Abstract In this paper we introduce a new one parameter generalization of the Pell quaternions – P (r, n)-Pell quaternions. We give some of their properties, among others the Binet formula, convolution identity and the generating function.
{"title":"On Some Combinatorial Properties of P (r, n)-Pell Quaternions","authors":"D. Bród, A. Szynal-Liana","doi":"10.2478/tmmp-2020-0027","DOIUrl":"https://doi.org/10.2478/tmmp-2020-0027","url":null,"abstract":"Abstract In this paper we introduce a new one parameter generalization of the Pell quaternions – P (r, n)-Pell quaternions. We give some of their properties, among others the Binet formula, convolution identity and the generating function.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"77 1","pages":"1 - 12"},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45183283","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 propose an efficient multivariate encryption scheme based on permutation polynomials over finite fields. We single out a commutative group ℒ(q, m) of permutation polynomials over the finite field Fqm. We construct a trapdoor function for the cryptosystem using polynomials in ℒ(2, m), where m =2k for some k ≥ 0. The complexity of encryption in our public key cryptosystem is O(m3) multiplications which is equivalent to other multivariate public key cryptosystems. For decryption only left cyclic shifts, permutation of bits and xor operations are used. It uses at most 5m2+3m – 4 left cyclic shifts, 5m2 +3m + 4 xor operations and 7 permutations on bits for decryption.
{"title":"A Public Key Cryptosystem Using a Group of Permutation Polynomials","authors":"Rajesh P. Singh, B. K. Sarma, A. Saikia","doi":"10.2478/tmmp-2020-0013","DOIUrl":"https://doi.org/10.2478/tmmp-2020-0013","url":null,"abstract":"Abstract In this paper we propose an efficient multivariate encryption scheme based on permutation polynomials over finite fields. We single out a commutative group ℒ(q, m) of permutation polynomials over the finite field Fqm. We construct a trapdoor function for the cryptosystem using polynomials in ℒ(2, m), where m =2k for some k ≥ 0. The complexity of encryption in our public key cryptosystem is O(m3) multiplications which is equivalent to other multivariate public key cryptosystems. For decryption only left cyclic shifts, permutation of bits and xor operations are used. It uses at most 5m2+3m – 4 left cyclic shifts, 5m2 +3m + 4 xor operations and 7 permutations on bits for decryption.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"77 1","pages":"139 - 162"},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47773931","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 show that the Diophantine equation ax +(a +2)y = z2, where a ≡ 5 (mod 42) and a ∈ ℕ has no solution in non-negative integers.
{"title":"On the Diophantine Equation ax +(a +2)y = z2 Where a ≡ 5 (mod 42)","authors":"Rakporn Dokchann, A. Pakapongpun","doi":"10.2478/tmmp-2020-0030","DOIUrl":"https://doi.org/10.2478/tmmp-2020-0030","url":null,"abstract":"Abstract In this paper, we show that the Diophantine equation ax +(a +2)y = z2, where a ≡ 5 (mod 42) and a ∈ ℕ has no solution in non-negative integers.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"77 1","pages":"39 - 42"},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47922536","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 Multimodal biometric systems are nowadays considered as state of the art subject. Since identity establishment in everyday situations has become very significant and rather difficult, there is a need for reliable means of identification. Multimodal systems establish identity based on more than one biometric trait. Hence one of their most significant advantages is the ability to provide greater recognition accuracy and resistance against the forgery. Many papers have proposed various combinations of biometric traits. However, there is an inferior number of solutions demonstrating the use of fingerprint and finger vein patterns. Our main goal was to contribute to this particular field of biometrics. In this paper, we propose OpenFinger, an automated solution for identity recognition utilizing fingerprint and finger vein pattern which is robust to finger displacement as well as rotation. Evaluation and experiments were conducted using SDUMLA-HMT multimodal database. Our solution has been implemented using C++ language and is distributed as a collection of Linux shared libraries. First, fingerprint images are enhanced by means of adaptive filtering where Gabor filter plays the most significant role. On the other hand, finger vein images require the bounding rectangle to be accurately detected in order to focus just on useful biometric pattern. At the extraction stage, Level-2 features are extracted from fingerprints using deep convolutional network using a popular Caffe framework. We employ SIFT and SURF features in case of finger vein patterns. Fingerprint features are matched using closed commercial algorithm developed by Suprema, whereas finger vein features are matched using OpenCV library built-in functions, namely the brute force matcher and the FLANN-based matcher. In case of SIFT features score normalization is conducted by means of double sigmoid, hyperbolic tangens, Z-score and Min-Max functions. On the side of finger veins, the best result was obtained by a combination of SIFT features, brute force matcher with scores normalized by hyperbolic tangens method. In the end, fusion of both biometric traits is done on a score level basis. Fusion was done by means of sum and mean methods achieving 2.12% EER. Complete evaluation is presented in terms of general indicators such as FAR/FRR and ROC.
{"title":"Openfinger: Towards a Combination of Discriminative Power of Fingerprints and Finger Vein Patterns in Multimodal Biometric System","authors":"I. Kovác, Pavol Marák","doi":"10.2478/tmmp-2020-0012","DOIUrl":"https://doi.org/10.2478/tmmp-2020-0012","url":null,"abstract":"Abstract Multimodal biometric systems are nowadays considered as state of the art subject. Since identity establishment in everyday situations has become very significant and rather difficult, there is a need for reliable means of identification. Multimodal systems establish identity based on more than one biometric trait. Hence one of their most significant advantages is the ability to provide greater recognition accuracy and resistance against the forgery. Many papers have proposed various combinations of biometric traits. However, there is an inferior number of solutions demonstrating the use of fingerprint and finger vein patterns. Our main goal was to contribute to this particular field of biometrics. In this paper, we propose OpenFinger, an automated solution for identity recognition utilizing fingerprint and finger vein pattern which is robust to finger displacement as well as rotation. Evaluation and experiments were conducted using SDUMLA-HMT multimodal database. Our solution has been implemented using C++ language and is distributed as a collection of Linux shared libraries. First, fingerprint images are enhanced by means of adaptive filtering where Gabor filter plays the most significant role. On the other hand, finger vein images require the bounding rectangle to be accurately detected in order to focus just on useful biometric pattern. At the extraction stage, Level-2 features are extracted from fingerprints using deep convolutional network using a popular Caffe framework. We employ SIFT and SURF features in case of finger vein patterns. Fingerprint features are matched using closed commercial algorithm developed by Suprema, whereas finger vein features are matched using OpenCV library built-in functions, namely the brute force matcher and the FLANN-based matcher. In case of SIFT features score normalization is conducted by means of double sigmoid, hyperbolic tangens, Z-score and Min-Max functions. On the side of finger veins, the best result was obtained by a combination of SIFT features, brute force matcher with scores normalized by hyperbolic tangens method. In the end, fusion of both biometric traits is done on a score level basis. Fusion was done by means of sum and mean methods achieving 2.12% EER. Complete evaluation is presented in terms of general indicators such as FAR/FRR and ROC.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"77 1","pages":"109 - 138"},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42389686","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 bihyperbolic balancing and Lucas-balancing numbers. We give some of their properties, among others the Binet formula, Catalan, Cassini, d’Ocagne identities and the generating function.
{"title":"On the Combinatorial Properties of Bihyperbolic Balancing Numbers","authors":"D. Bród, A. Szynal-Liana, I. Włoch","doi":"10.2478/tmmp-2020-0029","DOIUrl":"https://doi.org/10.2478/tmmp-2020-0029","url":null,"abstract":"Abstract In this paper, we introduce bihyperbolic balancing and Lucas-balancing numbers. We give some of their properties, among others the Binet formula, Catalan, Cassini, d’Ocagne identities and the generating function.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"77 1","pages":"27 - 38"},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45345693","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 representations of rational numbers defined in terms of numeral systems that are certain generalizations of the classical q-ary numeral system.
摘要本文讨论了有理数的表示，有理数是根据数论系统定义的，数论系统是经典q元数论系统的某些推广。
{"title":"A Note on Expansions of Rational Numbers by Certain Series","authors":"S. Serbenyuk","doi":"10.2478/tmmp-2020-0032","DOIUrl":"https://doi.org/10.2478/tmmp-2020-0032","url":null,"abstract":"Abstract This paper deals with representations of rational numbers defined in terms of numeral systems that are certain generalizations of the classical q-ary numeral system.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"77 1","pages":"53 - 58"},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46635555","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 independent sequences with continuous asymptotic distribution functions. We construct a compact metric space with Borel probability measure. We use its properties to prove the central limit theorem for independent sequences with continuous distribution functions.
{"title":"Central Limit Theorem and the Distribution of Sequences","authors":"M. Paštéka","doi":"10.2478/tmmp-2020-0031","DOIUrl":"https://doi.org/10.2478/tmmp-2020-0031","url":null,"abstract":"Abstract The paper deals with independent sequences with continuous asymptotic distribution functions. We construct a compact metric space with Borel probability measure. We use its properties to prove the central limit theorem for independent sequences with continuous distribution functions.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"77 1","pages":"43 - 52"},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47502946","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 a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers H2n are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers Hn. As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung ∑n=1∞(Hnn)2=17π4360sumlimits_{n = 1}^infty {{{left( {{{{H_n}} over n}} right)}^2} = {{17{pi ^4}} over {360}}} together with its closely related alternating cousin are given. New proofs for other closely related quadratic Euler-type sums that are known in the literature are also obtained.
{"title":"Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers","authors":"S. Stewart","doi":"10.2478/tmmp-2020-0034","DOIUrl":"https://doi.org/10.2478/tmmp-2020-0034","url":null,"abstract":"Abstract In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers H2n are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers Hn. As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung ∑n=1∞(Hnn)2=17π4360sumlimits_{n = 1}^infty {{{left( {{{{H_n}} over n}} right)}^2} = {{17{pi ^4}} over {360}}} together with its closely related alternating cousin are given. New proofs for other closely related quadratic Euler-type sums that are known in the literature are also obtained.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"77 1","pages":"73 - 98"},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47926895","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 investigate Fibonacci type sequences defined by kth order linear recurrence. Based on their companion matrix and its graph interpretation we determine multinomial and binomial formulas for these sequences. Moreover we present a graphical rule for calculating the words of these sequences from the Pascal’s triangle.
{"title":"On Some Multinomial Sums Related to the Fibonacci Type Numbers","authors":"I. Włoch, A. Włoch","doi":"10.2478/tmmp-2020-0035","DOIUrl":"https://doi.org/10.2478/tmmp-2020-0035","url":null,"abstract":"Abstract In this paper we investigate Fibonacci type sequences defined by kth order linear recurrence. Based on their companion matrix and its graph interpretation we determine multinomial and binomial formulas for these sequences. Moreover we present a graphical rule for calculating the words of these sequences from the Pascal’s triangle.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"77 1","pages":"99 - 108"},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41784837","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 Aspace X is said to have the absolutely strongly star --Hurewicz (ASSH) property if for each sequence (𝒰n : n ∈ )of opencovers of X and each dense subset Y of X, there is a sequence (Fn : n ∈ ) of finite subsets of Y such that for each x ∈ X, {n ∈ : x ∉ St(Fn, 𝒰n)}∈ , where is the proper admissible ideal of . In this paper, we investigate the relationship between the ASSH property and other related properties and study the topological properties of the ASSH property. This paper generalizes several results of Song [25] to the larger class of spaces having the ASSH properties.
文摘Aspace X据说绝对强烈恒星--Hurewicz(屁股H)财产如果每个序列(𝒰n: n∈)opencovers每个稠密子集X和Y的X有一个序列(Fn: n∈)的有限子集为每个X∈X, Y, {n∈:X∉圣(Fn𝒰n)}∈,是适当的容许的理想的地方。本文研究了ASSH属性与其他相关属性之间的关系，并研究了ASSH属性的拓扑性质。本文将Song[25]的几个结果推广到具有ASSH性质的更大的一类空间。
{"title":"The Absolutely Strongly Star-Hurewicz Property with Respect to an Ideal","authors":"Sumit Singh, B. Tyagi, M. Bhardwaj","doi":"10.2478/tmmp-2020-0020","DOIUrl":"https://doi.org/10.2478/tmmp-2020-0020","url":null,"abstract":"Abstract Aspace X is said to have the absolutely strongly star --Hurewicz (ASSH) property if for each sequence (𝒰n : n ∈ )of opencovers of X and each dense subset Y of X, there is a sequence (Fn : n ∈ ) of finite subsets of Y such that for each x ∈ X, {n ∈ : x ∉ St(Fn, 𝒰n)}∈ , where is the proper admissible ideal of . In this paper, we investigate the relationship between the ASSH property and other related properties and study the topological properties of the ASSH property. This paper generalizes several results of Song [25] to the larger class of spaces having the ASSH properties.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"76 1","pages":"81 - 94"},"PeriodicalIF":0.0,"publicationDate":"2020-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44575885","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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