arXiv subjects

Petr Klimov

Publications and source records attributed to Petr Klimov.

3 recordsLinked to original sources

arXiv2026

Universal Fibonacci sequences and UFS-groupoids

In a binary groupoid $(G, *)$, a Fibonacci sequence is a recurrent sequence defined by $f_1 = a, f_2 = b, \ldots, f_n = f_{n - 2} * f_{n - 1}$. A universal Fibonacci sequence (UFS) is a singly or doubly infinite sequence whose set of suffixes coincides precisely with the set of all Fibonacci sequences in the groupoid. This paper studies UFS-groupoids, i.e., groupoids that admit a universal Fibonacci sequence. It is shown that every nontrivial UFS-groupoid is at most countable, locally cyclic, and non-power-associative; that the right cancellation property and the right quasigroup property hold for all pairs of elements except possibly one and two, respectively; that no neutral element or zero element exists; and that there is at most one idempotent element. It is proved that any UFS-groupoid whose universal Fibonacci sequence is not doubly infinite strictly preperiodic is cyclic. It has also been proved that the class of UFS-groupoids is closed under taking subgroupoids and homomorphic images, but is not closed under finite direct products. The structure of subgroupoids of UFS-groupoids is described. A complete classification of UFS-groupoids is given in terms of the cardinality of $G$ and the periodicity of the universal Fibonacci sequences. Finite UFS-groupoids are described combinatorially via de Bruijn sequences. The number of distinct UFS-groupoids on a finite set is determined, and explicit constructions are provided for both finite and infinite cases across all periodicity classes, including embeddings of UFS-groupoids as subgroupoids into other UFS-groupoids and infinitely generated UFS-groupoids.

math.GR↗

arXiv2026

On the Construction of Recursively Differentiable Quasigroups and an Example of a Recursive $[4,2,3]_{26}$-Code

In 1998, E. Couselo, S. Gonz\'alez, V. T. Markov, and A. A. Nechaev introduced the notions of recursive codes and recursively differentiable quasigroups. They conjectured that recursive MDS codes of dimension $2$ and length $4$ exist over every finite alphabet of size $q \not\in \{2, 6\}$, and verified this conjecture in all cases except $q \in \{14, 18, 26, 42\}$. In 2008, V. T. Markov, A. A. Nechaev, S. S. Skazhenik, and E. O. Tveritinov resolved the case $q=42$ by providing an explicit construction. The present paper settles the outstanding case $q=26$. The construction rests upon methods for producing recursively differentiable quasigroups and recursive MDS codes via perfect cyclic Mendelsohn designs. Moreover, we sharpen several known bounds concerning the existence of recursively $n$-differentiable quasigroups of small orders.

cs.IT↗

arXiv2023

On Translation-Invariant Matrix Product States and advances in MPS representations of the $W$-state

This work is devoted to the study Translation-Invariant (TI) Matrix Product State (MPS) representations of quantum states with periodic boundary conditions (PBC). We pursue two directions: we introduce new methods for constructing TI MPS representations of a certain class of TI states and study their optimality in terms of their bond dimension. We pay particular attention to the $n$-party $W$-state and construct a TI MPS representation of bond dimension $\left \lfloor \dfrac{n}{2} \right \rfloor +1$ for it. We further study properties of this class and show that we can can always achieve a bond dimension of $n$ for TI MPS representation of states in this class. In the framework of studying optimality of TI MPS representations with PBC, we study the optimal bond dimension $d(\psi)$ for a given state $\psi$. In particular we introduce a deterministic algorithm for the search of $d(\psi)$ for an arbitary state. Using numerical methods, we verify the optimality of our previous construction for the $n$-party $W$-state for small $n$.

quant-ph↗