For a positive semi-definite biquadratic forms F in (3, 3) variables, we prove that if F has a finite number but at least 7 real zeros Z(F ), then it is not a sum of squares. We show also that if F has at least 11 zeros, then it has infinitely many real zeros and it is a sum of squares. It can be seen as the counterpart for biquadratic forms as the results of Choi, Lam and Reznick for positive ...

First year undergraduates usually learn about classical Rolle’s theorem saying that between two consecutive zeros of a smooth univariate function f one can always find at least one zero of its derivative f . In this paper we study a generalization of Rolle’s theorem dealing with zeros of higher derivatives of smooth univariate functions enjoying a natural additional property. Namely, we call a ...

where l > 0 and {dk0, dk1} is a so-called symmetrically coherent pair, with dk0 or dk1 the classical Gegenbauer measure (x−1) dx, a > −1. If dk1 is the Gegenbauer measure, then Sn has n different, real zeros. If dk0 is the Gegenbauer measure, then Sn has at least n−2 different, real zeros. Under certain conditions Sn has complex zeros. Also the location of the zeros of Sn with respect to Gegenb...

Let [Formula: see text] be a real and non-principal Dirichlet character, its text]-function let generic prime number. We prove the following result: If for some partial sums change sign only finite number of integers text], then there exists such that has no zeros in half plane text].

In this paper, an attempt is made to include division by zero in ordinary arithmetic. Counting or measuring done get the value of multiples, powers, quotients, sums and differences zeros. Zero divided taken be equal one. There are many infinities zeroes. The biggest all cannot imagined smallest also. An define a mathematical structure from set real numbers, zeros infinities. Some fallacies have...

Let G(z) be a real entire function of order less than 2 with only real zeros. Then we classify certain distributions functions F such that the convolution (G ∗ dF )(z) = ∫∞ −∞G(z − is) dF (s) has only real zeros.

Based on a new explicit upper bound for the number of zeros of exponential polynomials in a horizontal strip, we obtain a uniform upper bound for the number of zeros of solutions to an ordinary differential equation near its Fuchsian singular point, provided that any two distinct characteristic exponents at this point have distinct real parts. The latter result implies that a Fuchsian different...