Academic literature on the topic 'Ramanujan sums'
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Journal articles on the topic "Ramanujan sums"
Chan, T. H., and A. V. Kumchev. "On sums of Ramanujan sums." Acta Arithmetica 152, no. 1 (2012): 1–10. http://dx.doi.org/10.4064/aa152-1-1.
Full textLaohakosol, Vichian, Pattira Ruengsinsub, and Nittiya Pabhapote. "Ramanujan sums via generalized Möbius functions and applications." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–34. http://dx.doi.org/10.1155/ijmms/2006/60528.
Full textFujisawa, Yusuke. "On sums of generalized Ramanujan sums." Indian Journal of Pure and Applied Mathematics 46, no. 1 (February 2015): 1–10. http://dx.doi.org/10.1007/s13226-015-0103-1.
Full textTóth, László. "Sums of products of Ramanujan sums." ANNALI DELL'UNIVERSITA' DI FERRARA 58, no. 1 (December 24, 2011): 183–97. http://dx.doi.org/10.1007/s11565-011-0143-3.
Full textFowler, Christopher F., Stephan Ramon Garcia, and Gizem Karaali. "Ramanujan sums as supercharacters." Ramanujan Journal 35, no. 2 (July 17, 2013): 205–41. http://dx.doi.org/10.1007/s11139-013-9478-y.
Full textKiuchi, Isao. "Sums of averages of generalized Ramanujan sums." Journal of Number Theory 180 (November 2017): 310–48. http://dx.doi.org/10.1016/j.jnt.2017.03.026.
Full textKIUCHI, Isao. "On Sums of Averages of Generalized Ramanujan Sums." Tokyo Journal of Mathematics 40, no. 1 (June 2017): 255–75. http://dx.doi.org/10.3836/tjm/1502179227.
Full textChen, Guangyi, Sridhar Krishnan, and Tien D. Bui. "Matrix-Based Ramanujan-Sums Transforms." IEEE Signal Processing Letters 20, no. 10 (October 2013): 941–44. http://dx.doi.org/10.1109/lsp.2013.2273973.
Full textBERNDT, BRUCE C., and PING XU. "An Integral Analogue of Theta Functions and Gauss Sums in Ramanujan's Lost Notebook." Mathematical Proceedings of the Cambridge Philosophical Society 147, no. 2 (September 2009): 257–65. http://dx.doi.org/10.1017/s0305004109002552.
Full textCooper, Shaun, and Michael Hirschhorn. "Sums of Squares and Sums of Triangular Numbers." gmj 13, no. 4 (December 2006): 675–86. http://dx.doi.org/10.1515/gmj.2006.675.
Full textDissertations / Theses on the topic "Ramanujan sums"
Souza, Matheus Bernardini de. "A equação de Ramanujan-Nagell e algumas de suas generalizações." reponame:Repositório Institucional da UnB, 2013. http://repositorio.unb.br/handle/10482/13573.
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O objetivo deste trabalho é mostrar algumas técnicas para resolução de equações diofantinas. Métodos algébricos são ferramentas de grande utilidade para a resolução da equação equation x2 + 7 = yn, em que y = 2 ou Y é ímpar. O uso do método hipergeométrico traz um resultado recente (de 2008) no estudo da equação x2 + 7 =2n. m e técnicas algébricas garantem uma condição necessária para que essa última equação tenha solução. _______________________________________________________________________________________ ABSTRACT
The objective of this work is to show some techniques for solving Diophantine equations. Algebraic methods are useful tools for solving the equation x2 + 7 = yn, where y = 2 or y is odd. The use of the hypergeometric method brings a recent result (from 2008) in the study of the equation x2 + 7 = 2n.m and algebraic techniques ensure a necessary condition for the last equation to have a solution.
Tenneti, Srikanth Venkata. "The Nested Periodic Subspaces: Extensions of Ramanujan Sums for Period Estimation." Thesis, 2018. https://thesis.library.caltech.edu/11029/14/tenneti-srikanth-venkata-thesis-V4.pdf.
Full textIn the year 1918, the Indian mathematician Srinivasa Ramanujan proposed a set of sequences called Ramanujan Sums as bases to expand arithmetic functions in number theory. Today, exactly a 100 years later, we will show that these sequences re-emerge as exciting tools in a completely different context: For the extraction of periodic patterns in data. Combined with the state-of-the-art techniques of DSP, Ramanujan Sums can be used as the starting point for developing powerful algorithms for periodicity applications.
The primary inspiration for this thesis comes from a recent extension of Ramanujan sums to subspaces known as the Ramanujan subspaces. These subspaces were designed to span any sequence with integer periodicity, and have many interesting properties. Starting with Ramanujan subspaces, this thesis first develops an entire family of such subspace representations for periodic sequences. This family, called Nested Periodic Subspaces due to their unique structure, turns out to be the least redundant sets of subspaces that can span periodic sequences.
Three classes of new algorithms are proposed using the Nested Periodic Subspaces: dictionaries, filter banks, and eigen-space methods based on the auto-correlation matrix of the signal. It will be shown that these methods are especially advantageous to use when the data-length is short, or when the signal is a mixture of multiple hidden periods. The dictionary techniques were inspired by recent advances in sparsity based compressed sensing. Apart from the l1 norm based convex programs currently used in other applications, our dictionaries can admit l2 norm formulations that have linear and closed form solutions, even when the systems is under-determined. A new filter bank is also proposed using the Ramanujan sums. This, named the Ramanujan Filter Bank, can accurately track the instantaneous period for signals that exhibit time varying periodic nature. The filters in the Ramanujan Filter Bank have simple integer valued coefficients, and directly tile the period vs time plane, unlike classical STFT (Short Time Fourier Transform) and wavelets, which tile the time-frequency plane. The third family of techniques developed here are a generalization of the classic MUSIC (MUltiple SIgnal Classification) algorithm for periodic signals. MUSIC is one of the most popular techniques today for line spectral estimation. However, periodic signals are not just any unstructured line spectral signals. There is a nice harmonic spacing between the lines which is not exploited by plain MUSIC. We will show that one can design much more accurate adaptations of MUSIC using Nested Periodic Subspaces. Compared to prior variants of MUSIC for the periodicity problem, our approach is much faster and yields much more accurate results for signals with integer periods. This work is also the first extension of MUSIC that uses simple integer valued basis vectors instead of using traditional complex-exponentials to span the signal subspace. The advantages of the new methods are demonstrated both on simulations, as well as real world applications such as DNA micro-satellites, protein repeats and absence seizures.
Apart from practical contributions, the theory of Nested Periodic Subspaces offers answers to a number of fundamental questions that were previously unanswered. For example, what is the minimum contiguous data-length needed to be able to identify the period of a signal unambiguously? Notice that the answer we seek is a fundamental identifiability bound independent of any particular period estimation technique. Surprisingly, this basic question has never been answered before. In this thesis, we will derive precise expressions for the minimum necessary and sufficient datalengths for this question. We also extend these bounds to the context of mixtures of periodic signals. Once again, even though mixtures of periodic signals often occur in many applications, aspects such as the unique identifiability of the component periods were never rigorously analyzed before. We will present such an analysis as well.
While the above question deals with the minimum contiguous datalength required for period estimation, one may ask a slightly different question: If we are allowed to pick the samples of a signal in a non-contiguous fashion, how should we pick them so that we can estimate the period using the least number of samples? This question will be shown to be quite difficult to answer in general. In this thesis, we analyze a smaller case in this regard, namely, that of resolving between two periods. It will be shown that the analysis is quite involved even in this case, and the optimal sampling pattern takes an interesting form of sparsely located bunches. This result can also be extended to the case of multi-dimensional periodic signals.
We very briefly address multi-dimensional periodicity in this thesis. Most prior DSP literature on multi-dimensional discrete time periodic signals assumes the period to be parallelepipeds. But as shown by the artist M. C. Escher, one can tile the space using a much more diverse variety of shapes. Is it always possible to account for such other periodic shapes using the traditional notion of parallelepiped periods? An interesting analysis in this regard is presented towards the end of the thesis.
Lu, Keng-Shih, and 盧耕世. "The Values and Applications of Ramanujan Sum in Signal Processing." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/55864420289974395280.
Full text國立臺灣大學
電信工程學研究所
102
This thesis contains two main parts: Part I and Part II. In Part I we give a summary of an arithmetic function, Ramanujan sum, and its applications in signal processing. In Part II we introduce a novel method for harmonic analysis of damped oscillators. Ramanujan sum is an arithmetic function, which is recently applied to signal processing. Among its applications, the most important one is the Ramanujan Fourier transform, which uses the unique periodic property of Ramanujan sum to create a Fourier-transform-like frequency transform. This transform can successfully extract some integer periodic components which Fourier transform cannot extract from the signals. In Part I, we mainly discuss about the physical meaning of this mathematical transform. According to previous works, we observed some advantages and disadvantages of Ramanujan Fourier transform. Based on these points, we conclude that there is something to improve on Ramanujan Fourier transform. Then, we introduce the concept of intrinsic integer-periodic functions. We also define a new RS map by considering time shift in the Ramanujan Fourier transform. The intrinsic integer-periodic functions can be regarded to be pure in periodic component, and we prove in mathematics that RS map presents the intrinsic integer-periodic components of a general signal on the column of the map. That is to say, RS map is equivalent to the intrinsic integer-periodic function decomposition, and with this decomposition by RS map, we can explain the physical meaning of Ramanujan Fourier transform. In Part II, we start with the equivalence between Pade approximation of Z-transform and Prony analysis and end up to propose a new algorithm to improve Prony analysis. By Pade approximation, we can analysis the poles and zeros of a discrete signal, and by the equivalent property mentioned above, those poles analyzed are the bases obtained by Prony analysis. In the pole-zero plot, most poles appear together with zeros, but we know that for the main bases of oscillators, their poles do not come up with zeros. Thus, we based on the idea in a previous work that we can remove pole-zero pairs from the complex plane so that we can use Prony analysis with the remaining poles to recover the original signal. We design several methods to recognition paired poles and zeros and do many experiments. The results show that our method work better than Prony analysis in noisy environment.
Lai, Hsiang-Ling, and 賴香伶. "On a Generalization of Ramanujan's Sum." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/05509714889395483432.
Full text國立彰化師範大學
數學系所
99
In this thesis, we study the generalized Ramanujan's sum and the generalized Von Sterneck function by S.-S. Huang and W.-S. Peng. We derive the multiplicativity and some interesting identities of the sum and the function, and establish elegant orthogonal relations.
Chang, Kuo-Wei, and 張國韋. "Ramanujan’s Sum and Its Application to SignalProcessing and Period Estimation." Thesis, 2018. http://ndltd.ncl.edu.tw/handle/d87gv5.
Full text國立臺灣大學
電信工程學研究所
106
This thesis presents two applications of Ramanujan’s sum in the domain of signal processing. The first one is integer zero autocorrelation sequence construction, which is useful in modern communication system. The concept is to transform this number theoretic problem into a constant amplitude signal construction problem, by Fourier transform and the integer property of Ramanujan’s sum. The second application is 1D and 2D period estimation. The periodic signal is separated into sub-period signals, just like filter bank. Each sub-period is a factor of the length. Finally we use all phase FFT to enhance the period estimation. All phase FFT use phase information to estimate frequency. The advantage is the frequency is non integer. Finally we propose some other applications of all phase FFT, such as chirp signal pitch tracking and fast 2D frequency estimation.
Books on the topic "Ramanujan sums"
Alladi, Krishnaswami, Frank Garvan, and Ae Ja Yee. Ramanujan 125: International conference to commemorate the 125th anniversary of Ramanujan's birth, Ramanujan 125, November 5--7, 2012, University of Florida, Gainesville, Florida. Providence, Rhode Island: American Mathematical Society, 2014.
Find full textTóth, László, Michael Th Rassias, and Helmut Maier. Recent Progress on Topics of Ramanujan Sums and Cotangent Sums Associated with the Riemann Hypothesis. World Scientific Publishing Co Pte Ltd, 2022.
Find full textBook chapters on the topic "Ramanujan sums"
McCarthy, Paul J. "Ramanujan Sums." In Introduction to Arithmetical Functions, 70–113. New York, NY: Springer New York, 1986. http://dx.doi.org/10.1007/978-1-4613-8620-9_2.
Full textBerndt, Bruce. "Sums of squares and sums of triangular numbers." In Number Theory in the Spirit of Ramanujan, 55–84. Providence, Rhode Island: American Mathematical Society, 2006. http://dx.doi.org/10.1090/stml/034/03.
Full textChen, Guangyi, Sridhar Krishnan, Weihua Liu, and Wenfang Xie. "Sparse Signal Analysis Using Ramanujan Sums." In Intelligent Computing Theories and Technology, 450–56. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-39482-9_52.
Full textLi, W. "Elliptic curves, Kloosterman sums, and Ramanujan graphs." In AMS/IP Studies in Advanced Mathematics, 179–90. Providence, Rhode Island: American Mathematical Society, 1997. http://dx.doi.org/10.1090/amsip/007/09.
Full textDe, Debaprasad, K. Gaurav Kumar, Archisman Ghosh, and M. K. Naskar. "Ramanujan Sums and Signal Processing: An Overview." In Lecture Notes in Electrical Engineering, 391–412. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-10-8234-4_34.
Full textLaohakosol, Vichian, and Pussadee Yangklan. "Generalized Unitary Convolution, Ramanujan Sums and Applications." In Lecture Notes in Networks and Systems, 69–92. Singapore: Springer Nature Singapore, 2022. http://dx.doi.org/10.1007/978-981-19-5181-7_6.
Full textHanumanthachari, J. "An extended Nagell Totient and related Ramanujan sums." In Number Theory, 115–29. Berlin, Heidelberg: Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/bfb0075755.
Full textMariconda, Carlo, and Alberto Tonolo. "Cauchy and Riemann Sums, Factorials, Ramanujan Numbers and Their Approximations." In UNITEXT, 579–618. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-03038-8_14.
Full textAndrews, George E., and Bruce C. Berndt. "Divisor Sums." In Ramanujan's Lost Notebook, 213–37. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-4081-9_9.
Full textBerndt, Bruce C. "Sums of Powers, Bernoulli Numbers, and the Gamma Function." In Ramanujan’s Notebooks, 150–80. New York, NY: Springer New York, 1985. http://dx.doi.org/10.1007/978-1-4612-1088-7_8.
Full textConference papers on the topic "Ramanujan sums"
Pei, Soo-Chang, and Chia-Chang Wen. "Legendre Ramanujan Sums transform." In 2015 23rd European Signal Processing Conference (EUSIPCO). IEEE, 2015. http://dx.doi.org/10.1109/eusipco.2015.7362519.
Full textAbraham, Deepa, and Manju Manuel. "Image Transmultiplexing using Ramanujan Sums." In 2020 12th International Conference on Computational Intelligence and Communication Networks (CICN). IEEE, 2020. http://dx.doi.org/10.1109/cicn49253.2020.9242550.
Full textChen, Guangyi, Sridhar Krishnan, and Wenfang Xie. "Ramanujan sums-wavelet transform for signal analysis." In 2013 International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR). IEEE, 2013. http://dx.doi.org/10.1109/icwapr.2013.6599326.
Full textZhou, Lina, Zulin Wang, Jiadong Shang, Lei Zhao, and Song Pan. "A Programmable Approach to Evaluate Ramanujan Sums." In 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013). Paris, France: Atlantis Press, 2013. http://dx.doi.org/10.2991/iccsee.2013.9.
Full textKrishnaprasad, P., and Ajeesh Ramanujan. "Ramanujan sums based image kernels for computer vision." In 2016 International Conference on Electrical, Electronics, and Optimization Techniques (ICEEOT). IEEE, 2016. http://dx.doi.org/10.1109/iceeot.2016.7754803.
Full textGhosh, Archisman, K. Gaurav Kumar, Debaprasad De, Arnab Raha, and Mrinal Kanti Naskar. "Energy-Efficient Edge Detection using Approximate Ramanujan Sums." In 2020 21st International Symposium on Quality Electronic Design (ISQED). IEEE, 2020. http://dx.doi.org/10.1109/isqed48828.2020.9137002.
Full textJaiswal, Priyansh, Noman Ahmed Ansari, Anup Kumar Mandpura, and Neeta Pandey. "An Iterative Method for Image Denoising Based on Ramanujan Sums." In 2022 International Conference on Intelligent Technologies (CONIT). IEEE, 2022. http://dx.doi.org/10.1109/conit55038.2022.9847857.
Full textSajikumar, S., and S. Vinod. "Ramanujan Sums-Wavelet Transform: A New Approach to Signal Processing." In 2022 Third International Conference on Intelligent Computing Instrumentation and Control Technologies (ICICICT). IEEE, 2022. http://dx.doi.org/10.1109/icicict54557.2022.9917837.
Full textYang, Qinbiao, Zulin Wang, and Qin Huang. "An Efficient Non-Uniform Multi-Tone System Based On Ramanujan Sums." In 2017 2nd Joint International Information Technology, Mechanical and Electronic Engineering Conference (JIMEC 2017). Paris, France: Atlantis Press, 2017. http://dx.doi.org/10.2991/jimec-17.2017.79.
Full textVaidyanathan, P. P. "Multidimensional Ramanujan-sum expansions on nonseparable lattices." In ICASSP 2015 - 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2015. http://dx.doi.org/10.1109/icassp.2015.7178655.
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