Can Menzerath’s law be a criterion of complexity in communication?

Menzerath’s law is a quantitative linguistic law which states that, on average, the longer is a linguistic construct, the shorter are its constituents. In contrast, Menzerath-Altmann’s law (MAL) is a precise mathematical power-law-exponential formula which expresses the expected length of the linguistic construct conditioned on the number of its constituents. In this paper, we investigate the anatomy of MAL for constructs being word tokens and constituents being syllables, measuring its length in graphemes. First, we derive the exact form of MAL for texts generated by the memoryless source with three emitted symbols, which can be interpreted as a monkey typing model or a null model. We show that this null model complies with Menzerath’s law, revealing that Menzerath’s law itself can hardly be a criterion of complexity in communication. This observation does not apply to the more precise Menzerath-Altmann’s law, which predicts an inverted regime for sufficiently range constructs, i.e., the longer is a word, the longer are its syllables. To support this claim, we analyze MAL on data from 21 languages, consisting of texts from the Standardized Project Gutenberg. We show the presence of the inverted regime, not exhibited by the null model, and we demonstrate robustness of our results. We also report the complicated distribution of syllable sizes with respect to their position in the word, which might be related with the emerging MAL. Altogether, our results indicate that Menzerath’s law—in terms of correlations—is a spurious observation, while complex patterns and efficiency dynamics should be rather attributed to specific forms of Menzerath-Altmann’s law.

Adaptive numerical integration of exponential finite elements for a phase field fracture model

Phase field models for fracture are energy-based and employ a continuous field variable, the phase field, to indicate cracks. The width of the transition zone of this field variable between damaged and intact regions is controlled by a regularization parameter. Narrow transition zones are required for a good approximation of the fracture energy which involves steep gradients of the phase field. This demands a high mesh density in finite element simulations if 4-node elements with standard bilinear shape functions are used. In order to improve the quality of the results with coarser meshes, exponential shape functions derived from the analytic solution of the 1D model are introduced for the discretization of the phase field variable. Compared to the bilinear shape functions these special shape functions allow for a better approximation of the fracture field. Unfortunately, lower-order Gauss-Legendre quadrature schemes, which are sufficiently accurate for the integration of bilinear shape functions, are not sufficient for an accurate integration of the exponential shape functions. Therefore in this work, the numerical accuracy of higher-order Gauss-Legendre formulas and a double exponential formula for numerical integration is analyzed.

Comparative Analysis of the Kinematics Solution Based on the DH Method and Screw Theory

The premise of analyzing and researching robot technology is to establish a proper mathematical model and then to solve it with kinematics. In this study, a self-developed humanoid hydraulically driven dual-arm robot is taken as the research object, and the DH (Denavit–Hartenberg) parameter method and the rotational exponential formula (POE) are used to solve the kinematics of the robot. The calculation results are verified by simulation. The advantages and disadvantages of the two methods are analyzed. The differences between the two methods are compared. It lays a foundation for other scholars to choose mathematical models when analyzing the mechanism in the future.

Numerical Picard Iteration Methods for Simulation of Non-Lipschitz Stochastic Differential Equations

In this paper, we present splitting approaches for stochastic/deterministic coupled differential equations, which play an important role in many applications for modelling stochastic phenomena, e.g., finance, dynamics in physical applications, population dynamics, biology and mechanics. We are motivated to deal with non-Lipschitz stochastic differential equations, which have functions of growth at infinity and satisfy the one-sided Lipschitz condition. Such problems studied for example in stochastic lubrication equations, while we deal with rational or polynomial functions. Numerically, we propose an approximation, which is based on Picard iterations and applies the Doléans-Dade exponential formula. Such a method allows us to approximate the non-Lipschitzian SDEs with iterative exponential methods. Further, we could apply symmetries with respect to decomposition of the related matrix-operators to reduce the computational time. We discuss the different operator splitting approaches for a nonlinear SDE with multiplicative noise and compare this to standard numerical methods.

Gravitational wave polarization in exponential forms of f(R) gravity with coupling to matter

We investigate polarizations of gravitational waves in generalized theories of gravity whose action contains a nonminimal coupling (NMC) between an exponential function of [Formula: see text] gravity and the matter field. Specifically, we consider two classes of such models. In the first case, we assume an explicit NMC between the matter Lagrangian density and curvature while in the second case, we consider the exponential [Formula: see text] gravity coupled to the trace of the stress-energy tensor of a scalar field. Using the Newman–Penrose quantities, we show that these models can be considered as suitable choices in order to produce new polarization modes.

Construction of conformal maps based on the locations of singularities for improving the double exponential formula

The double exponential formula, or DE formula, is a high-precision integration formula using a change of variables called a DE transformation; it has the disadvantage that it is sensitive to singularities of an integrand near the real axis. To overcome this disadvantage, Slevinsky & Olver (2015, On the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. SIAM J. Sci. Comput., 37, A676–A700) attempted to improve the formula by constructing conformal maps based on the locations of singularities. Based on their ideas, we construct a new transformation formula. Our method employs special types of the Schwarz–Christoffel transformation for which we can derive the explicit form. The new transformation formula can be regarded as a generalization of DE transformations. We confirm its effectiveness by numerical experiments.