modelling phase transitions in alloys abstract we present a numerical method for solving a system of two nonlinear evolution equations of order four and two known as the cahn hilliard allen cahn system the simulation results give a first intuition about the stationary states form and stability

dynamical properties of s gap shifts and other shift spaces abstract we study the dynamical properties of certain shift spaces to help study these properties we introduce two new classes of shifts namely boundedly supermultiplicative bsm shifts and balanced shifts it turns out that any almost specified shift is both bsm and balanced and any balanced shift is bsm however as we will demonstrate there are examples of shifts which are bsm but not balanced we also study the measure theoretic properties of balanced shifts we show that a shift space admits a gibbs state if and only if it is balanced restricting ourselves to s gap shifts we relate certain dynamical properties of an s gap shift to combinatorial properties from expansions in non integer bases this identification allows us to use the machinery from expansions in non integer bases to give straightforward constructions of s gap shifts with certain desirable properties we show that for any q 0 1 there is an s gap shift which has the specification property and entropy q we also use this identification to address the question for a given q 0 1 how many s gap shifts exist with entropy q for certain exceptional values of q there is a unique s gap shift with this entropy

strong solutions for two dimensional nonlocal cahn hilliard navier stokes systems abstract a well known diffuse interface model for incompressible isothermal mixtures of two immiscible fluids consists of the navier stokes system coupled with a convective cahn hilliard equation in some recent contributions the standard cahn hilliard equation has been replaced by its nonlocal version the corresponding system is physically more relevant and mathematically more challenging indeed the only known results are essentially the existence of a global weak solution and the existence of a suitable notion of global attractor for the corresponding dynamical system defined without uniqueness in fact even in the two dimensional case uniqueness of weak solutions is still an open problem here we take a step forward in the case of regular potentials first we prove the existence of a unique strong solution in two dimensions then we show that any weak solution regularizes in finite time uniformly with respect to bounded sets of initial data this result allows us to deduce that the global attractor is the union of all the bounded complete trajectories which are strong solutions we also demonstrate that each trajectory converges to a single equilibrium provided that the potential is real analytic and the external forces vanish

existence uniqueness and l q estimates for the stokes problem in an exterior domain as is well known the study of the properties of solutions to the stokes system plays an essential role in the mathematical theory of viscous fluid flows governed by the navier stokes equations it is not art overstatement to say that existence uniqueness and a priori estimates in sobolev spaces for the stokes system are fundamental in every basic question related to the navier stokes equations such as existence regularity asymptotic behavior in time etc see e g ladyzhenskaya 1969 solonni ov 1977 and te am 1977 so far however the stokes system has been fully analyzed only for bounded domains see sobolevskn 1960 ladvznenskaya 1959 solonnu ov 1960 vorovlch yot oowcr 1961 cattabrma 1961 in particular in the paper of cattabriga a complete treatment is given in the sobotev spaces w l 1 1 q c for space dimension n 3 cattabriga s results essentially state that if 2 is a bounded domain in r a of class c m m max l 2 2 l 1 then given f e wt q y2 3 gc wt l q o q e wl 2 1 q q 8o a 1 q there exists one and only one distributional solution v c w1 z q q 3 p e wt l q o to the stokes system

consistent models of cahn hilliard gurtin equations with neumann boundary conditions abstract in this paper we prove the equivalence between two formulations for generalizations of the cahn hilliard equation based on constitutive equations proposed by gurtin in a parallelepiped associated with mixed periodic neumann boundary conditions it is important to note that in the case of the classical cahn hilliard equation both formulations are useful for the study of the existence and uniqueness of solutions furthermore the interest for taking neumann boundary conditions appears in particular when one takes into account the deformations of the material indeed in that case the order parameter and the displacement are coupled through the boundary conditions as it is generally expected we then obtain some existence and uniqueness results

hausdorff dimension of univoque sets and devil s staircase abstract we fix a positive integer m and we consider expansions in arbitrary real bases q 1 over the alphabet 0 1 m we denote by u q the set of real numbers having a unique expansion completing many former investigations we give a formula for the hausdorff dimension d q of u q for each q 1 furthermore we prove that the dimension function d 1 0 1 is continuous and has bounded variation moreover it has a devil s staircase behavior in q where q denotes the komornik loreti constant although d q d q for all q q we have d 0 a e in q during the proofs we improve and generalize a theorem of erd s et al on the existence of large blocks of zeros in expansions and we determine for all m the lebesgue measure and the hausdorff dimension of the set u of bases in which x 1 has a unique expansion

an energetic variational approach for the cahn hilliard equation with dynamic boundary condition model derivation and mathematical analysis the cahn hilliard equation is a fundamental model that describes phase separation processes of binary mixtures in recent years several types of dynamic boundary conditions have been proposed in order to account for possible short range interactions of the material with the solid wall our first aim in this paper is to propose a new class of dynamic boundary conditions for the cahn hilliard equation in a rather general setting the derivation is based on an energetic variational approach that combines the least action principle and onsager s principle of maximum energy dissipation one feature of our model is that it naturally fulfills three important physical constraints conservation of mass dissipation of energy and force balance relations next we provide a comprehensive analysis of the resulting system of partial differential equations under suitable assumptions we prove the existence and uniqueness of global weak strong solutions to the initial boundary value problem with or without surface diffusion furthermore we establish the uniqueness of the asymptotic limit as t to infty and characterize the stability of local energy minimizers for the system

critical spaces for quasilinear parabolic evolution equations and applications abstract we present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations the approach is based on maximal l p regularity in time weighted function spaces it is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance applications to the vorticity equations for the navier stokes problem convection diffusion equations the nernst planck poisson equations in electro chemistry chemotaxis equations the mhd equations and some other well known parabolic equations are given

convergence of euler stokes splitting of the navier stokes equations we consider approximation by partial time steps of a smooth solution of the navier stokes equations in a smooth domain in two or three space dimensions with no slip boundary condition for small k 0 we alternate the solution for time k of the inviscid euler equations with tangential boundary condition and the solution of the linear stokes equations for time k with the no slip condition imposed we show that this approximation remains bounded in h2 p and is accurate to order k in lp for p the principal difficulty is that the initial state for each stokes step has tangential velocity at the boundary generated during the euler step and thus does not satisfy the boundary condition for the stokes step the validity of such a fractional step method or splitting is an underlying principle for some computational methods 1994 john wiley sons inc

periodic solutions of non isothermal phase separation models with constraint abstract in this paper we consider the existence of time periodic solutions to the non isothermal phase separation models with constraints for the order parameter and the dirichlet boundary condition for the temperature we firstly address some approximate problems by the leray schauder fixed point theorem then the periodic solutions of the original problem are obtained by establishing some a priori estimates and performing a limiting procedure

fuzzy sets and membership functions abstract in this paper we try to provide an analytical basis for the theory of fuzzy sets the main idea is to introduce the the notions of consensus sets and subjective variables which strongly resemble respectively the events and random variables in the probability theory the resemblance is a natural consequence from the fact that one of the most important roles of the fuzzy set theory is to present a method for the analysis of the uncertainty originated in human subjectivity while the probability theory for that of randomness

new thoughts on old results of r t seeley in this paper we show that the lp realization of a vector valued elliptic boundary value problem j admits a bounded calculus on lp g e 1 p provided the top order coefficients of are holder continuous here g denotes a domain in n with compact c2m boundary and e a banach space of class our proof is based on an abstract perturbation result for operators admitting bounded calculus and kernel estimates for the solution of j

a cahn hilliard equation with singular diffusion in the present work we address a class of cahn hilliard equations characterized by a singular diffusion term the problem is a simplified version with constant mobility of the cahn hilliard de gennes model of phase separation in binary incompressible isothermal mixtures of polymer molecules it is proved that for any final time t the problem admits a unique energy type weak solution defined over 0 t for any 0 such solution is classical in the sense of belonging to a suitable holder class over t and enjoys the property of being separated from the singular values corresponding to pure phases

on systems of cahn hilliard and allen cahn equations considered as gradient flows in hilbert spaces abstract we study systems of allen cahn and cahn hilliard equations with the mobility coefficients depending on c and c we interpret these systems of equations as gradient flows in hilbert spaces with a densely defined riemannian metric in particular we study gradient flows curves of maximal slope of the form t u l u s u f where s is a nonconvex functional l u s u is the strong weak closure of the subgradient of s and f is a time dependent right hand side the article generalizes the results by rossi and savare 36 to this setting and applies for systems of multiple phases derived by heida malek and rajagopal 20 19 in a simplified form more generally we will show that a certain class of reaction diffusion equations coming from a modeling approach by rajagopal and srinivasa 32 or by mielke 27 are automatically subject to the presented theory of curves of maximal slope

convergence to equilibrium for the cahn hilliard equation with a logarithmic free energy abstract in this paper we investigate the asymptotic behavior of the nonlinear cahn hilliard equation with a logarithmic free energy and similar singular free energies we prove an existence and uniqueness result with the help of monotone operator methods which differs from the known proofs based on approximation by smooth potentials moreover we apply the lojasiewicz simon inequality to show that each solution converges to a steady state as time tends to infinity

existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations a nonlinear parabolic equation of the fourth order is analyzed the equation is characterized by a mobility coefficient that degenerates at 0 existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a priori estimates if a viscosity term is added and additional conditions on the nonlinear terms are assumed then it is proved that any weak solution becomes instantaneously strictly positive this in particular implies uniqueness for strictly positive times and further time regularization properties the long time behavior of the problem is also investigated and the existence of trajectory attractors and under more restrictive conditions of strong global attractors is shown

regularity and separation from potential barriers for a non local phase field system abstract we show that solutions of a two phase model involving a non local interactive term become more regular immediately after the moment they separate from the pure phases this result allows us to prove stronger convergence to equilibria a new proof of the separation property is also given

nonlinear aspects of the cahn hilliard equation abstract this paper treats phase separation within the context of the phenomenological cahn hilliard equation c t m c f c k 2 c where c is the concentration m c is the mobility and f c is homogeneou s free energy which is assumed here to be a fourth degree polynomial natural boundary conditions are introduced the full set of equilibrium solutions is specified a comparison theorem for stability criteria which was postulated by langer is proved here within the framework of the natural boundary conditions energy methods are used to define and estimate the limit of monotonic global stability it is pointed out that within the parameter region where the uniform homogeneous state is the only equilibrium solution there may still exist some internal excitable region in which the homogeneous solution possesses growing fluctuations furthermore a periodic instability is shown to exist in the metastable region in addition to the well known nucleation instability

isogeometric analysis of the navier stokes cahn hilliard equations with application to incompressible two phase flows abstract in this work we provide a unified and comparative description of the most prominent phase field based two phase flow models and present our numerical results of the application of galerkin based isogeometric analysis iga to incompressible navier stokes cahn hilliard nsch equations in velocity pressure phase field chemical potential formulation for the approximation of the velocity and pressure fields lbb compatible non uniform rational b spline spaces are used which can be regarded as smooth generalizations of taylor hood pairs of finite element spaces the one step scheme is used for the discretization in time the static and rising bubble in addition to the nonlinear rayleigh taylor instability flow problems are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme

the cahn hilliard equation with time dependent constraint abstract we propose an abstract variational inequality formulation of the cahn hilliard equation with a time dependent constraint we introduce notions of strong and weak solutions and prove that a strong solution if it exists is a weak solution and that the existence of a unique weak solution holds under an appropriate time dependence condition on the constraint we also show that the weak solution is a strong solution under appropriate assumptions on the data our abstract results can be applied to various concrete problems

finite orbits in multivalued maps and bernoulli convolutions abstract bernoulli convolutions are certain measures on the unit interval depending on a parameter between 1 and 2 in spite of their simple definition they are not yet well understood we study their two dimensional density which exists by a theorem of solomyak to each bernoulli convolution there is an interval d called the overlap region and a map which assigns two values to each point of d and one value to all other points of 0 1 there are two types of finite orbits of these multivalued maps which correspond to zeros and potential singularities of the density respectively orbits which do not meet d belong to an ordinary map called transformation and exist for all 1 6182 they were studied by erdos joo komornik sidorov de vries and others as points with unique addresses and by jordan shmerkin and solomyak as points with maximal local dimension in the two dimensional view these orbits form address curves related to the milnor thurston itineraries in one dimensional dynamics the curves depend smoothly on the parameter and represent quantiles of all corresponding bernoulli convolutions finite orbits which intersect d have a network like structure and can exist only at perron parameters their points are intersections of extended address curves and can have finite or countable number of addresses as found by sidorov for an uncountable number of parameters the central point 1 2 has only two addresses the intersection of periodic address curves can lead to singularities of the measures we give examples which are not pisot or salem parameters

on the continuity of the hausdorff dimension of the univoque set abstract in a recent paper komornik et al 2017 19 proved a long conjectured formula for the hausdorff dimension of the set u q of numbers having a unique expansion in the non integer base q and showed that this hausdorff dimension is continuous in q unfortunately their proof contained a gap which appears difficult to fix this article gives a completely different proof of these results using a more direct combinatorial approach

universal attractor for some singular phase transition systems a singular parabolic system describing the thermal diffusion in a substance possibly subject to a phase transition is introduced the physical process is described by the variables absolute temperature and order parameter the latter may have or not conserved total mass with respect to time in both cases after recalling and sometimes improving some known well posedness results the long time behavior of the system is studied it is shown that the process is dissipative and the compact universal attractor is constructed it turns out to attract the trajectories of the system in a rather strong metric which is strictly linked to the constraints imposed to both variables the techniques used in the proofs seem likely to be applied to other types of evolution systems containing maximal monotone nonlinearities

unconditionally energy stable large time stepping method for the l2 gradient flow based ternary phase field model with precise nonlocal volume conservation abstract we consider numerical approximations for solving a new allen cahn type ternary phase field model we first formulate a phase field model from an energetic variational formulation based on the l 2 gradient flow and add three nonlocal lagrange multipliers to the system to conserve the volume for each phase then by combining the sav approach with the stabilization technique where a crucial linear stabilization term is added to enhance the stability and keep the required accuracy thus allowing large time steps we arrive at a decoupled linear non iterative and volume conserved scheme this scheme is unconditionally energy stable and requires solving only four linear second order elliptic equations with constant coefficients we further prove energy stability and present numerous 2d and 3d numerical simulations to show accuracy and stability

multiple expansions of real numbers with digits set 0 1 q for q 1 we consider expansions in base q with digits set left 0 1 q right let mathcal u q be the set of points which have a unique q expansion for k 2 3 ldots aleph 0 let mathcal b k be the set of bases q 1 for which there exists x having precisely k different q expansions and for q in mathcal b k let mathcal u q k be the set of all such x s which have exactly k different q expansions in this paper we show that begin aligned mathcal b aleph 0 2 infty quad text and quad mathcal b k q c infty quad text for any quad k ge 2 end aligned where q c approx 2 32472 is the appropriate root of x 3 3x 2 2x 1 0 moreover we show that for any integer k ge 2 and any q in mathcal b k the hausdorff dimensions of mathcal u q k and mathcal u q are the same i e begin aligned dim h mathcal u q k dim h mathcal u q quad text for any quad k ge 2 end aligned finally we conclude that the set of points having a continuum of q expansions has full hausdorff dimension

structure functions with fuzzy states systems are modeled in reliability theory by means of structure functions which show how the system state of performance depends on the state of all its components in this paper we consider that the space of states allowed for both the system and its components is an appropriate family of fuzzy numbers basic concepts in classical crisp binary and multistate models in reliability theory are in this way generalized some particular consequences relative to component importance are also analyzed

stabilization of solutions to a fitzhugh nagumo type system we consider a bistable reaction diffusion system arising in the theory of phase transitions it appears in several physical contexts such as thin magnetic films and the microphase separation in diblock copolymer melts mathematically it takes the form of an allen cahn equation coupled to an elliptic equation this system possesses a lyapunov functional which represents the gibbs free energy of the phase separation problem we study the large time behavior of the solution orbits and use the fact that the problem has a gradient structure to prove their stabilization by means of a version of ojasiewicz inequality

on a cahn hilliard allen cahn system coupled with a type iii heat equation and singular potentials abstract our aim in this article is to study the asymptotic behavior of a cahn hilliard allen cahn system coupled with a heat equation based on the type iii heat conduction law with singular potentials we also show further regularity results and we prove a strict separation property from the pure states in one space dimension

a solution algorithm for fuzzy linear programming with piecewise linear membership functions abstract this paper deals with the fuzzy linear program with continuous piecewise linear membership functions a fuzzy linear program with piecewise linear membership functions has been considered by hannan and nakamura hannan s method can solve the problems only when all membership functions are concave in the range 0 1 and nakamura s method needs to use the linear programming technique repetitionally we propose a technique to solve the problem using a standard linear programming when membership functions are strictly quasiconcave and the minimum operator is adopted for aggregating fuzzy goals the repetitional use of linear programming technique is not required in our method

entropy topological transitivity and dimensional properties of unique q expansions let m be a positive integer and q in 1 m 1 we consider expansions of real numbers in base q over the alphabet 0 ldots m in particular we study the set mathcal u q of real numbers with a unique q expansion and the set mathbf u q of corresponding sequences it was shown by komornik kong and li that the function h which associates to each q in 1 m 1 the topological entropy of mathcal u q is a devil s staircase in this paper we explicitly determine the plateaus of h and characterize the bifurcation set mathscr e of q s where the function h is not locally constant moreover we show that mathscr e is a cantor set of full hausdorff dimension we also investigate the topological transitivity of a naturally occurring subshift mathbf v q sigma which has a close connection with open dynamical systems finally we prove that the hausdorff dimension and box dimension of mathcal u q coincide for all q in 1 m 1

intersecting random translates of invariant cantor sets given two cantor setsx andy in 0 1 invariant under the mapx b x mod 1 the hausdorff dimension of x t y is constant almost everywhere whenx y are defined by admissible digits in baseb and more generally by sofic systems we compute this dimension in terms of the largest lyapunov exponent of a random product of matrices the results are extended to higher dimensions and multiple intersections

number representation using generalized β transformation we study non standard number systems with negative base b instead of the ito sadahiro definition based on the transformation t b of the interval b b 1 1 b 1 into itself we suggest a generalization using an interval l l 1 with l 1 0 such numeration systems share many properties of positive base numeration introduced by renyi although the proofs are not always straightforward in this paper we focus on the description of admissible digit strings and their periodicity we address the question of the description of reference strings used in the admissibility condition we give examples which contradict a result of gora and show that in this aspect the negative base numeration significantly differs from the renyi numeration

on a model for phase separation in binary alloys driven by mechanical effects this work is concerned with the mathematical analysis of a system of partial differential equations modeling the effect of phase separation driven by mechanical actions in binary alloys like tin lead solders the system combines the quasistationary balance of linear momentum with a fourth order evolution equation of cahn hilliard type for the phase separation and it is highly nonlinearly coupled existence and uniqueness results are shown tel 0382505631 email bonetti dimat unipv it

on a diffuse interface model for two phase flows of viscous incompressible fluids with matched densities we study a diffuse interface model for the flow of two viscous incompressible newtonian fluids of the same density in a bounded domain the fluids are assumed to be macroscopically immiscible but a partial mixing in a small interfacial region is assumed in the model moreover diffusion of both components is taken into account this leads to a coupled navier stokes cahn hilliard system which is capable of describing the evolution of droplet formation and collision during the flow we prove the existence of weak solutions of the non stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities which ensures in contrast to the usual case of a smooth free energy density that the concentration stays in the physical reasonable interval furthermore we find that unique strong solutions exist in two dimensions globally in time and in three dimensions locally in time moreover we show that for any weak solution the concentration is uniformly continuous in space and time because of this regularity we are able to show that any weak solution becomes regular for large times and converges as t to a solution of the stationary system these results are based on a regularity theory for the cahn hilliard equation with convection and singular potentials in spaces of fractional time regularity as well as on maximal regularity of a stokes system with variable viscosity and forces in l2 0 hs s in 0 frac12 which are new themselves

existence of weak solutions for a diffuse interface model for viscous incompressible fluids with general densities we study a diffuse interface model for the flow of two viscous incompressible newtonian fluids in a bounded domain the fluids are assumed to be macroscopically immiscible but a partial mixing in a small interfacial region is assumed in the model moreover diffusion of both components is taken into account in contrast to previous works we study the general case that the fluids have different densities this leads to an inhomogeneous navier stokes system coupled to a cahn hilliard system where the density of the mixture depends on the concentration the velocity field is no longer divergence free and the pressure enters the equation for the chemical potential we prove existence of weak solutions for the non stationary system in two and three space dimensions

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

on the initial boundary value problem for a bingham fluid in a three dimensional domain the initial boundary value problem associated with the motion of a bingham fluid is considered the existence and uniqueness of strong solution is proved under a certain assumption on the data it is also shown that the solution exists globally in time when the data are small and that the solution converges to a periodic solution if the external force is time periodic o introduction the purpose of this paper is to establish the existence of strong solutions to a variational inequality which describes the motion of a bingham fluid in a bounded three dimensional domain a bingham fluid is a rigid viscoplastic fluid which is governed by a special constitutive law such that it moves like a rigid body if a certain function of the stresses does not reach the yield limit and it behaves like a viscous fluid when the yield limit is reached since the motion is governed by two entirely different stress strain relations depending on the state of stresses the conservation of momentum is expressed in terms of a variational inequality so that one can avoid the difficulty of separating the fluid zone and the 1 rlglcl zone the initial boundary value problem we shall study is formulated as 0 l it uj u a u w u b u u w j w j u f w a in 0 t for each test function w such that v w 0 in q and w 0 on aq 0 2 v u 0 in q x 0 t 0 3 u 0 on aq x 0 t 0 4 8 x 0 vo x in q here q is a bounded domain in r3 with smooth boundary aq a x t denotes the velocity of the fluid and f x t stands for external force we assume that the density the yield limit and the viscosity are positive constants in particular the received by the editors august 15 1986 1980 mathematics subject classijication 1985 revision primary 35q20 35b10 35b40 35b65 secondary 76a05 76d99 this research was supported by the air force office of scientific research under grant afosr86 0085 and by nsf grant dms 8521848 r 1987 american mathematical society 0002 9947 87 1 00 25 per page

arithmetics in number systems with a negative base we study the numeration system with a negative base introduced by ito and sadahiro we focus on arithmetic operations in the sets fin b and z b of numbers having finite resp integer b expansions we show that fin b is trivial if b is smaller than the golden ratio 12 1 5 for b 12 1 5 we prove that fin b is a ring only if b is a pisot or salem number with no negative conjugates we prove the conjecture of ito and sadahiro that fin b is a ring if b is a quadratic pisot number with positive conjugate for quadratic pisot units we determine the number of fractional digits that may appear when adding or multiplying two b integers

on univoque and strongly univoque sets abstract much has been written about expansions of real numbers in noninteger bases particularly for a finite alphabet 0 1 and a real number base 1 1 the so called univoque set of numbers which have a unique expansion in base has garnered a great deal of attention in recent years motivated by recent applications of expansions to bernoulli convolutions and a certain class of self affine functions we introduce the notion of a strongly univoque set we study in detail the set d of numbers which are univoque but not strongly univoque our main result is that d is nonempty if and only if the number 1 has a unique nonterminating expansion in base and in that case d is uncountable we give a sufficient condition for d to have positive hausdorff dimension and show that on the other hand there are infinitely many values of for which d is uncountable but of hausdorff dimension zero

a free energy stable nodal discontinuous galerkin approximation with summation by parts property for the cahn hilliard equation abstract we present a nodal discontinuous galerkin dg scheme for the cahn hilliard equation that satisfies the summation by parts simultaneous approximation term sbp sat property the latter permits us to show that the discrete free energy is bounded and as a result the scheme is provably stable the scheme and the stability proof are presented for general curvilinear three dimensional hexahedral meshes we use the bassi rebay 1 br1 scheme to compute interface fluxes and a first order implicit explicit imex scheme to integrate in time we provide a semi discrete stability study and a fully discrete proof subject to a positivity condition on the solution lastly we test the theoretical findings using numerical cases that include two and three dimensional problems

uncertainty quantification of squeal instability under two fuzzy interval cases abstract automotive brake systems are always subjected to various kinds of uncertain factors and the uncertainty analyses of brake squeal have been investigated for decades however most of the existing methods of squeal analysis are merely effective in tackling single uncertain case in this study a unified method is developed for the uncertainty quantification of disc brake squeal which is able to handle two fuzzy interval cases due to subjective opinions or limited data in the first fuzzy interval case the uncertain brake parameters are expressed as either fuzzy variables or interval variables which exist in brake systems simultaneously and independently whereas the uncertain brake parameters are all treated as interval variables but their lower and upper bounds are expressed as fuzzy variables in the second fuzzy interval case in the developed method the first fuzzy interval case is equivalently treated as a special case of the second one on the basis of fuzzy boundary interval variables an uncertainty quantification model under two fuzzy interval cases is established in which the unified uncertain response is computed with the aid of the combination of level cut strategy taylor series expansion subinterval analysis and monte carlo simulation the unified method is then extended to quantify the uncertainty in disc brake squeal analysis the studies of several numerical examples demonstrate that the proposed method is able to quantify the uncertainty of squeal instability effectively in addition the proposed method can also help to carry out reliability analysis and optimization to reduce the likelihood of squeal occurrence

an efficient method combining active learning kriging and monte carlo simulation for profust failure probability abstract for more and more complicated engineering structures it is a challenge to efficiently estimate the profust failure probability based on the probability inputs and fuzzy state assumption by combining active learning kriging with monte carlo simulation ak mcs an efficient method is proposed to estimate the profust failure probability firstly the profust failure probability is transformed into an integral of the classical failure probability by introducing a variable related to the fuzzy state assumption this integral is further reorganized as a weighted sum of a series of classical failure probabilities by gaussian quadrature and the series of the classical failure probabilities have the similar limit state function constructions constrained by different thresholds secondly mcs is used according to the probability input distribution to generate the sample pool in which the active learning kriging is used to establish the surrogates of the series of similar limit state functions with different thresholds an improved learning function is proposed by minimizing the u learning function minima corresponding to all limit state functions so that the candidate with the largest effect on the surrogating quality of all limit states can be selected as a training point to update the kriging model once the updating process of the kriging model converges all limit state functions can be identified by the kriging model and the profust failure probability can be estimated by using the kriging model without any extra model evaluation several examples are used to demonstrate the feasibility of the proposed strategy for estimating the profust failure probability

ergodicity of stochastic cahn hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises abstract we study the asymptotic properties of the stochastic cahn hilliard equation with the logarithmic free energy by establishing different dimension free harnack inequalities according to various kinds of noises the main characteristics of this equation are the singularities of the logarithmic free energy at 1 and 1 and the conservation of the mass of the solution in its spatial variable both the space time colored noise and the space time white noise are considered for the highly degenerate space time colored noise the asymptotic log harnack inequality is established under the so called essentially elliptic conditions and the harnack inequality with power is established for non degenerate space time white noise

system failure engineering and fuzzy methodology an introductory overview abstract system failure engineering may encompass reliability safety security maintainability risk and quality control in this paper we pay a unified view i e a failure oriented view to system failure engineering the notion of failure can be represented in terms of fuzzy sets and widely interpreted we then discuss various application aspects of fuzzy methodology in system failure engineering which include fuzzy methodology in fault diagnosis in probist systems in structural reliability in software reliability in human reliability in safety engineering in security engineering in risk engineering and in quality control as well as profust posbist and posfust reliability theories engineering case studies and mathematical problems raised by applications of fuzzy methodology in system failure engineering are also addressed in this paper

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

regularity of the obstacle problem for the parabolic biharmonic equation we consider the obstacle problem for the parabolic biharmonic equation we study the problem via an implicit time discretization we prove the existence of a unique solution and discuss its regularity property

asymptotic limits and optimal control for the cahn hilliard system with convection and dynamic boundary conditions abstract in this paper we study initial boundary value problems for the cahn hilliard system with convection and nonconvex potential where dynamic boundary conditions are assumed for both the associated order parameter and the corresponding chemical potential while recent works addressed the case of viscous cahn hilliard systems the pure nonviscous case is investigated here in its first part the paper deals with the asymptotic behavior of the solutions as time approaches infinity it is shown that the limit of any trajectory can be characterized in terms of stationary solutions provided the initial data are sufficiently smooth the second part of the paper deals with the optimal control of the system by the fluid velocity results concerning existence and first order necessary optimality conditions are proved here we have to restrict ourselves to the case of everywhere defined smooth potentials in both parts of the paper we start from corresponding known results for the viscous case derive sufficiently strong estimates that are uniform with respect to the positive viscosity parameter and then let the viscosity tend to zero to establish the sought results for the nonviscous case

unique expansions of real numbers it was discovered some years ago that there exist non integer real numbers q 1 for which only one sequence ci of integers ci 0 q satisfies the equality i 1 ciq i 1 the set of such univoque numbers has a rich topological structure and its study revealed a number of unexpected connections with measure theory fractals ergodic theory and diophantine approximation in this paper we consider for each fixed q 1 the set uq of real numbers x having a unique representation of the form i 1 ciq i x with integers ci belonging to 0 q we carry out a detailed topological study of these sets for instance we characterize their closures and we determine those bases q for which uq is closed or even a cantor set we also study the set uq consisting of all sequences ci of integers ci 0 q such that i 1 ciq i uq we determine the numbers r 1 for which the map q uq defined on 1 is constant in a neighborhood of r and the numbers q 1 for which uq is a subshift or a subshift of finite type

some computations on the spectra of pisot and salem numbers properties of pisot numbers have long been of interest one line of questioning initiated by erdos joo and komornik in 1990 is the determination of l q for pisot numbers q where l q inf y y e0 e1q1 enqn ei 1 0 y 0 although the quantity l q is known for some pisot numbers q there has been no general method for computing l q this paper gives such an algorithm with this algorithm some properties of l q and its generalizations are investigated a related question concerns the analogy of l q denoted a q where the coefficients are restricted to 1 in particular for which non pisot numbers is a q nonzero this paper finds an infinite class of salem numbers where a q 0

a d conversion with imperfect quantizers this paper analyzes mathematically the effect of quantizer threshold imperfection commonly encountered in the circuit implementation of analog to digital a d converters such as pulse code modulation pcm and sigma delta sigmadelta modulation sigmadelta modulation which is based on coarse quantization of oversampled redundant samples of a signal enjoys a type of self correction property for quantizer threshold errors bias that is not shared by pcm although classical sigmadelta modulation is inferior to pcm in the rate distortion sense this robustness feature is believed to be one of the reasons why sigmadelta modulation is preferred over pcm in a d converters with imperfect quantizers motivated by these facts other encoders are constructed in this paper that use redundancy to obtain a similar self correction property but that achieve higher order accuracy relative to bit rate compared to classical sigmadelta more precisely two different types of encoders are introduced that exhibit exponential accuracy in the bit rate in contrast to the polynomial type accuracy of classical sigmadelta while possessing the self correction property

efficient global reliability analysis for nonlinear implicit performance functions many engineering applications are characterized by implicit response functions that are expensive to evaluate and sometimes nonlinear in their behavior making reliability analysis difficult this paper develops an efficient reliability analysis method that accurately characterizes the limit state throughout the random variable space the method begins with a gaussian process model built from a very small number of samples and then adaptively chooses where to generate subsequent samples to ensure that the model is accurate in the vicinity of the limit state the resulting gaussian process model is then sampled using multimodal adaptive importance sampling to calculate the probability of exceeding or failing to exceed the response level of interest by locating multiple points on or near the limit state more complex and nonlinear limit states can be modeled leading to more accurate probability integration by concentrating the samples in the area where accuracy is important i e in the vicinity of the limit state only a small number of true function evaluations are required to build a quality surrogate model the resulting method is both accurate for any arbitrarily shaped limit state and computationally efficient even for expensive response functions this new method is applied to a collection of example problems including one that analyzes the reliability of a microelectromechanical system device that current available methods have difficulty solving either accurately or efficiently

convergence to equilibrium for the cahn hilliard equation with dynamic boundary conditions this paper is concerned with the asymptotic behavior of solution to the cahn hilliard equation 0 1 u t u u u3 x t r subject to the following dynamic boundary conditions 0 2 s u u hs gsu 1 sut t 0 x 0 t 0 x and the initial condition 0 3 u t 0 u0 x x where is a bounded domain in rn n 3 with smooth boundary and s 0 s 0 gs 0 hs are given constants is the tangential laplacian operator and is the outward normal direction to the boundary this problem has been considered in the recent paper by racke and zheng adv differential equations 8 1 2003 83 where the global existence and uniqueness were proved in a very recent manuscript by pruss racke and zheng konstanzer schrift math inform 189 2003 the results on existence of global attractor and maximal regularity of solution have been obtained in this paper convergence of solution of this problem to an equilibrium as time goes to infinity is proved

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

quasi periodic β expansions and cut languages abstract motivated by the analysis of neural net models between integer and rational weights we introduce a so called cut language over a real digit alphabet which contains finite expansions i e base representations of the numbers less than a given threshold we say that an infinite expansion is eventually quasi periodic if its tail sequence formed by the numbers whose representations are obtained by removing leading digits contains an infinite constant subsequence we prove that a cut language is regular iff its threshold is a quasi periodic number whose all expansions are eventually quasi periodic by showing that altogether they have a finite number of tail values for algebraic bases we prove that there is an eventually quasi periodic expansion with an infinite number of tail values iff there is a conjugate of on the unit circle for transcendental combined with algebraic digits a expansion is eventually quasi periodic iff it has a finite number of tail values for a pisot base and digits from the smallest field extension q over rational numbers including we show that any number from q is quasi periodic in addition we achieve a dichotomy that a cut language is either regular or non context free and we show that any cut language with rational parameters is context sensitive

the nonlocal cahn hilliard equation with singular potential well posedness regularity and strict separation property abstract we consider the nonlocal cahn hilliard equation with singular potential and constant mobility well posedness and regularity of weak solutions are studied then we establish the validity of the strict separation property in dimension two further regularity results as well as the existence of regular finite dimensional attractors and the convergence of a weak solution to a single equilibrium are also provided finally regularity results and the strict separation property are also proven for the two dimensional nonlocal cahn hilliard navier stokes system with singular potential

bayesian sparse polynomial chaos expansion for global sensitivity analysis polynomial chaos expansions are frequently used by engineers and modellers for uncertainty and sensitivity analyses of computer models they allow representing the input output relations of computer models usually only a few terms are really relevant in such a representation it is a challenge to infer the best sparse polynomial chaos expansion of a given model input output data set in the present article sparse polynomial chaos expansions are investigated for global sensitivity analysis of computer model responses a new bayesian approach is proposed to perform this task based on the kashyap information criterion for model selection the efficiency of the proposed algorithm is assessed on several benchmarks before applying the algorithm to identify the most relevant inputs of a double diffusive convection model

conservative stochastic cahn hilliard equation with reflection we consider a stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average the equation is driven by the derivative in space of a space time white noise and contains a double laplacian in the drift due to the lack of the maximum principle for the double laplacian the standard techniques based on the penalization method do not yield existence of a solution we propose a method based on infinite dimensional integration by parts formulae obtaining existence and uniqueness of a strong solution for all continuous nonnegative initial conditions and detailed information on the associated invariant measure and dirichlet form

fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context abstract conventional reliability theory is based on the binary state assumption and the probability assumption namely 1 the systems are precisely defined as functioning or failed 2 the system behavior is fully characterized in the context of probability measures however the probability assumption is no longer reasonable in the case of small size of samples since the term probability stems from the sample generality in this way possibility measures should be used to characterize the sample particularity in this paper the binary state assumption is reserved and the possibility assumption is taken in place of the probability assumption thus a concept of fuzzy reliability i e posbist reliability is introduced and a theory of fuzzy reliability i e posbist reliability theory is established various conclusions which are contrary to statements in conventional reliability theory i e probist reliability theory are then drawn

profust reliability of a gracefully degradable system abstract research in traditional reliability theory is based mainly on probist reliability which uses a binary state assumption and classical reliability distributions in the present paper the binary state assumption has been replaced by a fuzzy state assumption thereby leading to profust reliability estimates of a powerloom plant which is modelled as a two unit gracefully degradable system results of bowles and palaez application of fuzzy logic to reliability engineering proc ieee 83 3 1995 435 449 have been deduced as a particular case of results presented here it is also recognized that estimation of system parameters such as failure rates is vital in reliability estimation available methods for such estimation do not cover the underlying uncertainty in the failure data collection involving human judgment evaluation and decision in this paper we introduce a new approach based on fuzzy set theory to estimate such system parameters

differentiability of a two parameter family of self affine functions abstract this paper highlights an unexpected connection between expansions of real numbers to noninteger bases so called expansions and the infinite derivatives of a class of self affine functions precisely we extend okamoto s function itself a generalization of the well known functions of perkins and katsuura to a two parameter family f n a n n 1 n 1 a 1 we first show that for each x f n a x is either 0 or undefined we then extend okamoto s theorem by proving that for each n depending on the value of a relative to a pair of thresholds the set x f n a x 0 is either empty uncountable but lebesgue null or of full lebesgue measure we compute its hausdorff dimension in the second case the second result is a characterization of the set d a x f n a x which enables us to closely relate this set to the set of points which have a unique expansion in the typically noninteger base 1 a recent advances in the theory of expansions are then used to determine the cardinality and hausdorff dimension of d a which depend qualitatively on the value of a relative to a second pair of thresholds

the conserved penrose fife system with temperature dependent memory abstract a nonlinear parabolic system of penrose fife type with a singular evolution term arising from modelling dynamic phenomena of the nonisothermal diffusive phase separation is studied here we consider the evolution of a material in which the heat flux is a superposition of two different contributions one part is proportional to the spacial gradient of the inverse of the absolute temperature while the other agrees with the gurtin pipkin law introduced in the theory of materials with thermal memory the phase transition here is described through the evolution of the conserved order parameter which may represent the density or concentration of some substance it is shown that an initial boundary value problem for the resulting state equations has a unique solution

time dependent failure possibility analysis under consideration of fuzzy uncertainty abstract the problem which is actually being addressed in this paper includes two parts one is establishing the time dependent failure possibility tdfp model to measure the safety degree for structure under the fuzzy uncertainty over a given time intervals the other is presenting two methods to estimate the tdfp involving the double loop nested optimization method dlom and the efficient single loop optimization method slom for measuring the safety degree of the structure under the fuzzy uncertainty in the given time interval the tdfp defined as the possibility of the performance less than zero under the fuzzy uncertainty in the given time interval is proposed based on the possibility theory of the safety measure for evaluating the proposed tdfp the dlom is established firstly by the definition of tdfp but the computational cost of the dlom is too high to solve the problems of the practical engineering thus the slom which transforms the estimation of the tdfp to the time independent failure possibility by the extreme value is established to save the computational cost the equivalency between the dlom and the slom is strictly proved the comparison between tdfp and the time dependent failure probability is completely in the theory and implementation to show the advantages of the proposed tdfp after the detailed implementation of the proposed tdfp is given several examples are used to verify the rationality of the established tdfp and the efficiency of the slom