A few remarks on the values of the Bernoulli polynomials at rational arguments and some relations with ζ(2k + 1)

A problem posed by Lehmer in 1938 asks for simple closed formulae for the values of the even Bernoulli polynomials at rational arguments in terms of the Bernoulli numbers. We discuss this problem based on the Fourier expansion of the Bernoulli polynomials. We also give some necessary and sufficient conditions for ζ(2k + 1) be a rational multiple of π2k+1.

Exact properties of an integrated correlator in $$ \mathcal{N} $$ = 4 SU(N) SYM

We present a novel expression for an integrated correlation function of four superconformal primaries in SU(N) $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills ($$ \mathcal{N} $$ N = 4 SYM) theory. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. In this paper the correlator is re-expressed as a sum over a two dimensional lattice that is valid for all N and all values of the complex Yang-Mills coupling $$ \tau =\theta /2\pi +4\pi i/{g}_{\mathrm{YM}}^2 $$ τ = θ / 2 π + 4 πi / g YM 2 . In this form it is manifestly invariant under SL(2, ℤ) Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the SU(N) correlator to the SU(N + 1) and SU(N − 1) correlators. For any fixed value of N the correlator can be expressed as an infinite series of non-holomorphic Eisenstein series, $$ E\left(s;\tau, \overline{\tau}\right) $$ E s τ τ ¯ with s ∈ ℤ, and rational coefficients that depend on the values of N and s. The perturbative expansion of the integrated correlator is an asymptotic but Borel summable series, in which the n-loop coefficient of order (gYM/π)2n is a rational multiple of ζ(2n + 1). The n = 1 and n = 2 terms agree precisely with results determined directly by integrating the expressions in one-loop and two-loop perturbative $$ \mathcal{N} $$ N = 4 SYM field theory. Likewise, the charge-k instanton contributions (|k| = 1, 2, . . .) have an asymptotic, but Borel summable, series of perturbative corrections. The large-N expansion of the correlator with fixed τ is a series in powers of $$ {N}^{\frac{1}{2}-\mathrm{\ell}} $$ N 1 2 − ℓ (ℓ ∈ ℤ) with coefficients that are rational sums of $$ E\left(s;\tau, \overline{\tau}\right) $$ E s τ τ ¯ with s ∈ ℤ + 1/2. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider the ’t Hooft topological expansion of large-N Yang-Mills theory in which $$ \lambda ={g}_{\mathrm{YM}}^2N $$ λ = g YM 2 N is fixed. The coefficient of each order in the 1/N expansion can be expanded as a series of powers of λ that converges for |λ| < π2. For large λ this becomes an asymptotic series when expanded in powers of $$ 1/\sqrt{\lambda } $$ 1 / λ with coefficients that are again rational multiples of odd zeta values, in agreement with earlier results and providing new ones. We demonstrate that the large-λ series is not Borel summable, and determine its resurgent non-perturbative completion, which is $$ O\left(\exp \left(-2\sqrt{\lambda}\right)\right) $$ O exp − 2 λ .

Rational arrangement of multiple exciplex hosts for highly efficient white organic light-emitting diodes

Designing a novel exciplex host and arranging rational multiple-exciplex hosts in emissive layers.Two and three color WOLEDs based on multiple exciplex hosts exhibited maximum EQEs of 30.9% and 27.0%, respectively. Remarkedly, their EQEs remained at 22.1% and 21.3% at a high luminance of 5000 cd m−2.

Infinitely many periodic solutions for a semilinear wave equation with x-dependent coefficients

This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with an approximation argument, we prove that there exist infinitely many periodic solutions whenever the period is a rational multiple of the length of the spatial interval. The proof is essentially based on the spectral properties of the wave operator with x-dependent coefficients.

X-ray dynamical diffraction analogues of the integer and fractional Talbot effects

The X-ray integer and fractional Talbot effect is studied under two-wave dynamical diffraction conditions in a perfect crystal, for the symmetrical Laue case of diffraction. The fractional dynamical diffraction Talbot effect is studied for the first time. A theory of the dynamical diffraction integer and fractional Talbot effect is given, introducing the dynamical diffraction comb function. An expression for the dynamical diffraction polarization-sensitive Talbot distance is established. At the rational multiple depths of the Talbot depth the wavefield amplitude for each dispersion branch is a coherent sum of the initial distributions, shifted by rational multiples of the object period and having its own phases. The simulated dynamical diffraction Talbot carpet for the Ronchi grating is presented.

Existence of multiple periodic solutions to a semilinear wave equation with x-dependent coefficients

This paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with saddle point reduction technique, we obtain the existence of at least three periodic solutions whenever the period is a rational multiple of the length of the spatial interval. Our method is based on a delicate analysis for the asymptotic character of the spectrum of the wave operator with x-dependent coefficients, and the spectral properties play an essential role in the proof.

Normalized rational semiregular graphs

Let G be a graph and let A and D be the adjacency matrix of G and diagonal matrix of vertex degrees of G respectively. If each vertex degree is positive, then the normalized adjacency matrix of G is \hat{A} = D^(−1/2)AD^(−1/2). A classification is given of those graphs for which the all eigenvalues of the normalized adjacency matrix are integral. The problem of determining those graphs G for which \lambda \in Q for each eigenvalue of \hat{A}(G) is considered. These graphs are called normalized rational. It will be shown that a semiregular bipartite graph G with vertex degrees r and s is normalized rational if and only if every eigenvalue of A is a rational multiple of (rs)^{1/2}. This result will be used to classify the values of n for which the semiregular graph (with vertex degrees 2 and n − 1) obtained from subdividing each edge of K_n is normalized rational. Necessary conditions for the k-uniform complete hypergraph on n vertices to be normalized rational are also given. Finally, conditions for the incidence graphs of Steiner triple and quadruple systems to be normalized rational are given.

Beyond the Basel problem: Euler’s derivation of the general formula for ζ (2n)

The problem of finding a closed-form evaluation ofbaffled the pioneers of calculus such as Leibniz and James Bernoulli and, following the latter’s promulgation of the problem, it became known as the Basel problem after his home town (which was also Euler’s birthplace). Euler’s early sensational success in solving the Basel problem by identifyingis extremely well-documented. In this paper, we give the full details of his subsequent derivation of the general formulawhere (Bn) is a sequence of ‘strange constants’. Euler’s polished account of his discovery, in which he popularised the designation of the strange constants as ‘Bernoulli numbers’, appears in Chapter 5 of Volume 2 of his great textbookInstitutiones calculi differentialis[1; E212]: see [2] for an online English translation. Here, we will focus on his initial step-by-step account which appeared in his paper with Eneström number E130, written c1739, carrying the rather nondescript titleDe seriebus quibusdam considerationes, ‘Considerations about certain series’. (For convenience, we will just use ‘Eneström numbers’ when referencing Euler’s work: all are readily available on-line at [1].) Euler’s proof is notable for its early, sophisticated and incisive use of generating functions and for his brilliant insight that the sequence (Bn) occurring in the coefficients of the general ζ(2n) formula (1) also occurs in the Euler-Maclaurin summation formula and in the Maclaurin expansion of. By retracing Euler’s original path, we shall not only be able to admire the master in full creative flow, but also appreciate the role played by recurrence relations such aswhich, as our ample list of references (which will be reviewed later) suggests, have been rediscovered over and over again in the literature. Moreover, our historical approach makes it clear that, while deriving (2) is relatively straightforward (and may be used to calculate ζ(2n) recursively as a rational multiple of ζ2n), it is establishing the connection between ζ(2n) and the Bernoulli numbers that was for Euler the more difficult step. Even today, this step presents pedagogical challenges depending on one’s starting definition for the Bernoulli numbers and what identities satisfied by them one is prepared to assume or derive.

Special Values of Class Group L-Functions for CM Fields

Let H be the Hilbert class field of a CM number field K with maximal totally real subfield F of degree n over ℚ. We evaluate the second term in the Taylor expansion at s = 0 of the Galoisequivariant L-function ΘS∞(s) associated to the unramified abelian characters of Gal(H/K). This is an identity in the group ring C[Gal(H/K)] expressing Θ (n)S∞ (0) as essentially a linear combination of logarithms of special values ﹛Ψ (zσ)﹜, where Ψ: ℍn →ℝ is a Hilbert modular function for a congruence subgroup of SL2 (OF) and ﹛zσ : σ ∈ Gal(H/K)﹜ are CM points on a universal Hilbert modular variety. We apply this result to express the relative class number hH/hK as a rational multiple of the determinant of an (hK − 1) × (hK − 1) matrix of logarithms of ratios of special values Ψ (zσ), thus giving rise to candidates for higher analogs of elliptic units. Finally, we obtain a product formula for Ψ (zσ) in terms of exponentials of special values of L-functions.