the weak metric approximation property we introduce and investigate the weak metric approximation property of banach spaces which is strictly stronger than the approximation property and at least formally weaker than the metric approximation property among others we show that if a banach space has the approximation property and is 1 complemented in its bidual then it has the weak metric approximation property we also study the lifting of the weak metric approximation property from banach spaces to their dual spaces this enables us in particular to show that the subspace of c0 constructed by johnson and schechtman does not have the weak metric approximation property

cortical surface based analysis ii inflation flattening and a surface based coordinate system the surface of the human cerebral cortex is a highly folded sheet with the majority of its surface area buried within folds as such it is a difficult domain for computational as well as visualization purposes we have therefore designed a set of procedures for modifying the representation of the cortical surface to i inflate it so that activity buried inside sulci may be visualized ii cut and flatten an entire hemisphere and iii transform a hemisphere into a simple parameterizable surface such as a sphere for the purpose of establishing a surface based coordinate system r 1999 academic press

quantitative evaluation of three cortical surface flattening methods during the past decade several computational approaches have been proposed for the task of mapping highly convoluted surfaces of the human brain to simpler geometric objects such as a sphere or a topological disc we report the results of a quantitative comparison of freesurfer circlepack and lscm with respect to measurements of geometric distortion and computational speed our results indicate that freesurfer performs best with respect to a global measurement of metric distortion whereas lscm performs best with respect to angular distortion and best in all but one case with a local measurement of metric distortion freesurfer provides more homogeneous distribution of metric distortion across the whole cortex than circlepack and lscm lscm is the most computationally efficient algorithm for generating spherical maps while circlepack is extremely fast for generating planar maps from patches

applying tensor based morphometry to parametric surfaces can improve mri based disease diagnosis many methods have been proposed for computer assisted diagnostic classification full tensor information and machine learning with 3d maps derived from brain images may help detect subtle differences or classify subjects into different groups here we develop a new approach to apply tensor based morphometry to parametric surface models for diagnostic classification we use this approach to identify cortical surface features for use in diagnostic classifiers first with holomorphic 1 forms we compute an efficient and accurate conformal mapping from a multiply connected mesh to the so called slit domain next the surface parameterization approach provides a natural way to register anatomical surfaces across subjects using a constrained harmonic map to analyze anatomical differences we then analyze the full riemannian surface metric tensors which retain multivariate information on local surface geometry as the number of voxels in a 3d image is large sparse learning is a promising method to select a subset of imaging features and to improve classification accuracy focusing on vertices with greatest effect sizes we train a diagnostic classifier using the surface features selected by an l1 norm based sparse learning method stability selection is applied to validate the selected feature sets we tested the algorithm on mri derived cortical surfaces from 42 subjects with genetically confirmed williams syndrome and 40 age matched controls multivariate statistics on the local tensors gave greater effect sizes for detecting group differences relative to other tbm based statistics including analysis of the jacobian determinant and the largest eigenvalue of the surface metric our method also gave reasonable classification results relative to the jacobian determinant the pair of eigenvalues of the jacobian matrix and volume features this analysis pipeline may boost the power of morphometry studies and may assist with image based classification

the approximation property in terms of the approximability of weak weak continuous operators abstract by a well known result of grothendieck a banach space x has the approximation property if and only if for every banach space y every weak weak continuous compact operator t x y can be uniformly approximated by finite rank operators from x y we prove the following metric version of this criterion x has the approximation property if and only if for every banach space y every weak weak continuous weakly compact operator t x y can be approximated in the strong operator topology by operators of norm t from x y as application easier alternative proofs are given for recent criteria of approximation property due to lima nygaard and oja

on the laplace beltrami operator and brain surface flattening in this paper using certain conformal mappings from uniformization theory the authors give an explicit method for flattening the brain surface in a way which preserves angles from a triangulated surface representation of the cortex the authors indicate how the procedure may be implemented using finite elements further they show how the geometry of the brain surface may be studied using this approach

constructing conformal maps of triangulated surfaces abstract we describe a means of computing the uniformizing conformal map from a triangulated surface whose triangles are realized as euclidean triangles in r 3 onto a fundamental domain in the unit disc d plane c or sphere s 2 mapping such triangulated surfaces arises in a number of applications such as conformal brain flattening we use the circle packing technique of bowers hurdal stephenson et al to first create a quasiconformal approximation to the conformal map then we apply a discrete form of conformal welding to reduce the distortion and converge to the conformal map in the limit

the convex approximation property of banach spaces abstract we study the convex approximation property of banach spaces to provide a unified approach to various approximation properties including besides the classical ones e g the positive approximation property of banach lattices and the approximation property for pairs of banach spaces our main results concern lifting of metric and weak metric approximation properties from banach spaces to their dual spaces as an easy application it follows that if x or x has the radon nikod m property then the approximation property of x defined by a convex subset of conjugate compact operators containing 0 in particular the positive approximation property of x is metric

cortical cartography using the discrete conformal approach of circle packings cortical flattening algorithms are becoming more widely used to assist in visualizing the convoluted cortical gray matter sheet of the brain metric based approaches are the most common but suffer from high distortions conformal or angle based algorithms are supported by a comprehensive mathematical theory the conformal approach that uses circle packings is versatile in the manipulation and display of results in addition it offers some new and interesting metrics that may be useful in neuroscientific analysis and are not available through numerical partial differential equation conformal methods in this paper we begin with a brief description of cortical flat mapping from data acquisition to map displays including a brief review of past flat mapping approaches we then describe the mathematics of conformal geometry and key elements of conformal mapping we introduce the mechanics of circle packing and discuss its connections with conformal geometry using a triangulated surface representing a cortical hemisphere we illustrate several manipulations available using circle packing methods and describe the associated ensemble conformal features ecfs we conclude by discussing current and potential uses of conformal methods in neuroscience and computational anatomy

brain surface conformal parameterization with the ricci flow in brain mapping research parameterized 3 d surface models are of great interest for statistical comparisons of anatomy surface based registration and signal processing here we introduce the theories of continuous and discrete surface ricci flow which can create riemannian metrics on surfaces with arbitrary topologies with user defined gaussian curvatures the resulting conformal parameterizations have no singularities and they are intrinsic and stable first we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves secondly we conformally parameterize each cortical surface to a parameter domain with a user designed gaussian curvature arrangement in the parameter domain a shape index based on conformal invariants is computed and inter subject cortical surface matching is performed by solving a constrained harmonic map we illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data to map statistical differences in cortical morphometry we studied brain asymmetry in 14 healthy control subjects we used a manifold version of hotelling s t2 test applied to the jacobian matrices of the surface parameterizations a permutation test along with the cumulative distribution of p values were used to estimate the overall statistical significance of differences the results show our algorithm s power to detect subtle group differences in cortical surfaces

near isometric flattening of brain surfaces flattened representations of brain surfaces are often used to visualize and analyze spatial patterns of structural organization and functional activity here we present a set of rigorous criteria and accompanying test cases with which to evaluate flattening algorithms that attempt to preserve shortest path distances on the original surface we also introduce a novel flattening algorithm that is the first to satisfy all of these criteria and demonstrate its ability to produce accurate flat maps of human and macaque visual cortex using this algorithm we have recently obtained results showing a remarkable unexpected degree of consistency in the shape and topographic structure of visual cortical areas within humans and macaques as well as between these two species

smooth stabilizing feedback for a class of mechanical systems with a pitchfork bifurcation abstract we consider the problem of local stabilization of a class of mechanical systems depending on a physical parameter and which exhibit an open loop pitchfork bifurcation furthermore at the critical value of the parameter the tangent linearization of the system contains two zero uncontrollable modes our goal is to find a nonlinear stabilizing feedback law

the weak metric approximation property and ideals of operators abstract we study the weak metric approximation property introduced by lima and oja we show that a banach space x has the weak metric approximation property if and only if f y x the space of finite rank operators is an ideal in w y x the space of weakly compact operators for all banach spaces y

multivariate tensor based morphometry on surfaces application to mapping ventricular abnormalities in hiv aids here we developed a new method called multivariate tensor based surface morphometry tbm and applied it to study lateral ventricular surface differences associated with hiv aids using concepts from differential geometry and the theory of differential forms we created mathematical structures known as holomorphic one forms to obtain an efficient and accurate conformal parameterization of the lateral ventricular surfaces in the brain the new meshing approach also provides a natural way to register anatomical surfaces across subjects and improves on prior methods as it handles surfaces that branch and join at complex 3d junctions to analyze anatomical differences we computed new statistics from the riemannian surface metrics these retain multivariate information on local surface geometry we applied this framework to analyze lateral ventricular surface morphometry in 3d mri data from 11 subjects with hiv aids and 8 healthy controls our method detected a 3d profile of surface abnormalities even in this small sample multivariate statistics on the local tensors gave better effect sizes for detecting group differences relative to other tbm based methods including analysis of the jacobian determinant the largest and smallest eigenvalues of the surface metric and the pair of eigenvalues of the jacobian matrix the resulting analysis pipeline may improve the power of surface based morphometry studies of the brain

absolutely summing operators on c 0 1 as a tree space and the bounded approximation property let x be a banach space for describing the space p c 0 1 x of absolutely summing operators from c 0 1 to x in terms of the space x itself we construct a tree space l1tree x on x it consists of special trees in x which we call two trunk trees we prove that p c 0 1 x is isometrically isomorphic to l1tree x as an application we characterize the bounded approximation property bap and the weak bap in terms of x valued sequence spaces

local feedback stabilization and bifurcation control ii stationary bifurcation abstract local feedback stabilization and bifurcation control of nonlinear systems are studied for the case in which the critical linearized system possesses a simple zero eigenvalue sufficient conditions are obtained for local stabilizability of the equilibrium point at criticality and for local stabilizability of bifurcated equilibria these conditions involve assumptions on the controllability of the critical mode for the linearized system explicit stabilizing feedback controls are constructed the projection method of analysis of stationary bifurcations is employed this work complements an earlier study by the same authors systems control lett 7 1986 11 17 of stabilization and bifurcation control in the hopf bifurcation case of two pure imaginary eigenvalues of the linearized system at criticality

stabilization of nonlinear systems with uncontrollable linearization the problem of local stabilization of nonlinear control systems with linearizations that contain uncontrollable modes on the imaginary axis is considered a methodology of designing a stabilizing control is investigated it involves the following steps 1 reduction of the stability problem to the stability of the center manifold system 2 simplification of the vector field on the center manifold using the theory of normal forms and 3 finding conditions under which the simplified vector field is asymptotically stable three cases of degeneracies in the linearized system are treated and sufficient conditions for the existence of stabilizing controls are given in each case a theorem is presented regarding the robustness of the above control strategies

brain surface conformal parameterization using riemann surface structure in medical imaging parameterized 3 d surface models are useful for anatomical modeling and visualization statistical comparisons of anatomy and surface based registration and signal processing here we introduce a parameterization method based on riemann surface structure which uses a special curvilinear net structure conformal net to partition the surface into a set of patches that can each be conformally mapped to a parallelogram the resulting surface subdivision and the parameterizations of the components are intrinsic and stable their solutions tend to be smooth functions and the boundary conditions of the dirichlet problem can be enforced conformal parameterization also helps transform partial differential equations pdes that may be defined on 3 d brain surface manifolds to modified pdes on a two dimensional parameter domain since the jacobian matrix of a conformal parameterization is diagonal the modified pde on the parameter domain is readily solved to illustrate our techniques we computed parameterizations for several types of anatomical surfaces in 3 d magnetic resonance imaging scans of the brain including the cerebral cortex hippocampi and lateral ventricles for surfaces that are topologically homeomorphic to each other and have similar geometrical structures we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other finally we present an automatic sulcal landmark location algorithm by solving pdes on cortical surfaces the landmark detection results are used as constraints for building conformal maps between surfaces that also match explicitly defined landmarks

the strong approximation property and the weak bounded approximation property abstract we show that the strong approximation property strong ap respectively strong cap and the weak bounded approximation property respectively weak bcap are equivalent for every banach space this gives a negative answer to oja s conjecture as a consequence we show that each of the spaces c 0 and l 1 has a subspace which has the ap but fails to have the strong ap

stabilization of a class of nonlinear systems by a smooth feedback control we consider the stabilizability problem for systems of the form x f x ub in the neighborhood of an equilibrium point of f x first by means of center manifold theory a lower order system is introduced if this system is stabilizable then so is the original system results on the stabilization of the low order system are presented no resort is taken to liapunov theory the relation between stabilizability and controllability is discussed

local feedback stabilization and bifurcation control i hopf bifurcation abstract local bifurcation control problems are defined and employed in the study of the local feedback stabilization problem for nonlinear systems in critical cases sufficient conditions are obtained for the local stabilizability of general nonlinear systems whose linearizations have a pair of simple nonzero imaginary eigenvalues the conditions show in particular that generically these nonlinear critical systems can be stabilized locally even if the critical modes are uncontrollable the analysis also yields a direct method for computing stabilizing feedback controls use is made of bifurcation formulae which require only a series expansion of the vector field

ideals of operators and the metric approximation property abstract we prove that a banach space x has the metric approximation property if and only if f y x the space of all finite rank operators is an ideal in l y x the space of all bounded operators for every banach space y moreover x has the shrinking metric approximation property if and only if f x y is an ideal in l x y for every banach space y similar results are obtained for u ideals and the corresponding unconditional metric approximation properties

the discrete geodesic problem we present an algorithm for determining the shortest path between a source and a destination on an arbitrary possibly nonconvex polyhedral surface the path is constrained to lie on the surface and distances are measured according to the euclidean metric our algorithm runs in time o n log n and requires o n2 space where n is the number ofedges ofthe surface afterwe run our algorithm the distance from the source to any other destination may be determined using standard techniques in time o log n by locating the destination in the subdivision created by the algorithm the actual shortest path from the source to a destination can be reported in time o k log n where k is the number of faces crossed by the path the algorithm generalizes to the case of multiple source points to build the voronoi diagram on the surface where n is now the maximum of the number of vertices and the number of sources