A Hybrid Search Algorithm for the Whitehead Minimization Problem

The Whitehead Minimization problem is a problem of finding elements of the minimal length in the automorphic orbit of a given element of a free group. The classical algorithm of Whitehead that solves the problem depends exponentially on the group rank. Moreover, it can be easily shown that exponential blowout occurs when a word of minimal length has been reached and, therefore, is inevitable except for some trivial cases. In this paper we introduce a deterministic Hybrid search algorithm and its stochastic variation for solving the Whitehead minimization problem. Both algorithms use search heuristics that allow one to find a length-reducing automorphism in polynomial time on most inputs and significantly improve the reduction procedure. The stochastic version of the algorithm employs a probabilistic system that decides in polynomial time whether or not a word is minimal. The stochastic algorithm is very robust. It has never happened that a non-minimal element has been claimed to be minimal

Heuristics for The Whitehead Minimization Problem

In this paper we discuss several heuristic strategies which allow one to solve the Whitehead's minimization problem much faster (on most inputs) than the classical Whitehead algorithm. The mere fact that these strategies work in practice leads to several interesting mathematical conjectures. In particular, we conjecture that the length of most non-minimal elements in a free group can be reduced by a Nielsen automorphism which can be identified by inspecting the structure of the corresponding Whitehead Graph

Conjugacy in Baumslag's group, generic case complexity, and division in power circuits

The conjugacy problem belongs to algorithmic group theory. It is the following question: given two words x, y over generators of a fixed group G, decide whether x and y are conjugated, i.e., whether there exists some z such that zxz^{-1} = y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the complexity of the conjugacy problem for two prominent groups: the Baumslag-Solitar group BS(1,2) and the Baumslag(-Gersten) group G(1,2). The conjugacy problem in BS(1,2) is TC^0-complete. To the best of our knowledge BS(1,2) is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group G(1,2) is an HNN-extension of BS(1,2). We show that the conjugacy problem is decidable (which has been known before); but our results go far beyond decidability. In particular, we are able to show that conjugacy in G(1,2) can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs conjugacy in G(1,2) can be decided efficiently. In contrast, we show that under a plausible assumption the average case complexity of the same problem is non-elementary. Moreover, we provide a lower bound for the conjugacy problem in G(1,2) by reducing the division problem in power circuits to the conjugacy problem in G(1,2). The complexity of the division problem in power circuits is an open and interesting problem in integer arithmetic.Comment: Section 5 added: We show that an HNN extension G = < H, b | bab^-1 = {\phi}(a), a \in A > has a non-amenable Schreier graph with respect to the base group H if and only if A \neq H \neq

Synthetic strategy toward furyl- and benzofuryl-containing building blocks for organic materials

Received: 17.05.22. Revised: 23.06.22. Accepted: 25.06.22. Available online: 12.07.22.A synthetic approach to furyl- and benzofuryl-containing building blocks utilizing easily accessible substrates is reported. Cascade acidcatalyzed reactions of 2-methylfuran with α,β-unsaturated carbonyl compounds or salicyl alcohols followed by oxidation afford function-alized furans and benzofurans, respectively. Synthetic potential of the obtained products was demonstrated by synthesizing hetarylsubstituted heterocycles, which may find an application in materials chemistry.The research was supported by the Perm Research and Education Centre for Rational Use Subsoil, 2022

Discriminating Groups

A group G is termed discriminating if every group separated by G is discriminated by G. In this paper we answer several questions concerning discrimination which arose from [2]. We prove that a finitely generated equationally Noetherian group G is discriminating if and only if the quasivariety generated by G is the minimal universal class containing G. Among other results, we show that the non-abelian free nilpotent groups are non-discriminating. Finally we list some open problems concerning discriminating groups

EXTENDED COREY-CHAYKOVSKY REACTION AS A PATHWAY FOR THE SYNTHESIS OF SUBSTITUTED FURANS

This work was supported by RSF № 21-73-10063

Cryo-EM structures and binding of mouse and human ACE2 to SARS-CoV-2 variants of concern indicate that mutations enabling immune escape could expand host range.

Investigation of potential hosts of the severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) is crucial to understanding future risks of spillover and spillback. SARS-CoV-2 has been reported to be transmitted from humans to various animals after requiring relatively few mutations. There is significant interest in describing how the virus interacts with mice as they are well adapted to human environments, are used widely as infection models and can be infected. Structural and binding data of the mouse ACE2 receptor with the Spike protein of newly identified SARS-CoV-2 variants are needed to better understand the impact of immune system evading mutations present in variants of concern (VOC). Previous studies have developed mouse-adapted variants and identified residues critical for binding to heterologous ACE2 receptors. Here we report the cryo-EM structures of mouse ACE2 bound to trimeric Spike ectodomains of four different VOC: Beta, Omicron BA.1, Omicron BA.2.12.1 and Omicron BA.4/5. These variants represent the oldest to the newest variants known to bind the mouse ACE2 receptor. Our high-resolution structural data complemented with bio-layer interferometry (BLI) binding assays reveal a requirement for a combination of mutations in the Spike protein that enable binding to the mouse ACE2 receptor

Equivalent temperature of nonlinear-optical crystals interacting with laser radiation

Temperature calibrated piezoelectric resonances of internal acoustic vibration modes of a nonlinear-optical crystal during its heating by high-power laser radiation are used for noncontact highly accurate measurements of both the non-uniform temperature distribution in the crystal volume and in the surrounding air. A novel notion of equivalent temperature of a crystal heated by laser radiation is introduced in laser physics. The true non-uniform crystal thermodynamic temperature at a given laser power is substituted by the measured equivalent crystal temperature, which is constant at that laser power. Using appropriate laser heating model the measured value of the equivalent crystal temperature allows one to calculate the unknown linear and nonlinear optical absorption coefficients as well as the heat transfer coefficient of the crystal with the surrounding air