Dirichlet series associated to sum-of-digits functions

We study the Dirichlet series [Formula: see text], where [Formula: see text] is the sum of the base-[Formula: see text] digits of the integer [Formula: see text], and [Formula: see text], where [Formula: see text] is the summatory function of [Formula: see text]. We show that [Formula: see text] and [Formula: see text] have analytic continuations to the plane [Formula: see text] as meromorphic functions of order at least 2, determine the locations of all poles, and give explicit formulas for the residues at the poles. We give a continuous interpolation of the sum-of-digits functions [Formula: see text] and [Formula: see text] to non-integer bases using a formula of Delange, and show that the associated Dirichlet series have a meromorphic continuation at least one unit left of their abscissa of absolute convergence.

Application of transport-based metric for continuous interpolation between cryo-EM density maps

<abstract><p>Cryogenic electron microscopy (cryo-EM) has become widely used for the past few years in structural biology, to collect single images of macromolecules "frozen in time". As this technique facilitates the identification of multiple conformational states adopted by the same molecule, a direct product of it is a set of 3D volumes, also called EM maps. To gain more insights on the possible mechanisms that govern transitions between different states, and hence the mode of action of a molecule, we recently introduced a bioinformatic tool that interpolates and generates morphing trajectories joining two given EM maps. This tool is based on recent advances made in optimal transport, that allow efficient evaluation of Wasserstein barycenters of 3D shapes. As the overall performance of the method depends on various key parameters, including the sensitivity of the regularization parameter, we performed various numerical experiments to demonstrate how MorphOT can be applied in different contexts and settings. Finally, we discuss current limitations and further potential connections between other optimal transport theories and the conformational heterogeneity problem inherent with cryo-EM data.</p></abstract>

Measures describing curvilinear short fiber distributions

In this article, we discuss measures for fibers having a curvilinear shape. This is the case, for example, for man-made cellulose fibers having a weak stiffness. The fibers are bent during the injection molding process of short fiber reinforced plastics. For this purpose, $$\mu $$ μ -CT data can be evaluated and several measures can be introduced defining the geometrical orientation of the fibers. These measures are the length, a mean curvature, and the mean torsion. Furthermore, a mean orientation of a fiber and a mean deviation to a straight line can be defined. Additionally, to these measures, which are based on a continuous interpolation of given data points, discretized quantities only considering the data points are compared. Finally, the distributions of these measures at real $$\mu $$ μ -CT data are provided.

On the paradoxical evolution of the number of photons in a new model of interpolating Hamiltonians

We introduce a new Hamiltonian model which interpolates between the Jaynes–Cummings model (JCM) and other types of such Hamiltonians. It works with two interpolating parameters, rather than one as traditional. Taking advantage of this greater degree of freedom, we can perform continuous interpolation between the various types of these Hamiltonians. As applications, we discuss a paradox raised in literature and compare the time evolution of the photon statistics obtained in the various interpolating models. The role played by the average excitation in these comparisons is also highlighted.

Haar Wavelet Based Implementation Method of the Non–integer Order Differentiation and its Application to Signal Enhancement

Non–integer order differentiation is changing application of traditional differentiation because it can achieve a continuous interpolation of the integer order differentiation. However, implementation of the non–integer order differentiation is much more complex than that of integer order differentiation. For this purpose, a Haar wavelet-based implementation method of non–integer order differentiation is proposed. The basic idea of the proposed method is to use the operational matrix to compute the non–integer order differentiation of a signal through expanding the signal by the Haar wavelets and constructing Haar wavelet operational matrix of the non–integer order differentiation. The effectiveness of the proposed method was verified by comparison of theoretical results and those obtained by another non–integer order differential filtering method. Finally, non–integer order differentiation was applied to enhance signal.

A CONTINUOUS INTERPOLATION BETWEEN CONSERVATIVE AND DISSIPATIVE SOLUTIONS FOR THE TWO-COMPONENT CAMASSA–HOLM SYSTEM

We introduce a novel solution concept, denoted ${\it\alpha}$-dissipative solutions, that provides a continuous interpolation between conservative and dissipative solutions of the Cauchy problem for the two-component Camassa–Holm system on the line with vanishing asymptotics. All the ${\it\alpha}$-dissipative solutions are global weak solutions of the same equation in Eulerian coordinates, yet they exhibit rather distinct behavior at wave breaking. The solutions are constructed after a transformation into Lagrangian variables, where the solution is carefully modified at wave breaking.

ON CONTINUOUS ASSUMED GRADIENT ELEMENTS OF SECOND ORDER

This paper presents and compares continuous assumed gradient (CAG) methods when applied to structural elasticity. CAG elements are finite elements where the strain, i.e., the deformation gradient, is replaced by a C0-continuous interpolation. Similar approaches are found in nodal integration and SFEM. Recently, interpolation schemes for a continuous assumed deformation gradient were proposed for first order tetrahedral and hexahedral finite elements. These schemes try to balance accuracy and numerical efficiency. At the same time, the stability of the interpolation with respect to hourglassing and spurious low energy modes is ensured. This paper recalls the fundamentals of CAG elements, i.e., the formulation and linearization. Furthermore, it extends the approach to second order finite elements. Examples prove convergence and accuracy of the quadratic elements. Two interpolation schemes, one being supported by finite element nodes and interior points and the other being a higher-order tensor-product polynomial, are identified to be most accurate.