Invariant Plane Research Articles

Moduli spaces of vortex knots for an exact fluid flow

The moduli spaces S(D) of non-isotopic vortex knots are introduced for the ideal fluid flows in invariant domains D. The analogous moduli spaces of the magnetic fields B knots are defined. We derive and investigate new exact fluid flows (and analogous plasma equilibria) satisfying the Beltrami equation which have nested invariant balls Bk3 with radii Rk ≈ (k + 1) π, k⟶∞. The first flow is z-axisymmetric; the other ones do not possess any rotational symmetries. The axisymmetric flow has an invariant plane z = 0. Due to an involutive symmetry of the flow, its vortex knots in the invariant half-spaces z > 0 and z < 0 are equivalent. It is demonstrated that the moduli space 𝒮(ℝ3) for the derived fluid flow in ℝ3 is naturally isomorphic to the set of all rational numbers p/q in the interval J1:0.25<q<M̃1≈0.5847, where q is the safety factor. For the fluid flow in the first invariant ball B13, it is shown that all values of the safety factor q belong to a small interval of length ℓ ≈ 0.1261. It is established that only torus knots Kp,q with 0.25 < p/q < 0.5847 are realized as vortex knots for the constructed flow in ℝ3. Each torus knot Kp,q with 0.25 < p/q < 0.5 is realized on countably many invariant tori Tk2 located between the invariant spheres Sk2 and Sk+12, while torus knots with 0.5<p/q<M̃1 are realized only on finitely many invariant tori. The moduli spaces Sm(Ba3)(m=1,2,…) of vortex knots are constructed for some axisymmetric steady fluid flows that are solutions to the boundary eigenvalue problem for the curl operator on a ball Ba3.

THE INCLINATION OF THE PLANETARY SYSTEM RELATIVE TO THE SOLAR EQUATOR MAY BE EXPLAINED BY THE PRESENCE OF PLANET 9

ABSTRACT We evaluate the effects of a distant planet, commonly known as planet 9, on the dynamics of the giant planets of the solar system. We find that the dynamics of the giant planets can be decomposed into a classic Lagrange–Laplace dynamics relative to their own invariant plane and a slow precession of said plane relative to the total angular momentum vector of the solar system, including planet 9. Under specific configurations for planet 9, this precession can explain the current tilt of ∼6° between the invariant plane of the giant planets and the solar equator. An analytical model is developed to map the evolution of the inclination of the inner giant planets’ invariant plane as a function of the planet 9's mass and orbital elements, and numerical simulations of the equations of motion are performed to validate our analytical approach. The longitude of the ascending node of planet 9 is found to be linked to the longitude of the ascending node of the giant planets’ invariant plane, which also constrains the longitude of the node of planet 9 on the ecliptic. Some of the planet 9 configurations that allow the explanation of the current solar tilt are compatible with those proposed to explain the orbital confinement of distant Kuiper Belt objects. This work gives an elegant explanation for the current tilt between the invariant plane of the inner giant planets and the solar equator and also adds new constraints to the orbital elements of planet 9.

A review of some elements for the history of mechanical twinning centred on its German origins until Otto Mügge’s K 1 and K 2 invariant plane notation

There is a specific nomenclature for the so-called mechanical twins or twins obtained by mechanical deformation. This nomenclature, (K 1, η 1, K 2, η 2), is due to Otto Mugge in 1889. The German mineralogist and crystallographer Mugge (1858–1932) rationalised twinning observations made by himself and by other German scientists since 1859 such as Friedrich Pfaff, Heinrich Wilhelm Dove, Eduard Reusch and Heinrich Baumhauer, not to forget a formal contribution by William Thomson and Peter Guthrie Tait noticed by Theodor Liebisch who informed Mugge about it, and further classifications by Arrien Johnsen, one of Mugge’s pupils. The presentation of this scientific development also mentions the medieval alchemist Pseudo-Geber and the Renaissance metallurgist Vannoccio Biringuccio who ‘heard’ the cry of the tin now known to be due to deformation twinning, the first models of progressive twinning involving some local atomic rearrangement, with Woldemar Voigt in 1898, Yacov Il’ich Frenkel and Tatyana Kontorova in 1938 and the first drawings of a twin(ning) dislocation by Vladimirskii in 1947 and by Charles Frank and Jan van der Merwe two years later.

Understanding the Rocksalt-to-Wurtzite phase transformation through microstructural analysis of (Al,Sc)N epitaxial thin films

Rocksalt-to-wurtzite structural phase transitions in semiconducting materials (such as III–V nitrides, ZnO, CdSe, and others) have been studied for several decades. Almost all experimental works related to this phase transition involve diamond anvil cells to apply hydrostatic pressure, and as a result, direct observation of the microstructural transformation during the phase transition has not been possible. In this article, we have addressed and uncovered the intimate microstructural details and epitaxial relationships between phases by capturing what is essentially a thin-film snapshot of the transformation after growth of AlxSc1-xN films with a composition chosen to be close to the equilibrium phase boundary between wurtzite and rocksalt. The results support the hypothesis that the transformation is triggered by defects at rs-{01¯1} growth fronts that offer a nearly invariant plane with respect to the parallel w-{21¯1¯0} planes. The intermediate crystal structures and their epitaxial relationships are consistent with theoretical models that predict a transformation pathway involving homogeneous orthorhombic shear strain.

Crystallographic analysis of the martensitic transformation in medium-carbon steel with packet martensite

Based on X-ray diffraction studies of the martensite texture in a single martensite packet, exact orientation relationships between the orientations of martensite crystallites and the original austenite single crystal in medium-carbon steel 37KhN3A have been determined to be as follows: (011)α||(1; 0.990; 1.009)γ to an accuracy of \( \pm 0.15^\circ ,{\left[ {01\overline 1 } \right]_\alpha }||{\left[ {1;1.163; - 2.133} \right]_\gamma }\) to an accuracy of ±0.15°. It has been shown that the orientation relationships proved to be almost the same as in the Fe–31% Ni alloy with a twinned martensite with close lattice parameters. Therefore, the conclusion has been drawn that the mechanism of the lattice deformation upon the martensitic transformation is the same in both alloys. It is described as follows. The lattice deformation occurs by shear on the (111) plane in the \({\left[ {11\overline 2 } \right]_{_\gamma }}\) direction and is accompanied by an additional change in the dimensions in the mutually perpendicular directions \({\left[ {11\overline 2 } \right]_{_\gamma }},\left[ {111} \right],\;and{\left[ {1\overline 1 0} \right]_{_\gamma }}\). The invariantlattice deformation is implemented by slip in martensite on the planes of the (112)α type in the direction \({\left[ {\overline 1 \overline 1 1} \right]_\alpha }\). One of the 24 crystallographically equivalent variants of the transformation mechanism has been considered. Apart from this type of deformation, an additional deformation of martensite is possible that does not change its orientation. It has been shown that the orientation of the martensite crystallite calculated via the phenomenological theory of the martensitic transformations (PTMT) differs by approximately 1° from the experimentally determined orientation. This refers to both the lath and twinned martensite. In the twinned martensite, the invariant plane obtained in the PTMT calculations and the habit plane coincide. In lath martensite of 37KhN3A steel, the invariant plane of the martensite crystal obtained in PTMT calculations deviates by ~25° from the orientation of the surface of the martensite plate (habit plane), which is close to the (111)γ plane. An explanation of this phenomenon is given.

Interface stress evolution of martensitic transformation in MnCu alloys: A phase-field study

Martensitic transformations (MTs) in shape memory alloys (SMAs) usually give rise to internal elastic stress, because of the shape change from austenite to martensite. The stress condition of the interface plays a key role in the thermoelastic transformation behaviour and shape memory effect. In order to investigate the stress field originated from MT, a three-dimensional (3D) phase-field model was applied to the face-centered cubic structure to face-centered tetragonal structure (fcc-to-fct) MT in a MnCu alloy. The following results were obtained: The interface stresses on the invariant habit plane arise with the thickening of martensite plate; When the habit plane is not an invariant plane, the stress state around the interface changes abruptly; The formation of transformation twin is resulted from the fluctuant interface stress, the macroscopic habit plane has not been well obtained, and the elastic strain energy increases with the growth of twinning martensite; Martensitic block containing several twinning domains can be generated from a single-variant embryo, and the fluctuant elastic stresses have retained in the block.

Symmetric periodic orbits and uniruled real Liouville domains

A real Liouville domain is a Liouville domain with an exact anti-symplectic involution. The authors call a real Liouville domain uniruled if there exists an invariant finite energy plane through every real point. Asymptotically, an invariant finite energy plane converges to a symmetric periodic orbit. In this note, they work out a criterion which guarantees uniruledness for real Liouville domains.

Experimental and theoretical evidence of displacive martensite in an intermetallic Mo-containing γ-TiAl based alloy

In this study the martensitic transformation behavior of a Mo-bearing γ-TiAl based alloy was investigated. Therefore, a homogenization treatment within the single β-phase field region followed by water quenching has been carried out, whereby the majority of the disordered β-phase transforms into hexagonal α2′-martensite during cooling. Since the β to α transformation in the intermetallic β-solidifying Ti–44Al–3Mo–0.1B alloy (at%) occurs at very high temperatures causing enhanced diffusional processes, a very locally diffusion-controlled transformation together with a displacive, hence purely martensitic transformation take place. This work investigates the displacive martensite formation in the high-temperature regime using state-of-the-art experimental methods as well as modelling concepts from the phenomenological theory of martensite crystallography. The high temperature at which the transformation takes place suggests the preference of plastic slip over twinning. This fact has also been verified by transmission electron microscopy, as no twinning has been observed after generation of single martensite variants forming an invariant interface plane with the initial β-lattice. Such invariant interfaces are formally possible according to the phenomenological theory of martensite crystallography, if the Bain strains of the martensite variants are superimposed by additional simple shear.

Bianchi IX dynamics in bouncing cosmologies: homoclinic chaos and the BKL conjecture

We examine the dynamics of a Bianchi IX model with three scale factors on a 4-dim Lorentzian brane embedded in a 5-dim conformally flat empty bulk with a timelike extra dimension. The matter content is a pressureless perfect fluid restricted to the brane, with the embedding consistently satisfying the Gauss–Codazzi equations. The 4-dim Einstein equations on the brane reduce to a 6-dim Hamiltonian dynamical system with additional terms (due to the bulk–brane interaction) that avoid the singularity and implement nonsingular bounces in the model. We examine the complex Bianchi IX dynamics in its approach to the neighborhood of the bounce which replaces the cosmological singularity of general relativity. The phase space of the model presents (i) two critical points (a saddle-center–center and a center–center–center) in a finite region of phase space, (ii) two asymptotic de Sitter critical points at infinity, one acting as an attractor to late-time acceleration and (iii) a 2-dim invariant plane, which together organize the dynamics of the phase space. The saddle-center–center engenders in the phase space the topology of stable and unstable 4-dim cylinders R × S3, where R is a saddle direction and S3 is the center manifold of unstable periodic orbits, the latter being the nonlinear extension of the center–center sector. By a proper canonical transformation the degrees of freedom of the dynamics are separated into one degree connected with the expansion/contraction of the scales of the model, and two rotational degrees of freedom associated with the center manifold S3. The typical dynamical flow is thus an oscillatory mode about the orbits of the invariant plane. The stable and unstable cylinders are spanned by oscillatory orbits about the separatrix towards the bounce, leading to the homoclinic transversal intersection of the cylinders, as shown numerically in two distinct simulations. The homoclinic intersection manifold has the topology of R × S2 consisting of homoclinic orbits biasymptotic to the center manifold S3. This behavior defines a chaotic saddle associated with S3, indicating that the intersection points of the cylinders have the nature of a Cantor set with compact support S2. This is an invariant signature of chaos in the model. We discuss the connection between these properties of the dynamics, namely the oscillatory approach to the bounce together with its chaotic behavior, and analogous features present in the BKL conjecture in general relativity.

Comprehensive analysis of the dilatation during bainitic transformation under stress

The thermal simulation experiments on bainitic transformation were conducted on Gleeble 3800. The microstructures of specimens were examined by scanning electron microscopy. The main purpose is to investigate the relationship between the dilatation of specimens and bainitic transformation under stress. Results show that the radial strain cannot represent the real amount of bainitic transformation under stress because the radial strain contains both shear and dilatational components of invariant plane strain. In this case, the volume strain, which eliminates the influence of the shear components, should be used to analyze the amount of bainitic transformation under stress. In addition, the radial strain should not be used for the investigation of the kinetics of bainitic transformation under stress. For bainitic transformation without stress, the radial strain of the specimen can represent the amount of bainite, and can be used for the analysis of the kinetics of bainitic transformation; so it is not necessary to measure the volume strain. The present study clarifies the relationship between dilatation and the bainitic transformation under stress, and provides a useful reference to the analysis of bainitic transformation under stress.

Evolutionary competition between boundedly rational behavioral rules in oligopoly games

In this paper, we propose an evolutionary model of oligopoly competition where agents can select between different behavioral rules to make decisions on productions. We formalize the model as a general class of evolutionary oligopoly games and then we consider an example with two specific rules, namely Local Monopolistic Approximation and Gradient dynamics. We provide several results on the global dynamic properties of the model, showing that in some cases the attractor of the system may belong to an invariant plane where only one behavioral rule is adopted (monomorphic state). The attractors on the invariant planes can be either strong attractors or weak attractors. However, we also explain why the system can be in a state of Evolutionary Stable Heterogeneity, where it is more profitable for the agents to employ both heuristics in the long term (polymorphic state).

On Lotka--Volterra Equations with Identical Minimal Intrinsic Growth Rate

We investigate the long-run behavior of time-dependent Lotka--Volterra equations $({\rm E}_{\phi}):\frac{dx_i}{dt}=x_i(1+\phi(t)+\sum_{j=1}^na_{ij}x_j)$, $i=1,\ldots,n$, on the positive orthant. It is proved that the system $({\rm E}_{\phi})$ is decomposed into $({\rm E}_0)$ and the logistic equation $({\rm L}):\frac{dg}{dt}=g(1+\phi(t)-g)$ in the sense that $({\rm D}):\Phi(t)=g(t)\Psi(\int_0^tg(s)ds)$, where $\Phi(\cdot)$, $\Psi(\cdot)$, and $g(\cdot)$ are the solutions of $({\rm E}_{\phi})$, $({\rm E}_0)$, and (L), respectively. Suppose that $\phi(t)$ is periodic with the mean value ${\cal M}\{\phi\}>-1$. Then the existence of stable equilibrium, periodic solution, and chaotic motion for $({\rm E}_0)$ implies the existence of stable periodic solution, quasiperiodic solution, and chaotic motion for $({\rm E}_{\phi})$, respectively. The complete dynamical classification for 3-dimensional competitive system $({\rm E}_0)$ is provided in terms of competitive coefficients. There are 37 topological classes, in 34 of which any trajectory converges to an equilibrium. Among the remaining three classes, the first has a heteroclinic cycle attracting all positive points except the ray joining the origin and the positive equilibrium; the second possesses a family of periodic orbits attracting all positive points; the third class, where a family of periodic orbits, an asymptotically stable equilibrium, a heteroclinic cycle, and an invariant noncoordinate plane coexist, is a new type we find for the first time. It is shown that for 3-dimensional periodic competitive system $({\rm E}_{\phi})$, any trajectory for Poincaré mapping tends to a fixed point, a periodic orbit, an invariant minimal closed curve, or a heteroclinic cycle. All their attracting domains, which are all cone-like, are exactly described via (D) and carrying simplex theory. The dynamics can be completely classified into 37 classes corresponding to the autonomous systems. All counterparts hold when $\phi(t)$ is another minimal function, such as a quasiperiodic, almost periodic, or almost automorphic function, via (D), the classification above, and skew-product flow theory.

The Frank-Bilby Equation and Invariant-plane Interfaces

At equilibrium the tensor strains and rotations produced by the network of dislocations in a semi-coherent interphase interface are equal and opposite to those arising from the coherent terraces. We express this condition using the Frank-Bilby equation. All strains and rotations are partitioned between the two phases in a manner dependent on the relative compliance of the phases. Consequently, the proposition that a parent-martensite habit plane is an invariant plane of the shape transformation is only exact in the absence of rotational distortions. Rotational distortions are inevitably produced by lattice-invariant deformation (slip or twinning), and also, generally, by transformation dislocations (disconnections).

Preferential Morphology of Self-accommodation Microstructure in Ti-Ni-Pd Shape Memory Alloy

The formation frequency of junction planes (JPs) and orientation relationships across the JPs in self-accommodation microstructure in Ti-25Ni-25Pd was investigated by TEM and SEM-EBSD. Each martensite plates followed the invariant plane condition and identified as habit plane variants (HPV). The self-accommodation microstructure consisted of four kinds of HPV and contained three kinds of JPs. The formation frequency of each JP was 14% for {011}B19 compound twin, 36% for <111>B19type I twin and 50% for <211>B19type II twin. The formation frequency was explained by the magnitude of the incompatibility at the JPs.

Dynamical Analysis of Raychaudhuri Equations Based on the Localization Method of Compact Invariant Sets

In this paper, we examine the localization problem of compact invariant sets of Raychaudhuri equations with nonzero parameters. The main attention is attracted to the localization of periodic/homoclinic orbits and homoclinic cycles: we prove that there are neither periodic/homoclinic orbits nor homoclinic cycles; we find heteroclinic orbits connecting distinct equilibrium points. We describe some unbounded domain such that nonescaping to infinity positive semitrajectories which are contained in this domain have the omega-limit set located in the boundary of this domain. We find a locus of other types of compact invariant sets respecting three-dimensional and two-dimensional invariant planes. Besides, we describe the phase portrait of the system obtained from the Raychaudhuri equations by the restriction on the two-dimensional invariant plane.

Opening previously impossible avenues for phase transformation in innovative steels by atom probe tomography

After decades of debate on the mechanism for the formation of bainite, it is accepted that bainite grows via a displacive mechanism, i.e. as plate shaped transformation products exhibiting an invariant plane strain surface relief effect. But there is still much discussion on the diffusion or diffusionless nature of bainite. The purpose of this atom probe tomography study was to track the atom distributions during the bainite reaction in steels with different carbon and silicon contents. The steels were transformed over a wide range of temperatures (200–525°C) to elucidate the role of reaction rate and diffusion in the formation of bainite with and without cementite precipitation.

COMPACT PLANETARY SYSTEMS PERTURBED BY AN INCLINED COMPANION. I. VECTORIAL REPRESENTATION OF THE SECULAR MODEL

The non-resonant secular dynamics of compact planetary systems are modeled by a perturbing function which is usually expanded in eccentricity and absolute inclination with respect to the invariant plane. Here, the expressions are given in a vectorial form which naturally leads to an expansion in eccentricity and mutual inclination. The two approaches are equivalent in most cases, but the vectorial one is specially designed for those where a quasi-coplanar system tilts as a whole by a large amount. Moreover, the vectorial expressions of the Hamiltonian and of the equations of motion are slightly simpler than those given in terms of the usual elliptical elements. We also provide the secular perturbing function in vectorial form expanded in semimajor axis ratio allowing for arbitrary eccentricities and inclinations. The interaction between the equatorial bulge of a central star and its planets is also provided, as is the relativistic periapse precession of any planet induced by the central star. We illustrate the use of this representation for following the secular oscillations of the terrestrial planets of the solar system, and for Kozai cycles as may take place in exoplanetary systems.

Motion of particles on a $$z=2$$ z = 2 Lifshitz black hole background in 3 $$+$$ + 1 dimensions

The object of study is the geodesic structure of a \(z=2\) Lifshitz black hole in 3+1 space–time dimensions, which is an exact solution to the Einstein-scalar-Maxwell theory. The motion of massless and massive particles in this background is researched using the standard Lagrangian procedure. Analytical expressions are obtained for radial and angular motions of the test particles, where the polar trajectories are given in terms of the \(\wp \)-Weierstraß elliptic function. It will be demonstrated that an external observer can see that photons with radial motion arrive at spatial infinity in a finite coordinate time. For particles with non-vanished angular momentum, the motion is studied on the invariant plane \(\phi = \pi /2\) and, it is shown that bounded orbits are not allowed for this space–time on this plane. These results are consistent with other recent studies on Lifshitz black holes.

Biometric system for measuring gait and fall characteristics captured on video.

This paper presents a methodology that quantifies gait and fall characteristics from video of real-life fall events. The method consists in selecting on-screen the points on the ground where the feet are in contact with the ground. The essence of the method lies in establishing a transformation from the video frames to the "real world." In projected images, geometric properties such as lengths, angles, and parallelism are not preserved; thus, concepts of projective geometry are applied, namely homography. Because the ground is an invariant plane, using this plane for homography results in a constant transformation. The homographic transformation relies on the accuracy in the selection of on-screen points. An optimization algorithm that minimizes the errors caused by inaccurate on-screen point selection improves the results of the homographic transformation. Experimental trials are conducted at three walking velocities (slow, preferred, and fast) using two video cameras and a GAITRite walkway system. Spatial parameters of two independent video analyses are compared with the GAITRite system, yielding a limit of agreement of step length from -2.12 cm to 2.03 cm. Temporal parameters are less confident due to the existence of dropped frames in the video footage. This method is then used to analyze two real fall events as demonstrative cases. First, the gait characteristics are analyzed before imbalance, and subsequently, the characteristics of stepping are analyzed during the fall. In particular, we propose the stepping/impact angle as the metric that quantifies how much stepping affected the direction of the fall.

The microstructure of lath martensite in quenched 9Ni steel

Many of the most useful structural steels have dislocated lath martensitic structures. The microstructures of these steels are complex since each prior austenite grain contains as many as 24 different crystallographic variants of the γ(fcc)–α′(bcc) transformation. Recent research, using electron backscatter diffraction (EBSD), has significantly clarified the “block-and-packet” structure of lath martensite in low-carbon steel. The blocks are bivariant composites of two transformation variants, with the three distinct blocks stacked so that all six of the possible variants of the packet are used. The present work was undertaken to complete the description of this structure and identify its underlying causes. We address these issues in two steps. First, we present an EBSD characterization of lath martensite in low-carbon 9Ni steel. The results show that all packets have the bivariant block structure, and, with the proper notation, the full hierarchical structure has a simple pattern that is easily described and visualized. The results are readily explained on the basis of two assumptions: (i) the bivariant block, in contrast to a single-variant block, has an α′–γ invariant plane very near {011}α″||{111}γ, which permits the plate-shaped blocks to stack without significant strain to form a packet; (ii) the transformation goes to completion, with the consequence that the net strain in the prior austenite grain is a simple dilatation, and polygranular bodies can transform martensitically without significant residual stress. In a companion paper we use an appropriate modification of the crystallographic theory of lath martensite to validate the first of these assumptions.