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...

When G has size n and g ∈ G, for any character χ of G we have χ(g)n = χ(gn) = χ(1) = 1, so the values of χ lie among the nth roots of unity in S1. More precisely, the order of χ(g) divides the order of g (which divides #G). Characters on finite abelian groups were first studied in number theory, since number theory is a source of many interesting finite abelian groups. For instance, Dirichlet u...

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...

We discuss the smallest algebraic number field which contains the nth roots of unity and which may be reached from the rational field Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this field over Q is φ(m), where m is the smallest multiple of n divisible by each prime factor of φ(m). The prime factors of m/n are precisely the primes not dividing n but which do d...

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...

We consider a sequence of independent and identicaly distributed (iid) random variables with absolutely continuous distribution function F(x) and probability density function (pdf) f(x). Let Rnl be the largest observation after observing nth record and R(ns) be the smallest observation after observing the nth record. Then we say Wnr = Rnl− R(ns), n > 1, as the nth record range. We will c...

To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zero-free annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of the nth degree into two factors balanced in the degrees and with the zero sets separated by the ba...

In this article we develop a general method by which one can explicitly evaluate certain sums of nth powers products d≥1 elementary trigonometric functions evaluated at m=(m1,…,md)-th roots unity. Our approach is to first identify the individual terms in expression under consideration as eigenvalues discrete Laplace operator associated graph whose vertices form d-dimensional torus Gm depends on...

This work reviews from a unified viewpoint the concepts underlying “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems” (nth-CASAM-L) and Nonlinear (nth-CASAM-N) methodologies. The practical application of nth-CASAM-L methodology is illustrated an OECD/NEA reactor physics benchmark, while nth-CASAM-N nonlinear model dynamics that...

In this paper, we study the nth-bias optimality problem for finite continuous-time Markov decision processes (MDPs) with a multichain structure. We first provide nth-bias difference formulas for two policies and present some interesting characterizations of an nth-bias optimal policy by using these difference formulas. Then, we prove the existence of an nth-bias optimal policy by using nth-bias...