We study the problem of placing effective upper bounds for the number of zeroes of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of a given dimension $n$ having $m$ singular points. As a function of $n,m$, this bound turns out to be double exponential in the precise sense explained in the paper.As a corollary, we obtain a solution of the so-called restricted infinitesimal...

In this paper I discuss quantum systems whose Hamiltonians are non-Hermitian but whose energy levels are all real and positive. Such theories are required to be symmetric under $𝒞𝒫𝒯$, but not symmetric under $𝒫$ and $𝒯$ separately. Recently, quantum mechanical systems having such properties have been investigated in detail. In this paper I extend the results to quantum field theories. Among the systems that I discuss are $-{\phi }^4$ and $i{\phi }^3$ theories. These theories all have unexpected and remarkable properties. I discuss...

We introduce and discuss a new class of (multivalued analytic) transcendental functions which still share with algebraic functions the property that the number of their isolated zeros can be explicitly counted. On the other hand, this class is sufficiently rich to include all periods (integral of rational forms over algebraic cycles).

For a Fuchsian system $dY/dx=({\sum }_{j=}^p\left(A_j\right)/(x-t_j))Y$, (F) $t₁,t₂,...,t_p$ being distinct points in ℂ and $A₁,A₂,...,A_p\in M(n\times n;ℂ)$, the number α of accessory parameters is determined by the spectral types $s\left(A₀\right),s\left(A₁\right),...,s\left(A_p\right)$, where $A₀=-{\sum }_{j=1}^p{A}_j$. We call the set $z=(z₁,z₂,...,z_{\alpha })$ of α parameters a regular coordinate if all entries of the $A_j$ are rational functions in z. It is not yet known that, for any irreducibly realizable set of spectral types, a regular coordinate does exist. In this paper we study a process of obtaining a new regular coordinate from a given one by a coalescence of eigenvalues of the matrices...

Dans cet article, nous montrons que la notion analytique d’exposants développée par Levelt pour les systèmes différentiels linéaires en une singularité régulière s’interprète algébriquement en termes d’invariants de réseaux, relatifs à un réseau stable maximal que nous appelons « réseau de Levelt ». Nous obtenons en particulier un encadrement pour la somme des exposants des systèmes n’ayant que des singularités régulières sur ${\mathbb{P}}^1(ℂ$).

Analysis of a non-classically damped engineering structure, which is subjected to an external excitation, leads to the solution of a system of second order ordinary differential equations. Although there exists a large variety of powerful numerical methods to accomplish this task, in some cases it is convenient to formulate the explicit inversion of the respective quadratic fundamental system. The presented contribution uses and extends concepts in matrix polynomial theory and proposes an implementation...

We are concerned with the uniqueness problem for solutions to the second order ODE of the form $x^{\text{'}\text{'}}+f(x,t)=0$, subject to appropriate initial conditions, under the sole assumption that $f$ is non-decreasing with respect to $x$, for each $t$ fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable. The interest in this problem comes, among other, from the study of oscillations of lumped parameter systems with implicit...