The key here was that each function oscillated at a different spatial scale , and the functions were all orthogonal to each other (so that the upper bound involved a factor of rather than ). One can replicate this example for the Heisenberg group without much difficulty. Indeed, if we let...
I. Progressions of length 4 in finite field geometries“. The main objective in both papers is to bound the quantity for a vector space over a finite field of characteristic greater than , where is defined as the cardinality of the largest subset of that does not contain an arithmetic ...
While the ball does indeed bounce an infinite number of times, with the bounces getting smaller and smaller in height, that doesn’t mean it keeps bouncing forever. The bounces get smaller and smaller not only in height, but also in duration, and the sum of their durations converges to a...
Notes: The upper bound refers to where one assumes there is no measurement error in the test used to identify high-ability children. Our central estimate assumes test reliability of 0.7. Figures refer to the difference in the outcome between high-ability children from high-income backgrounds and...
BA & LH: [In our studies] through interviews, the voices of equity-focused leaders provided examples of practices that contributed to remedying past inequities and attaining a more just future in education. Lead the Change (LtC): What are some of the ideas you hope the field of Educational...
No matter what the band does, however, Don’t You Forget About Me and The Breakfast Club will forever be intertwined. We won’t forget about Simple Minds for sure. Stats: Number 1 in the US, number 6 in Australia, number 7 in the UK (although it remained on the UK charts from ...
The upper bound of two here for the number of solutions in the region (2) is best possible, due to the infinite family of solutions to the equation coming from , and is the Fibonacci number. The appearance of the quantity in Theorem 1 may be familiar to readers that are acquainted ...
for any putative factorization, we obtain an upper bound thanks to the Stirling approximation. At one point, Erdös, Selfridge, and Straus claimed that this upper bound was asymptotically sharp, in the sense that as ; informally, this means we can split into factors that are (mostly) app...
In the mean zero case, it becomes more efficient to use an inverse Littlewood-Offord theorem of Rudelson and Vershynin to obtain (with the normalisation that the entries of have unit variance, so that the eigenvalues of are with high probability), giving the bound for (one also has good...
In this paper we improve the upper bound to come closer to the lower bound: Theorem 1 For any , and any infinite-dimensional , there exist operators obeying (1) such that One can probably improve the exponent somewhat by a modification of the methods, though it does not seem likely th...