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New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function
, 1996
"... Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n ..."
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Cited by 35 (1 self)
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Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular Cfractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the ηfunction identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing
Solving constrained Pell equations
 Math. Comp
, 1998
"... Abstract. Consider the system of Diophantine equations x 2 − ay 2 = b, P (x, y) =z 2,whereP is a given integer polynomial. Historically, such systems have been analyzed by using Baker’s method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an i ..."
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Cited by 23 (0 self)
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Abstract. Consider the system of Diophantine equations x 2 − ay 2 = b, P (x, y) =z 2,whereP is a given integer polynomial. Historically, such systems have been analyzed by using Baker’s method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to the cases P (x, y) =cy 2 + d and P (x, y) =cx + d, which arise when looking for integer points on an elliptic curve with a rational 2torsion point. 1.
On the zeros of polynomials with restricted coefficients
 Illinois J. Math
, 1997
"... It is proved that a polynomial p of the form has at most c # n zeros inside any polygon with vertices on the unit circle, where the constant c depends only on the polygon. Furthermore, if # (0, 1) then every polynomial p of the above form has at most c/# zeros inside any polygon w ..."
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Cited by 19 (6 self)
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It is proved that a polynomial p of the form has at most c # n zeros inside any polygon with vertices on the unit circle, where the constant c depends only on the polygon. Furthermore, if # (0, 1) then every polynomial p of the above form has at most c/# zeros inside any polygon with vertices on the circle =1#},where the constant c depends only on the number of vertices of the polygon. It is also shown that there is an absolute constant c such that every polynomial of the form an  arg(z)##}.
Matched spectralnulls codes for partial response channels
 IEEE Trans. Inform. Theory
, 1991
"... AbstractA new family of codes is described that improve the reliability of digital communication over noisy, partialresponse channels. The codes are intended for use on channels where the input alphabet size is limited. These channels arise in the context of digital data recording and certain data ..."
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Cited by 18 (6 self)
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AbstractA new family of codes is described that improve the reliability of digital communication over noisy, partialresponse channels. The codes are intended for use on channels where the input alphabet size is limited. These channels arise in the context of digital data recording and certain data transmission applications. The codescalled matchedspectralnull codessatisfy the property that the frequencies at which the code power spectral density vanishes correspond precisely to the frequencies at which the channel transfer function is zero. It is shown that matchedspectralnul1 sequences provide a distance gain on the order of 3 dB and higher for a broad class of partialresponse channels, including many of those of primary interest in practical applications. The embodiment of the matchedspectralnull coded partialresponse system incorporates a slidingblock code and a Viterbi detector based upon a reducedcomplexity trellis structure, both derived from canonical diagrams that characterize spectralnull sequences. The detectors are shown to achieve the same asymptotic average performance as maximumlikelihood sequencedetectors, and the slidingblock codes exclude quasicatastrophic trellis sequences in order to reduce the required path memory length and improve “worstcase ” detector performance. Several examples are described in detail. Index TermsSpectralnull codes, partialresponse channels.
LittlewoodType Problems On [0,1]
"... A highlight of the paper states that there are absolute constants c1 > 0 and c2 > 0 such that exp , c1 # n # # inf 0#=p#Fn #p# [0,1] # exp , c2 # n # for every n # 2, where Fn denotes the set of polynomials of degree at most n with coe#cients from {1, 0, 1}. This Chebyshevtype prob ..."
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Cited by 16 (12 self)
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A highlight of the paper states that there are absolute constants c1 > 0 and c2 > 0 such that exp , c1 # n # # inf 0#=p#Fn #p# [0,1] # exp , c2 # n # for every n # 2, where Fn denotes the set of polynomials of degree at most n with coe#cients from {1, 0, 1}. This Chebyshevtype problem is closely related to the question of how many zeros a polynomial from the above classes can have at 1. We also give essentially sharp bounds for this problem. Among others we prove that there is an absolute constant c>0 such that every polynomial p of the form p(x)= n X j=0 a j x j , a j #1, a 0 =a n =1,a j #C has at most c # n real zeros. This improves the old bound c # n log n givenbySchur in 1933 as well as more recent related bounds of Bombieri and Vaaler, and up to the constant c this is the best possible result.
An Investigation of Bounds for the Regulator of Quadratic Fields
 Experimental Mathematics
, 1995
"... This paper considers the following problems: How large, and how small, can R get? And how often? The answer is simple for the problem of how small R can be, but seems to be extremely difficult for the question of how large ..."
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Cited by 12 (6 self)
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This paper considers the following problems: How large, and how small, can R get? And how often? The answer is simple for the problem of how small R can be, but seems to be extremely difficult for the question of how large
Acceleration of Euclidean algorithm and rational number reconstruction
 SIAM Journal on Computing
, 2003
"... Abstract. We accelerate the known algorithms for computing a selected entry of the extended Euclidean algorithm for integers and, consequently, for the modular and numerical rational number reconstruction problems. The acceleration is from quadratic to nearly linear time, matching the known complexi ..."
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Cited by 11 (3 self)
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Abstract. We accelerate the known algorithms for computing a selected entry of the extended Euclidean algorithm for integers and, consequently, for the modular and numerical rational number reconstruction problems. The acceleration is from quadratic to nearly linear time, matching the known complexity bound for the integer gcd, which our algorithm computes as a special case.