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The Encyclopedia of Integer Sequences
"... This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs ..."
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Cited by 631 (15 self)
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This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other
An elementary problem equivalent to the Riemann hypothesis
 Amer. Math. Monthly
"... ABSTRACT. The problem is: Let Hn = n∑ n ≥ 1, that with equality only for n = 1. j=1 1 j d ≤ Hn + exp(Hn)log(Hn), ..."
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Cited by 21 (2 self)
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ABSTRACT. The problem is: Let Hn = n∑ n ≥ 1, that with equality only for n = 1. j=1 1 j d ≤ Hn + exp(Hn)log(Hn),
On Robin’s criterion for the Riemann Hypothesis
, 2006
"... Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality σ(n): = ∑ dn d < eγn log log n is satisfied for n ≥ 5041, where γ denotes the Euler(Mascheroni) constant. We show by elementary methods that if n ≥ 37 does not satisfy Robin’s criterion it must be e ..."
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Cited by 5 (3 self)
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Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality σ(n): = ∑ dn d < eγn log log n is satisfied for n ≥ 5041, where γ denotes the Euler(Mascheroni) constant. We show by elementary methods that if n ≥ 37 does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that n must be divisible by a fifth power> 1. As consequence we obtain that RH holds true iff every natural number divisible by a fifth power> 1 satisfies Robin’s inequality.
Eight Hateful Sequences
, 2008
"... Sequences (the OEIS) [12] contains 140000 sequences. Here are eight of them, suggested by the theme of the Eighth Gathering: they are all infinite, and all ’ateful in one way or another. I hope you like ’em! Each one is connected with an interesting unsolved problem. Since this is a 15minute talk, ..."
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Cited by 1 (0 self)
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Sequences (the OEIS) [12] contains 140000 sequences. Here are eight of them, suggested by the theme of the Eighth Gathering: they are all infinite, and all ’ateful in one way or another. I hope you like ’em! Each one is connected with an interesting unsolved problem. Since this is a 15minute talk, I can’t give many details—see the entries in the OEIS for more information, and for links to related sequences. 1. Hateful or Beastly Numbers The most hateful sequence of all! These are the numbers that contain the string 666 in their decimal expansion:
Multidimensional Filter Bank Signal Reconstruction From Multichannel Acquisition
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
"... We study the theory and algorithm for an optimal use of multidimensional signal reconstruction from multichannel acquisition using a filter bank setup. Suppose that we have an Nchannel convolution system in M dimensions. Instead of taking all the data and applying multichannel deconvolution, we fi ..."
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We study the theory and algorithm for an optimal use of multidimensional signal reconstruction from multichannel acquisition using a filter bank setup. Suppose that we have an Nchannel convolution system in M dimensions. Instead of taking all the data and applying multichannel deconvolution, we first reduce the collected data set by an integer M × M sampling matrix D and then search for synthesis filters which could perfectly reconstruct the input discrete signal. We determine the existence of perfect reconstruction (PR) systems for given Laurent polynomial analysis filters with some sampling matrices and some Laurent polynomial synthesis polyphase matrices. We present an efficient algorithm to find a sampling matrix with maximum sampling rate and a Laurent polynomial PR synthesis polyphase matrix for given Laurent polynomial analysis filters. Finally, once a particular Laurent polynomial PR synthesis polyphase matrix is found, we can characterize all Laurent polynomial PR synthesis matrices and find an optimal one according to design criteria including robust reconstruction in the presence of noise.
Superabundant Numbers and the Riemann Hypothesis
"... The function σ(n) = ∑ dn d is the sum of divisors function, so for example σ(12) = 28. In 1913 Gronwall proved that lim sup n→∞ σ(n) e γ n log log n where γ ≈ 0.57721 is Euler’s constant (see [4, Theorem 323]). This says that the maximal size of σ(n) is roughly e γ n log log n. The following theo ..."
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The function σ(n) = ∑ dn d is the sum of divisors function, so for example σ(12) = 28. In 1913 Gronwall proved that lim sup n→∞ σ(n) e γ n log log n where γ ≈ 0.57721 is Euler’s constant (see [4, Theorem 323]). This says that the maximal size of σ(n) is roughly e γ n log log n. The following theorem of Robin [7, Theorem 2] gives a more refined version of this upper bound. Theorem 1 For n ≥ 3 we have σ(n) e γ n log log n
Notes on the Riemann hypothesis and abundant
, 2005
"... RH abundant.tex (‘notes ’ option) typeset in pdfLATEX on a linux system ⋆ In [1], Lagarias showed the equivalence of the Riemann hypothesis (RH) to a condition on harmonic sums, namely RH ⇔ σ(n) � Hn+exp(Hn) log(Hn) ∀n ⋆ Robin [3] had already shown that ..."
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RH abundant.tex (‘notes ’ option) typeset in pdfLATEX on a linux system ⋆ In [1], Lagarias showed the equivalence of the Riemann hypothesis (RH) to a condition on harmonic sums, namely RH ⇔ σ(n) � Hn+exp(Hn) log(Hn) ∀n ⋆ Robin [3] had already shown that
Remarks on Robin’s and Nicolas Inequalities
, 2009
"... Nicolas Conjecture is disproved. The Robin Conjecture follows. To V. I. Arnold, the greatest mathematician of all times. I believe this to be false. There is no evidence whatever for it (unless one counts that it is always nice when any function has only real roots). One should not believe things fo ..."
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Nicolas Conjecture is disproved. The Robin Conjecture follows. To V. I. Arnold, the greatest mathematician of all times. I believe this to be false. There is no evidence whatever for it (unless one counts that it is always nice when any function has only real roots). One should not believe things for which there is no evidence. In the spirit of this anthology I should also record my feelings that there is no imaginable reason why it should be true. J. E. Littlewood on Riemnann’s Conjecture... We are all in our own eyes a failure: after all, we haven’t proved Fermat’s Last Theorem, nor Riemann’s Conjecture.
LAURENT POLYNOMIAL INVERSE MATRICES AND MULTIDIMENSIONAL PERFECT RECONSTRUCTION SYSTEMS
, 2008
"... We study the invertibility of Mvariate polynomial (respectively: Laurent polynomial) matrices of size N by P. Such matrices represent multidimensional systems in various settings including filter banks, multipleinput multipleoutput systems, and multirate systems. Given an N × P polynomial matrix H ..."
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We study the invertibility of Mvariate polynomial (respectively: Laurent polynomial) matrices of size N by P. Such matrices represent multidimensional systems in various settings including filter banks, multipleinput multipleoutput systems, and multirate systems. Given an N × P polynomial matrix H(z) of degree at most k, we want to find a P × N polynomial (resp.: Laurent polynomial) left inverse matrix G(z) of H(z) such that G(z)H(z) = I. We provide computable conditions to test the invertibility and propose algorithms to find a particular inverse. The main result of this thesis is to prove that when N −P ≥ M, then H(z) is generically invertible; whereas when N − P < M, then H(z) is generically noninvertible. Based on this fact, we provide some applications and propose a faster algorithm to find a particular inverse of a Laurent polynomial matrix. The next main topic we are interested is the theory and algorithms for the optimal use of multidimensional signal reconstruction from multichannel acquisition using a filter bank setup. Suppose that we have an Nchannel convolution system in M dimensions. Instead of taking all the data and applying multichannel deconvolution, we can first reduce the collected data set by an integer M × M sampling matrix D and still perfectly reconstruct the signal with a synthesis polyphase matrix. First, we determine the existence of perfect reconstruction systems for given finite impulse response (FIR) analysis filters with some sampling