Unique paired vs edge-vertex minimum dominating sets in trees
We prove that the class of trees with unique minimum edge-vertex dominating sets is equivalent to the class of trees with unique minimum paired dominating sets.
math.CO↗
arXiv subjects
Michał Zakrzewski
Publications and source records attributed to Michał Zakrzewski.
Emergence of the polydeterminant in QCD
A generalization of the determinant appears in particle physics in effective Lagrangian interaction terms that model the chiral anomaly in Quantum Chromodynamics (PRD 97 (2018) 9, 091901 PRD 109 (2024) 7, L071502), in particular in connection to mesons. This \textit{polydeterminant function}, known in the mathematical literature as a mixed discriminant, associates $N$ distinct $N\times N$ complex matrices into a complex number and reduces to the usual determinant when all matrices are taken as equal. Here, we explore the main properties of the polydeterminant applied to (quantum) fields by using a formalism and a language close to high-energy physics approaches. We discuss its use as a tool to write down novel chiral anomalous Lagrangian terms and present an explicit illustrative model for mesons. Finally, the extension of the polydeterminant as a function of tensors is shown.
math-ph↗
Variations on hypergeometric functions
We prove new integral formulas for generalized hypergeometric functions and their confuent variants. We apply them, via stationary phase formula, to study WKB expansions of solutions: for large argument in the confuent case and for large parameter in the general case. We also study variations of hypergeometric functions for small perturbations of hypergeometric equations, i.e., in expansions of solutions in powers of a small parameter. Next, we present a new proof of a theorem due to Wasow about equivalence of the Airy equation with its perturbation; in particular, we explain that this result does not deal with the WKB solutions and the Stokes phenomenon. Finally, we study hypergeometric equations, one of second order and another of third order, which are related with two generating functions for MZVs, one $Δ_2 (λ)$ for $ζ(2, \ldots , 2)$'s and another $Δ_3 (λ)$ for $ζ(3, \ldots , 3)$'s; in particular, we correct a statement from [ZZ3] that the function $Δ_3(λ)$ admits a regular WKB expansion.
math.CA↗
Low-degree spanning trees of $2$-edge-connected graphs in linear time
We present a simple linear-time algorithm that finds a spanning tree $T$ of a given $2$-edge-connected graph $G$ such that each vertex $v$ of $T$ has degree at most $\lceil \frac{°_G(v)}{2}\rceil + 1$.
cs.DS↗