Meaning and structural dynamics in poetry: a computational perspective

This work addresses Sri Aurobindo’s mantric poem, Savitri, with a computational linguistics approach. This is one of the longest poems ever written in English. We build the connectivity matrix between all main word pairs and analyse its structure. Concepts emerge as directions that better explain the variance of the data in the hyperspace of words. When projected to the low dimensional space of concepts, the vector of attention as the reader moves through the text shows a large correlation across sections of the poem, thus acting the future and the past over again. These findings suggest that the mathematical structure of Savitri is and reflects a substrate for the author’s main ideas, facilitating the reader’s understanding of the poem’s meaning via its long-range dynamical correlations. Acknowledging an irreducible essence to poetry, future studies on the relationship between words and sounds, and sounds and ideas may provide invaluable hints of the origin of language and its intimate relationship with the evolution of human consciousness.

On Active Vibration Absorption in Motion Control of a Quadrotor UAV

Conventional dynamic vibration absorbers are physical control devices designed to be coupled to flexible mechanical structures to be protected against undesirable forced vibrations. In this article, an approach to extend the capabilities of forced vibration suppression of the dynamic vibration absorbers into desired motion trajectory tracking control algorithms for a four-rotor unmanned aerial vehicle (UAV) is introduced. Nevertheless, additional physical control devices for mechanical vibration absorption are unnecessary in the proposed motion profile reference tracking control design perspective. A new dynamic control design approach for efficient tracking of desired motion profiles as well as for simultaneous active harmonic vibration absorption for a quadrotor helicopter is then proposed. In contrast to other control design methods, the presented motion tracking control scheme is based on the synthesis of multiple virtual (nonphysical) dynamic vibration absorbers. The mathematical structure of these physical mechanical devices, known as dynamic vibration absorbers, is properly exploited and extended for control synthesis for underactuated multiple-input multiple-output four-rotor nonlinear aerial dynamic systems. In this fashion, additional capabilities of active suppression of vibrating forces and torques can be achieved in specified motion directions on four-rotor helicopters. Moreover, since the dynamic vibration absorbers are designed to be virtual, these can be directly tuned for diverse operating conditions. In the present study, it is thus demonstrated that the mathematical structure of physical mechanical vibration absorbers can be extended for the design of active vibration control schemes for desired motion trajectory tracking tasks on four-rotor aerial vehicles subjected to adverse harmonic disturbances. The effectiveness of the presented novel design perspective of virtual dynamic vibration absorption schemes is proved by analytical and numerical results. Several operating case studies to stress the advantages to extend the undesirable vibration attenuation capabilities of the dynamic vibration absorbers into trajectory tracking control algorithms for nonlinear four-rotor helicopter systems are presented.

Representing Geographic Space as a Hierarchy of Recursively Defined Subspaces for Computing the Degree of Order

As Christopher Alexander discovered, all space or matter – either organic or inorganic – has some degree of order in it according to its structure and arrangement. The order refers to a kind of structural character, called living structure, which is defined as a mathematical structure that consists of numerous substructures with an inherent hierarchy. Across the hierarchy, there are far more small substructures than large ones, while on each level of the hierarchy the substructures are more or less similar in size. In this paper we develop a new approach to representing geographic space as a hierarchy of recursively defined subspaces for computing the degree of order. A geographic space is first represented as a hierarchy of recursively defined subspaces, and all the subspaces are then topologically represented as a network for computing the degree of order of the geographic space, as well as that of its subspaces. Unlike conventional geographic representations, which are mechanical in nature, this new geographic representation is organic, conceived, and developed under the third view of space; that is, space is neither lifeless nor neutral, but a living structure capable of being more living or less living. Thus, the order can also be referred to as life, beauty, coherence, or harmony. We applied the new representation to three urban environments, 253 patterns, and 35 black-white strips to verify it and to demonstrate advantages of the new approach and the new kind of order. We further discuss the implications of the approach and the order on geographic information science and sustainable urban planning.

On the derivation of a nonlinear generalized Langevin equation

We recast the Zwanzig's derivation of a non linear generalized Langevin equation (GLE) for a heavy particle interacting with a heat bath in a more general framework showing that it is necessary to readjust the Zwanzig's definitions of the kernel matrix and noise vector in the GLE in order to be able performing consistently the continuum limit. As shown by Zwanzig, the non linear feature of the resulting GLE is due to the non linear dependence of the equilibrium map by the heavy particle variables. Such an equilibrium map represents the global equilibrium configuration of the heat bath particles for a fixed (instantaneous) configuration of the system. Following the same derivation of the GLE, we show that a deeper investigation of the equilibrium map, considered in the Zwanzig's Hamiltonian, is necessary. Moreover, we discuss how to get an equilibrium map given a general interaction potential. Finally, we provide a renormalization procedure which allows to divide the dependence of the equilibrium map by coupling coefficient from the dependence by the system variables yielding a more rigorous mathematical structure of the non linear GLE.

Developing a Spatial Framework for Ship Vulnerability Assessment Based on Category Theory

Ships vulnerability analysis is one of the most important issues in today’s research, to reduce damage and increase safety. To increase the safety of ships, the effective parameters of the vulnerability of ships, the impact of each of them, and the relationship between these parameters should be identified to formulate different scenarios to analyze the vulnerability of ships. This process leads to the formation of simulation models to assess the risk of vessels. The creation of a spatial conceptual framework is needed to create integrated vulnerability models. The most important innovation of this research is the development of a spatial framework for analyzing ships’ vulnerability based on category theory. A framework that can be used to model the various scenarios of ships’ vulnerability from a variety of perspectives. To provide this framework, objects, operators, relationships, and assumptions for vulnerability analysis have been developed. To better express and convey the concepts, the spatial framework of the vulnerability analysis is presented in the form of category theory, which is a mathematical structure. The category theory is a good tool for expressing and creating a mathematical structure for objects and complex relationships in the real world, where other tools do not have this ability. The benefits of the built-in framework have been described with an integrated, precise mathematical structure that can be generalized to other subjects. Studies show that the developed framework is capable of modeling different scenarios for vulnerability analysis to find the best solution to reduce vulnerability.

Language Modeling with Reduced Densities

This work originates from the observation that today's state-of-the-art statistical language models are impressive not only for their performance, but also---and quite crucially---because they are built entirely from correlations in unstructured text data. The latter observation prompts a fundamental question that lies at the heart of this paper: What mathematical structure exists in unstructured text data? We put forth enriched category theory as a natural answer. We show that sequences of symbols from a finite alphabet, such as those found in a corpus of text, form a category enriched over probabilities. We then address a second fundamental question: How can this information be stored and modeled in a way that preserves the categorical structure? We answer this by constructing a functor from our enriched category of text to a particular enriched category of reduced density operators. The latter leverages the Loewner order on positive semidefinite operators, which can further be interpreted as a toy example of entailment.

A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases

A method of the Riemann–Hilbert problem is employed for Zhang’s conjecture 2 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in a zero external magnetic field. In this work, we first prove that the 3D Ising model in the zero external magnetic field can be mapped to either a (3 + 1)-dimensional ((3 + 1)D) Ising spin lattice or a trivialized topological structure in the (3 + 1)D or four-dimensional (4D) space (Theorem 1). Following the procedures of realizing the representation of knots on the Riemann surface and formulating the Riemann–Hilbert problem in our preceding paper [O. Suzuki and Z.D. Zhang, Mathematics 9 (2021) 776], we introduce vertex operators of knot types and a flat vector bundle for the ferromagnetic 3D Ising model (Theorems 2 and 3). By applying the monoidal transforms to trivialize the knots/links in a 4D Riemann manifold and obtain new trivial knots, we proceed to renormalize the ferromagnetic 3D Ising model in the zero external magnetic field by use of the derivation of Gauss–Bonnet–Chern formula (Theorem 4). The ferromagnetic 3D Ising model with nontrivial topological structures can be realized as a trivial model on a nontrivial topological manifold. The topological phases generalized on wavevectors are determined by the Gauss–Bonnet–Chern formula, in consideration of the mathematical structure of the 3D Ising model. Hence we prove the Zhang’s conjecture 2 (main theorem). Finally, we utilize the ferromagnetic 3D Ising model as a platform for describing a sensible interplay between the physical properties of many-body interacting systems, algebra, topology, and geometry.

Analytical 1D transfer functions for layered soils

Transfer functions are constantly used in both Seismology and Geotechnical Earthquake Engineering to relate seismic displacement at different depths within strata. In the context of Diffusive Theory, they also appear in the expression of the imaginary part of 1D Green's functions. In spite of its remarkable importance, their mathematical structure is not fully understood yet, except in the simplest cases of two or three layers at most. This incomplete understanding, in particular as to the effect of increasing number of layers, hinders progress in some areas, as researchers have to resort to expensive and less conclusive numerical parametric studies. This text presents the general form of transfer functions for any number of layers, overcoming the above issues. Owing to the formal connection between seismic wave propagation and other phenomena that, in essence, represent different instances of wave propagation in a linear-elastic medium, one can extend the results derived elsewhere [Garcia-Suarez, Joaquin. 2021. “Trace Spectrum of 1D Transfer Matrices for Wave Propagation in Layered Media.” engrXiv. June 24. doi:10.31224/osf.io/ygt8z] in the context of longitudinal wave propagation in modular rods to seismic response of stratified sites. The knowledge of the general closed-form expression of the transfer functions allows to analytically characterize the long-wavelength asymptotics of the horizontal-to-vertical spectral ratio for any number of layers.

Approximation of Functions in Multi-criterial Synthesis of Composite Materials

A prerequisite for the synthesis of composite materials as complex systems is the principles of the control paradigm of Peace and the effectiveness of mathematics (for any reality and any given (not absolute) accuracy, there is a mathematical structure that describes this reality with this accuracy; the converse is also true (homomorphism, arbitrarily close to isomorphism between reality and mathematical structures)).The proposed methodology for managing the identification process (design of composites) includes the process of human choice: the probabilistic nature of the control; the main reason for the inadequacy of a purely analytical research procedure. Here, the optimization of the control of the properties of the composite is carried out experimentally on the model as a result of the approximation of the response function: not the generalized functional is approximated, but the particular criteria of which it consists. The development of composite materials is carried out on the basis of evaluating the parameters of the formation of operational properties. The parameters of each of the kinetic processes of the formation of the physical and mechanical characteristics of the material were taken as particular criteria. Kinetic processes are asymptotic for the composites under study and contain extremum and inflection points. A method is used to approximate multidimensional table-defined functions by generalized polynomials of a particular form. In the parametric identification of kinetic processes, their parameters are considered basic. Approximating models of the main properties are presented. Vector optimization of properties (selection of recipes, technologies and methods of material quality control) is carried out by overcoming ambiguities of goals using linear convolution, introducing benchmarks, building Pareto sets, etc. The expediency of using a systematic approach (the hierarchical structure of properties and the hierarchical structure of the composite proper) to the design of building materials as complex systems is shown. The research results are introduced as prototypes of new identification systems in the development of composite materials with adjustable structure and properties, in contrast to the replication of reference applied developments of identification theory in various industries.

Modeling and parameter identification of thin, tightly rolled dielectric elastomer actuators

Due to their large deformation, high energy density, and high compliance, dielectric elastomer actuators (DEAs) have found a number of applications in several areas of mechatronics and robotics. Among the many types of DEAs proposed in the literature, rolled DEAs (RDEAs) represent one of the most popular configurations. RDEAs can be effectively used as compact muscle-like actuators for soft robots, since they allow eliminating the need for external motors or compressors while providing at the same time a flexible and lightweight structure with self-sensing capabilities. To effectively design and control complex RDEA-driven systems and robots, accurate and numerically efficient mathematical models need to be developed. In this work, we propose a novel lumped-parameter model for silicone-based, thin and tightly rolled DEAs. The model is grounded on a free-energy approach, and permits to describe the electro-mechanically coupled response of the transducer with a set of nonlinear ordinary differential equations. After deriving the constitutive relationships, the model is validated by means of an extensive experimental campaign, conducted on three RDEA specimens having different geometries. It is shown how the developed model permits to accurately predict the effects of several parameters (external load, applied voltage, actuator geometry) on the RDEA electro-mechanical response, while maintaining an overall simple mathematical structure.