MINIMAL FOLIATIONS FOR THE HIGH-DIMENSIONAL FRENKEL-KONTOROVA MODEL

For the high-dimensional Frenkel-Kontorova(FK)model on lattices,we study the existence of minimal foliations by depinning force.We introduce the tilted gradient flow and define the depinning force as the critical value of the external force under which the average velocity of the system is zero.Then,the depinning force can be used as the criterion for the existence of minimal foliations for the FK model on a Z^(d)lattice for d>1.

Extended Group Foliation Method and Functional Separation of Variables to Nonlinear Wave Equations

Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.

Functional Separable Solutions to Nonlinear Diffusion Equations by Group Foliation Method

We consider the functional separation of variables to the nonlinear diffusion equation with source and convection term： ut = （A（x）D（u）ux）x ＋ B（x）Q（u）, Ax ≠ 0. The functional separation of variables to this equation is studied by using the group foliation method. A classification is carried out for the equations which admit the function separable solutions. As a consequence, some solutions to the resulting equations are obtained.

Resolving the Routine Presence of Kyanite, Andalusite and Sillimanite across a Region using Foliation Intersection/Inflection Axes Preserved in Porphyroblasts, Petrographic Observations and Thermobarometry

Constraints from P-T pseudosections （MnNCKFMASH system）, foliation intersection/ inflection axes preserved in porphyroblasts （FIAs）, mineral assemblages and textural relationships for rocks containing all three Al2 SiO5 polymorphs indicate a kyanite→ andalusite→ sillimanite sequential formation at different times rather than stable coexistence at the Al2SiO5 triple point. All three Al2SiO5 polymorphs grew in the Chl, Bt, Ms, Grt, St, Pl and Crd bearing Ordovician Clayhole Schist in Balcooma, northeastern Australia separately along a looped P-T-t-D path that swaps from clockwise to anticlockwise in the tectono-metamorphic history of the region. Kyanite grew during crustal thickening in an Early Silurian Orogenic event followed by decompression/heating, andalusite and fibrolitic sillimanite growth during Early Devonian exhumation.

FIAs(Foliation Intersection/Inflection Axes) Preserved in Porphyroblasts,the DNA of Deformation:A Solution to the Puzzle of Deformation and Metamorphism in the Colorado,Rocky Mountains,USA

FIAs have been used extensively for more than a decade to unravel deformation and metamorphic puzzles. Orogenic processes developing early during the history or orogenesis challenge scientists because compositional layering in rocks always reactivates where multiple deformations have occurred, leaving little evidence of the history of foliation development preserved in the matrix. The foothills of the Rocky Mountains in Colorado, USA contain a succession of four FIA sets （trends） that would not have been distinguishable if porphyroblasts had not grown during the multiple deformation events that affected these rocks or if they had rotated as these events took place. They reveal that both the partitioning of deformation and the location of isograds changed significantly as the deformation proceeded.

Role of stress,slenderness and foliation on large anisotropic deformations at deep underground excavations

High stress concentrations around underground excavations can result in significant damage to deep hard-rock mines.These conditions can be the result of stopping activities,blasting,seismicity,or other mining activities.Large anisotropic deformation and excavation closure,especially under high-stress conditions,are expected if the excavation is located in a foliated or thin-bedded rock mass.In this research,the behaviour of excavations under deep and high-stress conditions was investigated and categorised.The main purpose was to enhance the existing knowledge of managing large anisotropic deformations and to help prepare suitable measures for handling such contingencies.Numerical simulations using the distinct element method(DEM)and model calibration were performed to reproduce the anisotropic deformation of an ore drive based on the collected field data.Then,the roles of key factors(i.e.stress ratio,slenderness ratio,foliation orientation,and foliation considering excavation orientation)on the large deformation and damage depth of the excavations were investigated.This study found that increasing both the stress ratio and slenderness ratio induced linear increases in wall closure and damage depth,whereas increasing the foliation angle first increases the deformation and damage depth and then reduces them both before and after 45.The wall closure and damage thickness decreased with increasing orientation intercept.The deformation and damage levels were classified based on these factors.

Formation of Foliations and their Related Minerals from Diagenetic to Medium-grade Metamorphic Rocks:A Case Study of the Hongyanjing and Liao-Ji Backarc Basins,China

Deciphering the relationship between polyphase tectonic foliations and their associated mineral assemblages is significant in understanding the process from diagenesis to low-/medium-/high-grade metamorphism.It can provide information related to strain,metamorphic conditions and overprinting relationships and so help reveal the tectonic evolution of orogenesis.In this study,we predominately focus on the formation of foliations and their related minerals,as developed in two separate basins.First of all,two stages of axial plane cleavages(S1 and S2)were recognized in the Hongyanjing inter-arc basin,the formation of the S1 axial plane cleavage is associated with mica rotation and elongation in mudstones in the local area.The pencil structure of S2 formed during the refolding phase,the minerals in the sedimentary rocks not changing their shape and orientation.Secondly,in the Liao-Ji backarc basin,foliations include diagenetic foliation(bedding parallel foliation),tectonic S1 foliation(secondary foliation or axial plane cleavage of S0 folding)and crenulation cleavage(S2).The formation mechanism of foliation changes from mineral rotation or elongation and mineral solution transfer in S1 to crystal-plastic deformation,dynamic recrystallization and micro-folding in S2.Many index metamorphic minerals formed from low-grade to medium-grade consist of biotites,garnets,staurolite and kyanite,constituting a typical Barrovian metamorphic belt.Accordingly,a new classification of foliation is presented in this study.The foliations can be divided into continuous and disjunctive foliations,based on the existence of microlithons,detectable with the aid of a microscope.Disjunctive foliation can be further sub-divided into spaced foliation and crenulation cleavage,according to whether(or not)crenulation(micro-folding)is present.The size of the mineral grains is also significant for classification of the foliations.

A Note on Totally Real Riemannian Foliations with Parallelized Mean Curvature Vectors in a Complex Projective Space

We discussed a totally real Riemannian foliations with parallel mean curvature on a complex projective space.We carried out the divergence of a vector field on it and obtained a formula of Simons’type.

Anisotropic shearing mechanism of Kangding slate:Experimental investigation and numerical analysis

The shear mechanical behavior is regarded as an essential factor affecting the stability of the surrounding rocks in underground engineering.The shear strength and failure mechanisms of layered rock are significantly affected by the foliation angles.Direct shear tests were conducted on cubic slate samples with foliation angles of 0°,30°,45°,60°,and 90°.The effect of foliation angles on failure patterns,acoustic emission(AE)characteristics,and shear strength parameters was analyzed.Based on AE characteristics,the slate failure process could be divided into four stages:quiet period,step-like increasing period,dramatic increasing period,and remission period.A new empirical expression of cohesion for layered rock was proposed,which was compared with linear and sinusoidal cohesion expressions based on the results made by this paper and previous experiments.The comparative analysis demonstrated that the new expression has better prediction ability than other expressions.The proposed empirical equation was used for direct shear simulations with the combined finite-discrete element method(FDEM),and it was found to align well with the experimental results.Considering both computational efficiency and accuracy,it was recommended to use a shear rate of 0.01 m/s for FDEM to carry out direct shear simulations.To balance the relationship between the number of elements and the simulation results in the direct shear simulations,the recommended element size is 1 mm.

The topology of foliations formed by the generic K-orbits of a subclass of the indecomposable MD_5-groups

The paper is a continuation of the authors' works in 2007. We consider the MD_5-foliations associated to connected and simply connected MD_5-groups such that their Lie algebras have 4-dimensional commutative derived ideals. In this study, we give a topological classification of all considered MD_5-foliations. A description of these foliations by certain fibrations or suitable actions of R2 and the Connes' C*-algebras of the foliations which come from fibrations are also given.

Nano-texture of penetrative foliation in metamorphic rocks

By observing four samples obtained from Jiangxi Province, China, under the scanning electron mi-croscope (SEM), we discovered that nano-particle layers were commonly formed on sliding planes of the penetrative foliation in metamorphic rocks. We also successfully reproduced this phenomenon with a tri-axial pressure experiment. Having gone through the granulitization-alienation-partition in the shear sliding process, the nano-particles (40-95 nm in diameter) display different individual shapes and dis-tinct layered textures. This nano-confinement layer is essentially a frictional-viscous stripe with vis-cous-elastic deformation. In the micro-domain stripe, activities in structural stress field-rheological physical field-geochemical field are very dynamic, corresponding to the three stages (i.e., shear sliding strengthening-weakening-exfoliating) of the foliation development in metamorphism rocks. As such, the viscous-elastic deformation behavior helps shed light on the understanding of the micro-dynamic mechanism of the structural shearing.

Invariant algebraic curves and rational first integrals of holomorphic foliations in CP(2)

In this paper, using the tools of algebraic geometry we provide sufficient conditions for a holomor-phic foliation in CP(2) to have a rational first integral. Moreover, we obtain an upper bound of the degreesof invariant algebraic curves of a holomorphic foliation in CP(2). Then we use these results to prove that anyholomorphic foliation of degree 2 does not have cubic limit cycles.

Invariant foliations for parabolic equations

It is proved for parabolic equations that under certain conditions the weak (un-) stable manifolds possess invariant foliations, called strongly (un-) stable foliations. The relevant results on center manifoids are generalized to weak hyperbolic manifolds

The Minkowski norm and Hessian isometry induced by an isoparametric foliation on the unit sphere

Let M_(t) be an isoparametric foliation on the unit sphere(S^(n−1)(1),g^(st))with d principal curvatures.Using the spherical coordinatesinduced by M_(t),we construct a Minkowski norm with the representation F=r√2f(t),which generalizes the notions of(α,β)-norm and(α1,α2)-norm.Using the technique of the spherical local frame,we givean exact and explicit answer to the question when F=r√2 f(t)really defines a Minkowski norm.Using the similar technique,we study the Hessian isometry Φ between two Minkowski norms induced by M_(t),which preservesthe orientation and fixes the spherical ξ-coordinates.There aretwo ways to describe this Φ,either by a system of ODEs,or by its restriction toany normal plane for M_(t),which is then reduced to a Hessian isometry between Minkowski norms on R^(2) satisfying certain symmetry and(d)-properties.When d>2,we prove that this Φ can be obtained by gluing positive scalar multiplications and compositions of the Legendre transformation and positive scalar multiplications,so it must satisfy the(d)-property for any orthogonal decomposition R^(n)=V'+V'',i.e.,for any nonzero x=x'+x'' and Φ(x)=x=x'+x''with x',x'∈V'and x'',x''∈V'',we have g_(x)^(F1)(x'',x)=g_(x)^(F2)x(x'',x).As byproducts,we prove the following results.On the indicatrix(S_(F,g)),where F is a Minkowski norm induced by M_(t) and g is the Hessian metric,the foliation N_(t)=S_(F)∩R>_(0)M_(0) is isoparametric.Laugwitz Conjecture is valid for a Minkowski norm F induced by M_(t),i.e.,if its Hessian metric g is flat on R^(n)\{0}with n>2,then F is Euclidean.

Foliations on the tangent bundle of Finsler manifolds

Let M be a smooth manifold with Finsler metric F,and let T M be the slit tangent bundle of M with a generalized Riemannian metric G,which is induced by F.In this paper,we prove that (i) (M,F) is a Landsberg manifold if and only if the vertical foliation F V is totally geodesic in (T M,G);(ii) letting a:= a(τ) be a positive function of τ=F 2 and k,c be two positive numbers such that c=2 k(1+a),then (M,F) is of constant curvature k if and only if the restriction of G on the c-indicatrix bundle IM (c) is bundle-like for the horizontal Liouville foliation on IM (c),if and only if the horizontal Liouville vector field is a Killing vector field on (IM (c),G),if and only if the curvature-angular form Λ of (M,F) satisfies Λ=1-a 2/R on IM (c).

Foliations associated to harmonic maps on some complex two ball quotients

This article is an attempt to understand harmonic and holomorphic maps between two bounded symmetric domains in special situations. We study foliations associated to a lattice-equivariant harmonic map of small rank from a complex ball to another. The result is related to rigidity of some complex two ball quotients.Some open questions are raised as well.

FOLIATIONS ON CR MANIFOLD

In this paper, we prove that there exist CR foliation and complex foliation on abstractCR manifolds and obtain some properties of these foliations.

Smoothness of invariant manifolds and foliations for in nite dimensional random dynamical systems--Dedicated to Professor Shantao Liao

In this paper,we investigate the smoothness of invariant manifolds and foliations for random dynamical systems with nonuniform pseudo-hyperbolicity in Hilbert spaces.We discuss on the effect of temperedness and the spectral gaps in the nonuniform pseudo-hyperbolicity so as to prove the existence of invariant manifolds and invariant foliations,which preserve the CN,τ;(ω)Holder smoothness of the random system in the space variable and the measurability of the random system in the sample point.Moreover,we also prove that the stable foliation is CN-1;(ω)in the base point.

A class of metrics and foliations on tangent bundle of Finsler manifolds

Let （M, F） be a Finsler manifold, and let TMo be the slit tangent bundle of M with a generalized Riemannian metric G, which is induced by F. In this paper, we extract many natural foliations of （TMo, G） and study their geometric properties. Next, we use this approach to obtain new characterizations of Finsler manifolds with positive constant flag curvature. We also investigate the relations between Levi-Civita connection, Cartan connection, Vaisman connection, vertical foliation, and Reinhart spaces.

Higgs bundles over foliation manifolds

In this paper,we consider the stability,semi-stability and canonical metric structures on transverse Higgs bundles over a class of foliation manifolds.Also a transversal Bogomolov inequality is obtained.