In this paper, we are concerned with conditions under which [p(T)]⁎=p‾(T⁎), where p(z) is a one-variable complex polynomial, and T an unbounded, densely defined, linear operator. Then, deal the validity of identities σ(AB)=σ(BA), A B two unbounded operators. The equations (TT⁎)⁎=TT⁎ (T⁎T)⁎=T⁎T, defined closable operator, also studied. particular interest will be paid to equation T⁎T=p(T) its va...

antimicrobial activity of the polar and nonpolar extracts of the roots and aerial parts of peucedanum ruthenicum m. bieb., johreniopsis seseloides (c. a. mey.) pimenov and cervaria cervariifolia (c. a. mey.) pimenov were examined against gram-positive and gram-negative bacteria. the polar extracts of the roots and aerial parts displayed no activity against bacteria, while the nonoplar extracts ...

Let C0 X,X be the set of all continuous self-mappings on a topological space X. Let f ∈ C0 X,X . For integer n ≥ 0, define the nth iterate of f by f f ◦ fn−1 and f0 id, where id denotes the identity mapping on X and ◦ denotes the composition of mappings. An equation with iteration as its main operation is simply called an iterative equation. Founded on the problem of iterative roots, the proble...

We study sums of the form ∑ ζ R(ζ), where R is a rational function and the sum is over all nth roots of unity ζ (often with ζ = 1 excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit formula for ∏ ζ(1− xR(ζ)). Multisection ca...

ABSTRACT. For parameters n, l and r, we consider the problem of maximizing the determinant of an l × l Vandermonde matrix V, with nodes selected from the set Ωn of nth roots of unity, but avoiding a forbidden subset R of size r. An asymptotically tight lower bound is given for the expected value of ln |detV | in case the nodes are selected uniformly at random from Ωn/R. We apply our result to g...

Our goal in this section will be to collect some standard examples of finite groups. The main emphasis will be on realizing them as subgroups of GL(n,R) or GL(n,C). Cyclic groups: For a natural number n, let Z/nZ be the standard cyclic group of order n. We denote its elements by 0, 1, . . . , n − 1, and 1 is a generator. Another model for the cyclic group of order n is the nth roots of unity, o...

Then qk(t) = log det (I +Kk)− log det (I +Kk−1) is a solution to the system of equations. If ρ is supported in the set of nth roots of unity then the solution is n-periodic in that qk+n = qk for all k. The main interest is in the limit t→ 0+, while the t→ ∞ asymptotics are straightforward. The asymptotics for the qk are known once those for the determinants are, and since ρ is arbitrary there i...

We characterise nth order ODEs for which the space of solutions M is equipped with a particular paraconformal structure in the sense of [2], that is a splitting of the tangent email [email protected] email [email protected]

where we in introduce the standard notation e(x) = exp(2πix). Question 1.1. What is the magnitude of | ∑ x∈S e(x)|? By the trivial bound, we find that | ∑ x∈S e(x)| ≤ N , with equality whenever the terms are all equal. One can of course make the sum much smaller–simply sum the Nth roots of unity. In between these two extremes–completely in phase and maximally out of phase–we should consider wha...