693 research outputs found
Ramanujan Sums for Image Pattern Analysis
Ramanujan Sums (RS) have been found to be very successful in signal processing recently. However, as far as we know, the RS have not been applied to image analysis. In this paper, we propose two novel algorithms for image analysis, including moment invariants and pattern recognition. Our algorithms are invariant to the translation, rotation and scaling of the 2D shapes. The RS are robust to Gaussian white noise and occlusion as well. Our algorithms compare favourably to the dual-tree complex wavelet (DTCWT) moments and the Zernike's moments in terms of correct classification rates for three well-known shape datasets
On Plouffe's Ramanujan Identities
Recently, Simon Plouffe has discovered a number of identities for the Riemann
zeta function at odd integer values. These identities are obtained numerically
and are inspired by a prototypical series for Apery's constant given by
Ramanujan: Such sums follow from a general relation given by Ramanujan, which is
rediscovered and proved here using complex analytic techniques. The general
relation is used to derive many of Plouffe's identities as corollaries. The
resemblance of the general relation to the structure of theta functions and
modular forms is briefly sketched.Comment: 19 pages, 3 figures; v4: minor corrections; modified intro; revised
concluding statement
Uncovering Ramanujan's "Lost" Notebook: An Oral History
Here we weave together interviews conducted by the author with three
prominent figures in the world of Ramanujan's mathematics, George Andrews,
Bruce Berndt and Ken Ono. The article describes Andrews's discovery of the
"lost" notebook, Andrews and Berndt's effort of proving and editing Ramanujan's
notes, and recent breakthroughs by Ono and others carrying certain important
aspects of the Indian mathematician's work into the future. Also presented are
historical details related to Ramanujan and his mathematics, perspectives on
the impact of his work in contemporary mathematics, and a number of interesting
personal anecdotes from Andrews, Berndt and Ono
Classical elliptic hypergeometric functions and their applications
General theory of elliptic hypergeometric series and integrals is outlined.
Main attention is paid to the examples obeying properties of the "classical"
special functions. In particular, an elliptic analogue of the Gauss
hypergeometric function and some of its properties are described. Present
review is based on author's habilitation thesis [Spi7] containing a more
detailed account of the subject.Comment: 42 pages, typos removed, references update
Vertex-algebraic structure of the principal subspaces of certain A_1^(1)-modules, II: higher level case
We give an a priori proof of the known presentations of (that is,
completeness of families of relations for) the principal subspaces of all the
standard A_1^(1)-modules. These presentations had been used by Capparelli,
Lepowsky and Milas for the purpose of obtaining the classical Rogers-Selberg
recursions for the graded dimensions of the principal subspaces. This paper
generalizes our previous paper.Comment: 26 pages; v2: minor revisions, to appear in Journal of Pure and
Applied Algebr
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