Motivated by Wilmshurst’s conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy continuation to compute a numerical approximation of each zero and Smale’s alpha-theory to certify the results. Using this approach, we provide new examples of harmo...

We describe a computation which shows that the Riemann zeta function f(s) has exactly 75,000,000 zeros of the form a + it in the region 0 < t < 32,585,736.4; all these zeros are simple and lie on the line o = Hi. (A similar result for the first 3,500,000 zeros was established by Rosser, Yohe and Schoenfeld.) Counts of the number of Gram blocks of various types and the number of failures of "Ros...

Let p > 3 be a prime. We consider j-zeros of Eisenstein series Ek of weights k = p−1+Mp(p−1) with M,a ≥ 0 as elements of Qp. If M = 0, the j-zeros of Ep−1 belong to Qp(ζp2−1) by Hensel’s Lemma. Call these j-zeros p-adic liftings of supersingular j-invariants. We show that for every such lifting u there is a j-zero r of Ek such that ordp(r − u) > a. Applications of this result are considered. Th...

The paper studies the local zero spacings of deformations of the Riemann ξ-function under certain averaging and differencing operations. For real h we consider the entire functions Ah(s) := 1 2 (ξ(s + h) + ξ(s− h)) and Bh(s) = 1 2i (ξ(s+ h)− ξ(s− h)) . For |h| ≥ 1 2 the zeros of Ah(s) and Bh(s) all lie on the critical line R(s) = 1 2 and are simple zeros. The number of zeros of these functions ...

Prior results on input reconstruction for multi-input, multi-output discretetime linear systems are extended by defining l-delay input and initial-state observability. This property provides the foundation for reconstructing both unknown inputs and unknown initial conditions, and thus is a stronger notion than l-delay left invertibility, which allows input reconstruction only when the initial s...

We state precise results on the complexity of a classical bisectionexclusion method to locate zeros of univariate analytic functions contained in a square. The output of this algorithm is a list of squares containing all the zeros. It is also a robust method to locate clusters of zeros. We show that the global complexity depends on the following quantities: the size of the square, the desired p...

We describe extensive computations which show that Riemann's zeta function f(s) has exactly 200,000,001 zeros of the form a + it in the region 0 < t < 81,702,130.19; all these zeros are simple and he on the line a = j. (This extends a similar result for the first 81,000,001 zeros, established by Brent in Math. Comp., v. 33, 1979, pp. 1361-1372.) Counts of the numbers of Gram blocks of various t...

We investigate monotonicity properties of extremal zeros of orthogonal polynomials depending on a parameter. Using a functional analysis method we prove the monotonicity of extreme zeros of associated Jacobi, associated Gegenbauer and q-Meixner-Pollaczek polynomials. We show how these results can be applied to prove interlacing of zeros of orthogonal polynomials with shifted parameters and to d...

Fredholm determinant asymptotics of convolution operators on large finite intervals with rational symbols having real zeros are studied. The explicit asymptotic formulae obtained can be considered as a direct extension of the Ahiezer-Kac formula to symbols with real zeros.

The key idea of this contribution is the partial compensation non-minimum phase zeros or unstable poles. Therefore integer-order zero/pole split into a product fractional-order pseudo zeros/poles. amplitude and response these terms derived to include compensators loop-shaping design. Such can be generalized conjugate complex zeros/poles, also implicit applied. In case zero, its leads higher mar...