9) Cl 1 (θ) = − log 2 sin θ 2 , |θ| < 2π.Writing others out,

In this note, we give an exact formula for a general family of rational zeta series involving the coefficient ζ(2n) in terms of Hurwitz zeta values. This formula generalizes two previous formulas from a paper in [5]. Our method will involve derivatives polynomials for the cotangent function.

In this paper, we study pairs of oscillators that are indirectly coupled via active (excitable) cells. We introduce a scalar phase model for coupled oscillators and excitable cells. We first show that one excitable and one oscillatory cell will exhibit phase locking at a variety of m : n patterns. We next introduce a second oscillatory cell and show that the only attractor is synchrony between the oscillators. We will also study the robustness to heterogeneity when the excitable cell fires or is quiescent. We next examine the dynamics when the oscillators are coupled via two excitable cells. In this case, the dynamics are very complicated with many forms of bistability and, in some cases, chaotic behavior. We also apply weak coupling analysis to this case and explain some of the degeneracies observed in the bifurcation diagram. Further, we look at pairs of oscillators coupled via long chains of excitable cells and show that small differences in the frequency of the oscillators makes their locking more robust. Finally, we demonstrate many of the same phenomena seen in the phase model for a gap-junction coupled system of Morris-Lecar neurons.

In this paper, we investigate the integral of x n log m (sin(x)) for natural numbers m and n. In doing so, we recover some well-known results and remark on some relations to the log-sine integral Ls (n) n+m+1 (θ). Later, we use properties of Bell polynomials to find a closed expression for the derivative of the central binomial and shifted central binomial coefficients in terms of polygamma functions and harmonic numbers.π 3, k ∈ N.Here, we will focus on a similar integral, F (n, m, z) = z 0x n sin 2m (x) dx.Further, we can define

In this paper, we give elementary proofs of Zagier's formula for multiple zeta values involving Hoffman elements and its odd variant due to Murakami. Zagier's formula was a key ingredient in the proof of Hoffman's conjecture. Moreover, using the same approach, we prove Murakami's formula for multiple t-values. This formula is essential in proving a Brown type result which asserts that each multiple zeta value is a Q-linear combination of multiple t-values of the same weight involving 2's and 3's.