Equation and solutionedit the equation is and solutions in natural numbers n and x. The technique is implemented in software such as ecognition or the orfeo. Guergachi, pattern classification of signals using fisher kernels, mathematical problems in engineering, vol. In 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. Circular projection for pattern recognition request pdf. Seshu iyer, published in the collected papers of srinivasa ramanujan 2, would be moved by the. Ramanujan was born in his grandmothers house in erode, a small village about 400 km southwest of madras. Ramanujan images that are useful for samudrik, phrenology, palmistry hand reading, astrology and other methods of prediction. Ramanujan observed that this is the smallest number with this property. In mathematics, in the field of number theory, the ramanujannagell equation is a particular exponential diophantine equation. Srinivasa ramanujan, the greatest indian mathematician of modern times.

Krishnan, effective dysphonia detection using feature dimension reduction and kernel density estimation for patients with. However, as far as we know, the rs have not been applied to image analysis. Cohen and derive the weighted averages corresponding to the. Duration july 25 august 6, 2016 number of participants for the course will be limited to fifty. Recovering ramanujans notebooks from the archives of the university of madras. Our algorithms are invariant to the translation, rotation and scaling of the 2d shapes. On certain trigonometrical sums and their applications in the theory of numbers. Anyone who ever heard of srinivasa ramanujan and reads the compelling ragstointellectualriches story of ramanujan contained in the two notices, one by g. The original matrix based ramanujan sums transform rst by truncating the ramanujan sums series is nonorthogonal and lack of fast algorithm, the. Engineering and computer science computer science and software engineering. Ramanujans approximate formula, developed in 1918, helped him spot that numbers ending in 4 or 9 have a partition number divisible by 5, and he found similar rules for. Here was a man who could work out modular equations and theoremsto orders unheard o.

Ramanujan sumswavelet transform for signal analysis. Ramanujans mock theta functions and some recent developments. A large number of monolithic and highly taskoriented software solutions has been proposed to tackle the problems that occur in each step of bioimage analysis. Ramanujan sums for image pattern analysis international. Onos team found that if you add the corresponding outputs together, the total approaches 4, a relatively small number. Pattern recognition is the automated recognition of patterns and regularities in data. Previous attempts have only provided approximations or relied on crazy infinite sums, says ken ono at emory university in atlanta, georgia. Ramanujan summation is a technique invented by the mathematician srinivasa ramanujan for assigning a value to infinite divergent series. We choose a generalization of ramanujan sum given by e. Shanshan yang, fang zheng, xin luo, suxian cai, yunfeng wu, kaizhi liu, meihong wu, jian chen, s. Ramanujans own work on partial sums and products of hypergeometric series have led to major development in the topic. In other words, the difference in the value of the two functions, ignoring.

Character sums, automorphic forms, equidistribution, and. Can be used on the most stubborn divergent series, but depends on your choice of a parameter. Character sums, automorphic forms, equidistribution, and ramanujan graphs 1 chingli chai2 and wenching winnie li3 version 3. We derive certain identities involving various known arithmetical functions and a generalized version of ramanujan sum. The ramanujanfourier spectrum of the dow jones index over years and of the coronal index of solar activity over 69 years are taken as illustrative examples. The face was automatically detected by special software. In the following few years, before an early death at age 32, ramanujan produced an exceptional output in mathematical analysis, number theory, infinite series and continued fractions. The rs are orthogonal in nature and therefore offer excellent energy conservation capability.

Wikipedia states that ramanujan sums and the riemann zeta function have the. Berndt1, geumlan choi, younseo choi2, heekyoung hahn, boon pin yeap, ae ja yee 3, hamza yesilyurt, and jinhee yi abstract. Image analysis is the extraction of meaningful information from images. Mathematics, which was the most influential book in ramanujans. The opening lines enumerate the list of things the father left behind as legacy. In addition to the expansions discussed in this article, ramanujans sums are used.

Page from ramanujans notebook stating his master theorem. We propose to perform the radon transform to the input 2d shape fx,y 1. International journal of wavelets, multiresolution and information. On an unrelated note, this is one of the best results given by ramanujan which is exceedingly difficult to prove and to date there is no proof available within the limits of hand calculation. Such sums are closely connected with special series for certain additive problems in number theory cf. Ramanujans first published paperwhen ramanujans mathematical friends didnt succeed in getting him a scholarship, ramanujan started looking for. New results arise from the use of this ramanujanfourier. T he world will soon remember the renowned indian mathematician srinivasa ramanujan once again, as a biopic on his life, named the man who knew infinity, is set to release this april. Hardy remarked to ramanujan that he traveled in a taxi cab with license plate 1729, which seemed a dull number. Fans of ramanujans mathematics are sure to be delighted by this book. Ramanujan, who lived a short but very productive life, continues to be an inspiration for mathematicians across the world, and his work has inspired a lot of research over the years. Pattern analysis and machine intelligence, ieee transactions on 125. Mathematical proof reveals magic of ramanujans genius. Deep meaning in ramanujans simple pattern sciencedirect.

In particular, ranks, cranks, and congruences for pn are in the spotlight. H hardy summed up his contibution in following words, the limitations of his knowledge were as startling as its profundity. Also important is to note that this is an early work of ramanujan which was done in india before he got in touch with the british mathematician g. Its whereabouts were unknown to all but a few mathematicians until it was rediscovered by george andrews in 1976, in a box of effects of g.

Ramanujan obtained many formulas involving ramanujan sums. The first three, notebooks 1, 2 and 3 were published in two volumes, as a photocopy edition of the original manuscripts, in his own handwriting in 1957 by the tata institute of. In this paper, we propose two novel algorithms for image analysis, including moment. Sum of numbers of any column 9 sum of diagonal elements 9 sum of any what an interesting square by srinivasa ramanujan an indian mathematician a magic square is an nxn matrix in which every row, column, and diagonal add up to the same number. If we infinitely increased the size of the table, we could find an infinite number of other paired cube sums. Other topics include the ramanujan taufunction, the rogersramanujan functions, highly composite numbers, and sums of powers of theta functions. Life and work of the mathemagician srinivasa ramanujan. In this paper, we provide an application of ramanujan sum expansions to periodic, quasiperiodic and complex time series, as a vital alternative to the fourier transform.

Circular projection for pattern recognition springerlink. Ramanujans lost notebook is the manuscript in which the indian mathematician srinivasa ramanujan recorded the mathematical discoveries of the last year 19191920 of his life. Ramanujan sums rs have been found to be very successful in signal processing recently. To this, ramanujan replied that 1729 was a very interesting number it was the smallest number expressible as the sum of cubes of two numbers in two different ways. It has applications in statistical data analysis, signal processing, image. Ramanujans extraordinary mathematical ability has become the stuff of legend. Some properties of bernoullis numbers journal of the indian mathematical society, iii, 1911, 219 234. Srinivasa ramanujan frs 22 december 1887 26 april 1920 was an indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions.

Ramanujan sums for image pattern analysis spectrum. Ramanujan independently discovered results of gauss, kummer and others on hypergeometric series. Application of the ramanujan fourier transform for the analysis of. Application of the ramanujan fourier transform for the analysis of secondary structure content. Certain weighted averages of generalized ramanujan sums. Pattern recognition software and techniques for biological image. Ramanujan summation is a way to assign a finite value to a divergent series. I wish they had explained more fully what is happening here. In this letter, we study the ramanujan sums rs transform by means of matrix multiplication. Ramanujan reminiscences his fathers death, and the merit and meaning in the speakers familylife. The bad news is that the first of these ramanujan triples. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including. Ramanujans number plus the number of the beast equals the sum of the first prime and the squares of the next 9 primes.

The original matrix based ramanujan sums transform rst by truncating the ramanujan sums series is nonorthogonal and lack of fast algorithm, the proposed lrst has orthogonal property and onlog. Computers are indispensable for the analysis of large amounts of data. The sums are quasiperiodic versus the time nof the resonance and aperiodic versus the order qof the resonance. These sets came to be referred to as the notebooks. Today, exactly a 100 years later, we will show that these sequences reemerge as exciting tools in a completely different context. Deep meaning in ramanujans simple pattern new scientist. Ramanujan sums analysis of longperiod sequences and 1f. What are the mathematical contributions of ramanujan. Perhaps his most famous work was on the number pn of partitions of an integer n into summands. Can someone please explain the concept of a ramanujan sum in easier language than wikipedia and its relation to this question. Ramanujans approximate formula, developed in 1918, helped him spot that numbers ending in 4 or 9 have a partition number divisible by 5, and he found similar rules for partition numbers divisible by. Srinivasa ramanujan was one of indias greatest mathematical geniuses. Short biography of srinivasa ramanujan srinivasa ramanujan large picture color head and shoulders 800 x 850 px.

Surprisingly, the detection of a singly periodic signal by the ramanujan sum analysis has not been considered before. Newest ramanujansummation questions mathematics stack. Publications signal analysis research group ryerson. Ramanujan sums for image pattern analysis request pdf.

In mathematics, ramanujans master theorem named after srinivasa ramanujan is a technique that provides an analytic expression for the mellin transform of an analytic function. In number theory, a branch of mathematics, ramanujans sum, usually denoted c q n, is a function of two positive integer variables q and n defined by the formula. Ramanujan summation allows you to manipulate sums without worrying about operations on infinity that would be considered wrong. Krishnan, discriminant nonstationary signal features clustering using hard and fuzzy cluster labeling, eurasip journal on advances in signal processing, nov. This earlier work remained quite ambiguous about the detection of isolated periods. Ramanujan recorded most of his results in four sets of loose leaf papers before he left for england in 1914. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series. Srinivasa ramanujan was an indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. You should attend if you are a student or faculty from academic institution interested to learn the influence of ramanujan in number theory. In this paper, we propose two novel algorithms for image analysis, including moment invariants and pattern recognition. Srinivasa ramanujan mentioned the sums in a 1918 paper. Proof of certain identities in combinatory analysis proceedings of the cambridge philosophical society, xix.

311 1099 729 1092 266 479 305 1401 1308 1250 1193 501 1476 337 519 799 1370 995 879 730 428 1108 634 989 1311 128 101 486 549 515 1093 325 1303 1187 1045 1421 289 1182 539 951 1105 24 339 712 780 483 178 1282 629