Hyperbolic Type Research Articles

Labeled four cycles and the $K(\pi ,1)$-conjecture for Artin groups

AbstractWe show that for a large class of Artin groups with Dynkin diagrams being a tree, the $K(\pi ,1)$ K ( π , 1 ) -conjecture holds. We also establish the $K(\pi ,1)$ K ( π , 1 ) -conjecture for another class of Artin groups whose Dynkin diagrams contain a cycle, which applies to some Artin groups whose Dynkin diagrams are of hyperbolic type. This is based on a new approach to the $K(\pi ,1)$ K ( π , 1 ) -conjecture for Artin groups.

Modelling pressure dynamics in a gas pipeline with an in-line sampling

Gas pipelines, whether main, field, or urban, often operate in non-stationary modes. Changes in the operating modes of pumping stations, equipment start-up and shutdown, an in-line sampling or pumping and various factors are causes of instability of pressure, velocity, gas flow rate. Main pipelines are complex engineering systems with the pipeline itself being the main element. Fail-safety is the major factor in the operation of this system. Ensuring operational safety requires studying the movement regimes of the transported medium, particularly study of the dynamics of pressure duringstart-up or shutdown and at specific extraction points. This article aims to build a mathematical model and study the pressure dynamics in a gas pipeline with a sampling point. The theory of non-stationary motion liquid in round pipes has been strongly developed in the works of I. A. Charnyj. These works consider a large complex of engineering tasks taking into account viscous properties of the transported medium and pipe resistance in the hydraulic approximation. In the article on the basis of I. A. Charnyj's researches on the motion of real liquid in circular pipes the equation in partial derivatives of hyperbolic type is compiled. The equation describes the unsteady pressure of a horizontal section of gas pipeline with an extraction point. Using the Dirac delta function allows the formulation of the problem in the form of a single equation. Pressures are set at the ends of a given section, and the initial velocity is related to the Dirac delta function. By applying the finite Fourier sine transform, the partial differential equation is transformed into an ordinary differential equation and solved. Solution of equation is vision of a solution to the initial task. Inverse transform formulas based on Fourier theory allowed us to proceed to the solution of this task. Explicit dependences for the dynamics of unsteady pressure are obtained. The qualitative analysis of the formulas indicates the wave motion of a medium during the initial phase of operation, transitioning into a stationary mode after a brief period. The duration of the transition period depends on factors such as the hydraulic resistance coefficient and the velocity of the transported medium. Coefficient of hydraulic resistance and the velocity of the transported medium are the main factors. An example is considered for a horizontal section under isothermal flow conditions. With assumed numerical parameters, the transition to a stationary state occurs approximately 17 minutes after the process begins, as illustrated in the graphs provided in the article. Engineers can use mathematical models of the liquid and gas motion through pipes in the design of pipelines, as well as in solving tasks that arise during their operation. These tasks include monitoring the condition of the pipeline system, optimizing the operation, accumulation capacity estimates and others.

Thermally tunable phonon–plasmon polariton modes at hexagonal boron nitride (hBN) and indium antimonide (InSb) interfaces

Abstract This work examines the propagation of thermally tunable phonon–plasmon modes at the interfaces of hexagonal boron nitride (hBN) and isotropic indium antimonide (InSb). Both theoretical modeling and numerical simulations are carried out to analyze the effect of temperature on surface wave behavior. hBN is realized as a polar material via the Lorentzian model, while InSb ismodeled as a temperature-sensitive material (TSM) in the framework of Drude’s model. The possible plasmon–phonon polaritonic interactionsarestudied for the TSM–elliptic type interface and TSM– hyperbolic type interface. It is reported that by varying the temperature, the surface modes can be tuned for the lower and upper Reststrahlen (RS) bands of hBN. The dispersion curve, effective mode index, propagation length, and phase speed are computed for each case under different temperatures. It is concluded that the hBN–InSb-based phonon–plasmon polariton modes are actively tuned by changing the external temperature in the lower and upper RS bands. Surface waves propagating across the interface can be modulated from the terahertz (THz) region to the infrared (IR)region by changing the temperature of InSb. This study will help researchers to design innovative thermo-optical sensors, plasmonic platforms, detectors, and surface waveguides in the THz and IR regions.

Modified F(R,T2)-Gravity Coupled with Perfect Fluid Admitting Hyperbolic Ricci Soliton Type Symmetry

In the present research note, we discuss the energy–momentum squared gravity model F(R,T2) coupled with perfect fluid. We obtain the equation of state for the perfect fluid in the F(R,T2)-gravity model. Furthermore, we deal with the energy–momentum squared gravity model F(R,T2) coupled with perfect fluid, which admits the hyperbolic Ricci solitons with a conformal vector field. We provide a clue in this series to determine the density and pressure in the radiation and phantom barrier periods, respectively. Also, we investigate the rate of change in hyperbolic Ricci solitons within the same vector field. In addition, we determine the different energy conditions, black holes and singularity conditions for perfect fluid attached to F(R,T2)-gravity in terms of hyperbolic Ricci solitons. Lastly, we deduce the Schrödinger equation for the potential Un with hyperbolic Ricci solitons in the F(R,T2)-gravity model coupled with perfect fluid and a phantom barrier.

Characterization of affine [formula omitted]-surfaces of hyperbolic type

In this note we extend the result from [14] and prove that if S is an affine non-toric Gm-surface of hyperbolic type that admits a Ga-action and X is an affine irreducible variety such that Aut(X) is isomorphic to Aut(S) as an abstract group, then X is a Gm-surface of hyperbolic type. Further, we show that a smooth Danielewski surface Dp={xy=p(z)}⊂A3, where p has no multiple roots, is determined by its automorphism group seen as an ind-group in the category of affine irreducible varieties.

Numerical modeling of transient water table in shallow unconfined aquifers: A hyperbolic theory and well-balanced finite volume scheme

We present a new methodology capable of modeling transient motion of shallow phreatic surface of groundwater in unconfined aquifers. This methodology is founded on a new and comprehensive theory for water table motion and a corresponding efficient numerical scheme. In the theoretical aspect, we derived a new set of governing equations constituted by a depth-averaged continuity equation and momentum equations based on unsteady Darcy’s law. The derived governing equations are of the hyperbolic type and possess stiff terms in the momentum equations due to the inertia motion in a characteristic time scale that is relatively shorter than the time scale of seepage motion. To effectively solve the derived hyperbolic system with stiff terms, in the numerical aspect, we utilize f-wave propagation algorithm, an explicit finite volume method, that can ensure numerical convergence and well-balancing solutions when momentum is rapidly relaxing to an equilibrium of steady state. Verification is successfully performed by comparing the results with analytic solutions to the classic problem of multidimensional spreading of a groundwater mound. This study demonstrates that the proposed methodology can accurately and satisfactorily simulate the spatiotemporal distribution of shallow water table and its wetting front in unconfined aquifers.

Solitary wave type solutions of nonlinear improved mKdV equation by modified techniques

In this paper, the improved and modified version of the Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM) are manipulated effectively and generously to determine the exact solitary wave soliton solutions of the improved modified KdV (mKdV) equation. The purpose of this study is to provide novel exact solutions to the improved mKdV equation. Specifically, we utilized IMSSEM and IGREMM to study different solutions of the nonlinear improved mKdV equation, focusing on exponential, trigonometric, and trigonometric hyperbolic type solutions. Furthermore, the plotting of various solutions for direct viewing analysis is provided in two and three-dimensional graphs. The new strategies are straightforward, quick, and efficient and have many other advantages, whereas, they provide the most accurate and unique solution to many other types of nonlinear partial differential equations (NPDEs), which usually arise in engineering and applied sciences. It should be noted that these methodologies are novel mathematical instruments that have shown to be the most effective mathematical tools for solving higher-order nonlinear partial differential equations in mathematical physics. Symbolic computation was used to validate all of the solutions that were established. Thus, it is also hoped that these techniques will ultimately reduce the cumbersome workload involved during the process of solutions to complicated NLPDEs.

A Fixed Point Theorem on Partial Metric Spaces of Hyperbolic Type

A Fixed Point Theorem on Partial Metric Spaces of Hyperbolic Type

Analysis of energy dissipation in hyperbolic problems influenced by internal and boundary control mechanisms

ABSTRACT This article primarily focuses on the rational stabilisation of the wave equation, supplied with a second-order dynamical boundary condition of hyperbolic type, while considering an additional internal damping mechanism within the specified ring. To achieve rational decay rates of the associated energy, it is imperative to exponentially stabilise a portion of the domain using a global Ingham's-type estimate. This paper will subsequently illustrate how this partially localised exponential stabilisation, combined with a Bessel analysis, leads to a rational decrease in the overall energy of the system considered.

A variational approach to a constrained hyperbolic Kirchhoff type problem with free boundary

A variational approach to a constrained hyperbolic Kirchhoff type problem with free boundary

Indirect stabilization of locally weakly coupled second‐order evolution systems of hyperbolic type

The purpose of this work is to investigate the stabilization of locally weakly coupled second‐order evolution equations of hyperbolic type, where only one of the two equations is directly damped. As such system cannot be exponentially stable, we are interested in polynomial energy decay rates. Our main contributions concern strong abstract and polynomial stability properties based on stability properties of two auxiliary problems: the sole damped equation and the second equation with a damping related to the coupling operator. The main novelty is that the polynomial energy decay rates are obtained in different important situations not covered before; let us mention the case of a coupling operator not partially coercive and not necessarily bounded. The main tools are the use of the frequency domain approach combined with new multipliers technique based on the solutions of the resolvent equations of the two auxiliary problems mentioned before. Finally, our abstract framework is illustrated by several concrete examples not treated before.

Static liquefaction as a form of material instability in element test simulations of granular soil

Static liquefaction is a form of unstable behaviour of granular soil. It is most common in saturated loose sands under monotonically loaded undrained conditions. Predicting static liquefaction using an elastic-plastic model that incorporates the non-associated plastic flow rule and strain hardening is possible. The article briefly describes the unstable behaviour of saturated sand in undrained conditions under a monotonic load. A simple elastic-plastic model with deviatoric hardening and a Drucker–Prager load surface is presented. The constitutive relationships were programmed in a Python script. Simulations of triaxial tests under mixed stress-strain control demonstrated the model’s ability to predict various undrained sand responses, including fully stable responses (no liquefaction) and partial and complete liquefaction under triaxial compression and tension. Predicting static liquefaction is possible by properly selecting the proportions of the parameters involved in plastic potential and loading functions and the parameter A used in the deviatoric hardening rule of hyperbolic type.

Development of a method for solving elliptic differential equations based on a nonlinear compact-polynomial scheme

A numerical method for solving elliptic differential equations (on unstructured computational grids) is described, which is based on a nonlinear compact-polynomial computational scheme. For this purpose, an approximate transition from the elliptic equation to the hyperbolic type of equations was carried out in the work. To find the solution of the hyperbolic version of the diffusion equation, a transition (in the vicinity of the computational cell Ωi) to a local curvilinear coordinate system was performed, a nonlinear compact-polynomial computational scheme was used (the "predictor" stage), and the control volume method was applied (the "corrector" stage). The results of initial testing (on problems of different physical nature) of the proposed numerical method are presented.

A hyperbolic Kac-Moody Calogero model

A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra AE3 with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincaré disk with a unique potential. Since the Weyl group of AE3 is a ℤ2 extension of the modular group PSL(2,ℤ), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of AE3, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincaré series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case we find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.

Solution forms for generalized hyperbolic cotangent type systems of p-difference equations

Due to the recent increasing interest in hyperbolic-cotangent types of scalar-or two-dimensional systems of difference equations and treatment of some particular states. This paper presents a natural extension of the p-dimensional of four-systems of this generalized type and treats general states. Which is an extension of Stevic's work (J. Inequal. Appl., 2021, 184 (2021)). We also show these systems are solvable by using appropriate variable transformations and obtaining systems of homogeneous linear difference equations with constant coefficients. Some numerical examples of these systems are presented.

The Mechanism of Energy Changes That Occur Depending on the Ratio of Force and Speed in The Example of Bicycle Ergometric Testing

Introduction: For the first time, in the aspect of biophysics, the reasons for the increase in the power of the threshold of anaerobic metabolism developed by the test person during functional diagnostics. Methods: This occurs with an increase in the pedaling frequency with which the specified load on a bicycle ergometer in the range from 40 to 140 rpm (0.73-2.56 m/s) is overcome, have been substantiated. Results: It was determined that the ratio of force and velocity in the studied range of pedaling frequencies (muscle contractile speed) corresponds to the hyperbolic type with displaced axes Conclusion: At the same time, with an increase in pedaling frequency, power increases in a cubic dependence, and the rate of oxygen consumption by the test subject decreases linearly in the process of overcoming the same fixed-power load set on a bicycle ergometer and vice versa.

Field and laboratory microplastics uptake by a freshwater shrimp.

Microplastics are widespread pollutants, but few studies have linked field prevalence in organisms to laboratory uptakes. Aquatic filter feeders may be particularly susceptible to microplastic uptake, with the potential for trophic transfer to higher levels, including humans. Here, we surveyed microplastics from a model freshwater shrimp, common caraidina (Caridina nilotica) inhabiting the Crocodile River in South Africa to better understand microplastic uptake rates per individual. We then use functional response analysis (feeding rate as a function of resource density) to quantify uptake rates by shrimps in the laboratory. We found that microplastics were widespread in C. nilotica, with no significant differences in microplastic abundances among sampled sites under varying land uses, with an average abundance of 6.2 particles per individual. The vast majority of microplastics found was fibres (86.1%). Shrimp microplastic accumulation patterns were slightly higher in the laboratory than the field, where shrimp exhibited a hyperbolic Type II functional response model under varying exposure concentrations. Maximum feeding rates of 20 particles were found over a 6 h feeding period, and uptake evidenced at even the lowest laboratory concentrations (~10 particles per mL). These results highlight that microplastic uptake is widespread in field populations and partly density dependent, with field concentrations corroborating uptake rates recorded in the laboratory. Further research is required to elucidate trophic transfer from these taxa and to understand potential physiological impacts.

Nonlinear long waves in shallow water for normalized Boussinesq equations

In this work, we investigate a partial differential equation (PDE) derived from a Boussinesq system of two equations that describe the wave motion in shallow water when the wavelength is small compared to wave amplitude. The Lie point symmetries of the PDE are utilized to reduce it to an ordinary differential equation (ODE) the travelling wave solutions of which are obtained. These solutions are of trigonometric, hyperbolic and exponential types. Graphical representation of some solutions exhibiting solitary and cnoidal wave phenomena are presented.

Free vibrations of variablesection beams taking rotational and frictional forces into account

Introduction. Beams are widely used in the construction industry as load-bearing structures of bridges, overpasses, coverings, slabs, stairs, equipment platforms, etc. In order to fully utilize the bearing capacity of such structures and to reduce the rate of material consumption, beams of variable cross-section along the length can be used. During operation, such structural elements are subjected to various types of vibrations, which determines the relevance of studying various aspects of vibrational motion.Aim. To apply numerical methods for studying the influence of rotational inertial forces in the presence of viscous frictional forces on free vibrations of variable-section beams. This area of research presents interest, since the calculations are directly related to the determination of frequencies and forms of natural vibrations of structures.Materials and methods. Free vibrations were described by a homogeneous partial differential equation of hyperbolic type. The methods of variable separation and finite differences were used. A discrete domain in the form of a set of uniform grid nodes and a homogeneous system of algebraic equations were introduced. The system of equations in the matrix-vector form was used.Results. The spectra of natural frequencies, damping coefficients, and eigenforms of beam vibrations are determined. It is shown that the coefficient matrix has a banded and pentadiagonal form. The matrix elements are functions of the characteristic index. The damping coefficient and the frequency of free vibrations are determined from a system of two nonlinear equations. The solution of the system of equations is found using the method of coordinate descent. An example of calculation of a welded I-beam is considered. Five elements of the spectra of damping coefficients and natural frequencies are calculated.Conclusions. State-of-the-art MATLAB solvers allows numerical and graphical methods to be combined. In the solved examples, the advantages of these methods were successfully applied to determine the eigenvalues of matrices and eigenfunctions. The high validity and accuracy of the results obtained confirm the simplicity and versatility of the methodology for determining the characteristics of free vibrations of beams of variable cross-section.

Phase trajectories and Chaos theory for dynamical demonstration and explicit propagating wave formation

This paper is subjected to study the nonlinear integrable model which is the (3+1)-dimensional Boussinesq equation which has a lot of applications in engineering and modern sciences. To find and examine the analytical exact solitary wave solutions of (3+1)-dimensional Boussinesq equation, a modified generalized exponential rational functional method is exerted. As a result, waves, singular periodic, hyperbolic, and trigonometric type solutions are obtained. These acquired solutions are more innovative and encouraging to researchers in their endeavor to study physical marvels. To illustrate how some selected exact solutions propagate, the graphical representation in 2D, Contour, and 3D of those solutions is provided with various parametric values. The considered equation is additionally transformed into the planar dynamical structure by applying the Galilean transformation. All potential phase portraits of the dynamical system are investigated using the theory of bifurcation. The Hamiltonian function of the dynamical system of differential equations is established to see that, the system is conservative over time. The presentation of energy levels through graphics provides valuable insights, and it demonstrates that the model has solutions that can be expressed in closed form. The periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D, and time series are also observable once the dynamical system is subjected to an external force. Meanwhile, the sensitivity of the derived solutions is carefully examined for a range of initial conditions.