fibonacci sequence
در نشریات گروه ریاضی-
In this paper, we propose a generalized Lucas matrix (a recursive matrix of higher order) obtained from the generalized Fibonacci sequences. We obtain their algebraic properties such as direct inverse calculation, recursive nature, etc. Then, we propose a modified public key cryptography using the generalized Lucas matrices as a key element that optimizes the keyspace construction complexity. Furthermore, we establish a key agreement for encryption-decryption with a combination of the terms of generalized Lucas sequences under the residue operation.Keywords: Affine-Hill Cipher, Cryptography, Fibonacci Sequence, Lucas Sequence, Lucas Matrix
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در ریاضی، شیمی، و علوم نانو برای هر مولکول مدلی به صورت گراف در نظر می گیرند به طوری که راس های آن گراف اتم های مولکول و یال های آن پیوندهای بین اتم ها است. شاخص های توپولوژیکی یک گراف پارامترهای عددی وابسته به گراف هستند که اعدادی منحصربه فرد بوده و نسبت به یکریختی گراف ها ثابت اند. بسیاری از شاخص های توپولوژیکی تعریف شده در شیمی ریاضی برحسب درجات ریوس گراف است. در این مقاله، ارتباط برخی از مقادیر این شاخص های توپولوژیکی و نسبت طلایی را بررسی و مطالعه خواهیم کرد.
کلید واژگان: شاخص توپولوژیکی، شاخص هوسویا، شاخص زاگرب، نسبت طلایی، دنباله فیبوناتچیIn mathematics, chemistry , and nanoscience a graph model is considered for each molecule, so that the vertices are the atoms of the molecule and the edges are the bonds between the atoms. Graph topological indices are numerical parameters dependent on the graph, which are unique numbers and are fixed with respect to the isomorphism of the graphs. Many topological indices in Mathematical Chemistry are in terms of degrees of vertices. In this article, we study the relationship between some values of these topological indices and the golden ratio.
Keywords: topological index, Hosoya index, Zagreb Index, golden ratio, Fibonacci sequence -
We consider a new class of square Fibonacci $(q+1)\times(q+1)$-matrices in public key cryptography. This extends previous cryptography using generalized Fibonacci matrices. For a given integer $q$, a $(q+1)\times(q+1)$ binary matrix $M_{q}$ is a matrix which nonzero entries are located either on the super diagonal or on the last row of the matrix. In this article, we have proposed a modified public key cryptography using such matrices as key in Hill cipher and key agreement for encryption-decryption of terms of $M_{q}$-matrix. In this scheme, instead of exchanging the whole key matrix, only a pair of numbers needed to be exchanged, which reduces the time complexity as well as the space complexity of the transmission and has a large key space.Keywords: Cryptography, Hill cipher, key exchange Elgamal, Fibonacci sequence, $M, {q}$-matrix
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فرض کنید یک گروه متناهی و یک مولد مرتب آن باشد. طول فیبوناتچی متناظر با مجموعه مولد که با نشان داده می شود، عبارت است از کوچکترین عدد صحیح مثبتی مانند به طوری که برای دنباله فیبوناتچی، و ، از اعضای ، داشته باشیم برای .در این مقاله، طول فیبوناتچی 2-سیلو زیرگروه های گروه متقارن را محاسبه می کنیم.
کلید واژگان: طول فیبوناتچی، دنباله فیبوناتچی، گروه متناهیCalculating and deriving the properties of the Fibonacci length of finite groups have been done since 1990, where the Fibonacci length of the finite group is defined to be the least integer such that for the Fibonacci sequence of the elements of the group , all of the equalities hold. In this paper we calculate Fibonacci length of 2- sylow subgroups of , by giving suitable generating sets for it.
Keywords: Fibonacci length, Fibonacci sequence, Finite groups -
In this paper, a class of new polynomials based on Fibonacci sequence using Newton interpolation is introduced. This target is performed once using Newton forward- divided- difference formula and another more using Newton backward- divided- difference formula. Some interesting results are obtained for forward and backward differences. The relationship between forward (and backward) differences and the Khayyam- Pascal's triangle are also examined.Keywords: Fibonacci sequence, Newton interpolation, Forward differences, Backward differences
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در این مقاله سعی بر یاد آوری این نکته داریم که هیچ دانشی بدون ریاضیات امکان پذیر نیست. ریاضیات علم نظر و پایه تفکر آدمی است و موضوع آن، یافتن، توصیف و درک نظمی است که در وضعیت‌های ظاهرا پیچیده و بی‌نظم نهفته است. نظریه‌های انتزاعی ریاضی بستگی مستقیمی با طبیعت دارند و می‌توانند برای تفسیر آن به‌کار روند. در این خصوص نوربرت وینر جمله زیبایی دارد: "به نظر می‌رسد که می‌توان از بام تا شام به تماشای ناز و کرشمه‌های عجیب و غریب آب نشست ولی آنچه در میان این همه زیبایی مرا به طرف خود می‌کشید، ریاضیات و فیزیک بود و آن قانون مندی‌های ریاضی، که همه این توده بی‌نظم و ناآرام آب را هدایت می‌کند" [3]. در این مقاله با مروری برهندسه فرکتال، دنباله فیبوناچی، تقارن و کاربرد آنها در علوم زیستی نظمی را توصیف می‌کنیم که در وضعیت‌های ظاهرا پیچیده بیولوژی نهفته است.
کلید واژگان: نباله فیبوناتچی، تقارن، علوم زیستی، هندسه ی کلاسیک، نسبت طلایی، هندسه ی فرکتالIn this article we try to remind the reader that no knowledge is possible without mathematics. Mathematics is the science of thought and the basis of human thought and it is about finding, describing and understanding the order that lies in seemingly complex and chaotic situations. Abstract mathematical theories are directly related to nature and can be used to interpret it. In this regard, Norbert Wiener has a beautiful saying: “It seems that one can sit from the morning to evening watching the cute and strange creams of water but in the midst of all this beauty, I was drawn to mathematics and physics and those mathematical laws, which govern all these disordered masses of water” [18]. In this paper, with a review of fractal geometry, the Fibonacci sequence, symmetry, we describe their application in the regular biological sciences which lies in the seemingly complex situations of biology.
Keywords: Fibonacci sequence, symmetry, biological sciences, classical geometry, golden ratio, fractal geometry -
Historically, mathematics and architecture have been associated with one another. Ratios are good example of this interconnection. The origin of ratios can be found in nature, which makes the nature so attractive. As an example, consider the architecture inspired by flowers which seems so harmonic to us. In the same way, the architectural plan of many well-known historical buildings such as mosques and bridges shows a rhythmic balance which according to most experts the reason lies in using the ratios. The golden ratio has been used to analyze the proportions of natural objects as well as building’s harmony. In this paper, after recalling the (mathematical) definition of the golden ratio, its ability to describe the harmony in the nature is discussed. When teaching mathematics in the schools, one may refer to this interconnection to encourage students to feel better with mathematics and deepen their understanding of proportion. At the end, the golden ratio decimals as well as its binary digits has been statistically examined to confirm their behavior as a random number generator.Keywords: Fibonacci sequence, golden ratio, golden rectangular, random number generator
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Let $(F_n)_{ngeq 0}$ be the Fibonacci sequence given by $F_0 = 0, F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for $n geq 0$. In this paper, we solve all powers of two which are sums of four Fibonacci numbers with a few exceptions that we characterize.Keywords: Linear forms in logarithm, Diophantine equations, Fibonacci sequence, Lucas sequence, perfect powers
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The purpose of the paper is to present a new generalization of the dual Fibonacci quaternions called the generalized dual bi-periodic Fibonacci quaternions. This new generalization allow us to state several number of dual quaternion sequences in a unique sequence. Furthermore, we give the generating function, the Binet formula, the norm value and some basic properties of these dual quaternions.
Keywords: Fibonacci sequence, bi-periodic Fibonacci sequence, quaternions, dual quaternions, dual Fibonacci quaternions
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