degree of a regular graph $k$, the number of verticies $v$ and the existence of cycles with length $n$?

No relation

https://en.m.wikipedia.org/wiki/Degree_(graph_theory)

In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex’s degree, for the two ends of the edge

https://en.m.wikipedia.org/wiki/Vertex_(graph_theory)

n mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices).

https://en.m.wikipedia.org/wiki/Cycle_(graph_theory)

n graph theory, a cycle in a graph is a non-empty trail in which the only repeated vertices are the first and last vertices.

analysis – How to show that $T_k(x_1,ldots ,x_n)$ is the only polynomial of degree $k$ with the specific properties?

Let $Usubset mathbb{R}^n$ be oen and $f:Urightarrow mathbb{R}$a $k$-times continusouly differentiable function.

Let $x_0in U$ be fixed.

The $k$-th Taylow polynomial of $f$ in $x_0$ is $$T_k(x_1,ldots ,x_n)=sum_{m=0}^kfrac{1}{m!}sum_{i_1=1}^n ldots sum_{i_m=1}^n frac{partial}{partial{x_{i_1}}}ldots frac{partial}{partial{x_{i_m}}}f(x_0)cdot x_{i_1}cdot ldots x_{i_m}$$
Show that $T_k(x_1,ldots ,x_n)$ is the only polynomial of degree $k$ such that $$T_k(0)=f(x_0) \ frac{partial}{partial{x_{i_1}}}ldots frac{partial}{partial{x_{i_m}}}T_k(0)=frac{partial}{partial{x_{i_1}}}ldots frac{partial}{partial{x_{i_m}}}f(x_0)$$ for all $min{1,ldots , k}$, $i_ellin {1, ldots ,n}$ and $ell in {1, ldots , m}$.

To show that $T_k$ satisfies these properties we just have to replace $x_i=0$ for all $i$ for the first property and for the second one we have to calculate these partial derivatives, right?

But how can we show the uniqueness?

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.

number theory – Can sufficient high degree polynomial sequences contain infinitely many primes?

I have a conjecture:

For any integer $N$, there exist an positive integer $n > N$ such that there exist a degree-$n$ polynomial $P(x)$ satisfying:the sequence $left{P(n) right}$ contains infinitely many primes.

The conjecture came to me when I was trying to solve a problem concerning irreducible polynomials and prime numbers. This seems reasonable since we have a known result for degree-$1$ polynomials(Dirichlet’s theorem). Furthermore, we can show that the sequence $left{n^2 + 1 right}$ contains infinitely many square-free integers by easy sift-method.

Thanks in advance.

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?

linear algebra – Inner product on homogenous polynomials of degree $d$ over a vector space

I was studying length and distance between points and an algebraic variety, so I crossed the following theorem:

Let $q:W times W to mathbb{K}$ be a fixed inner product on vector space $W$, then there is a unique inner product on $Sym^d W $ (homogenous polynomials of degree $d$ over $W$), such that
$$Q:Sym^d W times Sym^d W to mathbb{K}, Q(f^d,g^d)=q(f,g)^d.$$
I the difinition of $Q(f,g)$ is given by:
(we know there is coresspondance between homogenous polynomials of degree $d$ over $W$ and symmetric tensor of order $d$)

Consider a symmetric tensor $fin Sym^dW$, it may be seen in coordinate systems as
$$f = sum_{|alpha|=d}{d choose alpha}f_{alpha} x^{alpha},$$
where
$alpha = (alpha_1, dots , alpha_n) in mathbb{Z}^n_{geq 0}$, $|alpha| colon= alpha_1 + cdots + alpha_n$, $x^{alpha} colon= x_1^{alpha_1} dots x_n^{alpha_n}$ and ${d choose alpha} colon= frac{d!}{{alpha_1}! dots{alpha_n}!}$ is multinational coefficient. Then for $f, g in Sym^d W$, the inner product is defined as
begin{align*}
Q(sum_{|alpha|=d}{d choose alpha} f_{alpha} x^{alpha},sum_{|alpha|=d}{d choose alpha} g_{alpha}x^{alpha}) colon= sum_{|alpha|=d}{d choose alpha}f_{alpha}g_{alpha} = sum_{|alpha|=d} frac{d!}{{alpha_1}! dots{alpha_n}!}f_{{alpha_1} dots{alpha_n}}g_{{alpha_1} dots{alpha_n}}.
end{align*}

This definition holds the inner products’ conditions (it is easy to check). Now I am stuck on proving $Q(f^d,g^d)=q(f,g)^d$. Which inner product should be considered here? Is the q(f,g) is considered as differential operator? or any fixed inner product enough for proving the theorem?

Any comment and guidance are highly appreciated.

estimation theory – Spline Interpolation error of higher degree

It is well known that the interpolation error of a cubic spline has at best order $mathcal{O}(h^4)$ 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 $mathcal{O}(h^{p+1})$ ?

Is this known in Literature ? (I couldn’t seem to find it.)

learning – What master’s degree is recommended to study at a university for establishing a start-up?

I studied for 3 years computer science at a relatively good university, and at the same time I have several years of experience with broad and deep knowledge in many fields on the subject. I am a creative person and I am interested in expressing my Innovative ideas by setting up a start-up from scratch which will bring a technological solution to many companies in high-tech.

I wanted to ask in general what it is recommended to study for a master’s degree so that I can bring the product that I can build independently through my knowledge of computer science and experience to a successful start-up company?

nt.number theory – Explicit construction of division algebras of degree 3 over Q

In his book Introduction to arithmetic groups, Dave Witte Morris implicitly gives a construction of central division algebras of degree 3 over $mathbb{Q}$ in Proposition 6.7.4. More precisely, let $L/mathbb{Q}$ be a cubic Galois extension and $sigma$ a generator of its Galois group.If $p in mathbb{Z}^+$ and $p neq tsigma(t)sigma^2(t)$ for all $t in L$, then
$$ D=left{ begin{pmatrix}
x & y & z\
psigma(z) & sigma(x) & sigma(y)\
psigma^2(y) & psigma^2(z) & sigma^2(x)
end{pmatrix} :(x,y,z)in L^3 right}
$$

is a division algebra.

On page 145, just before Proposition 6.8.8, Morris claims that it is knows that every division algebra of degree 3 arises in this manner. This should follow from the fact that every central division algebra of degree 3 is cyclic. I could not find this explicit construction in my references (e.g. Pierce – Associative Algebras, though maybe I missed something) and I would like to know if there is a reference or a quick way to see that this exhausts all central division algebras of degree 3 over $mathbb{Q}$.