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Isaksen, Judy L.

doi:10.1080/10572259909364651

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On the Basis Of Computations Performed By Armira Shkembi And1

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2008

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“…Theorem 5 proves that near most multiples of a power of 2 we find a counter-example to the conjecture (see [4]) that r s ⇒ A(r, s) A(s, r) with both r − s and A(s, r) − A(r, s) asymptotically as large as possible since using the asymptotic form for A(r, s), these are both clearly as bad as they can possibly get. both c 1 and t 2 − r are within O(log r) of r/2, so there is a multiple of 2 N +O(log log r) within O(log r) of r/2.…”

Section: On the Basis Of Computations Performed By Armira Shkembi Andmentioning

confidence: 95%

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On lower bounds on the size of sums-of-squares formulas

Kane

2008

Journal of Number Theory

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For sums-of-squares formulas of the form x 2 1 + · · · + x 2 r y 2 1 + · · · + y 2 s = z 2 1 + · · · + z 2 t where the z i are bilinear functions of the x i and y i . Let L(r, s) denote the smallest possible value of t allowing such a formula to hold. We have two well-known lower bounds on the size of L(r, s). One was obtained independently by Hopf and Stiefel, and another by Atiyah. These bounds are given by requiring certain binomial coefficients be divisible by certain powers of 2. Although the behavior of the Hopf-Stiefel bound is fairly well understood, the Atiyah bound is not. In this paper we provide an efficient algorithm for computing the Atiyah bound and some results on which of the lower bounds is larger.

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Section: On the Basis Of Computations Performed By Armira Shkembi Andmentioning

confidence: 95%

“…I am thankful to Dan Isaksen for providing me with the topic for this paper and for providing reference [4]. This paper was written at the University of Minnesota, Duluth REU program under the supervision of Joseph Gallian, with support from the NSF (DMS 0447070) and NSA (H98230-04-1-0050).…”

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confidence: 97%

On lower bounds on the size of sums-of-squares formulas

Kane

2008

Journal of Number Theory

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On lower bounds on the size of sums-of-squares formulas

On lower bounds on the size of sums-of-squares formulas

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