Related papers: Polar complex numbers in n dimensions

We introduce a new class of holomorphic polynomials extending the classical Gould--Hopper to two complex variables. The considered polynomials include the $1$-D and $2$-D holomorphic and polyanalytic It\^o--Hermite polynomials as particular…

This thesis examins a generalisation of polar decompositions to indefinite inner product spaces. The necessary general theory is studied and some general results are given. The main part of the thesis focuses on polar decompositions with…

Polar spaces over finite fields are fundamental in combinatorial geometry. The concept of polar space was firstly introduced by F. Veldkamp who gave a system of 10 axioms in the spirit of Universal Algebra. Later the axioms were simplified…

Integrals involving derivatives of Legendre polynomials frequently arise in applications ranging from multipole expansions for processes involving electromagnetic probes to spectral methods in numerical physics. Despite their practical…

Suppose that we are given a formal power series of many variables with coefficients in $\mathbb{R}$ (or $\mathbb{C}$) and we want to compute its $n$-th (multiplicative) root. As can be expected coefficients of the root have to satisfy a…

We introduce the notion of the generalized-analytical function of the poly-number variable, which is a non-trivial generalization of the notion of analytical function of the complex variable and, therefore, may turn out to be fundamental in…

Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory,…

We introduce a spinorial version of the scattering equations, the \emph{polarized scattering equations}, that incorporates spinor polarization data. They lead to new formulae for tree-level scattering amplitudes in six dimensions that…

Using a natural representation of a $1/s$-concave function on $\mathbb{R}^d$ as a convex set in $\mathbb{R}^{d+1},$ we derive a simple formula for the integral of its $s$-polar. This leads to convexity properties of the integral of the…

We present the Polar framework for fully automating the analysis of classical and probabilistic loops using algebraic reasoning. The central theme in Polar comes with handling algebraic recurrences that precisely capture the loop semantics.…

We study integral representation of so-called $d$-dimensional Catalan numbers $C_{d}(n)$, defined by $[\prod_{p=0}^{d-1} \frac{p!}{(n+p)!}] (d n)!$, $d = 2, 3, ...$, $n=0, 1, ...$. We prove that the $C_{d}(n)$'s are the $n$th Hausdorff…

We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This…

We show that the n-fold integrals $\chi^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green…

For certain negative rational numbers k, called singular values, and associated with the symmetric group S_N on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter equals k. It was shown by…

Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.

We discuss the evaluation of certain d dimensional angular integrals which arise in perturbative field theory calculations. We find that the angular integral with n denominators can be computed in terms of a certain special function, the…

Loop calculations involve the evaluation of divergent integrals. Usually [1] one computes them in a number of dimensions different than four where the integral is convergent and then one performs the analytical continuation and considers…

We classify irreducible polar foliations of codimension $q$ on quaternionic projective spaces $\mathbb H P^n$, for all $(n,q)\neq(7,1)$. We prove that all irreducible polar foliations of any codimension (resp. of codimension one) on…

This paper introduces an extension to the normal distribution through the polar method to capture bimodality and asymmetry, which are often observed characteristics of empirical data. The later two features are entirely controlled by a…

Polar codes are the first error-correcting codes to provably achieve the channel capacity but with infinite codelengths. For finite codelengths the existing decoder architectures are limited in working frequency by the partial sums…