(1) Z(t) = χ−1/2( 1 2 + it)ζ( 1 2 + it), χ(s) = 2sπs−1 sin( 1 2 πs)Γ(1− s), so that the functional equation for ζ(s) has the form ζ(s) = χ(s)ζ(1 − s). Since χ(s)χ(1− s) = 1, it follows that |Z(t)| = |ζ( 1 2 + it)|, and that Z(t) is a real-valued function of t. The function Z(t) plays an important rôle in the theory of the distribution of zeros of ζ(s) on the “critical line” Re s = 1 2 (see e.g....

A theory of pump-probe spectroscopy is developed in which optical fields drive two-photon Raman transitions between ground states of an ensemble of three-level Λ atoms. Effects related to the recoil the atoms undergo as a result of their interactions with the fields are fully accounted for in this theory. The linear absorption coefficient of a weak probe field in the presence of two pump fields...

A theory of pump-probe spectroscopy is developed in which optical fields drive two-photon Raman transitions between ground states of an ensemble of three-level atoms. Effects related to the recoil the atoms undergo as a result of their interactions with the fields are fully accounted for in this theory. The linear absorption coefficient of a weak probe field in the presence of two pump fields o...

For a graph G on n vertices with chromatic number χ(G), the Nordhaus–Gaddum inequalities state that d2 √ ne ≤ χ(G) + χ(G) ≤ n + 1, and n ≤ χ(G) · χ(G) ≤ ⌊( n+1 2 )2⌋ . Much analysis has been done to derive similar inequalities for other graph parameters, all of which are integer-valued. We determine here the optimal Nordhaus–Gaddum inequalities for the circular chromatic number and the fraction...

10. Dirichlet characters and Dirichlet L-functions Definition 10.1. Let m ∈ N. A Dirichlet character modulo m is a function χ : N→ C that satisfies the following three conditions. (1) χ is periodic with period m, i.e. if a ≡ b (mod m) then χ(a) = χ(b). (2) χ is completely multiplicative, i.e. χ(ab) = χ(a)χ(b) for all a, b ∈ N. (3) χ(a) = 0 if and only if gcd(a,m) > 1. For m ∈ N we let (Z/mZ)× d...

Let χl (G), χ ′ l (G), χ ′′ l (G), and 1(G) denote, respectively, the list chromatic number, the list chromatic index, the list total chromatic number, and the maximum degree of a non-trivial connected outerplane graph G. We prove the following results. (1) 2 ≤ χl (G) ≤ 3 and χl (G) = 2 if and only if G is bipartite with at most one cycle. (2) 1(G) ≤ χ ′ l (G) ≤ 1(G) + 1 and χ ′ l (G) = 1(G) + ...

We study an electric quadrupole contribution to sum frequency generation (SFG) at air∕liquid interfaces in an electronically and vibrationally nonresonant condition. Heterodyne-detected electronic sum frequency generation spectroscopy of air∕liquid interfaces reveals that nonresonant χ((2)) (second-order nonlinear susceptibility) has a negative sign and nearly the same value for all eight liqui...