. The first result concerns branching programs having width (log n) O(1) . We give an \Omega\Gamma n log n= log log n) lower bound for the size of such branching programs computing almost any symmetric Boolean function and in particular the following explicit function: "the sum of the input variables is a quadratic residue mod p" where p is any given prime between n 1=4 and n 1=3 . This is a strengthening of previous nonlinear lower bounds obtained by Chandra, Furst, Lipton and by Pudl'ak. We mention that by iterating our method the result can be further strengthened to \Omega\Gamma n log n). The second result is a C n lower bound for read-once-only branching programs computing an explicit Boolean function. For n = \Gamma v 2 \Delta , the function computes the parity of the number of triangles in a graph on v vertices. This improves previous exp(c p n) lower bounds for other graph functions by Wegener and Z'ak. The result implies a linear lower bound for the space comp...

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Nonstationary signals appear often in real-life applications and many of them can be modeled as polynomial phase signals (PPS). High-order ambiguity function (HAF) was first introduced to estimate the parameters of a single component PPS, but has not been widely used for multi-component PPS because of its nonlinearity. Multi-component PPS arise for example, in Doppler radar applications when multiple objects are tracked simultaneously. In this paper, we show that HAF is virtually additive for multi-component PPS and suggest an algorithm to estimate their parameters. Numerical examples are presented to illustrate the theories. Keywords: Polynomial phase signal, chirp signal, high-order ambiguity function, radar, Weyl sum. Original manuscript was submitted to Signal Processing 6/96; revised 12/96, 12/97. y Research supported in part by the National Science Foundation, grant DMS--9307601. 1 1 Introduction Signals encountered in engineering applications such as communications, radar,...

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Let η be a quadratic irrationality. The variant of Hua Loo Keng’s problem involving primes such that a < {ηp 2} < b, where a and b are arbitrary real numbers of the interval (0,1), solved in this paper. 1. Introduction Suppose that k