Asymptotic Behavior of Critical Primitive Multi-Type Branching Processes with Immigration

Under natural assumptions a Feller-type diffusion approximation is derived for critical multi-type branching processes with immigration when the offspring mean matrix is primitive (in other words, positively regular). Namely, it is proved that a sequence of appropriately scaled random step functions formed from a sequence of critical primitive multi-type branching processes with immigration converges weakly toward a squared Bessel process supported by a ray determined by the Perron vector of the offspring mean matrix.     

Transition between the Lagrange and Hermite–Fejér interpolations. I (Trigonometric case)

We construct a special class of discrete φ-summations to describe the transition between the well known (trigonometric) Lagrange and Hermite–Fejér interpolations. We give general properties, operator norm and convergence order for these cases.     

On a normed version of a Rogers–Shephard type problem

A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of n-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities for the four types of volumes regularly used in the literature: Busemann, Holmes–Thompson, and Gromov’s mass and mass*. We examine the problem also for higher dimensions, and for centrally symmetric convex bodies.     

Adoption of the Numerical Method of Chebyshev Polynomials to the Stability Analysis of Delayed DEs

We propose a new way of the application of the numerical method based on Chebyshev approximation concerning the stability analysis of (periodic) delay‐differential equations. The technique is tested on the delayed Mathieu equation.