Related papers: On a quaternionic Picard theorem

We show sufficient and necessary conditions, in terms of some partial differential equations with variable coefficients, for a quaternionic function to admit a continuous derivative in a open set in the sense of C. Schwartz.

In this article we give an approach to define continuous functional calculus for bounded quaternionic normal operators defined on a right quaternionic Hilbert space.

Let $k$ be an algebraically closed field of characteristic zero and $P(x,y)\in k[x,y]$ be a polynomial which depends on all its variables. $P$ has an algebraic constraint if the set $\{(P(a,b),(P(a',b'),P(a',b),P(a,b')\,|\,a,a',b,b'\in k\}$…

We give a partial answer to a question attributed to Chris Miller on algebraic values of certain transcendental functions of order less than one. We obtain C(logH)^n bounds for the number of algebraic points of height at most H on certain…

A classical theorem of Herglotz states that a function $n\mapsto r(n)$ from $\mathbb Z$ into $\mathbb C^{s\times s}$ is positive definite if and only there exists a $\mathbb C^{s\times s}$-valued positive measure $d\mu$ on $[0,2\pi]$ such…

Let H be the real algebra of quaternions. The notion of regular function of a quaternionic variable recently presented by G. Gentili and D. C. Struppa developed into a quite rich theory. Several properties of regular quaternionic functions…

In this paper, we utilize various integral representations derived from the Fueter-Sce extension theorem, to introduce novel functional calculi tailored for quaternionic operators of sectorial type. Specifically, due to the different…

In this paper we introduce fractional powers of quaternionic operators. Their definition is based on the theory of slice-hyperholomorphic functions and on the $S$-resolvent operators of the quaternionic functional calculus. The integral…

We study automorphisms and invariants for the algebra $\mathbb{O}$ of octonions and octonionic slice regular functions $f:\mathbb{O} \to \mathbb{O}$.

We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing…

We employ tools from complex analysis to construct the $*$-logarithm of a quaternionic slice regular function. Our approach enables us to achieve three main objectives: we compute the monodromy associated with the $*$-exponential; we…

Let $\mathcal{H}$ be a complex Hilbert space and $T:\mathcal{H}\to \mathcal{H}$ be a contraction. Let $$A_nf=\frac{1}{n}\sum_{j=1}^nT^jf$$ for $f\in \mathcal{H}$. Let $(n_k)$ be a lacunary sequence, then there exists a constant $C_1>0$ such…

We discuss the use of the variational principle within quaternionic quantum mechanics. This is non-trivial because of the non commutative nature of quaternions. We derive the Dirac Lagrangian density corresponding to the two-component Dirac…

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ of variables if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:\Delta^*\to\Sigma^*$ is a non-erasing…

We prove the real non-attractive fixed point conjecture for complex polynomial and rational harmonic functions. A harmonic function $f=h+\overline{g}$ is polynomial (rational) if both $h$ and $g$ are polynomials (rational functions) of…

Property $\mathcal{O}$ for an arbitrary complex, Fano manifold $X$, is a statement about the eigenvalues of the linear operator obtained from the quantum multiplication of the anticanonical class of $X$. Conjecture $\mathcal{O}$ is a…

We study some harmonic properties of slice regular functions in one and several Clifford variables and give explicit formulas of the iterated Laplacian applied to slice regular functions and to their spherical derivative, which are new also…

We revisit, qualify, and objectively resolve the seemingly controversial question about what is the number of dimensional fundamental constants in Nature. For this purpose, we only assume that all we can directly measure are space and time…

We propose and study a generalized continued fraction algorithm that can be executed in an arbitrary imaginary quadratic field, the novelty being a non-restriction to the five Euclidean cases. Many hallmark properties of classical continued…

Slice Fueter-regular functions, originally called slice Dirac-regular functions, are generalized holomorphic functions defined over the octonion algebra $\mathbb{O}$, recently introduced by M. Jin, G. Ren and I. Sabadini. A function…