Related papers: A case of the generalized vanishing conjecture

Let $G$ be a topological Abelian semigroup with unit, let $E$ be a Banach space, and let $C(G,E)$ denote the set of continuous functions $f\colon G\to E$. A function $f\in C(G,E)$ is a generalized polynomial, if there is an $n\ge 0$ such…

In this short note, we will show the following weak evidence of S. Lang conjecture over function fields. Let f : X ---> Y be a projective and surjective morphism of algebraic varieties over an algebraically closed field k of characteristic…

Coefficients in the expansions of the form $\partial P_{n}(\lambda;z)}/\partial\lambda=\sum_{k=0}^{n}a_{nk}(\lambda)P_{k}(\lambda;z)$, where $P_{n}(\lambda;z)$ is the $n$th classical (the generalized Laguerre, Gegenbauer or Jacobi)…

We present a rigorous analytic proof of a generalized continued fraction (GCF) identity for the transcendental constant $8/\pi^2$, a result recently conjectured via the algorithmic framework of the Ramanujan Machine. Distinct from canonical…

In this work we give direct proofs of two theorems concerning explicitly defined polynomial vector fields connected to differentiation of hyperelliptic functions of any genus. We prove that the operators determining the fields commute, and…

One of the aims of this article is to provide a class of polynomial mappings for which the Jacobian conjecture is true. Also, we state and prove several global univalence theorems and present a couple of applications of them.

Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$ complex variables . Assume that this polynomial satisfies the property : \\ $|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain $\{(z_1,...,z_n) :…

We prove the $\Lambda$-variation diminishing property of the Bernstein and Kantorovich polynomials. Next we apply this result to characterize the space $C\Lambda BV_c$ as the closure of the space of polynomials in the $\Lambda BV$ norm. A…

We prove that the $\Phi^4$ theory is trivial for any values of the bare coupling constant $\lambda$ thus extending previous results referring to very strong couplings to the full range of values for this parameter. The method is based on…

Let $\Phi_{\Lambda_{n}}$ be the unique solution of the differential operator $L=\prod_{j=0}^{n}\left( \frac{d}{dx}-\lambda_{j}\right) $ such that $\Phi_{\Lambda_{n}}^{\left( j\right) }\left( 0\right) =0$ for $j=0,...,n-1,$ and…

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which…

Given an automorphism and an anti-automorphism of a semigroup of a Geometric Algebra, then for each element of the semigroup a (generalized) projection operator exists that is defined on the entire Geometric Algebra. A single fundamental…

Let $B$ be an integral domain over a field $K$ of characteristic 0. The derivation $\delta$ of $B[Y_d]=B[y_1,\ldots,y_d]$ is elementary if $\delta(B)=0$ and $\delta(y_i)\in B$, $i=1,\ldots,d$. Then the elements…

We introduce the sequence of generalized Gon\v{c}arov polynomials, which is a basis for the solutions to the Gon\v{c}arov interpolation problem with respect to a delta operator. Explicitly, a generalized Gon\v{c}arov basis is a sequence…

The authors previous derivation of a variational principle from the total work functional, as a generalization of the first variation of an action functional, is extended by deriving a corresponding generalization of the Hamiltonian…

In the following text for $p\in[1,\infty]$, nonzero cardinal number $\tau$, self--map $\varphi:\tau\to\tau$ if there exists $N\in\mathbb{N}$ such that $\varphi^{-1}(\alpha)$ has at most $N$ elements for each $\alpha<\tau$, and operators…

Let $\alpha: X \to Y$ be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under $\alpha$ is semistable if the genus of $Y$ is at least $1$ and stable if the genus of $Y$ is at least $2$.…

We introduce a generalization of bivariate Griffiths polynomials depending on an additional parameter $\lambda$. These $\lambda$-Griffiths polynomials are bivariate, bispectral and biorthogonal. For two specific values of the parameter…

We prove an implicit function theorem for non-commutative functions. We use this to show that if $p(X,Y)$ is a generic non-commuting polynomial in two variables, and $X$ is a generic matrix, then all solutions $Y$ of $p(X,Y)=0$ will commute…

A well known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a pseudodifferential projection vanishes. This statement is non-local and implies the regularity of the eta invariant at zero…