Mohyla Mathematical Journal最新文献

{"title":"Polynomial Representation of Binary Trees of Entropy Binary Codes","authors":"Denys Morozov","doi":"10.18523/2617-70804202120-23","DOIUrl":"https://doi.org/10.18523/2617-70804202120-23","url":null,"abstract":"An important component of streaming large amounts of information are algorithms for compressing information flow. Which in turn are divided into lossless compression algorithms (entropic) - Shannon, Huffman, arithmetic coding, conditional compression - LZW, and otherinformation cone injections and lossy compression algorithms - such as mp3, jpeg and others. \u0000It is important to follow a formal strategy when building a lossy compression algorithm. It can be formulated as follows. After describing the set of objects that are atomic elements of exchange in the information flow, it is necessary to build an abstract scheme of this description, which will determine the boundary for abstract sections of this scheme, which begins the allowable losses. \u0000Approaches to the detection of an abstract scheme that generates compression algorithms with allowable losses can be obtained from the context of the subject area. For example, an audio stream compression algorithm can divide a signal into simple harmonics and leave among them those that are within a certain range of perception. Thus, the output signal is a certain abstraction of the input, which contains important information in accordance with the context of auditory perception of the audio stream and is represented by less information. A similar approach is used in the mp3 format, which is a compressed representation. \u0000Unlike lossy compression algorithms, entropic compression algorithms do not require contextanalysis, but can be built according to the frequency picture. Among the known algorithms for constructing such codes are the Shannon-Fano algorithm, the Huffman algorithm and arithmetic coding. \u0000Finding the information entropy for a given Shannon code is a trivial task. The inverse problem, namely finding the appropriate Shannon codes that have a predetermined entropy and with probabilities that are negative integer powers of two, is quite complex. It can be solved by direct search, but a significant disadvantage of this approach is its computational complexity. This article offers an alternative technique for finding such codes.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132306677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Constructing the Mate of Cospectral 5-regular Graphs with and without a Perfect Matching","authors":"V. Solomko, V. Sobolev","doi":"10.18523/2617-70804202124-27","DOIUrl":"https://doi.org/10.18523/2617-70804202124-27","url":null,"abstract":"The problem of finding a perfect matching in an arbitrary simple graph is well known and popular in graph theory. It is used in various fields, such as chemistry, combinatorics, game theory etc. The matching of M in a simple graph G is a set of pairwise nonadjacent edges, ie, those that do not have common vertices. Matching is called perfect if it covers all vertices of the graph, ie each of the vertices of the graph is incidental to exactly one of the edges. By Koenig's theorem, regular bipartite graphs of positive degree always have perfect matching. However, graphs that are not bipartite need further research. \u0000Another interesting problem of graph theory is the search for pairwise nonisomorphic cospectral graphs. In addition, it is interesting to find cospectral graphs that have additional properties. For example, finding cospectral graphs with and without a perfect matching. \u0000The fact that for each there is a pair of cospectral connected k-regular graphs with and without a perfect matching had been investigated by Blazsik, Cummings and Haemers. The pair of cospectral connected 5-regular graphs with and without a perfect matching is constructed by using Godsil-McKay switching in the paper.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115011628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Equilibrium in a Symmetric Game of Resource Extraction with Coalitional Structure","authors":"I. Sylenko","doi":"10.18523/2617-70804202141-47","DOIUrl":"https://doi.org/10.18523/2617-70804202141-47","url":null,"abstract":"The game of resource extraction / capital accumulation is a stochastic nonzero-sum infinite horizon game, obtained as an extension of the well-known optimal growth model to m strategically competing players, who jointly posses a renewable resource. The existence of a Nash equilibrium in different, often symmetric, frameworks of the game received a significant attention in the scientific literature on the topic. The focus of this paper is to introduce the coalitional component to the symmetric problem. Specifically, we examine whether the game with a fixed coalitional structure admits stability against profitable coalitional deviations.It is assumed that the set of all players is partitioned into coalitions which do not intersect and remain consistent throughout the game. The members of each coalition are able to coordinate their actions and perform joint deviations in a cooperative manner. Such setting incorporates a natural concept of established social ties, which may reflect a potential context appearing in practical applications. The corresponding notion of equilibrium in the paper is expressed as a position, from which none of the set coalitions can deviate in a manner to increase a total reward of its members. Its existence is studied in the context of a certain symmetric resource extraction game model with unbounded utilities of the players. This model was studied in [12; 13], concluding a Stationary Markov Perfect Equilibrium existence in both symmetric and non-symmetric game structure. The first feature of the model is that the preferences of the players are considered to be isoelastic in the form of strictly concave power functions. Furthermore, the law of motion between states is set to follow a geometric random walk in relation to players' joint investments. We prove that the game within the formulated settings admits stability against profitable coalitional deviations for any partition on the set of agents. The method provides an algorithm for building the corresponding stationary strategies, which can be useful for practical purposes. Finally, we use two examples with different numerical configurations to illustrate possible patterns of how the individual rewards of the players vary depending on a coalitional structure, which is set at the beginning of the game.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121731806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Search Algorithm of the Number of Unfixed Points of Permutations from Sylow 2-subgroups Syl2(S2n) of Symmetric Groups S2n","authors":"V. Olshevska","doi":"10.18523/2617-70804202134-40","DOIUrl":"https://doi.org/10.18523/2617-70804202134-40","url":null,"abstract":"The Symmetric permutation group S2n is a classical algebraic object that is also used in Computer science, Coding theory, Statistics, etc. In particular, the coding theory considers codes defined on the symmetric group Sn or its subgroups. The research of permutation codes has been started from 1970s. These codes can be obtained with using different distances: Hamming, Ulam, Cailey, Levenshtein. The finding distance on permutations depends on their number of fixed or unfixed points. Therefore, it is natural to count the number of unfixed points in a certain group of permutations.In this paper, we consider the number of unfixed points of permutations that are elements of the Sylow 2-subgroup Syl2(S2n) of symmetric groups S2n. Leo Kaluzhnin used tables to represent the elements of these groups [8]. Volodymyr Nekrashevych represented permutations by their portraits [9]. We use algorithms that describe the connection between the permutation group Syl2(S2n) and the group of labeled binary rooted trees [10].An algorithm for finding the number of unfixed points for permutations of the Sylow 2-subgroup Syl2(S2n) of the symmetric group S2n is proposed in the article. An isomorphism between the group Syl2(S2n) and a group of labeled binary root trees was used to construct this algorithm. It is proved, that the algorithm of searching the number of unfixed point for permutations of the Sylow 2-subgroup Syl2(S2n) of the symmetric group S2n has complexity O(2n). In addition, the average number of steps of the algorithm for the Sylow 2-subgroup of the symmetric group S2n is found. The result for small n (n = 2, 3, 4) was verified with a program, that is written in the language of the computer algebra Sage.\u0000At the end of the article we find the number of permutations from Syl2(S2n ) that have a maximumnumber of unfixed points. The number of such permutations in the symmetric group S2n is well known.Obviously that this number is smaller for the Sylow 2-subgroup of the symmetric group Syl2(S2n ). Inthis case, we calculate the maximum number of unfixed points using a recursive formula.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131697356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Honey Encryption Applied to Schnorr Signature Scheme","authors":"Mariia Oliynyk","doi":"10.18523/2617-7080420213-6","DOIUrl":"https://doi.org/10.18523/2617-7080420213-6","url":null,"abstract":"The security of any cryptosystem mostly depends on the reliability of the protection of secret keys used in it. In particular, key generation procedure must give a variety of keys so that they cannot be picked up by a brute-force attack. Honey encryption is used as an additional barrier of cryptosystems' keys protection to slow down a brute-force attack. As in the case of \" white box cryptography'', different honey encryption schemes are considered depending on what the additional protection is aimed at. The need to additionally protect secret keys arises in remote access systems, when it is necessary to provide access to information to authorized users.The idea of encryption, which would provide adequate additional protection, is to build a system so that the attacker will not be able to recover the original text, even after searching through all possible options for passwords or keys. Based on the concept of lure systems, this system outputs messages that are difficult to distinguish from the true ones. Such messages are called honey, which, like the sweet substance, but in computer security terminology, \" drag '' the opponent into a dead end.This article constructs schemes for additional key protection of the Schnorr Signature Scheme, describes the pseudocodes of the corresponding algorithms, analyzes the complexity of a brute-force attack.This scheme requires additional protection against the a secret key because storing it in the open is unreliable. With the proposed encryption algorythm, we can \"hide'' a into a sequence of integers, and extract it back with the proposed decryption algorythm. If the sequence is entered by an attacker and does not contain a masked number a , decryption algorythm will return the result of multiplying the last number by the inverse q to the first prime number, if any, in the set, or the penultimate number in the set. Therefore, when trying to artificially pick up a secret key by a brute-force attack, the attacker will get a number similar to a but which he will not be able to use when trying to forge Alice's signature.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134548270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Antipodal graphs of diameter 4","authors":"Liudmyla Pronchuk","doi":"10.18523/2617-7080I2018P34-37","DOIUrl":"https://doi.org/10.18523/2617-7080I2018P34-37","url":null,"abstract":"The metric space (X, d) is called antipodal if for an arbitrary point x exists the one point y such that for an arbitrary point z of the set X the equality d(z, x) + d(z, y) = d(x, y) holds. In other words, the metric space (X, d) is called antipodal if for an arbitrary x there exists an antipode. For any connected undirected finite graph G we define the associated graph distance. The distance between two vertices v1 and v2 in G equals to the length of the shortest path between v1 and v2. A connected undirected finite graph G is called antipodal if its associated graph metric is antipodal. Hamming’s graphs, Johnson’s graphs and Coctail-party graphs are well-known constructions of antipodal graphs. In particular, the antipodal graphs of diameter 2 only are Cocktail-party graphs. Cocktail-party graph is the graph consisting of two rows of paired vertices in which all vertices but the paired ones are connected with a graph edge. In the article was characterized antipodal graphs of diameter 3. In this paper was showed that almost every graph is an induced subgraph of some antipodal graph P(G) of diameter 3. The construction P(G) used connected undirected finite semi-canonical graphs. Using an idea by Dragan Stevanovic from, we introduce the construction of graphs which allows to construct antipodal graphs of diameter 4. The basis of this construction is using semi-canonical graphs on a set of four vertices. A graph G is called semi-canonical if for any vertex v of this graph there is at least one vertex u of graph G, such that v and u are not connected by edge. There are only two semi-canonical graphs in a set of four vertices that are C4 and L4 the cycle and the path on 4 vertices correspondingly. So, we construct two antipodal graphs of diameter 4. We prove the antipodal of both of these graphs.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124690488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A discrete regularization method for hidden Markov models embedded into reproducing kernel Hilbert space","authors":"G. Kriukova","doi":"10.18523/2617-7080I2018P15-20","DOIUrl":"https://doi.org/10.18523/2617-7080I2018P15-20","url":null,"abstract":"Hidden Markov models are a well-known probabilistic graphical model for time series of discrete, partially observable stochastic processes. We consider the method to extend the application of hidden Markov models to non-Gaussian continuous distributions by embedding a priori probability distribution of the state space into reproducing kernel Hilbert space. Corresponding regularization techniques are proposed to reduce the tendency to overfitting and computational complexity of the algorithm, i.e. Nystr¨om subsampling and the general regularization family for inversion of feature and kernel matrices. This method may be applied to various statistical inference and learning problems, including classification, prediction, identification, segmentation, and as an online algorithm it may be used for dynamic data mining and data stream mining. We investigate, both theoretically and empirically, the regularization and approximation bounds of the discrete regularization method. Furthermore, we discuss applications of the method to real-world problems, comparing the approach to several state-of-the-art algorithms.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128558616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}