Hyperbolic Lines Research Articles

Lie–Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications

Abstract We propose a generalization of two classes of Lie–Hamilton systems on the Euclidean plane to two-dimensional curved spaces, leading to novel Lie–Hamilton systems on Riemannian spaces (flat 2-torus, product of hyperbolic lines, sphere and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes), as well as to semi-Riemannian spaces (Newtonian or non-relativistic spacetimes). The vector fields, Hamiltonian functions, symplectic form and constants of the motion of the Euclidean classes are recovered by a contraction process. The construction is based on the structure of certain subalgebras of the so-called conformal algebras of the two-dimensional Cayley–Klein spaces. These curved Lie–Hamilton classes allow us to generalize naturally the Riccati, Kummer–Schwarz and Ermakov equations on the Euclidean plane to curved spaces, covering both the Riemannian and Lorentzian possibilities, and where the curvature can be considered as an integrable deformation parameter of the initial Euclidean system.

A twistor construction of Hopf real hypersurfaces in complex hyperbolic space

It is very well known that Hopf real hypersurfaces in the complex projective space can be locally characterized as tubes over complex submanifolds. This also holds true for some, but not all, Hopf real hypersurfaces in the complex hyperbolic space. The main goal of this paper is to show, in a unified way, how to construct Hopf real hypersurfaces in the complex hyperbolic space from a horizontal submanifold in one of the three twistor spaces of the indefinite complex 2-plane Grassmannian with respect to the natural para-quaternionic Kähler structure. We also identify these twistor spaces with the sets of circles in totally geodesic complex hyperbolic lines in the complex hyperbolic space. As an application, we describe all classical Hopf examples. We also solve the remarkable and long-standing problem of the existence of Hopf real hypersurfaces in the complex hyperbolic space, different from the horosphere, such that the associated principal curvature is 2. We exhibit a method to obtain plenty of them.

Hyperbolic Genetic Mechanism of Human Evolution

The concept of “hyperbolic medicine” covers several aspects. Its main characteristic is that hyperbolic spatiotemporal curves found in nature generate hyperbolic mechanisms that influence human physiology. A bibliographic review of scientific works related to genetic aspects of human evolution and the structure of the genetic code has been carried out. Then, theoretically, the hyperbolic patterns that occur in human physiology and the genetic code have been studied, as a possible hyperbolic mechanism of human evolution. Images in nature are space-time hyperbolas and exist independently of the longitude and latitude of the Earth where they are observed. The lines of force of electromagnetic fields and many curves of processes that occur in human physiology are also hyperbolic curves. Electromagnetic fields have effects on human physiology through hyperbolic curves. The genetic code is shaped like a helix because when a current passes through it, a transverse magnetic field is generated, and its hyperbolic lines of force follow a counterclockwise helical path. The different types of evolution occur according to different mechanisms. The mutation is an evolution mechanism that is related to a stable and heritable alteration of DNA. Nature is hyperbolic, and this conditions human physiology and genetics to make it hyperbolic. The alteration of the genetic code is a mechanism of human evolution. This evolution may be conditioned by a change in the hyperbolic helical structure of the genetic code. The primary mechanism of evolution can be found in an alteration of the hyperbolic component of the genetic code, linked to nature.

Helical Hyperbolic Magnetic Field in Nerves, when they Act as Current Conductors

Knowledge of the magnetic field of a nerve is of interest for the application of external electromagnetic therapy to regenerate injured nerves. The aim of this work is to the theoretical study of the magnetic field generated in the nerves when they act as current conductors and thus determine how their hyperbolic curves are, within the concept of hyperbolic medicine. Using Internet search engines and various databases (Medline, Google Scholar, Researchgate, Scielo), a bibliographic review of scientific papers related to the electromagnetic field generated in the nerves, when they act as current conductors, has been carried out. For this work we use, theoretically, the right-hand rule in electromagnetism. So, we apply it to a model nerve, to determine what its hyperbolic magnetic field is like. The conclusions are: a) Hyperbolic curves are very common in nature and human physiology. The lines of force of an electromagnetic field act on human physiology through hyperbolic curves. b) Human physiology and a nerve can be divided into smaller fragments like a magnet do and maintain their same characteristics. c) A nerve fragment is a conducting wire of electric current that generates a transversal magnetic field, according to the right-hand rule in electromagnetism. That magnetic field has hyperbolic lines of force that follow a counterclockwise helical path.

Absolutely Feasible Synchronous Reluctance Machine Rotor Barrier Topologies with Minimal Parametric Complexity

The first step in the synchronous reluctance machine design is the selection of rotor flux barrier type. The literature provides various barrier construction methods with a common goal of reducing parametric complexity. However, too excessive simplification can lead to decreased performance, while overly complex geometries tend to increase optimization time. This paper presents a set of novel flux barrier construction methods with an increased degree of freedom and minimal geometrical complexity. The paper proposes four topologies based on circular, hyperbolic, and original Zhukovsky lines. When considering parametrization complexity, the original Zhukovsky type is the simplest, but it has barrier depth limitations. Other topologies have equal complexity. The paper proposes a novel Modified Zhukovsky variable depth type based on geometrical conformal mapping of the original Zhukovsky lines. The step-by-step construction of each topology is presented in a form of pseudo-code with detailed comments and illustrations. Overall, the presented research provides a valuable starting point for the designer who wants to investigate different smooth rotor barrier topologies.

Two kinds of real lines on real del Pezzo surfaces of degree 1

We show how the real lines on a real del Pezzo surface of degree 1 can be split into two species, elliptic and hyperbolic, via a certain distinguished, intrinsically defined, \({\text {Pin}}^-\)-structure on the real locus of the surface. We prove that this splitting is invariant under real automorphisms and real deformations of the surface, and that the difference between the total numbers of hyperbolic and elliptic lines is always equal to 16.

On cyclic quadrilaterals in euclidean and hyperbolic geometries

Four points ordered in the positive order on the unit circle determine the vertices of a quadrilateral, which is considered either as a euclidean or as a hyperbolic quadrilateral depending on whether the lines connecting the vertices are euclidean or hyperbolic lines. In the case of hyperbolic lines, this type of quadrilaterals are called ideal quadrilaterals. Our main result gives a euclidean counterpart of an earlier result on the hyperbolic distances between the opposite sides of ideal quadrilaterals. The proof is based on computations involving hyperbolic geometry. We also found a new formula for the hyperbolic midpoint of a hyperbolic geodesic segment in the unit disk. As an application of some geometric properties, we provided a euclidean construction of the symmetrization of random four points on the unit circle with respect to a diameter which preserves the absolute cross ratio of quadruples.

200-OR: Optimal Use of Measures of Insulin Secretion and Sensitivity in Predicting Diabetes: Beyond the Disposition Index

In persons without diabetes, the product of estimated insulin secretion and sensitivity, the disposition index (DI), is used as an index of beta cell function and a predictor of T2D. However, its use in predicting T2D disregards differences in insulin secretion and sensitivity among persons with the same DI. We studied diabetes incidence in 420 persons without diabetes followed for a median of 9.9 years during which time 146 (35%) developed diabetes. Insulin secretion was measured 3-5 min after a 25gm glucose bolus (AIR) and sensitivity (M) by the hyperinsulinemic-euglycemic clamp. The Figure plots AIR by M and shows bands representing a range of lowest to highest incidence. Different (hyperbolic) lines of constant DI are superimposed. A single DI line crosses several risk zones; subjects with the same DI but lower M had higher T2D incidence, suggesting that DI is not optimal for prediction. We also compared the performance of different Cox models in predicting T2D using the area under the receiver operating characteristic curve (AUC). In prediction by separate secretion and sensitivity variables, AUC=0.72, higher (p Disclosure E. Vazquez arreola: None. R. L. Hanson: None. C. Bogardus: None. W. C. Knowler: None.

An arithmetic count of the lines on a smooth cubic surface

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field$k$, generalizing the counts that over${\mathbf {C}}$there are$27$lines, and over${\mathbf {R}}$the number of hyperbolic lines minus the number of elliptic lines is$3$. In general, the lines are defined over a field extension$L$and have an associated arithmetic type$\alpha$in$L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group$\operatorname {GW}(k)$of$k$,\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]where$\operatorname {Tr}_{L/k}$denotes the trace$\operatorname {GW}(L) \to \operatorname {GW}(k)$. Taking the rank and signature recovers the results over${\mathbf {C}}$and${\mathbf {R}}$. To do this, we develop an elementary theory of the Euler number in$\mathbf {A}^1$-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

Justin Clarke-Doane* Morality and Mathematics.

This excellent book compares and contrasts morality and mathematics. Their similarities and differences are not what one might naïvely suppose, as the author demonstrates. The book is highly recommended to philosophers interested in both subjects, and to anyone who seeks a global understanding of how morality and mathematics fit into our belief system. In Chapter 1 the reader is introduced to the author’s understanding of two key terms: realism and objectivity. One overarching opinion presented in this book is that these notions are in tension, and here the motivation for this view is put forward. First, a general recipe for a ‘realist’ view of some subject matter F is devised, with candidate ingredients being examined one by one before being either included or rejected. In the end, the author advocates an understanding of ‘F-realism’ as the view that F-sentences should be interpreted at face-value, that they are commonly used to express beliefs and typically have truth values, that some atomic F-sentences (not merely tautologous compounds) are true, and finally that the F-truths hold independently of mind and language. Surprisingly, among the ingredients rejected by Clarke-Doane is objectivity. Though he denies that the term is being used here in a revisionary or technical sense, he also wishes to deny that ‘objectivity’ should be taken to mean the same as either intersubjectivity or mind-independence. Rather, objectivity here just means a kind of anti-pluralism: we can call the answer to a question an objective one just in case that answer is uniquely correct. Now the idea of treating a matter as objective just in cases where there is only ‘one right answer’ seems, at first blush, to accord with common usage. It is only when the author begins to push on this line that it shows itself to be rather more radical. To use his own example: it is not an objective matter whether the parallel postulate holds (as a pure mathematical conjecture), even if robust platonism is true, since it is true in Euclidean space but not in hyperbolic space (or rather, true of Euclidean lines, but not of hyperbolic lines). There is thus not a unique right answer to the question ‘Is the parallel postulate true?’ To go in for an illustrative moral example, error theorists about morality could plausibly be called moral objectivists (though importantly, not realists) in Clarke-Doane’s sense, given that they give one and only one answer — namely, ‘no’ — to any question at all of whether something is right, wrong, good, bad, etc. The idea of separating the notions of realism and objectivity, if cogent, is clearly one with a great deal of potential, ripe for application to more diverse areas of philosophical dispute. In one light, the present monograph might be seen as a proof-of-concept for a more general metaphilosophical project, the broader horizons of which are sketched in the Conclusion.

The Algebra of Projective Spheres on Plane, Sphere and Hemisphere

Numerous authors studied polarities in incidence structures or algebrization of projective geometry [1] [2]. The purpose of the present work is to establish an algebraic system based on elementary concepts of spherical geometry, extended to hyperbolic and plane geometry. The guiding principle is: “The point and the straight line are one and the same”. Points and straight lines are not treated as dual elements in two separate sets, but identical elements within a single set endowed with a binary operation and appropriate axioms. It consists of three sections. In Section 1 I build an algebraic system based on spherical constructions with two axioms: ab = ba and (ab)(ac) = a, providing finite and infinite models and proving classical theorems that are adapted to the new system. In Section Two I arrange hyperbolic points and straight lines into a model of a projective sphere, show the connection between the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry. In Section Three I create another model of a projective sphere in the Cartesian coordinate system of the plane, and give methods and techniques for using the model in the theory of functions.

Lagrangian analysis of surface transport patterns in the northern south China sea

The present work investigated the surface transport structures in the Northern South China Sea (NSCS) by estimating Lagrangian Coherent Structures (LCSs) in the altimetry-based Finite-Size Lyanpunov Exponents (FSLE) fields. The investigations covered the period of the two LEDS (Living-resource and Ecosystem Dynamics on the Slope) surveys: LEDS2014 (10–30 October 2014) and LEDS2015 (10–30 June 2015), with special focus on the survey area. FSLE analysis allowed us to describe the sampling area as divided regions characterized by different dynamical patterns. Backward longitude and latitude advection experiments were performed in addition to mapped LCSs to depict the pathway and origin of water parcels. The water masses of the sampling area were further categorized by their estimated origins to provide essential insight into the interpretation of the field measurements. The in situ measurements by cruises and satellite images were compared in order to validate the FSLE-derived estimation. The FSLE analysis results revealed completely different surface transport patterns between the two surveys. Cross-slope transport was only observed in LEDS2015 summer survey and was considered as a result of eddy entrainment. FSLE revealed the pathway of the Pearl River plume water from the shelf to the basin along the hyperbolic structure lines. No such cross-slope exchange was detected in LEDS2014 autumn survey while attractive LCS appeared at the inner edge of the survey area and no eddy structures were present.

High Platform Elevations Highlight the Role of Storms and Spring Tides in Salt Marsh Evolution

We combine sea level records and repeat lidar surveys at 8 sites in the United Kingdom and the United States to explore controls on marsh accretion. We compare marsh elevations relative to sea level as well as lidar-derived marsh accretion rates to simple 0-dimensional settling simulations in order to explore constraints on suspended sediment concentration and particle size. We find that the marsh platforms examined occupy a narrow range of elevations in the upper tidal frame, situated between Mean High Tide $MHT$ and the Observed Highest High Tide $OHHT$. Under sinusoidal tidal forcing, common in marsh accretion models, marshes at these elevations are never inundated, highlighting the inadequacy of sinusoidal forcing in numerical models of salt marshes. Forcing the model with year-long tidal records, deposition rates follow hyperbolic contour lines when expressed as a function of sediment concentration and median grain size. We also observe that when using a median sediment grain size $D_{50} = 50 \mu m$ and sediment concentrations derived from satellite data, modeled deposition rates are much lower than when using field data. We find that the deposition of coarse, concentrated sediment is necessary for platforms in the upper tidal frame to withstand sea level rise, suggesting a strong dependance on infrequent high-deposition events. This is particularly true for marshes that are very high in the tidal frame, making accretion increasingly storm-driven as marsh platforms gain elevation. Finally, we reflect on the capacity of marshes to regenerate after erosion events within a context of changing sediment supply conditions.

Пасивний метод позиціонування за інформацією далекоміра та цифрових повідомлень у форматі автоматичного залежного спостереження

Мета: представлені результати досліджень спрямовані на розробку нового методу позиціонування, що використовує сигнали DME в просторі та повідомлення ADS-B від користувачів повітряного простору. Запропонований підхід позиціонування базується на зборі та аналізі сигналів. Один літак, що взаємодіє з двома різними DME, є джерелом трьох гіперболічних ліній розташування, перетин яких може вказувати на розташування користувача. Методи дослідження: представлений підхід ґрунтується на використанні різницево-далекомірному методу, статистичних методах оцінки похибок та комп'ютерному моделюванні. Результати: запропонований метод виявлення координат літальних апаратів може зменшити навантаження на наземну мережу DME, оскільки він базується на пасивному спостереженні сигналів DME, присутніх у повітряному просторі. Обговорення: Підхід може бути застосований в алгоритмах обчислювальної системи літаководіння у якості альтернативного методу позиціонування до існуючих.

Cooperative Processing of Measurements in Pseudorange Radio-technical Finding System

Background/Objectives: The relevance of the current research is determined by the intensive development of multilateration systems which are widely used in time-positioning support systems. The authors propose to use the cooperative processing of coordinate information. Methods: In the present article a procedure of obtaining the ranging position lines with regard to each station is studied using a theory of redundant measurements which enables to implement cooperative processing. It is shown that using the least squares method, one can obtain the estimation of the required range-measurement parameters containing a minimum value of mean-square error. Findings: In our opinion, the new results of the study show that a multilateration system with one query station which forms a range line of position and with cooperative processing of Time Difference Of Arrival (TDOA) and Time Of Arrival (TOA), in some cases it is characterized by the improved accuracy compared to hyperbolic and elliptic systems. Measurement processing method studied in the present article enables to expand a system work area with respect to the accuracy for multilateration systems with short length base (distances between receiver stations). The offered procedures of radar data processing allow for a formation of range position lines that are indirectly obtained based on hyperbolic and elliptic lines of position. Cooperative processing of measurements in TOA and TDOA finding systems enables to improve the accuracy of the indirect range measurement. TDOA and TOA measurements do not improve the accuracy of the indirect range estimation with the offered method of measurement processing since the accuracy remains constant regardless of the number of stations. Application/ Improvements: The study results obtained could be used to upgrade the existing systems of aircraft position estimation and to make reasonable technical demands on the systems under development.

Effective evaluation of arterial pulse waveform analysis by two-dimensional stroke volume variation-stroke volume index plots.

Arterial pulse waveform analysis (APWA) with a semi-invasive cardiac output monitoring device is popular in perioperative hemodynamic and fluid management. However, in APWA, evaluation of hemodynamic data is not well discussed. In this study, we analyzed how we visually interpret hemodynamic data, including stroke volume variation (SVV) and stroke volume (SV) derived from APWA. We performed arithmetic estimation of the SVV-SV relationship and applied measured values to this estimation. We then collected measured values in six anesthesia cases, including three liver transplantations and three other types of surgeries, to apply them to this SVV-SVI (stroke volume variation index) plot. Arithmetic analysis showed that the relationship between SVV and SV can be drawn as hyperbolic curves. Plotting SVV-SV values in the semi-logarithmic scale showed linear correlations, and the slopes of the linear regression lines theoretically represented average mean cardiac contractility. In clinical measurements in APWA, plotting SVV and SVI values in the linear scale and the semi-logarithmic scale showed the correlations represented by hyperbolic curves and linear regression lines. The plots approximately shifted on the rectangular hyperbolic curves, depending on blood loss and blood transfusion. Arithmetic estimation is close to real measurement of the SVV-SV interaction in hyperbolic curves. In APWA, using SVV as an index of preload and the cardiac index or SVI derived from arterial pressure-based cardiac output as an index of cardiac function, is likely to be appropriate for categorizing hemodynamic stages as a substitute for Forrester subsets.

Radiative processes of uniformly accelerated entangled atoms

We study radiative processes of uniformly accelerated entangled atoms, interacting with an electromagnetic field prepared in the Minkowski vacuum state. We discuss the structure of the rate of variation of the atomic energy for two atoms travelling in different hyperbolic world lines. We identify the contributions of vacuum fluctuations and radiation reaction to the generation of entanglement as well as to the decay of entangled states. Our results resemble the situation in which two inertial atoms are coupled individually to two spatially separated cavities at different temperatures. In addition, for equal accelerations we obtain that one of the maximally entangled antisymmetric Bell state is a decoherence-free state.

Bifurcations and degenerate periodic points in a three dimensional chaotic fluid flow.

Analysis of the periodic points of a conservative periodic dynamical system uncovers the basic kinematic structure of the transport dynamics and identifies regions of local stability or chaos. While elliptic and hyperbolic points typically govern such behaviour in 3D systems, degenerate (parabolic) points also play an important role. These points represent a bifurcation in local stability and Lagrangian topology. In this study, we consider the ramifications of the two types of degenerate periodic points that occur in a model 3D fluid flow. (1) Period-tripling bifurcations occur when the local rotation angle associated with elliptic points is reversed, creating a reversal in the orientation of associated Lagrangian structures. Even though a single unstable point is created, the bifurcation in local stability has a large influence on local transport and the global arrangement of manifolds as the unstable degenerate point has three stable and three unstable directions, similar to hyperbolic points, and occurs at the intersection of three hyperbolic periodic lines. The presence of period-tripling bifurcation points indicates regions of both chaos and confinement, with the extent of each depending on the nature of the associated manifold intersections. (2) The second type of bifurcation occurs when periodic lines become tangent to local or global invariant surfaces. This bifurcation creates both saddle-centre bifurcations which can create both chaotic and stable regions, and period-doubling bifurcations which are a common route to chaos in 2D systems. We provide conditions for the occurrence of these tangent bifurcations in 3D conservative systems, as well as constraints on the possible types of tangent bifurcation that can occur based on topological considerations.

Binary codes of the symplectic generalized quadrangle of even order

Let $$q$$q be a prime power and $$W(q)$$W(q) be the symplectic generalized quadrangle of order $$q$$q. For $$q$$q even, let $$\mathcal {O}$$O (respectively, $$\mathcal {E}$$E, $$\mathcal {T}$$T) be the binary linear code spanned by the ovoids (respectively, elliptic ovoids, Tits ovoids) of $$W(q)$$W(q) and $$\Gamma $$Γ be the graph defined on the set of ovoids of $$W(q)$$W(q) in which two ovoids are adjacent if they intersect at one point. For $$\mathcal {A}\in \{\mathcal {E},\mathcal {T},\mathcal {O}\}$$A?{E,T,O}, we describe the codewords of minimum and maximum weights in $$\mathcal {A}$$A and its dual $$\mathcal {A}^{\perp }$$A?, and show that $$\mathcal {A}$$A is a one-step completely orthogonalizable code (Theorem 1.1). We prove that, for $$q>2$$q>2, any blocking set of $$PG(3,q)$$PG(3,q) with respect to the hyperbolic lines of $$W(q)$$W(q) contains at least $$q^2+q+1$$q2+q+1 points and equality holds if and only if it is a hyperplane of $$PG(3,q)$$PG(3,q) (Theorem 1.3). We deduce that a clique in $$\Gamma $$Γ has size at most $$q$$q (Theorem 1.4).

The solar type II radio bursts of 7 March 2012: Detailed simulation analyses

AbstractType II solar radio bursts are often indicators for impending space weather events at Earth. They are consequences of shock waves driven by coronal mass ejections (CMEs) that move outward from the Sun. We simulate such type II radio bursts by combining elaborate three‐dimensional (3‐D) magnetohydrodynamic (MHD) predictions of realistic CMEs near the Sun with an analytic kinetic radiation theory developed recently. The simulation approach includes the reconstruction of initial solar magnetic fields, the dimensioning of the initial flux rope of the CME with STEREO spacecraft data, and the launch of the CME into an empirical data‐driven corona and solar wind. In this paper, we simulate a complicated double CME event (a very fast CME followed by a slower CME without interaction) and the related coronal and interplanetary type II radio bursts that occurred on 7 March 2012. We extend our previous work to show harmonic and interplanetary emission as well as the simulation's surprising ability (for these events at least) for predicting emission for two closely spaced CMEs leaving the same active region. We demonstrate that the theory predicts well the observed fundamental and harmonic emission from ∼20 MHz to 50 kHz or from the high corona to near 1 AU. Specifically, the theory predicts flux, frequency, and time variations that are consistent with the presence or absence of observed type II emissions when interfering emissions are absent and are not inconsistent with observations when interfering type III bursts are present. The predicted and observed type II emission is predominantly fundamental for these two events. Harmonic emission occurs for the second CME only for a short time interval, when an extended shock has developed that can drive flank emission. The coronal and interplanetary emission follow closely hyperbolic lines in frequency‐time space, consisting of a succession of islands of emission with varying intensity. The islands develop due to competition between the shock moving through varying coronal and solar wind magnetic field structures (e.g., loops and streamers), growth of the driven radio source due to the spherical expansion of the shock, and movement of the active radio sources from the shock's nose to its flanks.