cyclic plasticity using prager s translation rule and both nonlinear kinematic and isotropic hardening theory validation and algorithmic implementation abstract finite element analysis of structures under elasto plastic nonproportional cyclic loadings is useful in seismic engineering fatigue analysis and ductile fracture usual models with nonlinear stress strain curves in cyclic behavior are based on mroz multisurface plasticity bounding surface models or models derived from the armstrong frederick rule these models depart from the associative prager s rule with the main purpose of modeling aspects of cyclic nonlinear hardening in this paper we develop a model for cyclic plasticity within the framework of the associative classical plasticity theory using prager s rule accounting for anisotropic nonlinear kinematic hardening coupled with nonlinear isotropic hardening we include the validation of the theory against several uniaxial and multiaxial cyclic experiments and an efficient fully implicit radial return algorithm the parameters of the model are obtained directly by a discretization of the uniaxial stress strain behavior remarkably both the presented theory and the computational algorithm automatically recover classical bi linear plasticity and the krieg and key algorithm if the user prescribed stress strain curve is bilinear

on the structure of conformal singularities in classical general relativity ii evolution equations and a conjecture of k p tod consideration is given to the cauchy problem for perfect fluid space times which evolve from an initial singularity of conformal type the evolution equations for the conformally transformed unphysical geometry are shown to be expressible as a first order symmetric hyperbolic system albeit with a singular forcing term it is concluded that the 3 metric on the initial hypersurface of the unphysical space time constitutes the freely specifiable initial data subject to penroses s weyl curvature hypothesis according to which the weyl tensor was initially zero it follows that the physical space time is robertson walker this may provide a basis for a new explanation for the large scale isotropy of the universe

self similarity of quasilattices in two dimensions ii the non bravais type n gonal quasilattice for pt i see ibid vol 22 p 193 1989 presents a systematic method of dividing an n gonal lattice into identical sublattices which are also n gonal lattices a non bravais type n gonal quasilattice in two dimensions is constructed with the projection method by assigning windows with different shapes sizes and or orientations to the sublattices the symmetries and the orientations of the windows are determined by the relative point symmetries of the sublattices to the n gonal lattice and windows assigned to equivalent sublattices differ from one another only in their orientations the sublattices are transformed among themselves by a volume conserving linear transformation induced by a complex pv unit in the n cyclotomic field a necessary and sufficient condition for a pv unit to be a complex self similarity ratio of a non bravais type quasilattice is presented it is shown that every non bravais type n gonal quasilattice has a self similarity characterised by pv unit

contracting asymptotics of the linearized lapse scalar field sub system of the einstein scalar field equations we prove an asymptotic stability result for a linear coupled hyperbolic elliptic system on a large class of singular background spacetimes in constant mean curvature gauge on the n torus at each spatial point these background spacetimes are perturbations of kasner like solutions of the einstein scalar field equations which are not required to be close to the homogeneous and isotropic case we establish the existence of a homeomorphism between cauchy data for this system and a set of functions naturally associated with the asymptotics in the contracting direction which we refer to as asymptotic data this yields a complete characterization of the degrees of freedom of all solutions of this system in terms of their asymptotics spatial derivative terms can in general not be fully neglected which yields a clarification of the notion of asymptotic velocity term dominance

on the choice of a conformal gauss gauge near the cylinder representing spatial infinity a convenient approach to analyze spatial infinity is to use a cylinder representation i and impose a gauge based on a congruence of conformal geodesics this so called conformal gauss gauge comes along with the freedom to specify initial data for the conformal geodesics such a gauge has been constructed from an ordinary cauchy surface and from past null infinity i respectively the purpose of this note is to compare these gauges near the critical set i where i touches i as it turns out that they are related in a somewhat unexpected intricate way a convenient approach to analyze spatial infinity is to use a cylinder representation i and impose a gauge based on a congruence of conformal geodesics this so called conformal gauss gauge comes along with the freedom to specify initial data for the conformal geodesics such a gauge has been constructed from an ordinary cauchy surface and from past null infinity i respectively the purpose of this note is to compare these gauges near the critical set i where i touches i as it turns out that they are related in a somewhat unexpected intricate way

the structure and uniqueness of generalized solutions of the spherically symmetric einstein scalar equations in a previous paper we proved the global existence of generalized solutions of the spherically symmetric einstein scalar field equations in the large in this paper we study the regularity properties of the spacetime and the scalar field corresponding to a generalized solution we also prove a uniqueness theorem which shows that a generalized solution is an extension of a classical solution

consistent derivation of the constitutive algorithm for plane stress isotropic plasticity part i theoretical formulation abstract a derivation of the projected algorithm for general isotropic three invariant plasticity models under plane stress conditions is presented it is obtained by consistently specializing the 3d formulation to the 2d subspace defined by the plane stress condition closed form intrinsic algorithm linearization and a novel expression of the consistent tangent tensor are provided these are also shown to directly emanate from the analogous quantities pertaining to the fully 3d case a detailed discussion of the proposed implementation along with a representative set of numerical examples is provided in the second part of this paper valoroso n rosati l 2008 consistent derivation of the constitutive algorithm for plane stress isotropic plasticity part ii computational issues international journal of solids and structures 46 92 124

a proof of price s law for the collapse of a self gravitating scalar field a well known open problem in general relativity dating back to 1972 has been to prove price s law for an appropriate model of gravitational collapse this law postulates inverse power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates it is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon in this paper we prove a well defined upper bound formulation of price s law for the collapse of a self gravitating scalar field with spherically symmetric initial data we also allow the presence of an additional gravitationally coupled maxwell field our results are obtained by a new mathematical technique for understanding the long time behavior of large data solutions to the resulting coupled non linear hyperbolic system of p d e s in 2 independent variables the technique is based on the interaction of the conformal geometry the celebrated red shift effect and local energy conservation we feel it may be relevant for the problem of non linear stability of the kerr solution when combined with previous work of the first author concerning the internal structure of charged black holes which had assumed the validity of price s law our results can be applied to the strong cosmic censorship conjecture for the einstein maxwell real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data under christodoulou s c0 formulation the conjecture is proven to be false

analysis of a bianchi like equation satisfied by the mars simon tensor the mars simon tensor mst which e g plays a crucial role to provide gauge invariant characterizations of the kerr nut a ds family satisfies a bianchi like equation in this paper we analyze this equation in close analogy to the bianchi equation in particular it will be shown that the constraints are preserved supposing that a generalized buchdahl condition holds this permits the systematic construction of solutions to this equation in terms of a well posed cauchy problem a particular emphasis lies on the asymptotic cauchy problem where data are prescribed on a space like i i e for 0 in contrast to the bianchi equation the mst equation is of fuchsian type at i for which existence and uniqueness results are derived

consistent derivation of the constitutive algorithm for plane stress isotropic plasticity part ii computational issues abstract the implementation of the projected algorithm and of the consistent tangent tensor for general isotropic three invariant elastoplastic models under plane stress conditions discussed in part i of this paper valoroso n rosati l 2008 consistent derivation of the constitutive algorithm for plane stress isotropic plasticity part i theoretical formulation international journal of solids and structures doi 10 1016 j ijsolstr 2008 08 012 is addressed the connections between the general three dimensional case and the plane stress problem are analyzed in detail and an algorithmic treatment taking full advantage of the isotropic properties of the model is presented in particular intrinsic matrix free expressions are provided for all steps of the stress computation scheme that allow one to carry out the numerical implementation in a way that is completely independent from the matrix representations the numerical performances of the present solution scheme are evaluated through representative numerical examples

on the structure of conformal singularities in classical general relativity consideration is given to big bang singularities which can be conformally transformed to a spacelike hypersurface with the conformal factor and conformal metric both smooth on the extended manifold a precise definition of such conformal singularities is proposed by analogy with the established definition of conformal infinity the energy tensor of the physical space time is assumed to have a form appropriate to an isentropic perfect fluid the smoothness condition implies that the adiabatic index of this fluid must tend at the singularity to the value 4 3 appropriate to a highly relativistic initial state the smoothness condition also facilitates a natural choice of conformal factor in terms of the synge index of the fluid with this choice the four dimensional curvature is uniquely determined at the initial hypersurface by the induced 3 metric independently of the equation of state subject to a condition on the rate at which the adiabatic index approaches its limiting value the study refines in the smooth case previous work of goode wainwright 1985 concerning isotropic singularities justification is given for the new terminology the work provides the foundation for a subsequent investigation of the cauchy problem for space times which originate from a big bang of conformal type

theory of self similarity of periodic approximants to a quasilattice ii the case of the dodecagonal quasilattice for pt i see ibid vol 24 p 4873 87 1991 the space groups of the periodic approximants pas to the dodecagonal quasilattice in two dimensions are investigated the bravais lattices of the pas are tetragonal p4mm rectangular pmm hexagonal p6mm and rhombic cmm the author presents several series of pas each of which is derived from a prototype pa by a successive application of the deflation and rescaling dar and the space group is common among the members each series is composed of two subseries the consecutive members in each subseries are related by the proper dar and the two subseries are related to each other by the improper dar

stability and instability of the cauchy horizon for the spherically symmetric einstein maxwell scalar field equations this paper considers a trapped characteristic initial value problem for the spherically symmetric einstein maxwell scalar field equations for an open set of initial data whose closure contains in particular reissner nordstrdata the future boundary of the maximal domain of development is found to be a light like surface along which the curvature blows up and yet the metric can be continuously extended beyond it this result is related to the strong cosmic censorship conjecture of roger penrose the principle of determinism in classical physics is expressed mathemat ically by the uniqueness of solutions to the initial value problem for certain equations of evolution indeed in the context of the einstein equations of general relativity where the unknown is the very structure of space and time uniqueness is equivalent on a fundamental level to the validity of this principle the question of uniqueness may thus be termed the issue of the predictability of the equation the present paper explores the issue of predictability in general relativity since the work of leray it has been known that for the einstein equations contrary to common experience uniqueness for the cauchy problem in the large does not generally hold even within the class of smooth solutions in other words uniqueness may fail without any loss in regularity such failure is thus a global phenomenon the central question is whether this violation of predictability may occur in solutions representing actual physical processes physical phenomena and concepts related to the general theory of relativity namely gravitational collapse black holes angular momentum etc must cer tainly come into play in the study of this problem unfortunately the math ematical analysis of this exciting problem is very difficult at present beyond reach for the vacuum einstein equations in the physical dimension conse

black holes without spacelike singularities it is shown that for small spherically symmetric perturbations of asymptotically flat two ended reissner nordstrom data for the einstein maxwell real scalar field system the boundary of the dynamic spacetime which evolves is globally represented by a bifurcate null hypersurface across which the metric extends continuously under additional assumptions it is shown that the hawking mass blows up identically along this bifurcate null hypersurface and thus the metric cannot be extended twice differentiably in fact it cannot be extended in a weaker sense characterized at the level of the christoffel symbols the proof combines estimates obtained in previous work with an elementary cauchy stability argument there are no restrictions on the size of the support of the scalar field and the result applies to both the future and past boundary of spacetime in particular it follows that for an open set in the moduli space of solutions around reissner nordstrom there is no spacelike component of either the future or the past singularity

a classification of special points of quasilattices in two dimensions a complete classification of the special points of octagonal decagonal and dodecagonal quasilattices in two dimensions 2d is presented they are obtained by projecting the special points of the starting n gonal lattice n 8 10 or 12 in 4d onto a 2d subspace a set of equivalent special points of a quasilattice are located on the centres of a single kind of tile in the quasiperiodic tiling associated with the quasilattice the point symmetry of the special points is identical to that of the tiles they form a quasilattice with the same point symmetry as that of the original quasilattice the new quasilattice is of a bravais type or non bravais type according to whether the point symmetry group of the special points is identical to the full point symmetry group of the quasilattice or its true subgroup respectively

on the consistency of nested surfaces models and their kinematic hardening rules abstract in this paper we briefly address the consistency of formulations for nested surfaces plasticity and their kinematic hardening translation rules some requirements for these formulations are then given it is shown and discussed that multilayer plasticity based on nested yield surfaces present some inconveniences when modelling multiaxial cyclic loading on the other hand the use of hardening surfaces instead of yield surfaces solves the problem partially it is also shown that multilayer plasticity based on the mroz kinematic rule yields clearly different multiaxial predictions when using the same uniaxial bilinear curve and different number of surfaces which is a priori inconsistent since the same monotonic stress strain curve should not result into a clearly different multiaxial behavior simply because of the discretization employed by the user it is demonstrated that in contrast with the mroz translation rule multilayer plasticity using the prager translation rule results in the same prediction regardless of the number of surfaces in which the hardening part of the bilinear curve is discretized this rule is also compliant with the principle of maximum dissipation it also allows for a vanishing elastic region without falling into theoretical or numerical difficulties hence it should not be discarded when developing advanced models

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

einstein spaces as attractors for the einstein flow in this paper we prove a global existence theorem in the direction of cosmological expansion for sufficiently small perturbations of a family of n 1 dimensional spatially compact spacetimes which generalizes the k 1 friedmann lema otre robertson walker vacuum spacetime this work extends the result from 3 the background spacetimes we consider are lorentz cones over negative einstein spaces of dimension n 3 we use a variant of the constant mean curvature spatially harmonic cmcsh gauge introduced in 2 an important difference from the 3 1 dimensional case is that one may have a nontrivial moduli space of negative einstein geometries this makes it necessary to introduce a time dependent background metric which is used to define the spatially harmonic coordinate system that goes into the gauge instead of the bel robinson energy used in 3 we here use an expression analogous to the wave equation type of energy introduced in 2 for the einstein equations in cmcsh gauge in order to prove energy estimates it turns out to be necessary to assume stability of the einstein geometry further for our analysis it is necessary to have a smooth moduli space fortunately all known examples of negative einstein geometries satisfy these conditions we give examples of families of einstein geometries which have nontrivial moduli spaces a product construction allows one to generate new families of examples our results demonstrate causal geodesic completeness of the perturbed spacetimes in the expanding direction and show that the scale free geometry converges toward an element in the moduli space of einstein geometries with a rate of decay depending on the stability properties of the einstein geometry

weak null singularities in general relativity we construct a class of spacetimes without symmetry assumptions satisfying the vacuum einstein equations with singular boundaries on two null hypersurfaces intersecting in the future on a 2 sphere the metric of these spacetimes extends continuously beyond the singularities while the christoffel symbols fail to be square integrable in a neighborhood of any point on the singular boundaries the construction shows moreover that the singularities are stable in a suitable sense these singularities are stronger than the impulsive gravitational spacetimes considered by luk rodnianski and are conjecturally the generic type of singularity in the interior of black holes arising from gravitational collapse

global existence of generalized solutions of the spherically symmetric einstein scalar equations in the large in this paper we study the global initial value problem for the spherically symmetric einstein scalar field equations in the large we introduce the concept of a generalized solution of our problem and taking as initial hypersurface a future light cone with vertex at the center of symmetry we prove without any restriction on the size of the initial data the global in retarded time existence of generalized solutions

how solvable is 2 1 dimensional einstein gravity in this paper the relationship between witten s approach to the 2 1 dimensional vacuum einstein equations for spatially compact space times and the conventional annowitt deser and misner adm hamiltonian approach is discussed it is argued especially for the space times with higher genus cauchy surfaces that neither approach is complete in itself witten s because it does not provide a technique even at the classical level for recovering the space time metric and the conventional approach because it provides no mechanism for solving a seemingly intractable set of hamilton equations it is also argued however that the two formulations are instead complementary in the sense that the wilson loops which play a key role in witten s approach provide at least in principle a mechanism for solving the reduced hamilton equations and thereby completing the picture at the classical level an example of this synthesis for the explicitly computable case of genus one hypersurfaces is provided the m

superquasicrystals self similar ordered structures with non crystallographic point symmetries we present a systematic method of constructing limit quasiperiodic structures with non crystallographic point symmetries such structures are different aperiodic ordered structures from quasicrystals and we call them superquasicrystals they are sections of higher dimensional limit periodic structures constructed on super bravais lattices we enumerate important super bravais lattices superquasicrystals with strong self similarities form an important subclass the simplest example is a two dimensional octagonal superquasicrystal

local propagation of impulsive gravitationalwaves in this paper we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves we construct such spacetimes as solutions to the characteristic initial value problem of the einstein vacuum equations with a data curvature delta singularity we show that in the resulting spacetime the delta singularity propagates along a characteristic hypersurface while away from that hypersurface the spacetime remains smooth unlike the known explicit examples of impulsive gravitational spacetimes this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions the arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of nonregular characteristic data 2015 wiley periodicals inc

self similarity of quasilattices in two dimensions iii inflation by a non unit pv number for pt ii see ibid vol 22 205 1989 it is shown that every n gonal quasilattice in two dimensions has self similarity with a complex self similarity ratio which is equal to a non unit pv number in the n cyclotomic field in this self similarity the original quasilattice can be inflated to a number of different quasilattices which are locally isomorphous to each other the number is equal to the algebraic norm of the relevant pv number in the n cyclotomic field several examples are presented for the cases of octagonal decagonal and dodecagonal quasilattices the difference between the present self similarity and that associated with a pv unit is clarified

the cauchy problem for quasi linear hyperbolic evolution problems with a singularity in the time consideration is given to abstract quasi linear first order hyperbolic evolution problems with a forcing term having an initial 1 t singularity in the time t the results are obtained in the context of a reflexive banach space existence and uniqueness of solutions are established in spaces of sufficiently t differentiable functions and stability with respect to perturbations of the initial data is proved the special case of symmetric hyperbolic systems of first order pdes is discussed the work completes a previous study by newman concerning penrose9s weyl curvature hypothesis in general relativity

generalised 2d penrose tilings structural properties usual penrose tilings are defined by a set of five phases gamma j which add up to an integer the authors study the general case where the phases add up to any number and show that these generalised tilings can be obtained with four tiles two thick rhombi and two thin rhombi with different colourings of the edges the consideration of generalised tilings is relevant for the study of phase distortions local structural transitions and therefore various defects of the usual things

self similarity of quasilattices in two dimensions i the n gonal quasilattice the self similarity of a two dimensional quasilattice with an n gonal point symmetry has been investigated for even n n or 8 it is shown that for any n an n gonal quasilattice has a self similarity characterised by a complex number tau on inflation the quasilattice is scaled by mod tau mod and subsequently rotated by arg tau tau is a pv unit of the n cyclotomic field q zeta zeta exp 2 pi i n tau satisfies i tau and tau 1 are both algebraic integers in q zeta ii mod tau mod 1 and iii mod tau mod 1 for any conjugate tau but tau the complex conjugate of tau in q zeta pv units are calculated for every n gonal quasilattice whose multiplicity m phi n 2 with phi being the eulerian function is less than 5 m 2 for n 8 10 12 m 3 for n 14 18 and m 4 for n 16 20 24 30 it is found that a quasilattice has two or more independent scales of self similarity if its multiplicity is larger than 2

a self similar dodecagonal quasiperiodic tiling of the plane in terms of squares regular hexagons and thin rhombi the author presents a dodecagonal quasiperiodic tiling of the plane in terms of three kinds of tiles a square a regular hexagon and a thin rhombus whose acute inner angles are 30 degrees the dodecagonal tiling is self similar with respect to a dilatation by 2 square root 3 it is associated with a dodecagonal quasiperiodic lattice which is obtained by a projection from a hyperhoneycomb lattice in four dimensions the hyperhoneycomb lattice is a non bravais lattice composed of four bravais sublattices and the author must assign windows with different orientations to different sublattices the author discusses in detail the relationships of the present method of obtaining a dodecagonal tiling to other methods

on the formation of trapped surfaces in a recent important breakthrough d christodoulou has solved a long standing problem of general relativity of evolutionary formation of trapped surfaces in the einstein vacuum space times he has identified an open set of regular initial conditions on an outgoing null hypersurface both finite and at past null infinity leading to a formation a trapped surface in the corresponding vacuum space time to the future of the initial outgoing hypersurface and another incoming null hypersurface with the prescribed minkowskian data in this paper we give a simpler proof for a finite problem by enlarging the admissible set of initial conditions and consistent with this relaxing the corresponding propagation estimates just enough that a trapped surface still forms we also reduce the number of derivatives needed in the argument from two derivatives of the curvature to just one more importantly the proof which can be easily localized with respect to angular sectors has the potential for further developments

implicit j2 bounding surface plasticity using prager s translation rule a bounding surface j2 plasticity model that uses prager s translation rule is presented the model preserves masing s rules and is developed from the same ideas as classical infinitesimal j2 plasticity resulting in the same formulation with the exception of the algorithm for the computation of the hardening function instead of utilizing a loading surface as in a previous formulation hardening surfaces are introduced the formulation is similar to that of multilayer plasticity using prager s rule presented in previous work an implicit algorithm based on the radial return concept is used and the consistent elastoplastic tangent is also developed in closed form examples illustrating anisotropic behaviour are presented and compared to that predicted by a multilayer j2 plasticity model the model is also applied to a soil dynamics problem to show the robustness of the algorithm and its applicability to complex loading copyright 2002 john wiley sons ltd

computational issues in large strain elasto plasticity an algorithm for mixed hardening and plastic spin in this paper an algorithm for large strain elasto plasticity with isotropic hyperelasticity based on the multiplicative decomposition is formulated the algorithm includes a possible constitutive equation for the plastic spin and mixed hardening in which the principal stress and principal backstress directions are not necessarily preserved it is shown that if the principal trial stress directions are preserved during the plastic flow as assumed in some algorithms a plastic spin is inadvertently introduced for the kinematic mixed hardening case if the formulation is performed in the principal stress space a rotation of the backstress is inadvertently introduced as well the consistent linearization of the algorithm is also addressed in detail copyright 2005 john wiley sons ltd

implicit plane stress algorithm for multilayer j2 plasticity using the prager ziegler translation rule a totally implicit algorithm for plane stress multilayer plasticity is presented the algorithm presents the plane stress version of the 3d plane strain model for multilayer plasticity presented in a previous work as in the 3d plane strain case the model is consistent with the principle of maximum dissipation and it may be considered as an extension of classical j2 plasticity for anisotropic non linear kinematic behaviour preserving masing s rules in order to obtain the asymptotic second order convergence of the newton algorithm the consistent tangent moduli are also given copyright 2003 john wiley sons ltd

the interior of charged black holes and the problem of uniqueness in general relativity we consider a spherically symmetric double characteristic initial value problem for the real einstein maxwell scalar field equations on the initial outgoing characteristic the data is assumed to satisfy the price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat cauchy surface we establish that the heuristic mass inflation scenario put forth by israel and poisson is mathematically correct in the context of this initial value problem in particular the maximal future development has a future boundary over which the space time is extendible as a c0 metric but along which the hawking mass blows up identically thus the space time is inextendible as a c1 metric in view of recent results of the author in collaboration with i rodnianski which rigorously establish the validity of price s law as an upper bound for the decay of scalar field hair the c0 extendibility result applies to the collapse of complete asymptotically flat spacelike initial data where the scalar field is compactly supported this shows that under christodoulou s c0 formulation the strong cosmic censorship conjecture is false for this system 2005 wiley periodicals inc

a regime of linear stability for the einstein scalar field system with applications to nonlinear big bang formation we linearize the einstein scalar field equations expressed relative to constant mean curvature cmc transported spatial coordinates gauge around members of the well known family of kasner solutions on 0 t the kasner solutions model a spatially uniform scalar field evolving in a typically spatially anisotropic spacetime that expands towards the future and that has a big bang singularity at t 0 we place initial data for the linearized system along t 1 t and study the linear solution s behavior in the collapsing direction t 0 our main result is the identification of a new form of approximate l monotonicity for the linear solutions that holds whenever the background kasner solution is sufficiently close to the friedmann lemaitre robertson walker flrw solution using the approximate monotonicity we derive sharp information about the asymptotic behavior of the linear solution as t 0 in particular we show that some of its time rescaled components converge to regular functions defined along t 0 in addition we motivate the preferred direction of the approximate monotonicity by showing that the cmctransported spatial coordinates gauge can be realized as a limiting version of a family of parabolic gauges for the lapse variable an approximate l monotonicity inequality also holds in the parabolic gauges but the corresponding parabolic pdes are locally well posed only in the direction t 0 in a companion article we use the linear stability results to prove a stable singularity formation result for the nonlinear equations specifically we show that the flrw solution is globally nonlinearly stable in the collapsing direction t 0 under small perturbations of its data at t 1