Let [Formula: see text] be an odd natural number. In 1738, Abraham de Moivre introduced a family of polynomials degree with rational coefficients, all which are solvable. So far, the Galois groups these have been investigated only for prime numbers and under special assumptions. We describe arbitrary in irreducible case, up to few exceptions. addition, we express zeros such polynomial as functi...

The large group delay of the high order FIR filters is unacceptable in some applications. So, recently, how to reduce the group delay of FIR filters has been studied intensively. To reduce the ringing in the time domain and to maximize the stopband attenuation, it is useful to design FIR filters with maximally flat characteristics in the passband and transmission zeros in the stopband. We prese...

We prove that the number γN of the zeros of a two-parameter simple random walk in its first N × N time steps is almost surely equal to N as N →∞. This is in contrast with our earlier joint effort with Z. Shi [4]; that work shows that the number of zero crossings in the first N × N time steps is N (3/2)+o(1) as N → ∞. We prove also that the number of zeros on the diagonal in the first N time ste...

Let χ be a real odd Dirichlet character of modulus d, and let L(s, χ) be the associated Dirichlet L-function. As a consequence of the work of Low and Purdy, it is known that if d ≤ 800 000 and d 6= 115 147, 357 819, 636 184, then L(s, χ) has no positive real zeros. By a simple extension of their ideas and the advantage of thirty years of advances in computational power, we are able to prove tha...

In this paper, we study polynomials with only real zeros based on the method of compatible zeros. We obtain a necessary and sufficient condition for the compatible property of two polynomials whose leading coefficients have opposite sign. As applications, we partially answer a question proposed by M. Chudnovsky and P. Seymour in the recent publication [M. Chudnovsky, P. Seymour, The roots of th...

For a simplicial complex or more generally Boolean cell complex ∆ we study the behavior of the f and h-vector under barycentric subdivision. We show that if ∆ has a non-negative h-vector then the h-polynomial of its barycentric subdivision has only simple and real zeros. As a consequence this implies a strong version of the Charney-Davis conjecture for spheres that are the subdivision of a Bool...

In factor analysis, a loading matrix is often rotated to simple target for its simplicity. For the purpose, Procrustes rotation minimizes discrepancy between and loadings using two types of approximation: 1) approximate zeros in by non-zeros loadings, 2) loadings. The central issue considered article that it equally treats approximation, while former more important simplifying matrix. Furthermo...

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 ...