numerical computations of coarsening in the one dimensional cahn hilliard model of phase separation time dependent solutions of the cahn hilliard equation are studied numerically in particular heteroclinic orbits which connect different equilibrium solutions at t and t are sought thus boundary value problems in space time are computed this computation requires an investigation of the stability of equilibria since projections onto the stable and unstable manifolds determine the boundary conditions at t and t this stability analysis is then followed by solution of the appropriate boundary value problem in space time the results obtained cannot be found by standard initial value simulations by specifying the two steady states at t appropriately it is possible to find orbits reflecting a given degree of coarsening over the time evolution this gives a clear picture of the dynamic coarsening admissible in the equation it also provides an understanding of orbits on the global attractor for the equation

computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

a double bubble assembly as a new phase of a ternary inhibitory system a ternary inhibitory system is a three component system characterized by two properties growth and inhibition a deviation from homogeneity has a strong positive feedback on its further increase in the meantime a longer ranging confinement mechanism prevents unlimited spreading together they lead to a locally self enhancing and self organizing process the model considered here is a planar nonlocal geometric problem derived from the triblock copolymer theory an assembly of perturbed double bubbles is mathematically constructed as a stable stationary point of the free energy functional triple junction a phenomenon in which the three components meet at a single point is a key issue addressed in the construction coarsening an undesirable scenario of excessive growth is prevented by a lower bound on the long range interaction term in the free energy the proof involves several ideas perturbation of double bubbles in a restricted class use of internal variables to remove nonlinear constraints local minimization in a restricted class formulated as a nonlinear problem on a hilbert space and reduction to finite dimensional minimization this existence theorem predicts a new morphological phase of a double bubble assembly

on stable nonconstant stationary solutions and mesoscopic patterns for fitzhugh nagumo equations in higher dimensions abstract fitzhugh nagumo equation has been studied extensively in the field of mathematical biology it has the mechanism of lateral inhibition which seems to play a big role in the pattern formation of plankton distribution we consider fitzhugh nagumo equation in high dimension and show the existence of stable nonconstant stationary solutions which have fine structures on a mesoscopic scale we construct spatially periodic stationary solutions moreover we compute the singular limit energy which suggests that the transition from planar structure to droplet pattern can occur when parameters change

fractals of forbidden words and approximating their box dimensions abstract this paper provides the construction of the fractals defined by more than two forbidden words some properties of the table structure in the hao s frame representation are emphasized because they are the key to drawing the fractals the recurrence formula are not only used in the enumeration but also applied to derive the characteristic equations furthermore a more efficient method of approximating their box dimensions is presented

viscous cahn hilliard equation ii analysis the viscous cahn hilliard equation may be viewed as a singular limit of the phase field equations for phase transitions it contains both the allen cahn and cahn hilliard models of phase separation as particular cases by specific choices of parameters it may be formulated as a one parameter say homotopy connecting the cahn hilliard 0 and allen cahn 1 models the limit 0 is singular in the sense that the smoothing property of the analytic semigroup changes from being of the type associated with second order operators to the type associated with fourth order operators the properties of the gradient dynamical system generated by the viscous cahn hilliard equation are studied as varies in 0 1 continuity of the phase portraits near equilibria is established independently of 0 1 and using this a piecewise uniform in time perturbation result is proved for trajectories finally the continuity of the attractor is established and in one dimension the existence and continuity of inertial manifolds shown and the flow on the attractor detailed

on the first and second variations of a nonlocal isoperimetric problem we consider a nonlocal perturbation of an isoperimetric variational prob lem the problem may be viewed as a mathematical paradigm for the ubiquitous phenom enon of energy driven pattern formation associated with competing short and long range interactions in particular it arises as a g limit of a model for microphase separation of diblock copolymers in this article we establish precise conditions for criticality and stability i e we explicitly compute the first and second variations we also present some applications

multifractal properties of hao s geometric representations of dna sequences abstract hao proposed a graphic representation of subsequence structure in dna sequences and computed fractal dimensions of such representations for factorizable languages in this study we extend hao s work in several directions 1 we generalize hao s scheme to accommodate sequences over an arbitrary finite number of symbols 2 we establish a direct correspondence between the statistical characterization of symbolic sequences via renyi entropy spectra and the multifractal characteristics renyi generalized dimensions of the sequences spatial representations 3 we show that for general symbolic dynamical systems the multifractal fh spectra in the sequence space endowed with commonly used metrics coincide with the fh spectra on hao s sequence representations 4 so far the connection between the hao s scheme and another well known subsequence visualization scheme jeffrey s chaos game representation cgr has been characterized only in very vague terms we show that the fractal dimension results for hao s visualization frames directly translate to jeffrey s cgr scheme

spectral analysis of planar transition fronts for the cahn hilliard equation abstract we consider the spectrum of the linear operator that arises upon linearization of the cahn hilliard equation in dimensions d 2 about a planar transition front a solution that depends on only one distinguished space variable and that has different values at in previous work the author has established conditions on this spectrum under which such planar transition fronts are asymptotically stable and we verify here that those conditions hold for all such waves arising in a general form of the cahn hilliard equation

variational bayesian inference for a nonlinear forward model variational bayes vb has been proposed as a method to facilitate calculations of the posterior distributions for linear models by providing a fast method for bayesian inference by estimating the parameters of a factorized approximation to the posterior distribution here a vb method for nonlinear forward models with gaussian additive noise is presented in the case of noninformative priors the parameter estimates obtained from this vb approach are identical to those found via nonlinear least squares however the advantage of the vb method lies in its bayesian formulation which permits prior information to be included in a hierarchical structure and measures of uncertainty for all parameter estimates to be obtained via the posterior distribution unlike other bayesian methods vb is only approximate in comparison with the sampling method of mcmc however the vb method is found to be comparable and the assumptions made about the form of the posterior distribution reasonable practically the vb approach is substantially faster than mcmc as fewer calculations are required some of the advantages of the fully bayesian nature of the method are demonstrated through the extension of the noise model and the inclusion of automatic relevance determination ard within the vb algorithm

combinatorial fiber tracking of the human brain this paper presents a novel fiber tracking algorithm termed combinatorial tracking which uses stochastic process modeling and global optimization algorithm for tractography combinatorial tracking is a probabilistic tracking algorithm that transforms the brain s white matter into a grid in which each voxel has 26 weighted connections with adjacent voxels we model the random walk on this graph using a markov chain model and suggest two approaches for fiber reconstruction in the first approach we find the most probable paths between two voxels with prior connectivity knowledge using a shortest path algorithm in the second approach the all pairs mean first passage time mfpt matrix m or hitting time as referred to in the spectral graph theory literature is calculated analytically we suggest that m can be interpreted as a global connectivity matrix and use it for fiber reconstruction we also introduce a simulation framework that can be used to calculate specific elements of the matrix m and show how it can be employed to select the target of a fiber in a high resolution diffusion tensor imaging dti dataset because any source and any target voxel can be connected combinatorial tracking permits true connectivity analysis overcoming the limitations of conventional tracking especially stopping criteria e g low fa we applied combinatorial tracking to a standard dti dataset and demonstrated the reconstruction of the cortico thalamic pathway the pyramidal decussation and the medial cerebellar peduncle fibers while the dti ellipsoid served as input for the algorithms any diffusion imaging based orientation density function odf can be used this framework can potentially turn diffusion imaging tractography into a true connectivity measure

simultaneous multi scale diffusion estimation and tractography guided by entropy spectrum pathways we have developed a method for the simultaneous estimation of local diffusion and the global fiber tracts based upon the information entropy flow that computes the maximum entropy trajectories between locations and depends upon the global structure of the multi dimensional and multi modal diffusion field computation of the entropy spectrum pathways requires only solving a simple eigenvector problem for the probability distribution for which efficient numerical routines exist and a straight forward integration of the probability conservation through ray tracing of the convective modes guided by a global structure of the entropy spectrum coupled with a small scale local diffusion the intervoxel diffusion is sampled by multi b shell multi q angle diffusion weighted imaging data expanded in spherical waves this novel approach to fiber tracking incorporates global information about multiple fiber crossings in every individual voxel and ranks it in the most scientifically rigorous way this method has potential significance for a wide range of applications including studies of brain connectivity

linear stability of steady states for thin film and cahn hilliard type equations we study the linear stability of smooth steady states of the evolution equation h t f h xxx x g h h x x ah under both periodic and neumann boundary conditions if a 0 we assume f 1 in particular we consider positive periodic steady states of thin film equations where a 0 and f g might have degeneracies such as f 0 0 as well as singularities like g 0

some mathematical aspects of the micro phase separation in diblock copolymers abstract a free energy functional of nonlocal type is considered that was originally introduced to describe the micro phase separation of diblock copolymer a mathematical framework is given to the issues of the scaling law for stationary states and the governing equation of morphology the scaling law is equivalent to finding a nice rescaling in order to have a well defined limiting interfacial problem which is free from the interfacial thickness and the total chain length of copolymer and the associated stationary problem becomes a morphology equation that governs the configuration of final patterns although the steady patterns become finer and finer in our scaling limit due to the mesoscopic nature a possible rigorous approach is presented to stability and morphological selection problems for them

a double bubble in a ternary system with inhibitory long range interaction the free energy of a ternary system with a self organization property includes an interface energy and a longer ranging inhibitory interaction energy in a planar domain if the two energies are properly balanced and two of the three constituents make up an equal but small fraction the free energy admits a local minimizer that is shaped like a perturbed double bubble most difficulties in the proof of this result are related to the triple junction phenomenon that the three constituents of the ternary system meet at a point two techniques are developed to deal with the triple junction first one defines restricted classes of perturbed double bubbles each perturbed double bubble in a restricted class is obtained from a standard double bubble by a special perturbation the two triple junction points of the standard double bubble can only move along the line connecting them in opposite directions and by the same distance the second technique is the use of the so called internal variables these variables derive from the more geometric quantities that describe perturbed double bubbles in restricted classes the advantage of the internal variables is that they are only subject to linear constraints and perturbed double bubbles in a restricted class represented by internal variables are elements of a hilbert space a local minimizer of the free energy in each restricted class is found as a fixed point of a nonlinear equation by a contraction mapping argument the second variation at the fixed point within its restricted class is positive this perturbed double bubble satisfies three of the four equations for critical points of the free energy the unsolved equation is the 120 degree angle condition at triple junction points then perform another minimization among the local minimizers from all restricted classes a minimum of minimizers emerges and solves all the equations for critical points

uniform energy distribution for an isoperimetric problem with long range interactions we study minimizers of a nonlocal variational problem the problem is a mathematical paradigm for the ubiquitous phenomenon of energy driven pattern formation induced by competing short and long range interactions the short range interaction is attractive and comes from an interfacial energy and the long range interaction is repulsive and comes from a nonlocal energy contribution in particular the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers a natural conjecture is that in all space dimensions minimizers are essentially periodic on an intrinsic scale however proving any periodicity result turns out to be a formidable task in dimensions larger than one in this paper we address a weaker statement concerning the distribution of energy for minimizers we prove in any space dimension that each component of the energy interfacial and nonlocal of any minimizer is uniformly distributed on cubes which are suciently large with respect to the intrinsic length scale moreover we also prove an l 1 bound on the optimal potential associated with the long range interactions this bound allows for an interesting interpretation note that the average volume fraction of the optimal pattern in a subsystem of size r fluctuates around the system average m the bound on the potential yields a rate of decay of these fluctuations as r tends to 1 this rate of decay is stronger than the one for a random checkerboard pattern in this sense the optimal pattern has less large scale variations of the average volume fraction than a pattern with a finite correlation

dti segmentation via the combined analysis of connectivity maps and tensor distances we describe a novel approach to extract the neural tracts of interest from a diffusion tensor image dti compared to standard streamline tractography existing probabilistic methods are able to capture fiber paths that deviate from the main tensor diffusion directions at the same time tensor clustering methods are able to more precisely delimit the border of the bundle to the best of our knowledge we propose the first algorithm which combines the advantages supplied by probabilistic and tensor clustering approaches the algorithm includes a post processing step to limit partial volume related segmentation errors we extensively test the accuracy of our algorithm on different configurations of a dti software phantom for which we systematically vary the image noise the number of gradients the geometry of the fiber paths and the angle between adjacent and crossing fiber bundles the reproducibility of the algorithm is supported by the segmentation of the corticospinal tract of nine patients additional segmentations of the corticospinal tract the arcuate fasciculus and the optic radiations are in accordance with anatomical knowledge the required user interaction is comparable to that of streamline tractography which allows for an uncomplicated integration of the algorithm into the clinical routine

energy levels of steady states for thin film type equations abstract we study the phase space of the evolution equationht f h hxxx x g h hx x by means of a dissipated energy here h x t 0 and at h 0 the coefficient functions f 0 and g can either degenerate to 0 or blow up to or tend to a nonzero constant we first show that all positive periodic steady states are saddles in the energy landscape with respect to zero mean perturbations if g f 0 or if the perturbations are allowed to have period longer than that of the steady state for power law coefficients f y yn and g y b ym for some b 0 we analytically determine the relative energy levels of distinct steady states for example with m n 1 2 and for suitable choices of the period and mean value we find three fundamentally different steady states the first is a constant steady state that is stable and is a local minimum of the energy the second is a positive periodic steady state that is linearly unstable and has higher energy than the constant steady state it is a saddle point for the energy the third is a periodic collection of droplet compactly supported steady states having lower energy than either the positive steady state or the constant one since the energy must decrease along every orbit these results significantly constrain the dynamics of the evolution equation

slow dynamics for the cahn hilliard equation in higher space dimensions the motion of bubbles it is known that the van der waals cahn hilliard w c h dynamics can be approximated by a quasi static stefan problem with surface tension it turns out that the stefan problem has a manifold of equilibria equal in dimension to that of the domain any sphere of fixed radius with interface contained in the domain is an equilibrium indistinguishable from the point of view of the perimeter functional we resolve this degeneracy by showing that at the w c h level this manifold is replaced by a quasi invariant stable manifold on which the typical solution preserves its bubble like shape until it reaches the boundary moreover we show that the bubble moves superslowly we also obtain an equation that determines those special spheres that correspond to equilibria at the w c h level our work establishes the phenomenon of superslow motion in higher space dimensions in the class of single interface solutions

spectral comparison principles for the cahn hilliard and phase field equations and time scales for coarsening we initiate a study of the instability properties of steady solutions of the cahn hilliard ch equation and the phase field pf system in terms of the coarseness of those solutions other spectral results are obtained as well steady state solutions for the ch equation with no flux or periodic boundary conditions are also stationary states for the bistable reaction diffusion equation and the pf system with constant temperature thus it is meaningful to linearize each of these equations about a common stationary solution and discuss the stability of this state in each dynamical process converting the linearized operators to self adjoint form and expressing eigenvalues in terms of modified rayleigh quotients enables us to relate the spectra of the three operators in particular we relate the indices of instability with respect to the three processes we also discuss the change in the spectrum of the linearized ch operator in one dimension when we change the stationary solution at which it is linearized we conjecture that the eigenvalues increase with the number of oscillations of the steady state for a given mass and prove that this is so in certain cases this lends credence to the idea that motion of these dynamical systems near coarse solutions takes place on longer time scales

plausibility tracking a method to evaluate anatomical connectivity and microstructural properties along fiber pathways abstract diffusion mri is a non invasive method that potentially gives insight into the brain s white matter structure regarding the pathway of connections and properties of the axons here we propose a novel global tractography method named plausibility tracking that provides the most plausible pathway modeled as a smooth spline curve between two locations in the brain compared to other tractography methods plausibility tracking combines the more complete connectivity pattern of probabilistic tractography with smooth tracks that are globally optimized using the fiber orientation density function and hence is relatively robust against local noise and error propagation it has been tested on phantom and biological data and compared to other methods of tractography plausibility tracking provides reliable local directions all along the fiber pathways which makes it especially interesting for tract based analysis in combination with direction dependent indices of diffusion mri in order to demonstrate this potential of plausibility tracking we propose a framework for the assessment and comparison of diffusion derived tissue properties this framework comprises atlas guided parameterization of tract representation and advanced bundle specific indices describing fiber density fiber spread and white matter complexity we explore the new method using real data and show that it allows for a more specific interpretation of the white matter s microstructure compared to rotationally invariant indices derived from the diffusion tensor

the dynamics of nucleation for the cahn hilliard equation when a constant metastable solution of the cahn hilliard equation is subjected to a spatially localized large amplitude perturbation a transition process may be triggered leading to a globally stable stationary solution in one space dimension the existence and instability of a third stationary solution with the same mass is proved a spike like solution called a canonical nucleus within the class of solutions which are even with respect to the center of the spike it has a one dimensional unstable manifold in addition the process of nucleation by formal arguments using two space scales and two timescales is described the last stage in the process can be approximated by a nonlinear stefan free boundary problem

probabilistic diffusion tractography with multiple fibre orientations what can we gain we present a direct extension of probabilistic diffusion tractography to the case of multiple fibre orientations using automatic relevance determination we are able to perform online selection of the number of fibre orientations supported by the data at each voxel simplifying the problem of tracking in a multi orientation field we then apply the identical probabilistic algorithm to tractography in the multi and single fibre cases in a number of example systems which have previously been tracked successfully or unsuccessfully with single fibre tractography we show that multi fibre tractography offers significant advantages in sensitivity when tracking non dominant fibre populations but does not dramatically change tractography results for the dominant pathways

a generalized phase field system the existence the estimate and the uniqueness of a solution to a general phase field system motivated by caginalp s model describing the phase changes are established the paper extends the results for the already studied types of nonlinearities related to the model

multiple subjects connectivity based parcellation using hierarchical dirichlet process mixture models we propose a hierarchical infinite mixture model approach to address two issues in connectivity based parcellations i choosing the number of clusters and ii combining data from different subjects in a bayesian setting we model voxel wise anatomical connectivity profiles as an infinite mixture of multivariate gaussian distributions with a dirichlet process prior on the cluster parameters this type of prior allows us to conveniently model the number of clusters and estimate its posterior distribution directly from the data an important benefit of using bayesian modelling is the extension to multiple subjects clustering via a hierarchical mixture of dirichlet processes data from different subjects are used to infer on class parameters and the number of classes at individual and group level such a method accounts for inter subject variability while still benefiting from combining different subjects data to yield more robust estimates of the individual clusterings

the eckhaus and zigzag instability criteria in gradient skew gradient dissipative systems abstract the concept of gradient skew gradient structure an extension of free energy is introduced in dissipative systems with this structure the eckhaus and zigzag instability criteria can be represented in terms of first integral and free energy per unit length for steady states moreover an exact relation between these criteria is presented these results are valid for steady states of arbitrary profile and amplitude which implies that the instability criteria are valid for steady states far from threshold furthermore our results support the marginal stability hypothesis for roll solutions in the wavenumber selection problem in gradient systems the derivation for the results is simple and rigorous

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

lyapunov function and spectrum comparison for a reaction diffusion system with mass conservation abstract we are dealing with a two component system of reaction diffusion equations with mass conservation in a bounded domain with the neumann boundary conditions we prove the global boundedness of the solution in l norm for t 0 under a condition and then the existence of a lyapunov function moreover by studying the linearized eigenvalue problem of a nonconstant equilibrium solution we provide a comparison theorem for the spectrum between the linearized operators of the system and an appropriate nonlocal scalar equation as an application of the comparison result we obtain that any stable equilibrium solution must be monotone if the space dimension is one it is also shown that a modified system with a new parameter which covers the present model possesses a lyapunov function

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

thermodynamically consistent models of phase field type for the kinetics of phase transitions abstract a general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space instead of a ginzburg landau free energy functional as used in formulating the cahn hilliard equation we use the analogous entropy functional model entropy functionals and the kinetic equations resulting from them are constructed for various cases phase transitions with and without a critical point and in the former case with or without a latent heat the class considered is general enough to include the entropy functionals for the ising model in mean field approximation the van der waals fluid and a simplified version of the density functional theory of freezing a case without critical point for which the energy is conserved but the order parameter is not provides a thermodynamically consistent derivation of the phase field equations studied by caginalp fix and others and also leads in a natural way to the lyapunov functional given by langer for these equations but the treatment also suggests that a modified version of the phase field equations might provide a more realistic model of freezing

metastable bubble solutions for the allen cahn equation with mass conservation in a multidimensional domain the slow motion behavior of internal layer solutions with spherical interfaces referred to as bubble solutions is analyzed for the nonlocal allen cahn equation with mass conservation this problem represents the simplest model for the phase separation of a binary mixture in the presence of a mass constraint the bubble is shown to drift exponentially slowly across the domain without change of shape toward the closest point on the boundary of the domain an explicit ordinary differential equation for the motion of the center of the bubble is derived by extending to a multidimensional setting the asymptotic projection method developed previously by the author to treat metastable problems in one spatial dimension an asymptotic formula for the time of collapse of the bubble against the boundary of the domain is derived in terms of the principal radii of curvature of the boundary at the initial contact point an analogy between slow bubble motion and the classical exit prob

morse index of layered solutions to the heterogeneous allen cahn equation abstract let u be a single layered radially symmetric unstable solution of the allen cahn equation 2 u u u a x 1 u over the unit ball with neumann boundary conditions we estimate the small eigenvalues of the linearized eigenvalue problem at u when is small as a consequence we prove that the morse index of u is asymptotically given by o 1 n 1 2 with a certain positive constant expressed in terms of parameters determined by the allen cahn equation our estimates on the small eigenvalues have many other applications for example they may be used in the search of other non radially symmetric solutions which will be considered in forthcoming papers

bootgraph probabilistic fiber tractography using bootstrap algorithms and graph theory bootstrap methods have recently been introduced to diffusion weighted magnetic resonance imaging to estimate the measurement uncertainty of ensuing diffusion parameters directly from the acquired data without the necessity to assume a noise model these methods have been previously combined with deterministic streamline tractography algorithms to allow for the assessment of connection probabilities in the human brain thereby the local noise induced disturbance in the diffusion data is accumulated additively due to the incremental progression of streamline tractography algorithms graph based approaches have been proposed to overcome this drawback of streamline techniques for this reason the bootstrap method is in the present work incorporated into a graph setup to derive a new probabilistic fiber tractography method called bootgraph the acquired data set is thereby converted into a weighted undirected graph by defining a vertex in each voxel and edges between adjacent vertices by means of the cone of uncertainty which is derived using the wild bootstrap a weight is thereafter assigned to each edge two path finding algorithms are subsequently applied to derive connection probabilities while the first algorithm is based on the shortest path approach the second algorithm takes all existing paths between two vertices into consideration tracking results are compared to an established algorithm based on the bootstrap method in combination with streamline fiber tractography and to another graph based algorithm the bootgraph shows a very good performance in crossing situations with respect to false negatives and permits incorporating additional constraints such as a curvature threshold by inheriting the advantages of the bootstrap method and graph theory the bootgraph method provides a computationally efficient and flexible probabilistic tractography setup to compute connection probability maps and virtual fiber pathways without the drawbacks of streamline tractography algorithms or the assumption of a noise distribution moreover the bootgraph can be applied to common dti data sets without further modifications and shows a high repeatability thus it is very well suited for longitudinal studies and meta studies based on dti

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

comparing dynamic causal models this article describes the use of bayes factors for comparing dynamic causal models dcms dcms are used to make inferences about effective connectivity from functional magnetic resonance imaging fmri data these inferences however are contingent upon assumptions about model structure that is the connectivity pattern between the regions included in the model given the current lack of detailed knowledge on anatomical connectivity in the human brain there are often considerable degrees of freedom when defining the connectional structure of dcms in addition many plausible scientific hypotheses may exist about which connections are changed by experimental manipulation and a formal procedure for directly comparing these competing hypotheses is highly desirable in this article we show how bayes factors can be used to guide choices about model structure both concerning the intrinsic connectivity pattern and the contextual modulation of individual connections the combined use of bayes factors and dcm thus allows one to evaluate competing scientific theories about the architecture of large scale neural networks and the neuronal interactions that mediate perception and cognition

the computational modelling of branching fine structures in constrained crystals we introduce a finite element method which is piecewise continuous on the microscopic scale of the spatial resolution h but discontinuous on the mesoscopic scale h 0 1 the method is designed to capture the morphology of needle twin structures frequently found in ferric and pseudo elastic crystals namely in uniaxial ferromagnets and au milieu of the austenite martensite interfaces the approach is based on a domain decomposition method that interpolates between the scale on order of the size of crystal and the microscopic scale of finite element approximation h the scale interpolation is enabled by incorporating frequency adaptivity the visualization and analysis of the computational results presented disclose microstructures corresponding to complex scaling laws we document that the fine structures obtained by the presented method are not visible using classical formulation of the underlying variational problem and using conforming approximation of admissible sets the proposed method is suitable for non smooth relaxation and optimization when the minimizers lack the often required c1 regularity and when they exhibit fractal behavior

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

joint reconstruction of white matter pathways from longitudinal diffusion mri data with anatomical priors we consider the problem of reconstructing white matter pathways in a longitudinal study where diffusion weighted and t1 weighted mr images have been acquired at multiple time points for the same subject we propose a method for joint reconstruction of a subject s pathways at all time points given the subject s entire set of longitudinal data we apply a method for unbiased within subject registration to generate a within subject template from the t1 weighted images of the subject at all time points we follow a global probabilistic tractography approach where the unknown pathway is represented in the space of this within subject template and propagated to the native space of the diffusion weighted images at all time points to compute its posterior probability given the images this ensures spatial correspondence of the reconstructed pathway among time points which in turn allows longitudinal changes in diffusion measures to be estimated consistently along the pathway we evaluate the reliability of the proposed method on data from healthy controls scanned twice within a month where no changes in white matter microstructure are expected between scans we evaluate the sensitivity of the method on data from huntington s disease patients scanned repeatedly over the course of several months where changes are expected between scans we show that reconstructing white matter pathways jointly using the data from all time points leads to improved reliability and sensitivity when compared to reconstructing the pathways at each time point independently

dt mri fiber tracking a shortest paths approach we derive a new fiber tracking algorithm for dt mri that parts with the locally ldquogreedyrdquo paradigm intrinsic to conventional tracking algorithms we demonstrate the ability to precisely reconstruct a diverse range of fiber trajectories in authentic and computer generated dt mri data for which well known conventional tracking algorithms are shown to fail our approach is to pose fiber tracking as a problem in computing shortest paths in a weighted digraph voxels serve as vertices and edges are included between neighboring voxels we assign probabilities weights to edges using a bayesian framework higher probabilities are assigned to edges that are aligned with fiber trajectories in their close proximity we compute optimal paths of maximum probability using computationally scalable shortest path algorithms the salient features of our approach are global optimality unlike conventional tracking algorithms local errors do not accumulate and one ldquowrong turnrdquo does not spell disaster a target point is specified a priori precise reconstruction is demonstrated for extremely low signal to noise ratio impartiality to which of two endpoints is used as a seed and faster computation times than conventional all paths tracking we can use our new tracking algorithm in either a single path tracking mode deterministic tracking or an all paths tracking mode probabilistic tracking

stability of spot and ring solutions of the diblock copolymer equation the convergence theory shows that under certain conditions the diblock copolymer equation has spot and ring solutions we determine the asymptotic properties of the critical eigenvalues of these solutions in order to understand their stability in two dimensions a threshold exists for the stability of the spot solution it is stable if the sample size is small and unstable if the sample size is large the stability of the ring solutions is reduced to a family of finite dimensional eigenvalue problems in one study no two interface ring solutions are found by the convergence method if the sample is small a stable two interface ring solution exists if the sample size is increased it becomes unstable if the sample size is increased further

a dti derived measure of cortico cortical connectivity we arm researchers with a simple method to chart a macroscopic cortico cortical connectivity network in living human subjects the researcher provides a diffusion magnetic resonance imaging mri data set and n cortical regions of interest in return we provide an n times n structural adjacency matrix sam quantifying the relative connectivity between all cortical region pairs we also return a connectivity map for each pair to enable visualization of interconnecting fiber bundles the measure of connectivity we devise is 1 free of length bias 2 proportional to fiber bundle cross sectional area and 3 invariant to an exchange of seed and target we construct a 3 d lattice scaffolding graph for white matter by drawing a link between each pair of voxels in a 26 voxel neighborhood for which their two respective principal eigenvectors form a sufficiently small angle the connectivity between a cortical region pair is then measured as the maximum number of link disjoint paths that can be established between them in the white matter graph we devise an efficient edmonds karp like algorithm to compute a conservative bound on the maximum number of link disjoint paths using both simulated and authentic diffusion tensor imaging data we demonstrate that the number of link disjoint paths as a measure of connectivity satisfies properties 1 3 unlike the fraction of intersecting streamlines the measure intrinsic to most existing probabilistic tracking algorithms finally we present connectivity maps of some notoriously difficult to track longitudinal and contralateral fasciculi

approximate bayesian inference with the weighted likelihood bootstrap we introduce the weighted likelihood bootstrap wlb as a way to simulate approximately from a posterior distribution this method is often easy to implement requiring only an algorithm for calculating the maximum likelihood estimator such as iteratively reweighted least squares in the generic weighting scheme the wlb is first order correct under quite general conditions inaccuracies can be removed by using the wlb as a source of samples in the sampling importance resampling sir algorithm which also allows incorporation of particular prior information the sir adjusted wlb can be a competitive alternative to other integration methods in certain models asymptotic expansions elucidate the second order properties of the wlb which is a generalization of rubin s bayesian bootstrap the calculation of approximate bayes factors for model comparison is also considered we note that given a sample simulated from the posterior distribution the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values a modification of this estimator that avoids instability is also noted these methods provide simple ways of calculating approximate bayes factors and posterior model probabilities for a very wide class of models

filtered multitensor tractography we describe a technique that uses tractography to drive the local fiber model estimation existing techniques use independent estimation at each voxel so there is no running knowledge of confidence in the estimated model fit we formulate fiber tracking as recursive estimation at each step of tracing the fiber the current estimate is guided by those previous to do this we perform tractography within a filter framework and use a discrete mixture of gaussian tensors to model the signal starting from a seed point each fiber is traced to its termination using an unscented kalman filter to simultaneously fit the local model to the signal and propagate in the most consistent direction despite the presence of noise and uncertainty this provides a causal estimate of the local structure at each point along the fiber using two and three fiber models we demonstrate in synthetic experiments that this approach significantly improves the angular resolution at crossings and branchings in vivo experiments confirm the ability to trace through regions known to contain such crossing and branching while providing inherent path regularization

connecting and merging fibres pathway extraction by combining probability maps probability mapping of connectivity is a powerful tool to determine the fibre structure of white matter in the brain probability maps are related to the degree of connectivity to a chosen seed area in many applications however it is necessary to isolate a fibre bundle that connects two areas a frequently suggested solution is to select curves which pass only through two or more areas this is very inefficient especially for long distance pathways and small areas in this paper a novel probability based method is presented that is capable of extracting neuronal pathways defined by two seed points a monte carlo simulation based tracking method similar to the probabilistic index of connectivity pico approach was extended to preserve the directional information of the main fibre bundles passing a voxel by combining two of these extended visiting maps arising from different seed points two independent parameters are determined for each voxel the first quantifies the uncertainty that a voxel is connected to both seed points the second represents the directional information and estimates the proportion of fibres running in the direction of the other seed point connecting fibre or face a third area merging fibre both parameters are used to calculate the probability that a voxel is part of the bundle connecting both seed points the performance and limitations of this dti based method are demonstrated using simulations as well as in vivo measurements

spectral analysis for transition front solutions in multidimensional cahn hilliard systems abstract we consider the spectrum associated with the linear operator obtained when a cahn hilliard system on r n is linearized about a planar transition front solution in the case of single cahn hilliard equations on r n it s known that under general physical conditions the leading eigenvalue moves into the negative real half plane at a rate 3 where is the fourier transform variable corresponding with components transverse to the wave moreover it has recently been verified that for single equations this spectral behavior implies nonlinear stability in the current analysis we establish that the same cubic rate law holds for a broad range of multidimensional cahn hilliard systems the analysis of nonlinear stability will be carried out separately

an application of the modular function in nonlocal variational problems using the modular function and its invariance under the action of a modular group and an heuristic reduction of a mathematical model we present a mathematical account of a hexagonal pattern selection observed in di block copolymer melts

minimality via second variation for a nonlocal isoperimetric problem we discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts we show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and in fact satisfy a quantitative isoperimetric inequality with respect to sets that are l 1 close the link with local minimizers for the diffuse interface ohta kawasaki energy is also discussed as a byproduct of the quantitative estimate we get new results concerning periodic local minimizers of the area functional and a proof via second variation of the sharp quantitative isoperimetric inequality in the standard euclidean case as a further application we address the global and local minimality of certain lamellar configurations

ring pattern solutions of a free boundary problem in diblock copolymer morphology abstract the cross section of a diblock copolymer in the cylindrical phase is made up of a large number of microdomains of small discs with high concentration of the minority monomers often several ring like microdomains appear among the discs we show that a ring like structure may exist as a stable solution of a free boundary problem derived from the ohta kawasaki theory of diblock copolymers the existence of such a stable single ring structure explains why rings exist for a long period of time before they eventually disappear or become discs in a diblock copolymer a variant of lyapunov schmidt reduction process is carried out that rigorously reduces the free boundary problem to a finite dimensional problem the finite dimensional problem is solved numerically a stability criterion on the parameters determines whether the ring solution is stable

oval shaped droplet solutions in the saturation process of some pattern formation problems the first stage in the transition from a coarse structure to a fine structure in many pattern formation problems involves the change of a standard geometric object such as a round disc to a less standard geometric object such as an oval shaped set on a generic domain two small oval shaped sets are found as solutions to a nonlocal geometric problem this problem arises as the singular limit of both the ohta kawasaki theory for diblock copolymers and the gierer meinhardt theory for morphogenesis in cell development the two sets have the same center which is a global minimum of the diagonal of the regular part of a green s function this minimum point may be regarded as a kind of center of the domain moreover the second derivatives of the regular part of the green s function define a major axis and a minor axis for the domain one of the oval shaped solutions is stable and aligns itself along the major axis the other oval shaped solution is unstable and is parallel to the minor axis

convex splitting method for the calculation of transition states of energy functional abstract among numerical methods for partial differential equations arising from steepest descent dynamics of energy functionals e g allen cahn and cahn hilliard equations the convex splitting method is well known to maintain unconditional energy stability for a large time step size in this work we show how to use the convex splitting idea to find transition states i e index 1 saddle points of the same energy functionals based on the iterative minimization formulation imf for saddle points 14 we introduce the convex splitting method to minimize the auxiliary functional at each cycle of the imf we present a general principle of constructing convex splitting forms for these auxiliary functionals and show how to avoid solving nonlinear equations the new numerical scheme based on the convex splitting method allows for large time step sizes the new methods are tested for the one dimensional ginzburg landau energy functional in the search of the allen cahn or cahn hilliard types of transition states we provide the numerical results of transition states for the two dimensional landau brazovskii energy functional for diblock copolymers

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

the dirichlet boundary problem for a nonlocal cahn hilliard equation abstract we study the existence uniqueness and continuous dependence on initial data of the solution for a nonlocal cahn hilliard equation with dirichlet boundary condition on a bounded domain under a nondegeneracy assumption the solutions are classical but when this is relaxed the equation is satisfied in a weak sense also we prove that there exists a global attractor in some metric space

the neumann boundary problem for a nonlocal cahn hilliard equation we study the existence uniqueness and continuous dependence on initial data of the solution to a nonlocal cahn hilliard equation on a bounded domain the equation generates a gradient flow for a free energy functional with nonlocal interaction also we apply a nonlinear poincare inequality to show the existence of an absorbing set in each constant mass affine space

metastable patterns in solutions of ut ϵ2uxx f u we consider the equation in question on the interval 0 x 1 having neumann boundary conditions with f u f u where f is a double well energy density with equal minima at u 1 the only stable states of the system are patternless constant solutions but given two phase initial data a pattern of interfacial layers typically forms far out of equilibrium the ensuing nonlinear relaxation process is extremely slow patterns persist for exponentially long times proportional to exp a l where a f 1 and l is the minimum distance between layers physically a tiny potential jump across a layer drives its motion we prove the existence and persistence of these metastable patterns and characterise accurately the equations governing their motion the point of view is reminiscent of center manifold theory a manifold parametrising slowly evolving states is introduced a neighbourhood is shown to be normally attracting and the parallel flow is characterised to high relative accuracy proofs involve a detailed study of the dirichlet problem spectral gap analysis and energy estimates

deterministic and probabilistic tractography based on complex fibre orientation distributions we propose an integral concept for tractography to describe crossing and splitting fibre bundles based on the fibre orientation distribution function odf estimated from high angular resolution diffusion imaging hardi we show that in order to perform accurate probabilistic tractography one needs to use a fibre odf estimation and not the diffusion odf we use a new fibre odf estimation obtained from a sharpening deconvolution transform sdt of the diffusion odf reconstructed from q ball imaging qbi this sdt provides new insight into the relationship between the hardi signal the diffusion odf and the fibre odf we demonstrate that the sdt agrees with classical spherical deconvolution and improves the angular resolution of qbi another important contribution of this paper is the development of new deterministic and new probabilistic tractography algorithms using the full multidirectional information obtained through use of the fibre odf an extensive comparison study is performed on human brain datasets comparing our new deterministic and probabilistic tracking algorithms in complex fibre crossing regions finally as an application of our new probabilistic tracking we quantify the reconstruction of transcallosal fibres intersecting with the corona radiata and the superior longitudinal fasciculus in a group of eight subjects most current diffusion tensor imaging dti based methods neglect these fibres which might lead to incorrect interpretations of brain functions

a nonlocal biharmonic operator and its connection with the classical analogue we consider a singular integral operator as a natural generalization to the biharmonic operator that arises in thin plate theory the operator is built in the nonlocal calculus framework defined in math models methods appl sci 23 03 493 540 2013 and connects with the recent theory of peridynamics this framework enables us to consider non smooth approximations to fourth order elliptic boundary value problems for these systems we introduce nonlocal formulations of the clamped and hinged boundary conditions that are well defined even for irregular domains we demonstrate the existence and uniqueness of solutions to these nonlocal problems and demonstrate their l 2 strong convergence to functions in w 2 2 as the nonlocal interaction horizon goes to zero for regular domains we identify these limits as the weak solutions of the corresponding classical elliptic boundary value problems as a part of our proof we also establish that the nonlocal laplacian of a smooth function is lipschitz continuous

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

the soliton stripe pattern in the seul andelman membrane abstract the seul andelman membrane is a system of two coupled fields the composition of one of the two a and b constitutive molecules and the height profile h of the flexible membrane the free energy of the system consists of two parts the first part is the usual ginzburg landau free energy of the second part is attributed to the bending of the membrane and the coupling of to h the coupling term models the tendency that the two molecular constituents display an affinity for regions of the membrane of different local curvature in a particular parameter range we prove the existence of the soliton stripe pattern using the limit theory in perturbative variational calculus this pattern modeled by one dimensional local minimizers of the free energy of the system consists of a rich and b rich stripes covering the membrane delineated by sharp domain walls the optimal spacing between domain walls is determined from the global minimizer of the limit

the hele shaw problem and area preserving curve shortening motions we prove existence locally in time of a solution of the following hele shaw problem given a simply connected curve contained in a smooth bounded domain find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature we show that this motion is a curve shortening motion which does not change the area of the region enclosed by the curve in case is the whole plane 2 we also show that if the initial curve is close to an equilibrium curve i e to a circle then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity

on the slow dynamics for the cahn hilliard equation in one space dimension we study the limiting behaviour of the solution of the cahn hilliard equation using energy type methods we assume that the initial data has a transition layer structure i e u epsilon approx pm 1 except near finitely many transition points we show that in the limit as epsilon rightarrow 0 the solution maintains its transition layer structure and the transition layers move slower than any power of epsilon

standing pulse solutions to fitzhugh nagumo equations reaction diffusion systems serve as relevant models for studying complex patterns in several fields of nonlinear sciences a localized pattern is a stable non constant stationary solution usually located far away from neighborhoods of bifurcation induced by turing s instability in the study of fitzhugh nagumo equations we look for a standing pulse with a profile staying close to a trivial background state except in one localized spatial region where the change is substantial this amounts to seeking a homoclinic orbit for a corresponding hamiltonian system and we utilize a variational formulation which involves a nonlocal term such a functional is referred to as helmholtz free energy in modeling microphase separation in diblock copolymers while its global minimizer does not exist in our setting of dealing with standing pulse the homoclinic orbit obtained here is a local minimizer extracted from a suitable topological class of admissible functions in contrast with the known results for positive standing pulses found in the literature a new technique is attempted by seeking a standing pulse solution with a sign change

a bayesian framework for global tractography we readdress the diffusion tractography problem in a global and probabilistic manner instead of tracking through local orientations we parameterise the connexions between brain regions at a global level and then infer on global and local parameters simultaneously in a bayesian framework this approach offers a number of important benefits the global nature of the tractography reduces sensitivity to local noise and modelling errors by constraining tractography to ensure a connexion is found and then inferring on the exact location of the connexion we increase the robustness of connectivity based parcellations allowing parcellations of connexions that were previously invisible to tractography the bayesian framework allows a direct comparison of the evidence for connecting and non connecting models to test whether the connexion is supported by the data crucially by explicit parameterisation of the connexion between brain regions we infer on a parameter that is shared with models of functional connectivity this model is a first step toward the joint inference on functional and anatomical connectivity

visual exploration of hardi fibers with probabilistic tracking high angular resolution diffusion imaging hardi is an effective method for characterizing complex neural fiber paths in the human brain however visualizing and analyzing the fibers is often challenging because of the complexity of the fiber orientation distribution function used to describe the crossing kissing and fanning fibers in this paper we propose a novel visual analytics approach to study brain fiber paths that allows users to explore fiber bundles to reveal the probability of fiber paths using a new visual classification method first we use a spherical deconvolution model for diffusion estimation and a bayesian theorem for fiber tractography second each fiber is subjected to a cluster analysis using pixel based visual encoding this result is shown in a pixel based visual representation where each pixel bar maps opacity color and length to the probability direction and length of a fiber fiber bundles can then be acquired via a two step classification routine that uses dbscan to group fibers based on similarities then the user can further refine fiber bundle selection using probabilistic information from the pixel bars therefore the proposed approach shows not only the shape but also the confidence of the fiber paths we demonstrate the resulting hardi fiber bundles and compare a brain with tumor and normal brain using our system experiments and an empirical user study verify the effectiveness of our approach

existence and stability of spherically layered solutions of the diblock copolymer equation the relatively simple ohta kawasaki density functional theory for diblock copolymer melts allows us to construct and analyze exact solutions to the euler lagrange equation by singular perturbation techniques first we consider a solution of a single sphere pattern that models a cell in the spherical morphology we show the existence of the sphere pattern and find a stability threshold so that if the sphere is larger than the threshold value the sphere pattern becomes unstable next we study a spherical lamellar pattern which may be regarded as a defective lamellar pattern we reduce the existence and the stability problems to some finite dimensional problems which are accurately solved with the help of a computer we find two thresholds only when the size of the sample is larger than the first threshold value does a spherical lamellar pattern exist this patten is stable only when the size of the sample is less than the second threshold value as the stability of the spherical lamellar pattern changes at the second threshold a bifurcating branch with a pattern of wriggled spherical interfaces appears the free energy of the latter pattern is lower than that of the first pattern a similar bifurcation phenomenon also occurs in the single sphere pattern at its stability threshold

on the phase diagram for microphase separation of diblock copolymers an approach via a nonlocal cahn hilliard functional we consider analytical and numerical aspects of the phase diagram for microphase separation of diblock copolymers our approach is variational and is based upon a density functional theory which entails minimization of a nonlocal cahn hilliard functional based upon two parameters which characterize the phase diagram we give a preliminary analysis of the phase plane that is we divide the plane into regions wherein a combination of analysis and numerics is used to describe minimizers in particular we identify a regime wherein the uniform disordered state is the unique global minimizer a regime wherein the constant state is linearly unstable and where numerical simulations are currently the only tool for characterizing the phase geometry and a regime of small volume fraction wherein we conjecture that small well separated approximately spherical objects are the unique global minimizer for this last regime we present an asymptotic analysis from the point of view of the energetics which will be comp

the spectrum of the torus profile to a geometric variational problem with long range interaction abstract the profile problem for the ohta kawasaki diblock copolymer theory is a geometric variational problem the energy functional is defined on sets in r 3 of prescribed volume and the energy of an admissible set is its perimeter plus a long range interaction term related to the newtonian potential of the set this problem admits a solution called a torus profile that is a set enclosed by an approximate torus of the major radius 1 and the minor radius q the torus profile is both axially symmetric about the z axis and reflexively symmetric about the x y plane there is a way to set up the profile problem in a function space as a partial differential integro equation the linearized operator l of the problem at the torus profile is decomposed into a family of linear ordinary differential integro operators l m where the index m 0 1 2 is called a mode the spectrum of l is the union of the spectra of the l m s it is proved that for each m when q is sufficiently small l m is positive definite 0 is an eigenvalue for both l 0 and l 1 due to the translation and rotation invariance as q tends to 0 more and more l m s become positive definite however no matter how small q is there is always a mode m of which l m has a negative eigenvalue this mode grows to infinity like q 3 4 as q 0