Related papers: Non-crossing Knight's Tour in 3-Dimension
Two algorithms for construction of all closed knight's paths of lengths up to 16 are presented. An approach for classification (up to equivalence) of all such paths is considered. By applying the construction algorithms and classification…
We extend quantum circuit cutting to heterogeneous registers comprising mixed-dimensional qudits. By decomposing non-local interactions into tensor products of local generalised Gell-Mann matrices, we enable the simulation and execution of…
In 3-dimensional Euclidean space there exist two exceptional polyhedra, the rhombic dodecahedron and the rhombic triacontahedron, the only known polytopes (besides polygons) that are edge-transitive without being vertex-transitive. We show…
Multi-dimensional quantum walks can exhibit highly non-trivial topological structure, providing a powerful tool for simulating quantum information and transport systems. We present a flexible implementation of a 2D optical quantum walk on a…
We present a novel approach to quantizing the length in noncommutative spaces with positional-dependent noncommutativity. The method involves constructing ladder operators that change the length not only along a plane but also along the…
The expected hitting time of discrete quantum walks on a hypercube (HC) is numerically known to be exponentially shorter than that of their classical analogs in terms of the scaling with the HC dimension. Recent numerical analyses…
There exists a proper holomorphic mapping between balls of different dimensions such that it does not extend continuously to the boundary. The aim of this paper is to show the same phenomenon occurs for pseudoconvex domains of different…
The $n$ queens problem considers the maximum number of safe squares on an $n \times n$ chess board when placing $n$ queens; the answer is only known for small $n$. Miller, Sheng and Turek considered instead $n$ randomly placed rooks,…
Obstacle crossing is an essential component of human locomotion, particularly for individuals with lower limb amputations who face elevated risks of imbalance and falls. While prior studies have explored this task, they often lack a…
Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to…
Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries of the tetrahedron in R^3, we use…
Quantum walks, in virtue of the coherent superposition and quantum interference, possess exponential superiority over its classical counterpart in applications of quantum searching and quantum simulation. The quantum enhanced power is…
We consider three kinds of quotients of the curve complex which are obtained by coning off uniformly quasi-convex subspaces: symmetric curve sets, non-maximal train track sets, and compression body disc sets. We show that the actions of the…
The main problem considered in this paper is construction and theoretical study of efficient $n$-point coverings of a $d$-dimensional cube $[-1,1]^d$. Targeted values of $d$ are between 5 and 50; $n$ can be in hundreds or thousands and the…
We study the high-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condition, namely, the path cannot take any step in the…
We explore the flow of quantum correlations in cluster states defined on ladder type graphs as measurements are done on qubits located on the nodes of the cluster. We focus on three qubits at the end of the ladder and compute the…
We will discuss all possible closures of a Euclidean line in various geometric spaces. Imagine the Euclidean traveller, who travels only along a Euclidean line. She will be travelling to many different geometric worlds, and our question…
We measure a triple quantum dot in the regime where three addition lines, corresponding to the addition of an electron to each of three dots, pass through each other. In particular, we probe the interplay between transport and the…
The Shortest Cycle Cover Conjecture of Alon and Tarsi asserts that the edges of every bridgeless graph with $m$ edges can be covered by cycles of total length at most $7m/5=1.400m$. We show that every cubic bridgeless graph has a cycle…
In this paper we find an upper bound for the probability that a $3$ dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball of radius $N$. For $d\ge 4$, it has been shown in…