Almost all derivative quivers of artinian biserial rings contain chains

A lower estimate for the number Mn of all labelled quivers with n–vertex parts of Artinian biserial rings is given and the asymptotic of the relation Mn/Bn, where Bn denotes the number of those quivers all connected components of which are cycles, is studied

Length of the inverse symmetric semigroup

The length of the lattice of subsemigroups of the inverse symmetric semigroup ISn is calculated

On finite monoids of cellular automata.

For any group G and set A, a cellular automaton over G and A is a transformation τ:AG→AGτ:AG→AG defined via a finite neighbourhood S⊆GS⊆G (called a memory set of ττ) and a local function μ:AS→Aμ:AS→A. In this paper, we assume that G and A are both finite and study various algebraic properties of the finite monoid CA(G,A)CA(G,A) consisting of all cellular automata over G and A. Let ICA(G;A)ICA(G;A) be the group of invertible cellular automata over G and A. In the first part, using information on the conjugacy classes of subgroups of G, we give a detailed description of the structure of ICA(G;A)ICA(G;A) in terms of direct and wreath products. In the second part, we study generating sets of CA(G;A)CA(G;A). In particular, we prove that CA(G,A)CA(G,A) cannot be generated by cellular automata with small memory set, and, when G is finite abelian, we determine the minimal size of a set V⊆CA(G;A)V⊆CA(G;A) such that CA(G;A)=⟨ICA(G;A)∪V⟩CA(G;A)=⟨ICA(G;A)∪V⟩

On the interplay between Babai and Černý’s conjectures

Motivated by the Babai conjecture and the Černý conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with n states in this class, we prove that the reset thresholds are upperbounded by 2n2 -6n + 5 and can attain the value (Formula presented). In addition, we study diameters of the pair digraphs of permutation automata and construct n-state permutation automata with diameter (formula presented). © Springer International Publishing AG 2017

Cross-connections of linear transformation semigroups

Cross-connection theory developed by Nambooripad is the construction of a regular semigroup from its principal left (right) ideals using categories. We use the cross-connection theory to study the structure of the semigroup Sing(V) of singular linear transformations on an arbitrary vector space V over a field K. There is an inbuilt notion of duality in the cross-connection theory, and we observe that it coincides with the conventional algebraic duality of vector spaces. We describe various cross-connections between these categories and show that although there are many cross-connections, upto isomorphism, we have only one semigroup arising from these categories. But if we restrict the categories suitably, we can construct some interesting subsemigroups of the variants of the linear transformation semigroup. © 2018, Springer Science+Business Media, LLC, part of Springer Nature

Сокращение ледников Восточного Алтая (Шапшальский центр) после максимума малого ледникового периода

Based on the analysis of remote data and field observations, we reconstructed the glaciation of the Shapshal Center (Eastern Altai) for the maximum of the Little Ice Age (LIA) and by the state of the glaciers as of 2001. At the maximum of the LIA, glaciation was represented by 358 glaciers with a total area of 84.43 km2. It was found 87% reduction of the total area of glaciers in the interval from the LIA maximum to 2015. During the reduction, valley glaciers disintegrated and glaciers in the Kargy River basin disappeared. The moraines of the LIA have low lake coverage (0.17% of area), therefore a probability of their breakthrough is low. We obtained data on the retreat of the Mushtuk Glacier (№ 78), the largest one of the Shapshal center, in five time slices from the LIA maximum. The highest retreat rates were reconstructed in the interval 1989–2001, but in the interval 2010–2016 the average rates decreased to 5 m/year. Changes in the mass balance index of the Mushtuk Glacier between from 1961 to 2018 were calculated. A sharp decrease in the mass balance in the 1990s and stabilization of values at a low level after 2001 were found. According to the calculations, the response time of the Mushtuk Glacier was about 9 years. If the current climatic conditions persist, there is reason to assume stabilization of glaciers in the coming years.Реконструированы ледники малоисследованного Шапшальского центра оледенения на Восточном Алтае в максимум малого ледникового периода, проанализирован характер их последующего сокращения. Детально рассмотрено сокращение крупнейшего ледника Шапшальского хребта по пяти временным срезам c 1955 по 2019 г. Проведены расчёты индекса баланса массы ледника и времени его климатического отклика

Состояние Шапшальского центра оледенения (Восточный Алтай) в 2015 году

Catalogues and maps of glaciers (for 2015) of Shapshal Glacier Center, located in the eastern part of the Russian Altai, have been created based on the first field glaciological observations and space images interpretation. In total 123 glaciers with the total area of 14.07 km2 have been allocated. In comparison with the data from the Glacier Inventory of the USSR (1955–1965), the total area of the glaciers has decreased by more than 2 times. The lower limit of glacier development is 2475 m, to the south-east of the region it rises by 1 km, the height of the firm line rises from 2860 m to 3460 m, respectively. Small glaciers prevail (70% of glaciers have an area less than 0.1 km2, the area of the largest glacier is 0.9 km2). In terms of quantity and area, cirque glaciers predominate, there are no valley glaciers. The largest numbers of glaciers have northern and northeastern exposure, with the largest areas of glaciers concentrated on the north-eastern slopes. The highest glaciation intensity has been detected on the eastern slope of the Skalistiy Ridge and the northeastern slope of the southern part of the Shapshalsky Ridge in the upper reaches of the Chon-Khem River, which are optimal for glaciers by a combination of mountain heights and position relative to moisture-bearing atmospheric flows. To the west of these areas, intensity of glacierization decreases due to lower mountain heights, to the east – due to lower precipitation. In general, with low (0.1 km−1 and less) intensity of glacierization, the Shapshal Centre is an area of dispersed glaciation, most glaciers of which are on the verge of disappearance.Получены новые карты и каталог ледников мало исследованного Шапшальского центра оледенения на Восточном Алтае. По состоянию на 2015 г. здесь насчитывалось 123 ледника суммарной площадью 14,07 км2. По сравнению с данными 1955–1965 гг. суммарная площадь ледников сократилась более чем вдвое. Преобладают малые каровые ледники северо-восточной и северной экспозиций. Площади ледников и интенсивность оледенения убывают с северо-запада на юго-восток