arXiv subjects

Klas Markstrom

Publications and source records attributed to Klas Markstrom.

3 recordsLinked to original sources

arXiv2026

Improved lower bounds for the maximum size of Condorcet domains

Condorcet domains are sets of linear orders with the property that, whenever voters' preferences are restricted to the domain, the pairwise majority relation (for an odd number of voters) is transitive and hence a linear order. Determining the maximum size of a Condorcet domain, sometimes under additional constraints, has been a longstanding problem in the mathematical theory of majority voting. The exact maximum is only known for $n\leq 8$ alternatives. In this paper we use a structural analysis of the largest domains for small $n$ to design a new inductive search method. Using an implementation of this method on a supercomputer, together with existing algorithms, we improve the size of the largest known domains for all $9 \leq n \leq 20$. These domains are then used in a separate construction to obtain the currently largest known domains for $21 \leq n \leq 25$, and to improve the best asymptotic lower bound for the maximum size of a Condorcet domain to $\Omega(2.198139^n)$. Finally, we discuss properties of the domains found and state several open problems and conjectures.

cs.DM↗

arXiv2010

A quantum Bose-Hubbard model with evolving graph as toy model for emergent spacetime

We present a toy model for interacting matter and geometry that explores quantum dynamics in a spin system as a precursor to a quantum theory of gravity. The model has no a priori geometric properties, instead, locality is inferred from the more fundamental notion of interaction between the matter degrees of freedom. The interaction terms are themselves quantum degrees of freedom so that the structure of interactions and hence the resulting local and causal structures are dynamical. The system is a Hubbard model where the graph of the interactions is a set of quantum evolving variables. We show entanglement between spatial and matter degrees of freedom. We study numerically the quantum system and analyze its entanglement dynamics. We analyze the asymptotic behavior of the classical model. Finally, we discuss analogues of trapped surfaces and gravitational attraction in this simple model.

gr-qc↗

arXiv2006

The 1-vertex transfer matrix and accurate estimation of channel capacity

The notion of a 1-vertex transfer matrix for multi-dimensional codes is introduced. It is shown that the capacity of such codes, or the topological entropy, can be expressed as the limit of the logarithm of spectral radii of 1-vertex transfer matrices. Storage and computations using the 1-vertex transfer matrix are much smaller than storage and computations needed for the standard transfer matrix. The method is applied to estimate the first 15 digits of the entropy of the 2-dimensional (0,1) run length limited channel. In order to compare the computational cost of the new method with the standard transfer matrix and have rigorous bounds to compare the estimates with a large scale computation of eigenvalues for the (0,1) run length limited channel in 2 and 3 dimensions have been carried out. This in turn leads to improvements on the best previous lower and upper bounds for that channel.

math-ph↗