Abstract Polybius cipher is a special substitution system widely used during history. In our research, we found several Polybius-like ciphers used in Czecho-slovakia and in the Slovak State from the first half of the 20th century. Various types of this cipher are described in the first Czechoslovak cryptanalysis manual “šifrovací systémy a návod k luštění kryptogramů” by plk. Josef Růžek. It can be also found in the official cryptology directive called G–VII–8 “šifrování” (encryption) from 1938, and another variant in the new version of the same directive from 1946. In this work, we focus on a special Polybius-like cipher inspired by three real ciphers used in Czechoslovakia and in the Slovak State. Two were used during WW2 and one right after the war. We will show how these ciphers were used and how they can be solved with a modern heuristic approach on a personal computer. We evaluate the effectiveness of the Hill-Climbing heuristic methods with restarts. We also investigated several different fitness functions and language models.
波利比乌斯密码是历史上广泛使用的一种特殊的代入系统。在我们的研究中，我们发现了几个类似波利比乌斯的密码，从20世纪上半叶开始在捷克斯洛伐克和斯洛伐克使用。这种密码的各种类型在捷克斯洛伐克密码分析手册“šifrovací system k k luštění kryptogramv”中有描述。约瑟夫·Růž艾克。它也可以在1938年称为G-VII-8“šifrování”(加密)的官方密码学指令中找到，以及1946年同一指令的新版本中的另一个变体。在这项工作中，我们专注于一种特殊的波利比乌斯式密码，灵感来自捷克斯洛伐克和斯洛伐克国家使用的三个真实密码。其中两个在二战期间使用，另一个在战后使用。我们将展示如何使用这些密码，以及如何在个人计算机上使用现代启发式方法来解决它们。我们通过重新启动来评估爬坡启发式方法的有效性。我们还研究了几种不同的适应度函数和语言模型。
{"title":"Cryptanalysis of a Special Polybius-Like Cipher Using Hill-Climbing","authors":"Eugen Antal","doi":"10.2478/tmmp-2023-0032","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0032","url":null,"abstract":"Abstract Polybius cipher is a special substitution system widely used during history. In our research, we found several Polybius-like ciphers used in Czecho-slovakia and in the Slovak State from the first half of the 20th century. Various types of this cipher are described in the first Czechoslovak cryptanalysis manual “šifrovací systémy a návod k luštění kryptogramů” by plk. Josef Růžek. It can be also found in the official cryptology directive called G–VII–8 “šifrování” (encryption) from 1938, and another variant in the new version of the same directive from 1946. In this work, we focus on a special Polybius-like cipher inspired by three real ciphers used in Czechoslovakia and in the Slovak State. Two were used during WW2 and one right after the war. We will show how these ciphers were used and how they can be solved with a modern heuristic approach on a personal computer. We evaluate the effectiveness of the Hill-Climbing heuristic methods with restarts. We also investigated several different fitness functions and language models.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"1 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138586071","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}
Abstract This paper presents a general unified approach to the notions of generalized closedness in topological spaces. The research concerning the notion of generalized closed sets in topological spaces was initiated by Norman Levine in 1970. In the succeeding years, the concepts of this type of generalizations have been investigated in many versions using the standard generalizations of topologies which has resulted in a large body of literature. However, the methods and results in the past years have become standard and lacking in innovation. The basic notion used in this conception is the closure operator designated by a family ℬ ⊆ 𝒫(X), which need not be a Kuratowski operator. Here, we introduce a general conception of natural extensions of families ℬ ⊆ 𝒫 (X), denoted by ℬ ᐊ 𝒦, which are determined by other families 𝒦 ⊆ 𝒫(X). Precisely, ℬ⊲𝒦={ A⊆X:A¯ℬ⊆A¯𝒦 }, mathcal{B} triangleleft mathcal{K} = left{ {A subseteq X:{{bar A}^mathcal{B}} subseteq {{bar A}^mathcal{K}}} right}, where (…)¯𝒜 {overline {left( ldots right)} ^mathcal{A}} denotes the closure operator designated by 𝒜 ⊆ 𝒫(X). We prove that the collection of all generalizations ℬ ᐊ 𝒦, where ℬ, 𝒦 ⊆ 𝒫 (X), forms a Boolean algebra. In this theory, the family of all generalized closed sets in a topological space X(𝒯 )is equal to 𝒞 ᐊ 𝒯, where 𝒞 is the family of all closed subsets of X. This concept gives tools that enable the systemizing and developing of the current research area of this topic. The results obtained in this general conception easily extend and imply well-known theorems as obvious corollaries. Moreover, they also give many new results concerning relationships between various types of generalized closedness studied so far in a topological space. In particular, we prove and demonstrate in a graph that in a topological space X(𝒯) there exist only nine different generalizations determined by the standard generalizations of topologies. The tools introduced in this paper enabled us to show that many generalizations studied in the literature are improper.
{"title":"A Unified Treatment of Generalized Closed Sets in Topological Spaces","authors":"Emilia Przemska","doi":"10.2478/tmmp-2023-0028","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0028","url":null,"abstract":"Abstract This paper presents a general unified approach to the notions of generalized closedness in topological spaces. The research concerning the notion of generalized closed sets in topological spaces was initiated by Norman Levine in 1970. In the succeeding years, the concepts of this type of generalizations have been investigated in many versions using the standard generalizations of topologies which has resulted in a large body of literature. However, the methods and results in the past years have become standard and lacking in innovation. The basic notion used in this conception is the closure operator designated by a family ℬ ⊆ 𝒫(X), which need not be a Kuratowski operator. Here, we introduce a general conception of natural extensions of families ℬ ⊆ 𝒫 (X), denoted by ℬ ᐊ 𝒦, which are determined by other families 𝒦 ⊆ 𝒫(X). Precisely, ℬ⊲𝒦={ A⊆X:A¯ℬ⊆A¯𝒦 }, mathcal{B} triangleleft mathcal{K} = left{ {A subseteq X:{{bar A}^mathcal{B}} subseteq {{bar A}^mathcal{K}}} right}, where (…)¯𝒜 {overline {left( ldots right)} ^mathcal{A}} denotes the closure operator designated by 𝒜 ⊆ 𝒫(X). We prove that the collection of all generalizations ℬ ᐊ 𝒦, where ℬ, 𝒦 ⊆ 𝒫 (X), forms a Boolean algebra. In this theory, the family of all generalized closed sets in a topological space X(𝒯 )is equal to 𝒞 ᐊ 𝒯, where 𝒞 is the family of all closed subsets of X. This concept gives tools that enable the systemizing and developing of the current research area of this topic. The results obtained in this general conception easily extend and imply well-known theorems as obvious corollaries. Moreover, they also give many new results concerning relationships between various types of generalized closedness studied so far in a topological space. In particular, we prove and demonstrate in a graph that in a topological space X(𝒯) there exist only nine different generalizations determined by the standard generalizations of topologies. The tools introduced in this paper enabled us to show that many generalizations studied in the literature are improper.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"188 1","pages":"121 - 154"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139295723","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}
Abstract The purpose of this paper is to introduce a few variants of generalized quasi-continuity of functions defined on a bitopological space and to study their mutual relationship. Moreover, some characterization of sectional quasi-continuous function and its continuity points are investigated.
{"title":"A Few Variants of Quasi-Continuity in Bitopological Spaces","authors":"M. Matejdes","doi":"10.2478/tmmp-2023-0022","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0022","url":null,"abstract":"Abstract The purpose of this paper is to introduce a few variants of generalized quasi-continuity of functions defined on a bitopological space and to study their mutual relationship. Moreover, some characterization of sectional quasi-continuous function and its continuity points are investigated.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"1 1","pages":"27 - 44"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139296464","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}
Abstract In this paper, we introduce strongly star g-compactness as a topological covering property and compare its structure to other topological properties that have analogous structures. The characteristics of a strongly star g-compact subset and strongly star g-compact subspace are looked at. Finally, some finite intersection-like characteristics that will result in some situations akin to strongly star g-compactness are presented.
{"title":"On Strongly Star g-Compactness of Topological Spaces","authors":"Prasenjit Bal, Susmita Sarkar","doi":"10.2478/tmmp-2023-0026","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0026","url":null,"abstract":"Abstract In this paper, we introduce strongly star g-compactness as a topological covering property and compare its structure to other topological properties that have analogous structures. The characteristics of a strongly star g-compact subset and strongly star g-compact subspace are looked at. Finally, some finite intersection-like characteristics that will result in some situations akin to strongly star g-compactness are presented.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"59 1","pages":"89 - 100"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139292217","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}
Abstract The paper deals with the strong porosity of some families of real functions continuous with respect to a given topology 𝒯 or 𝒜-continuous (i.e., continuous with respect to some special family 𝒜 of sets of the real line). Particularly, porosity of those families is investigated in space of the Baire 1 functions or in the space of the Baire 1 and Darboux functions.
{"title":"On Strong Porosity of Some Families of Functions","authors":"J. Hejduk, G. Ivanova, Renata Wiertelak","doi":"10.2478/tmmp-2023-0023","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0023","url":null,"abstract":"Abstract The paper deals with the strong porosity of some families of real functions continuous with respect to a given topology 𝒯 or 𝒜-continuous (i.e., continuous with respect to some special family 𝒜 of sets of the real line). Particularly, porosity of those families is investigated in space of the Baire 1 functions or in the space of the Baire 1 and Darboux functions.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"1 1","pages":"45 - 56"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139299093","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}
Abstract In the present paper, we introduce new types of generalized closed sets called ij -pre-generalized closed sets and study some of their properties in bi-topological spaces. Also, we use them to construct new types of separation axioms. Further, we introduce and study the concepts of pairwise operation pc-open sets and pairwise operation pc-separation axioms in bitopological spaces. Several interesting characterizations of different spaces are discussed. The relationships between these spaces are given.
摘要 本文引入了新型广义闭集（称为 ij 前广义闭集），并研究了它们在双拓扑空间中的一些性质。同时，我们还利用它们构建了新型分离公理。此外，我们还引入并研究了位拓扑空间中成对运算 pc 开集和成对运算 pc 分离公理的概念。我们讨论了不同空间的几个有趣特征。我们还给出了这些空间之间的关系。
{"title":"Operation Pc-Open Sets and Operation Pc-Separation Axioms in Bitopological Spaces","authors":"Fathi. H. Khedr, Osama Rashed Sayed","doi":"10.2478/tmmp-2023-0027","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0027","url":null,"abstract":"Abstract In the present paper, we introduce new types of generalized closed sets called ij -pre-generalized closed sets and study some of their properties in bi-topological spaces. Also, we use them to construct new types of separation axioms. Further, we introduce and study the concepts of pairwise operation pc-open sets and pairwise operation pc-separation axioms in bitopological spaces. Several interesting characterizations of different spaces are discussed. The relationships between these spaces are given.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"97 1","pages":"101 - 120"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139299551","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}
Abstract We study two particular modifications of the P-property of ideals and related cardinal invariants cof𝒥 (ℐ)and cov+(ℐ). We give some results on the existence of P (𝒥)-ideals or non-P (𝒥)-ideals regarding specific classes of ideals, particularly meager ideals on ω. We also provide values of the cardinal invariant cof𝒥 (ℐ) describing the smallest families ensuring P(𝒥) for particular critical ideals. Moreover, we obtain a simple way of proving strict inequalities Fin
摘要 我们研究了理想的 P 特性的两个特殊修正以及相关的心项不变式 cof𝒥 (ℐ)and cov+(ℐ)。我们给出了一些关于 P (𝒥)-ideals 或非 P (𝒥)-ideals 的存在的结果，这些结果涉及特定的理想类，尤其是 ω 上的微不足道的理想。我们还提供了红心不变量 cof𝒥 (ℐ) 的值，描述了确保特定临界理想的 P(𝒥) 的最小族。此外，对于任何 MAD 族𝒜，我们还可以用弱 P 理想概念得到一个简单的方法来证明严格不等式 Fin
{"title":"P-Like Properties of Meager Ideals and Cardinal Invariants","authors":"Adam Marton","doi":"10.2478/tmmp-2023-0025","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0025","url":null,"abstract":"Abstract We study two particular modifications of the P-property of ideals and related cardinal invariants cof𝒥 (ℐ)and cov+(ℐ). We give some results on the existence of P (𝒥)-ideals or non-P (𝒥)-ideals regarding specific classes of ideals, particularly meager ideals on ω. We also provide values of the cardinal invariant cof𝒥 (ℐ) describing the smallest families ensuring P(𝒥) for particular critical ideals. Moreover, we obtain a simple way of proving strict inequalities Fin <K 〈𝒜 〉 <K Fin × Fin for any MAD family 𝒜 using the weak P-ideal notion.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"93 1","pages":"73 - 88"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139300116","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}
Abstract This paper deals with density on the set of natural numbers and its connections to the distribution of sequences. Under the assumption of independence, some formulas are derived.
摘要 本文论述自然数集的密度及其与序列分布的联系。在独立性假设下，得出了一些公式。
{"title":"Integrability of Sequences","authors":"Milan Paštéka","doi":"10.2478/tmmp-2023-0030","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0030","url":null,"abstract":"Abstract This paper deals with density on the set of natural numbers and its connections to the distribution of sequences. Under the assumption of independence, some formulas are derived.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"28 1","pages":"169 - 180"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139293366","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}
Abstract The aim of this paper is to give necessary and sufficient conditions in terms of the Fourier Laguerre-Bessel transform 𝒲LBf of the function f to ensure that f belongs to the generalized Lipschitz classes Hαk (X) and hkα (X), where X =[0, +∞) × [0, +∞).
摘要 本文旨在给出函数 f 的傅立叶拉盖尔-贝塞尔变换 𝒲LBf 的必要条件和充分条件，以确保 f 属于广义的 Lipschitz 类 Hαk (X) 和 hkα (X)，其中 X =[0, +∞) × [0, +∞) 。
{"title":"Laguerre-Bessel Transform and Generalized Lipschitz Classes","authors":"L. Rakhimi, Abdelmajid Khadari, R. Daher","doi":"10.2478/tmmp-2023-0029","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0029","url":null,"abstract":"Abstract The aim of this paper is to give necessary and sufficient conditions in terms of the Fourier Laguerre-Bessel transform 𝒲LBf of the function f to ensure that f belongs to the generalized Lipschitz classes Hαk (X) and hkα (X), where X =[0, +∞) × [0, +∞).","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"4 1","pages":"155 - 168"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139298394","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}
Abstract When analyzing cell trajectories, we often have to deal with noisy data due to the random motion of the cells and possible imperfections in cell center detection. To smooth these trajectories, we present a mathematical model and numerical method based on evolving open-plane curve approach in the Lagrangian formulation. The model contains two terms: the first is the smoothing term given by the influence of local curvature, while the other attracts the curve to the original trajectory. We use the flowing finite volume method to discretize the advection-diffusion partial differential equation. The PDE includes the asymptotically uniform tangential redistribution of curve grid points. We present results for macrophage trajectory smoothing and define a method to compute the cell velocity for the discrete points on the smoothed curve.
{"title":"Macrophages Trajectories Smoothing by Evolving Curves","authors":"Giulia Lupi, Karol Mikula, Seol Ah Park","doi":"10.2478/tmmp-2023-0031","DOIUrl":"https://doi.org/10.2478/tmmp-2023-0031","url":null,"abstract":"Abstract When analyzing cell trajectories, we often have to deal with noisy data due to the random motion of the cells and possible imperfections in cell center detection. To smooth these trajectories, we present a mathematical model and numerical method based on evolving open-plane curve approach in the Lagrangian formulation. The model contains two terms: the first is the smoothing term given by the influence of local curvature, while the other attracts the curve to the original trajectory. We use the flowing finite volume method to discretize the advection-diffusion partial differential equation. The PDE includes the asymptotically uniform tangential redistribution of curve grid points. We present results for macrophage trajectory smoothing and define a method to compute the cell velocity for the discrete points on the smoothed curve.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"72 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135809967","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}
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