the enhanced principal rank characteristic sequence for skew symmetric matrices abstract the enhanced principal rank characteristic sequence epr sequence was originally defined for an n n real symmetric matrix or an n n hermitian matrix such a sequence is defined to be l 1 l 2 l n where l k is a s or n depending on whether all some or none of the matrix principal minors of order k are nonzero here we give a complete characterization of the attainable epr sequences for real skew symmetric matrices with the constraint that l k 0 if k is odd we show that nearly all epr sequences are attainable by skew symmetric matrices which is in contrast to the case of real symmetric or hermitian matrices for which many epr sequences are forbidden

linear algebra meets lie algebra the kostant wallach theory abstract in two languages linear algebra and lie algebra we describe the results of kostant and wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices in addition we present a brief introduction to basic notions in algebraic geometry integrable systems and lie algebra aimed at specialists in linear algebra the lie algebra description lets us indicate the significance of the kostant wallach results based on their parameterization of square complex matrices

relations between the minors of a generic matrix abstract it is well known that the plucker relations generate the ideal of relations of the maximal minors of a generic m n matrix in this paper we discuss the relations of t minors for t min m n we will exhibit minimal relations in degrees 2 non plucker in general and 3 and give some evidence for our conjecture that we have found the generating system of the ideal of relations the approach is through the representation theory of the general linear group

principal minors and diagonal similarity of matrices abstract let a b be n n matrices with entries in a field f our purpose is to show the following theorem suppose n 4 a is irreducible and for every partition of 1 2 n into subsets with 2 2 either rank a 2 or rank a 2 if a and b have equal corresponding principal minors of all orders then b or b t is diagonally similar to a

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

stabilizer information inequalities from phase space distributions the shannon entropy of a collection of random variables is subject to a number of constraints the best known examples being monotonicity and strong subadditivity it remains an open question to decide which of these laws of information theory are also respected by the von neumann entropy of many body quantum states in this article we consider a toy version of this difficult problem by analyzing the von neumann entropy of stabilizer states we find that the von neumann entropy of stabilizer states satisfies all balanced information inequalities that hold in the classical case our argument is built on the fact that stabilizer states have a classical model provided by the discrete wigner function the phase space entropy of the wigner function corresponds directly to the von neumann entropy of the state which allows us to reduce to the classical case our result has a natural counterpart for multi mode gaussian states which sheds some light on the general properties of the construction we also discuss the relation of our results to recent work by linden ruskai and winter the quantum entropy cone of stabiliser states e print arxiv 1302 5453

the principal rank characteristic sequence of a real symmetric matrix abstract given a vector u r 2 n the principal minor assignment problem asks when is there an n n matrix having its 2 n principal minors given by u this paper explores the following related problem given a sequence r 0 r 1 r n of 0s and 1s does there exist an n n real symmetric matrix that has a principal submatrix of rank k if and only if r k 1 for all 0 k n certain conditions are shown to be necessary in order for this question to have an affirmative answer several families of matrices are constructed to attain certain classes of sequences the problem is solved completely for n 6 and for 7 n 10 in the case of sequences beginning with 010

polynomial relations among principal minors of a 4x4 matrix abstract the image of the principal minor map for n n matrices is shown to be closed in the 19th century nanson and muir studied the implicitization problem of finding all relations among principal minors when n 4 we complete their partial results by constructing explicit polynomials of degree 12 that scheme theoretically define this affine variety and also its projective closure in p 15 the latter is the main component in the singular locus of the 2 2 2 2 hyperdeterminant

principal minor assignment isometries of hilbert spaces volumes of parallelepipeds and rescaling of sesqui holomorphic functions abstract in this article we consider the following equivalence relation on the class of all functions of two variables on a set x we will say that l m x x c are rescalings if there are non vanishing functions f g on x such that m x y f x g y l x y for any x y x we give criteria for being rescalings when x is a topological space and l and m are separately continuous or when x is a domain in c n and l and m are sesqui holomorphic a special case of interest is when l and m are symmetric and f g only has values 1 this relation between m and l in the case when x is finite and so l and m are square matrices is known to be characterized by the equality of the principal minors of these matrices we extend this result to the case when x is infinite as an application we characterize restrictions of isometries of hilbert spaces on weakly connected sets as the maps that preserve the volumes of parallelepipeds spanned by finite collections of vectors

the quasi principal rank characteristic sequence abstract a minor of a matrix is quasi principal if it is a principal or an almost principal minor the quasi principal rank characteristic sequence qpr sequence of an n n symmetric matrix is introduced which is defined as q 1 q 2 q n where q k is a s or n according as all some but not all or none of its quasi principal minors of order k are nonzero this sequence extends the principal rank characteristic sequences in the literature which only depend on the principal minors of the matrix a necessary condition for the attainability of a qpr sequence is established using probabilistic techniques a complete characterization of the qpr sequences that are attainable by symmetric matrices over fields of characteristic 0 is given

dynamic 3d models with local and global deformations deformable superquadrics the authors present a physically based approach to fitting complex three dimensional shapes using a novel class of dynamic models that can deform both locally and globally they formulate the deformable superquadrics which incorporate the global shape parameters of a conventional superellipsoid with the local degrees of freedom of a spline the model s six global deformational degrees of freedom capture gross shape features from visual data and provide salient part descriptors for efficient indexing into a database of stored models the local deformation parameters reconstruct the details of complex shapes that the global abstraction misses the equations of motion which govern the behavior of deformable superquadrics make them responsive to externally applied forces the authors fit models to visual data by transforming the data into forces and simulating the equations of motion through time to adjust the translational rotational and deformational degrees of freedom of the models model fitting experiments involving 2d monocular image data and 3d range data are presented

wavelet based segmentation on the sphere abstract segmentation a useful powerful technique in pattern recognition is the process of identifying object outlines within images there are a number of efficient algorithms for segmentation in euclidean space that depend on the variational approach and partial differential equation modelling wavelets have been used successfully in various problems in image processing including segmentation inpainting noise removal super resolution image restoration and many others wavelets on the sphere have been developed to solve such problems for data defined on the sphere which arise in numerous fields such as cosmology and geophysics in this work we propose a wavelet based method to segment images on the sphere accounting for the underlying geometry of spherical data our method is a direct extension of the tight frame based segmentation method used to automatically identify tube like structures such as blood vessels in medical imaging it is compatible with any arbitrary type of wavelet frame defined on the sphere such as axisymmetric wavelets directional wavelets curvelets and hybrid wavelet constructions such an approach allows the desirable properties of wavelets to be naturally inherited in the segmentation process in particular directional wavelets and curvelets which were designed to efficiently capture directional signal content provide additional advantages in segmenting images containing prominent directional and curvilinear features we present several numerical experiments applying our wavelet based segmentation method as well as the common k means method on real world spherical images including an earth topographic map a light probe image solar data sets and spherical retina images these experiments demonstrate the superiority of our method and show that it is capable of segmenting different kinds of spherical images including those with prominent directional features moreover our algorithm is efficient with convergence usually within a few iterations

hyperdeterminants as integrable discrete systems we give the basic definitions and some theoretical results about hyperdeterminants introduced by a cayley in 1845 we prove integrability understood as 4d consistency of a nonlinear difference equation defined by the 2 2 2 hyperdeterminant this result gives rise to the following hypothesis the difference equations defined by hyperdeterminants of any size are integrable we show that this hypothesis already fails in the case of the 2 2 2 2 hyperdeterminant

primitivity is hereditary for 2 structures abstract it is proved that each primitive 2 structure on n elements n 3 contains a primitive substructure on either n 1 or n 2 elements

gaussian process morphable models models of shape variations have become a central component for the automated analysis of images an important class of shape models are point distribution models pdms these models represent a class of shapes as a normal distribution of point variations whose parameters are estimated from example shapes principal component analysis pca is applied to obtain a low dimensional representation of the shape variation in terms of the leading principal components in this paper we propose a generalization of pdms which we refer to as gaussian process morphable models gpmms we model the shape variations with a gaussian process which we represent using the leading components of its karhunen loeve expansion to compute the expansion we make use of an approximation scheme based on the nystrom method the resulting model can be seen as a continuous analog of a standard pdm however while for pdms the shape variation is restricted to the linear span of the example data with gpmms we can define the shape variation using any gaussian process for example we can build shape models that correspond to classical spline models and thus do not require any example data furthermore gaussian processes make it possible to combine different models for example a pdm can be extended with a spline model to obtain a model that incorporates learned shape characteristics but is flexible enough to explain shapes that cannot be represented by the pdm we introduce a simple algorithm for fitting a gpmm to a surface or image this results in a non rigid registration approach whose regularization properties are defined by a gpmm we show how we can obtain different registration schemes including methods for multi scale or hybrid registration by constructing an appropriate gpmm as our approach strictly separates modeling from the fitting process this is all achieved without changes to the fitting algorithm to demonstrate the applicability and versatility of gpmms we perform a set of experiments in typical usage scenarios in medical image analysis and computer vision the model based segmentation of 3d forearm images and the building of a statistical model of the face to complement the paper we have made all our methods available as open source

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

a matrix description of weakly bipartitive and bipartitive families abstract the notions of weakly bipartitive and bipartitive families were introduced by montgolfier 2003 as a general tool for studying some decomposition of graphs and other combinatorial structures one way to construct such families comes from a result of loewy 1986 given an irreducible n n matrix a over a field the family of partitions x y of 1 n such that the submatrices a x y and a y x have a rank at most 1 is weakly bipartitive in this paper we show that this family is bipartitive when a is symmetric in the converse direction we prove that weakly bipartitive and bipartitive families are all obtained via the construction above

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

an eigenvalue approach evaluating minors for weighing matrices w n n 1 abstract in the present paper we concentrate our study on the evaluation of minors for weighing matrices w n n 1 theoretical proofs concerning their minors up to the order of n 4 n 4 are derived introducing an eigenvalue approach a general theorem specifying the analytical form of any n l n l minor is developed an application to the growth problem for weighing matrices is given

principal pivot transforms properties and applications the principal pivot transform ppt is a transformation of the matrix of a linear system tantamount to exchanging unknowns with the corresponding entries of the right hand side of the system the notion of the ppt is encountered in mathematical programming statistics and numerical analysis among other areas the purpose of this paper is to draw attention to the main properties and uses of ppts make some new observations and motivate further applications of ppts in matrix theory special consideration is given to ppts of matrices whose principal minors are positive 2000 elsevier science inc all rights reserved

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

a matricial proof of the symmetric exchange axiom for eigenvalues of principal submatrices of a complex hermitian matrix abstract in c r johnson b kroschel m omladic eigenvalue multiplicities in principal submatrices linear algebra appl 390 2004 111 120 a result constraining the eigenvalues of principal submatrices of complex hermitian matrices based upon matroid theory was given here we give a matricial proof of this result which also enables us to find a generalization of the original result

on the almost principal minors of a symmetric matrix abstract the almost principal rank characteristic sequence apr sequence of an n n symmetric matrix is introduced which is defined to be the string a 1 a 2 a n 1 where a k is either a s or n according as all some but not all or none of its almost principal minors of order k are nonzero in contrast to the other principal rank characteristic sequences in the literature the apr sequence of a matrix does not depend on principal minors the almost principal rank of a symmetric matrix b denoted by ap rank b is defined as the size of a largest nonsingular almost principal submatrix of b a complete characterization of the sequences not containing an a that can be realized as the apr sequence of a symmetric matrix over a field f is provided a necessary condition for a sequence to be the apr sequence of a symmetric matrix over a field f is presented it is shown that if b f n n is symmetric and non diagonal then rank b 1 ap rank b rank b with both bounds being sharp moreover it is shown that if b is symmetric non diagonal and singular and does not contain a zero row then rank b ap rank b

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

the p matrix problem is co np complete recently rohn and poljak proved that for interval matrices with rank one radius matrices testing singularity is np complete this paper will show that given any matrix family belonging to the class of matrix polytopes with hypercube domains and rank one perturbation matrices a class which contains the interval matrices testing singularity reduces to testing whether a certain matrix is not a p matrix it follows from this result that the problem of testing whether a given matrix is a p matrix is co np complete

on the toric algebra of graphical models we formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model or a log linear model or other more general exponential models for decomposable graphical models these conditions are equivalent to a set of conditional independence statements similar to the hammersley clifford theorem however we show that for nondecomposable graphical models they are not we also show that nondecomposable models can have nonrational maximum likelihood estimates these results are used to give several novel characterizations of decomposable graphical models

symmetric 0 1 matrices with inverses having two distinct values and constant diagonal abstract this paper examines 0 1 symmetric matrices for which the inverse of the matrix has all entries of the form for some and constant diagonal several constructions of such matrices are given as well as a strong connection to matrices that are invertible and have all principal submatrices of order n 2 singular

fundamental study the modular decomposition of countable graphs definition and construction in monadic second order logic we consider the notion of modular decomposition for countable graphs the modular decomposition of a graph given with an enumeration of its set of vertices can be defined by formulas of monadic second order logic another result is the definition of a representation of modular decompositions by low degree relational structures such relational structures can also be defined in the considered graph by monadic second order formulas

principal minor ideals and rank restrictions on their vanishing sets abstract all matrices we consider have entries in a fixed algebraically closed field k a minor of a square matrix is principal means it is defined by the same row and column indices we study the ideal generated by size t principal minors of a generic matrix and restrict our attention to locally closed subsets of its vanishing set given by matrices of a fixed rank the main result is a computation of the dimension of the locally closed set of n n rank n 2 matrices whose size n 2 principal minors vanish this set has dimension n 2 n 4

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

on lossless feedback delay networks lossless feedback delay networks fdns are commonly used as a design prototype for artificial reverberation algorithms the lossless property is dependent on the feedback matrix which connects the output of a set of delays to their inputs and the lengths of the delays both unitary and triangular feedback matrices are known to constitute lossless fdns however the most general class of lossless feedback matrices has not been identified in this contribution it is shown that the fdn is lossless for any set of delays if all irreducible components of the feedback matrix are diagonally similar to a unitary matrix the necessity of the generalized class of feedback matrices is demonstrated by examples of fdn designs proposed in literature

applications of stable polynomials to mixed determinants johnson s conjectures unimodality and symmetrized fischer products for n x n matrices a and b define eta a b sigma s det a s det b s where the summation is over all subsets of 1 n s is the complement of s and a s is the principal submatrix

the principal rank characteristic sequence over various fields abstract given an n n matrix its principal rank characteristic sequence is a sequence of length n 1 of 0s and 1s where for k 0 1 n a 1 in the kth position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix the principal rank characteristic sequences for symmetric matrices over various fields are investigated with all such attainable sequences determined for all n over any field with characteristic 2 a complete list of attainable sequences for real symmetric matrices of order 7 is reported

towards a theory of negative dependence the fkg theorem says that the positive lattice condition an easily checkable hypothesis which holds for many natural families of events implies positive association a very useful property thus there is a natural and useful theory of positively dependent events there is as yet no corresponding theory of negatively dependent events there is however a need for such a theory this paper unfortunately contains no substantial theorems its purpose is to present examples that motivate a need for such a theory give plausibility arguments for the existence of such a theory outline a few possible directions such a theory might take and state a number of specific conjectures which pertain to the examples and to a wish list of theorems

the method of controlled lagrangians energy plus force shaping we study the method of controlled lagrangians to stabilize mechanical systems connected with external forces the basic idea is that we transform by feedback a given controlled lagrangian system to another controlled lagrangian system with positive definite energy and a dissipative external force such that a dissipative feedback force stabilizes the closed loop system we derive matching conditions for energy plus force shaping that are more general and stronger than those in the literature we provide various easily verifiable criteria for stabilizability by the method of controlled lagrangians including a necessary and sufficient condition for stabilizability by energy shaping for all linear mechanical systems a necessary and sufficient condition for stabilizability by energy shaping for the class of all mechanical systems with one degree of underactuation and a sufficient condition for stabilizability by energy plus force shaping for the class of all mechanical systems with one degree of underactuation connected with external forces

on elliptic lax systems on the lattice and a compound theorem for hyperdeterminants a general elliptic n n matrix lax scheme is presented leading to two classes of elliptic lattice systems one which we interpret as the higher rank analogue of the landau lifschitz equations while the other class we characterize as the higher rank analogue of the lattice krichever novikov equation or adler s lattice we present the general scheme but focus mainly on the latter type of models in the case n 2 we obtain a novel lax representation of adler s elliptic lattice equation in its so called 3 leg form the case of rank n 3 is analyzed using cayley s hyperdeterminant of format yielding a multi component system of coupled 3 leg quad equations

spine segmentation in medical images using manifold embeddings and higher order mrfs we introduce a novel approach for segmenting articulated spine shape models from medical images a nonlinear low dimensional manifold is created from a training set of mesh models to establish the patterns of global shape variations local appearance is captured from neighborhoods in the manifold once the overall representation converges inference with respect to the manifold and shape parameters is performed using a higher order markov random field homrf singleton and pairwise potentials measure the support from the global data and shape coherence in manifold space respectively while higher order cliques encode geometrical modes of variation to segment each localized vertebra models generic feature functions learned from ground truth data assigns costs to the higher order terms optimization of the model parameters is achieved using efficient linear programming and duality the resulting model is geometrically intuitive captures the statistical distribution of the underlying manifold and respects image support clinical experiments demonstrated promising results in terms of spine segmentation quantitative comparison to expert identification yields an accuracy of 1 6 0 6 mm for ct imaging and of 2 0 0 8 mm for mr imaging based on the localization of anatomical landmarks

not all gkk τ matrices are stable abstract hermitian positive definite totally positive and nonsingular m matrices enjoy many common properties in particular a positivity of all principal minors b weak sign symmetry c eigenvalue monolonicity d positive stability the class of gkk matrices is defined by properties a and b whereas the class of nonsingular matrices by a and c it was conjectured that a b d d carlson j res nat bur standards sect b 78 1974 1 2 a c d g m engel and h schneider linear and multilinear algebra 4 1976 155 176 a b a property stronger than d r varga numerical methods in linear algebra 1978 pp 5 15 a b c d d hershkowitz linear algebra appl 171 1992 161 186 we describe a class of unstable gkk matrices thus disproving all four conjectures

theory of 2 structures part i clans basic subclasses and morphisms abstract the notion of a 2 structure is more general than the notion of a graph and less general than the notion of a relational structure this paper develops the theory of 2 structures and in particular it demonstrates that each 2 structure can be constructed from decomposed into three sorts of basic 2 structures this result is obtained through a hierarchical representation of 2 structures by trees

arithmetic mutually unbiased bases and complementary observables complementary observables in quantum mechanics may be viewed as frobenius structures in a dagger monoidal category such as the category of finite dimensional hilbert spaces over the complex numbers on the other hand their properties crucially depend on the discrete fourier transform and its associated quantum torus requiring only the finite fields that underlie mutually unbiased bases in axiomatic topos theory the complex numbers are difficult to describe and should not be invoked unnecessarily this paper surveys some fundamentals of quantum arithmetic using finite field complementary observables with a view considering more general axiom systems

theory of 2 structures part ii representation through labeled tree families abstract this paper which is the second part of a whole consisting of two parts continues the investigation of the theory of 2 structures initiated in part i in particular we investigate hierarchical representations of 2 structures through trees where the local relationship between direct descendants of an inner node of a tree is given through a 2 structure the main result of the paper is that for such hierarchical definition it suffices to consider trees where these local relationships are given by either a primitive 2 structure or a complete 2 structure or a linear 2 structure hence each 2 structure can be constructed from decomposed into primitive complete and linear 2 structures

the quadrifocal variety abstract multi view geometry is reviewed from an algebraic geometry perspective and multi focal tensors are constructed as equivariant projections of the grassmannian a connection to the principal minor assignment problem is made by considering several flatlander cameras the ideal of the quadrifocal variety is computed up to degree 8 and partially in degree 9 using the representations of gl 3 4 in the polynomial ring on the space of 3 3 3 3 tensors further representation theoretic analysis gives a lower bound for the number of minimal generators

self validating methods abstract we present some ideas on self validating sv methods this note is especially intended to give some background for the articles in this special volume of laa it is not intended to be a survey of the big variety of all possible aspects of sv methods but rather a summary of some basic concepts especially some common misconceptions are mentioned and explored

a shape based approach to the segmentation of medical imagery using level sets we propose a shape based approach to curve evolution for the segmentation of medical images containing known object types in particular motivated by the work of leventon grimson and faugeras 2000 we derive a parametric model for an implicit representation of the segmenting curve by applying principal component analysis to a collection of signed distance representations of the training data the parameters of this representation are then manipulated to minimize an objective function for segmentation the resulting algorithm is able to handle multidimensional data can deal with topological changes of the curve is robust to noise and initial contour placements and is computationally efficient at the same time it avoids the need for point correspondences during the training phase of the algorithm we demonstrate this technique by applying it to two medical applications two dimensional segmentation of cardiac magnetic resonance imaging mri and three dimensional segmentation of prostate mri

the hyperdeterminant and triangulations of the 4 cube peter huggins bernd sturmfels josephine yu and debbie s yusterabstract the hyperdeterminant of format 2 2 2 2 is a polynomialof degree 24 in 16 unknowns which has 2894276 terms we compute thenewton polytope of this polynomial and the secondary polytope of the 4 cube the 87959448 regular triangulations of the 4 cube are classi ed into25448 d equivalence classes one for each vertex of the newton polytope the 4 cube has 80876 coarsest regular subdivisions one for each facet of thesecondary polytope but only 268 of them come from the hyperdeterminant

model based deformable surface finding for medical images describes a new global shape parameterization for smoothly deformable three dimensional 3 d objects such as those found in biomedical images whose diversity and irregularity make them difficult to represent in terms of fixed features or parts this representation is used for geometric surface matching to 3 d medical image data such as from magnetic resonance imaging mri the parameterization decomposes the surface into sinusoidal basis functions four types of surfaces are modeled tori open surfaces closed surfaces and tubes this parameterization allows a wide variety of smooth surfaces to be described with a small number of parameters extrinsic model based information is incorporated by introducing prior probabilities on the parameters surface finding is formulated as an optimization problem results of the method applied to synthetic images and 3 d medical images of the heart and brain are presented

t structures t functions and texts the theory of 2 structures see e g ehrenfeucht and rozenberg 1990 provides a convenient framework for investigating various mathematical structures encountered in computer science this paper investigates a subclass of 2 structures called t structures which turned out to be important in the investigation of basic properties of 2 structures see ehrenfeucht and rozenberg 1992 we prove that t structures are a natural generalization of linear orders in particular we prove that a t structure can be represented by two linear orders based on this result the notion of a text is introduced which generalizes the notion of a word as used in formal language theory a text may be seen as a word with a structure spanned on it this structure may be a tree but it may also be more general than a tree basic properties of the class of texts corresponding to trees are investigated

region competition unifying snakes region growing and bayes mdl for multiband image segmentation we present a novel statistical and variational approach to image segmentation based on a new algorithm named region competition this algorithm is derived by minimizing a generalized bayes minimum description length mdl criterion using the variational principle the algorithm is guaranteed to converge to a local minimum and combines aspects of snakes balloons and region growing the classic snakes balloons and region growing algorithms can be directly derived from our approach we provide theoretical analysis of region competition including accuracy of boundary location criteria for initial conditions and the relationship to edge detection using filters it is straightforward to generalize the algorithm to multiband segmentation and we demonstrate it on gray level images color images and texture images the novel color model allows us to eliminate intensity gradients and shadows thereby obtaining segmentation based on the albedos of objects it also helps detect highlight regions

a k structure generalization of the theory of 2 structures the prime tree decomposition of graphs facilitates the solution of a number of important combinatorial problems the decomposition is also known as modular decomposition and substitution decomposition it has been generalized to 2 structures and to k ary relations but while both of these generalizations give the decomposition on graphs as a special case neither is a generalization of the other in this paper we propose a type of edge colored hypergraph which we will call a k structure we define the modular decomposition of k structures and generalize the essential algebraic properties this unifies the prime tree decompositions on 2 structures and k ary relations by giving them as special cases and extends the decomposition to k structures that are neither 2 structures nor k ary relations in addition we show that any indecomposable k structure on n 3 nodes contains a smaller indecomposable substructure that has at least n k nodes this is a generalization of a previously known result on graphs and 2 structures that is where k 2 the generalization to k ary relations that it gives as a special case is also new

an efficient algorithm for the symmetric principal minor assignment problem abstract we consider the problem of reconstructing a symmetric matrix from its principal minors which has several applications in information theory and statistical modeling we develop a theory of symmetric matrices with equal corresponding principal minors based on a simple equivalent property due to oeding 2011 10 we then use this theory to provide a method for choosing a canonical representative from the class of all symmetric matrices with specified principal minors finally we provide an efficient algorithm for computing this canonical representative given its principal minors as input

skew symmetric matrices and their principal minors abstract in this article we study the following problem given a positive integer k what is the relationship between two matrices with entries in a field k and having equal corresponding principal minors of order at most k our main theorem improves a result of loewy 13 for skew symmetric matrices all of whose off diagonal entries are nonzero

hyperdeterminantal relations among symmetric principal minors abstract the principal minors of a symmetric n n matrix form a vector of length 2 n we characterize these vectors in terms of algebraic equations derived from the 2 2 2 hyperdeterminant

principal minors part i a method for computing all the principal minors of a matrix an order o 2 n algorithm for computing all the principal minors of an arbitrary n n complex matrix is motivated and presented offering an improvement by a factor of n 3 over direct computation the algorithm uses recursive schur complementation and submatrix extraction storing the answer in a binary order an implementation of the algorithm in matlab is also given and practical considerations are discussed and treated accordingly

energy shaping for systems with two degrees of underactuation and more than three degrees of freedom we study the stabilizability via the method of energy shaping of a given lagrangian system with two degrees of underactuation and with n geq4 degrees of freedom by making use of the formal theory of pdes we derive an involutive system of pdes which governs energy shapability and thus deduce for the first time easily verifiable conditions under which energy shaping is guaranteed we illustrate our results with an example of a three cart one inverted pendulum system

detection of focal changes in human cortical thickness spherical wavelets versus gaussian smoothing subtle but progressive variations in human cortical thickness have been associated with the initial phases of prevalent neurological and psychiatric conditions but slight changes in cortical thickness at preclinical stages are typically masked by effects of the gaussian kernel smoothing on the cortical surface shape descriptors here we present the first study aimed at detecting changes in human cortical thickness maps by applying soft thresholding to multiresolution spherical wavelet coefficients in order to make gaussian and wavelet smoothing methods comparable the trade off between sensitivity and specificity was optimized to detect simulated thickness changes in various cortical areas of healthy elderly subjects results revealed a better sensitivity specificity trade off when using wavelet based methods as compared to gaussian smoothing in both the whole neocortex p 10 7 and cortical region based statistical analyses p 10 9 which was mainly due to the higher specificity obtained with the wavelet approach the lower smoothing introduced by wavelets and their adaptive properties may account for the enhanced specificity and sensitivity when compared with gaussian spatial filters these results strongly support the use of spherical wavelet methods to detect subtle variations in cortical thickness maps which may be crucial in better understanding the course of neuronal loss in normal aging and in finding early markers of cortical degeneration

the enhanced principal rank characteristic sequence abstract the enhanced principal rank characteristic sequence epr sequence of a symmetric n n matrix is a sequence l 1 l 2 l n where l k is a s or n according as all some or none of its principal minors of order k are nonzero such sequences give more information than the 0 1 pr sequences previously studied where basically the k th entry is 0 or 1 according as none or at least one of its principal minors of order k is nonzero various techniques including the schur complement are introduced to establish that certain subsequences such as nan are forbidden in epr sequences over fields of characteristic not two using probabilistic methods over fields of characteristic zero it is shown that any sequence of a s and s s ending in a is attainable and any sequence of a s and s s followed by one or more n s is attainable additional families of attainable epr sequences are constructed explicitly by other methods for real symmetric matrices of orders 2 3 4 and 5 all attainable epr sequences are listed with justifications

discrete line complexes and integrable evolution of minors based on the classical plcker correspondence we present algebraic and geometric properties of discrete integrable line complexes in cp3 algebraically these are encoded in a discrete integrable s

angular 2 structures abstract the theory of 2 structures forms a convenient framework for investigating graphs in this paper we investigate the class of angular 2 structures which results by requiring that the primitivity is forbidden on the lowest possible level i e it is required that no substructure on three elements is primitive the paper presents the basic theory of angular 2 structures and in particular the theory of primitive angular 2 structures we demonstrate that the notion of an angular 2 structure is a well chosen generalization of the notions of a symmetric graph and a partial order

area and length minimizing flows for shape segmentation a number of active contour models have been proposed that unify the curve evolution framework with classical energy minimization techniques for segmentation such as snakes the essential idea is to evolve a curve in two dimensions or a surface in three dimensions under constraints from image forces so that it clings to features of interest in an intensity image the evolution equation has been derived from first principles as the gradient flow that minimizes a modified length functional tailored to features such as edges however because the flow may be slow to converge in practice a constant hyperbolic term is added to keep the curve surface moving in the desired direction we derive a modification of this term based on the gradient flow derived from a weighted area functional with image dependent weighting factor when combined with the earlier modified length gradient flow we obtain a partial differential equation pde that offers a number of advantages as illustrated by several examples of shape segmentation on medical images in many cases the weighted area flow may be used on its own with significant computational savings

multiscale 3 d shape representation and segmentation using spherical wavelets this paper presents a novel multiscale shape representation and segmentation algorithm based on the spherical wavelet transform this work is motivated by the need to compactly and accurately encode variations at multiple scales in the shape representation in order to drive the segmentation and shape analysis of deep brain structures such as the caudate nucleus or the hippocampus our proposed shape representation can be optimized to compactly encode shape variations in a population at the needed scale and spatial locations enabling the construction of more descriptive nonglobal nonuniform shape probability priors to be included in the segmentation and shape analysis framework in particular this representation addresses the shortcomings of techniques that learn a global shape prior at a single scale of analysis and cannot represent fine local variations in a population of shapes in the presence of a limited dataset specifically our technique defines a multiscale parametric model of surfaces belonging to the same population using a compact set of spherical wavelets targeted to that population we further refine the shape representation by separating into groups wavelet coefficients that describe independent global and or local biological variations in the population using spectral graph partitioning we then learn a prior probability distribution induced over each group to explicitly encode these variations at different scales and spatial locations based on this representation we derive a parametric active surface evolution using the multiscale prior coefficients as parameters for our optimization procedure to naturally include the prior for segmentation additionally the optimization method can be applied in a coarse to fine manner we apply our algorithm to two different brain structures the caudate nucleus and the hippocampus of interest in the study of schizophrenia we show 1 a reconstruction task of a test set to validate the expressiveness of our multiscale prior and 2 a segmentation task in the reconstruction task our results show that for a given training set size our algorithm significantly improves the approximation of shapes in a testing set over the point distribution model which tends to oversmooth data in the segmentation task our validation shows our algorithm is computationally efficient and outperforms the active shape model algorithm by capturing finer shape details

towards a holistic cortical thickness descriptor heat kernel based grey matter morphology signatures in this paper we propose a heat kernel based regional shape descriptor that may be capable of better exploiting volumetric morphological information than other available methods thereby improving statistical power on brain magnetic resonance imaging mri analysis the mechanism of our analysis is driven by the graph spectrum and the heat kernel theory to capture the volumetric geometry information in the constructed tetrahedral meshes in order to capture profound brain grey matter shape changes we first use the volumetric laplace beltrami operator to determine the point pair correspondence between white grey matter and csf grey matter boundary surfaces by computing the streamlines in a tetrahedral mesh secondly we propose multi scale grey matter morphology signatures to describe the transition probability by random walk between the point pairs which reflects the inherent geometric characteristics thirdly a point distribution model is applied to reduce the dimensionality of the grey matter morphology signatures and generate the internal structure features with the sparse linear discriminant analysis we select a concise morphology feature set with improved classification accuracies in our experiments the proposed work outperformed the cortical thickness features computed by freesurfer software in the classification of alzheimer s disease and its prodromal stage i e mild cognitive impairment on publicly available data from the alzheimer s disease neuroimaging initiative the multi scale and physics based volumetric structure feature may bring stronger statistical power than some traditional methods for mri based grey matter morphology analysis

negative dependence and the geometry of polynomials we introduce the class of strongly rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter this class covers important models such as determinantal measures e g product measures and uniform random spanning tree measures and distributions for symmetric exclusion processes we show that strongly rayleigh measures enjoy all virtues of negative dependence and we also prove a series of conjectures due to liggett pemantle and wagner respectively moreover we extend lyons recent results on determinantal measures and we construct counterexamples to several conjectures of pemantle and wagner on negative dependence and ultra log concave rank sequences

primitive 2 structures with the n 2 property a fundamental notion in the theory of 2 structures is that of a primitive 2 structure in ehrenfeucht rozenberg 1990 it is proved that primitivity is hereditary in the sense that each primitive 2 structure on n elements where n 3 contains a primitive substructure on either n 1 or n 2 elements in this paper we determine the class of primitive 2 structures on n elements that do not contain primitive substructures on n 1 elements these 2 structures are said to satisfy the n 2 property we show that for each n 3 there is a restricted number of primitive 2 structures on n elements satisfying this property in fact for each n 4 there are four different reversible 2 structures up to isomorphism satisfying the n 2 property while for n odd there are five different 2 structures with this property

existence conditions of permanental and multivariate negative binomial distributions existence conditions of permanental distributions are deeply connected to existence conditions of multivariate negative binomial distributions the aim of this paper is twofold it answers several questions generated by recent works on this subject but it also goes back to the roots of this field and fixes existing gaps in older papers concerning conditions of infinite divisibility for these distributions

principal minors part ii the principal minor assignment problem the inverse problem of finding a matrix with prescribed principal minors is considered a condition that implies a constructive algorithm for solving this problem will always succeed is presented the algorithm is based on reconstructing matrices from their principal submatrices and schur complements in a recursive manner consequences regarding the overdeterminancy of this inverse problem are examined leading to a faster polynomial time version of the algorithmic construction care is given in the matlab implementation of the algorithms regarding numerical stability and accuracy

verification for existence of solutions of linear complementarity problems abstract recently g alefeld x chen and f potra numer math 83 1999 265 315 presented a verification method for solutions of linear complementarity problems lcps this paper is an attempt to obtain more useful information from the output of this verification method in particular existing results can only claim the nonexistence of solutions in a given interval we will use the farkas lemma to check if the interval contains a negative certification which shows the nonexistence of solutions in the whole space r n moreover we will study how to choose a good nonsingular matrix a in the interval operator for p 0 matrix linear complementarity problems we report numerical results to illustrate the efficiency of the proposed technique

characterization and complexity of uniformly nonprimitive labeled 2 structures abstract according to the clan decomposition theorem of ehrenfeucht and rozenberg 1990 each labeled 2 structure has a decomposition into three types of basic 2 structures complete linear and primitive this decomposition can be expressed as a node labeled tree the shape of the 2 structure our main interest is in the uniformly nonprimitive 2 structures which do not have primitive substructures every directed graph can be considered as a restricted 2 structure with only two labels arc no arc it is proved that forbidding primitivity in the 2 structures gives a unified approach to some well known classes of graphs viz the cographs and the transitive vertex series parallel graphs we also study the parallel complexity of the decomposition of 2 structures it is shown that there is a logcf algorithm which recognizes the uniformly nonprimitive 2 structures and constructs their shapes we prove also that for every mso monadic second order definable property of 2 structures there is a logcf algorithm to decide whether or not a uniformly nonprimitive 2 structure has that property

on matrices having equal corresponding principal minors abstract let a b be n n matrices with entries in a field f we say a and b satisfy property d if b or b t is diagonally similar to a it is clear that if a and b satisfy property d then they have equal corresponding principal minors of all orders the question is to what extent the converse is true there are examples which show the converse is not always true we modify the problem slightly and give conditions on a matrix a which guarantee that if b is any matrix which has the same principal minors as a then a and b will satisfy property d these conditions on a are formulated in terms of ranks of certain submatrices of a and the concept of irreducibility

nesting of prime substructures in k ary relations the modular decomposition of a graph or relation has a large number of combinatorial applications it divides the structure into a set of prime induced substructures which cannot be further decomposed recent work on graphs and k ary relations has focused on the discovery that prime induced substructures are densely nested when they occur lower bounds on the nesting density of prime substructures in graphs are used heavily in the only known linear time algorithm for directed graphs we improve on the previously known lower bounds for k ary relations and show that no further improvement is possible

hierarchical active shape models using the wavelet transform active shape models asms are often limited by the inability of relatively few eigenvectors to capture the full range of biological shape variability this paper presents a method that overcomes this limitation by using a hierarchical formulation of active shape models using the wavelet transform the statistical properties of the wavelet transform of a deformable contour are analyzed via principal component analysis and used as priors in the contour s deformation some of these priors reflect relatively global shape characteristics of the object boundaries whereas some of them capture local and high frequency shape characteristics and thus serve as local smoothness constraints this formulation achieves two objectives first it is robust when only a limited number of training samples is available second by using local statistics as smoothness constraints it eliminates the need for adopting ad hoc physical models such as elasticity or other smoothness models which do not necessarily reflect true biological variability examples on magnetic resonance images of the corpus callosum and hand contours demonstrate that good and fully automated segmentations can be achieved even with as few as five training samples

control via interconnection and damping assignment of linear time invariant systems a tutorial interconnection and damping assignment is a controller design methodology that regulates the behaviour of dynamical systems assigning a desired port hamiltonian structure to the closed loop a key step for the application of the method is the solution of the so called matching equation that in the case of nonlinear systems is a partial differential equation it has recently been shown that for linear systems the problem boils down to the solution of a linear matrix inequality that moreover is feasible if and only if the system is stabilisable making the method universally applicable it has also been shown that if we narrow the class of assignable structures e g to mechanical instead of the larger port hamiltonian the problem is still translated to a linear matrix inequality but now stabilisability is not sufficient to ensure its feasibility it is additionally required that the uncontrolled modes are simple and lie on the j axis which is consistent with the considered scenario of mechanical

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

perturbation theory without diagrams the polaron case higher order perturbative calculations in quantum field theory suffer from the factorial increase of the number of individual diagrams here i describe an approach which evaluates the total contribution numerically for finite temperature from the cumulant expansion of the corresponding observable followed by an extrapolation to zero temperature this method originally proposed by bogolyubov and plechko is applied to the calculation of higher order terms for the ground state energy of the polaron using state of the art multidimensional integration routines two new coefficients are obtained corresponding to a four and five loop calculation several analytical and numerical procedures have been implemented which were crucial for obtaining reliable results

principal minors and rhombus tilings the algebraic relations between the principal minors of a generic n n matrix are somewhat mysterious see e g lin and sturmfels 2009 j algebra 322 4121 31 we show however that by adding in certain almost principal minors the ideal of relations is generated by translations of a single relation the so called hexahedron relation which is a composition of six cluster mutations we give in particular a laurent polynomial parameterization of the space of n n matrices whose parameters consist of certain principal and almost principal minors the parameters naturally live on vertices and faces of the tiles in a rhombus tiling of a convex gon a matrix is associated to an equivalence class of tilings all related to each other by yang baxter like transformations by specializing the initial data we can similarly parameterize the space of hermitian symmetric matrices over or the quaternions moreover by further specialization we can parametrize the space of positive definite matrices over these rings this article is part of a special issue of journal of physics a mathematical and theoretical devoted to cluster algebras mathematical physics

genus zero surface conformal mapping and its application to brain surface mapping we developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one forms for surfaces with or without boundaries gu and yau 2002 gu and yau 2003 for genus zero surfaces our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map in this paper we apply the algorithm to the cortical surface matching problem we use a mesh structure to represent the brain surface further constraints are added to ensure that the conformal map is unique empirical tests on magnetic resonance imaging mri data show that the mappings preserve angular relationships are stable in mris acquired at different times and are robust to differences in data triangulation and resolution compared with other brain surface conformal mapping algorithms our algorithm is more stable and has good extensibility