In the first part of this paper, we give a classification of nontrivial left and right distributive hyperidentities satisfied in nontrivial divisible algebras. In the second part is characterized semigroups with hyperidentities of associativity with a singular functional variable.
Let (omega ) be the set of all nonnegative integers. Let P be a class of problems recognized by deterministic Turing machines, which run in polynomial time. It is known that effective enumeration of the sets of the class P (namely, ({{P}_{0}},{{P}_{1}}), …, ({{P}_{i}}), …) exists and thus ({mathbf{P}} = { {{P}_{i}},|,i in omega } .) Note that for each (i), ({{P}_{i}}) is a set of strings that are sequences of 0s and 1s. Based on available numbering of computably enumerable (c.e.) sets ({{{ {{W}_{i}}} }_{{i in omega }}}), a sequence of sets of non-negative numbers ({{hat {P}}_{i}}) is constructed such that there is an effective enumeration of them. Let us define ({mathbf{hat {P}}}) as follows: (~{mathbf{hat {P}}} = { {{hat {P}}_{i}},|,i in omega } ). It’s obvious that it is possible to define such relations between the elements of the set of mentioned strings and between the elements of the set of nonnegative integers that these two sets will be isomorphic (with respect to the relations in question). The article shows that it is possible to define such relations between the elements of ({mathbf{P}}) and between the elements of (hat {{mathbf{P}}}) that there will be homomorphic mappings from ({mathbf{P}}) to (hat {{mathbf{P}}}) and vice versa, from (hat {{mathbf{P}}}) to ({mathbf{P}}) (with respect to the relations in question). Based on the notions of T-mitoticity and T-autoreducibility, Ambos-Spies introduced the notions of P‑T-mitoticity, weakly P-T-mitoticity and P-T-autoreducibility. By analogy with the mentioned notions we introduce the notions of (hat {P})-T-mitoticity, weakly (hat {P})-T-mitoticity and (hat {P})-T-autoreducibility. It is proved in the article that the index sets {({text{z}},|,{{{text{W}}}_{{text{z}}}}) is ({{hat {P}}})-T-mitotic}, ({text{{ z}},|,{{{text{W}}}_{{text{z}}}}) is weakly ({{hat {P}}})-T-mitotic}, ({text{{ }}~{text{z}},|,{{{text{W}}}_{{text{z}}}}) is ({{hat {P}}})-T-autoreducible} and ({text{{ z}},|,{{{text{W}}}_{{text{z}}}} in {mathbf{hat {P}}}} ) are ({{{mathbf{Sigma }}}_{3}})-complete.
In 1980, Bondy conjectured a common generalization (depending on λ) of some well-known degree-sum conditions for a graph ensuring the existence of Hamilton cycles for λ = 1 (Ore, 1960) and dominating cycles for λ = 2 (Bondy, 1980) as special cases. The reverse (long-cycle) version of Bondy’s conjecture was proposed in 2001 due to Jung. The importance of these two conjectures in the field is motivated by the fact that they (as starting points) give rise to all (with few exceptions) further developments through various additional extensions and limitations. In this paper, we briefly outline all known notable achievements towards solving the problem: (i) confirmation (by the author) of Bondy’s and Jung’s conjectures for some versions that are very close to the original versions; and (ii) significant improvements (by the author) of results (i), inspiring a number of improved versions of original conjectures of Bondy and Jung. Next we derive a number of modifications from improvements in (ii), which are also very close to the original versions, but do not follow directly from the Bondy’s and Young’s conjectures. Finally, all results (both old and new) are shown to be best possible in a sense based on three types of sharpness, indicating the intervals in 0 < λ < δ + 1 where the result is sharp and the intervals where the result can be further improved, where δ denotes the minimum degree.
According to the hypothesis of abiogenesis, the simplest cellular, uncials, originated from chemical compounds that already existed in nature. Unfortunately, in spite of ongoing intensive research efforts, abiogenesis owns more difficulties and hopes than advances. That is why new hypotheses try to exempt its difficulties. Particularly, successful modeling of cognizing lets us assume that uncials were designed by some cognizers of the Universe, originated in nature as elementary recurrent classifiers, then evolved to attain the power of cognizing, at least, comparable with the highest human one, allowing them to design uncials analogous to the human design of robots nowadays. In parallel, molecular studying assumes that even elementary units of matter are able to communicate through the IDs of classifiers. And since the constituents of uncials are functionally analogous to those of cognizers, while communication is vital for cognizing, it is worth trying to promote the origin of constituents of cognizers by reaching in abiogenesis and communications. Thus, to promote origination of cognizers, we decompose the nuclei of cognizers to constituents, followed by examining the potential impact of constituents of uncials and molecular recurrent classifiers to the origin of functionally analogous ones of cognizers. Then recall algorithms of formation of 1-/2-place classifiers for possible clues to their origination. Finally, address to the origin of dynamicity of the nuclei of cognizers–doers, to trace dynamicity of doers to the dynamics of a variety of cases in sciences as a footstep to more general models.
Sequential data are ubiquitous and widely available in a range of applications in almost all areas. We aim at considering medical, metrological and motion capturing type applications in terms of sequential data analytics in general, and classification in particular. Two scenarios are considered. The first starts with a pass through the initial sequential data database, performing training/learning of set of classes–medical conditions of patients in case of the dynamic treatment regime problems. This learned procedure will be used during the automated classification of new patients. Before starting the next, second pass, we form a confusion matrix based on the learned classification algorithm, and we form a transition matrix, which can be obtained in two ways: by the original database and alternatively by the data classified by the trained algorithm. The second pass is designed to correct the original classification with help of an additional hidden Markov type model (HMM), based on the mentioned two matrices as transition and emission matrices. The database (set of trellises, the training set) has a lattice structure. A part of the trellis tracks end at the target class (important in dynamic treatment regime applications, sometime associated with the healthy class). The trained classification, applied to the training set, can change the set of tracks ending at the target class, which forms one of the performance indicators of this algorithm. The next scenario also is based on the HMM type model. If one takes a lattice track, treating it as a sequence of observations, then HMM can improve that sequence by generating a complementary sequence, similar to the sequence of Viterbi states of HMM. It can also change the set of tracks ending at the target class, which forms the next performance measure, this time for the HMM procedure. Convergence to the target class is characterized by the convergence of the degrees of the transition matrices to the simple special case of such matrices. Alternatively, by extracting the root of the convergent matrices, the corresponding characterization of the transition matrix can be obtained so that the convergence is guaranteed. This work is mostly methodological than innovative being a complementary part to our previous work on target class classification topics. In the experimental part of this work we considered a root-oriented directed acyclic graphs that correspond to the target class classification policy. On the model of this graphs, a random set of tracks is generated, forming a so-called synthetic training set, synthetic trellis.
An Erratum to this paper has been published: https://doi.org/10.1134/S105466182401022X
This work is devoted to methods for creating fast and accurate neural network algorithms for central processors, which were proposed by scientists of the V.L. Arlazarov’s scientific school. It outlines general principles and approaches to improving computational efficiency and discusses specific examples: tensor convolution decompositions that simplify convolutional neural networks; bounded nonlinear activation function ratio, which is calculated faster than exponential activation functions; and p-im2col convolution algorithm, which allows you to achieve a balance between computational efficiency and RAM consumption. Particular attention is paid to quantized (8- and 4-bit integer) neural networks, their training, implementation, and limitations on some central processor architectures, such as Elbrus.
The history of development and the results of research by scientific groups in the field of pattern recognition, signal analysis, and biotechnical systems are described. Teams associated with the St. Petersburg Electrotechnical University “LETI” have been considered.
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