on the cellular convexity of complexes in this paper we discuss cellular convexity of complexes a new definition of cellular convexity is given in terms of a geometric property then it is proven that a regular complex is celiularly convex if and only if there is a convex plane figure of which it is the cellular image hence the definition of cellular convexity by sklansky 7 is equivalent to the new definition for the case of regular complexes the definition of minsky and papert 4 is shown to be equivalent to our definition therefore au definitions are virtually equivalent it is shown that a regular complex is cellularly convex if and only if its minimum perimeter polygon does not meet the boundary of the complex a 0 n time algorithm is presented to determine the cellular convexity of a complex when it resides in n m cells and is represented by the run length code

discrete sets with minimal moment of inertia we analyze the moment of inertia i s relative to the center of gravity of finite plane lattice sets s we classify these sets according to their roundness a set s is rounder than a set t if i s

three dimensional shape analysis using local shape descriptors the three dimensional shape analysis problem is a very demanding test of shape analysis algorithms previous approaches to the problem have employed global features such as moments and fourier descriptors global features lack the capacity for solving the partial shape recognition problem in which only part of the unknown shape is available previous approaches to local shape analysis have employed structural syntactic methods but these methods have so far failed to solve the three dimensional problem this paper describes a hybrid structural statistical local shape analysis algorithm which is applied to the three dimensional problem

a three dimensional edge operator modern scanning techniques such as computed tomography have begun to produce true three dimensional imagery of internal structures the first stage in finding structure in these images like that for standard two dimensional images is to evaluate a local edge operator over the image if an edge segment in two dimensions is modeled as an oriented unit line segment that separates unit squares i e pixels of different intensities then a three dimensional edge segment is an oriented unit plane that separates unit volumes i e voxels of different intensities in this correspondence we derive an operator that finds the best oriented plane at each point in the image this operator which is based directly on the 3 d problem complements other approaches that are either interactive or heuristic extensions of 2 d techniques

random matrices in 2d laplacian growth and operator theory since it was first applied to the study of nuclear interactions by wigner and dyson almost 60 years ago random matrix theory rmt has developed into a field of its own within applied mathematics and is now essential to many parts of theoretical physics from condensed matter to high energy the fundamental results obtained so far rely mostly on the theory of random matrices in one dimension the dimensionality of the spectrum or equilibrium probability density in the last few years this theory has been extended to the case where the spectrum is two dimensional or even fractal with dimensions between 1 and 2 in this paper we review these recent developments and indicate some physical problems where the theory can be applied

multidimensional edge detection by hypersurface fitting one way to define operators for detecting edges in digital images is to fit a surface plane quadric to a neighborhood of each image point and take the magnitude of the gradient of the surface as an estimate of the rate of change of gray level in the image at that point this approach is extended to define edge detectors applicable to multidimensional arrays of data e g three dimensional arrays obtained by reconstruction from projections by locally fitting hypersurfaces to the data the resulting operators for hypersurfaces of degree 1 or 2 are closely analogous to those in the two dimensional case examples comparing some of these three dimensional operators with their twodimensional counterparts are given

spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions non linear sigma models that arise from the supersymmetric approach to disordered electron systems contain a non compact bosonic sector we study the model with target space h2 the two hyperboloid with isometry group su 1 1 and prove that in three dimensions moments of the fields are finite in the thermodynamic limit thus the non compact symmetry su 1 1 is spontaneously broken the bound on moments is compatible with the presence of extended states

patterns for multigrid equidistributed functions application to general parabolas and length estimation we show that for some special functions called k multigrid equidistributed functions we can compute the limit of the frequency of patterns in the discretization of their graph when the resolution tends to zero this result is applied to parabolas we deduce also that local length estimators almost never converge to the length for the parabolas

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

a new pt symmetric complex hamiltonian with a real spectrum we construct an isospectrum system in terms of a real and a complex potential to show that the underlying pt symmetric complex hamiltonian possesses a real spectrum which is shared by its real partner

a robust filtering algorithm for subpixel reconstruction of chain coded line drawings a robust algorithm is presented for smoothing and achieving subpixel accuracy in the reconstruction of chain coded line drawings the algorithm does not remove sharp corners and does not need a priori knowledge of curvature statistics a fast on line implementation can be achieved using a table look up a simplified algorithm can be used for reconstructing digitized polygons

delocalization for a class of random block band matrices we consider n times n hermitian random matrices h consisting of blocks of size m ge n 6 7 the matrix elements are i i d within the blocks close to a gaussian in the four moment matching sense but their distribution varies from block to block to form a block band structure with an essential band width m we show that the entries of the green s function g z h z 1 satisfy the local semicircle law with spectral parameter z e mathbf i eta down to the real axis for any eta gg n 1 using a combination of the supersymmetry method inspired by shcherbina j stat phys 155 3 466 499 2014 and the green s function comparison strategy previous estimates were valid only for eta gg m 1 the new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized

dynamical delocalization in random landau hamiltonians we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero in this article we prove the existence of dynamical delocalization for random landau hamiltonians near each landau level more precisely we prove that for these two dimensional hamiltonians there exists at least one energy e near each landau level such that e 1 where e the local transport exponent introduced in gk5 is a measure of the rate of transport in wave packets with spectral support near e since typically there is dynamical localization at the edges of each disordered broadened landau band this implies the existence of at least one dynamical mobility edge at each landau band namely a boundary point between the localization and delocalization regimes which we prove to converge to the corresponding landau level as either the magnetic field goes to infinity or the disorder goes to zero random landau hamiltonians are the subject of intensive study due to their links with the integer quantum hall e ect kli for which von klitzing received the 1985 nobel prize in physics they describe an electron moving in a very thin flat conductor with impurities under the influence of a constant magnetic field perpendicular to the plane of the conductor and play an important role in the understanding of the quantum hall e ect l aoa t h nt ku be avss bees laughlin s argument l as pointed out by halperin h uses the assumption that under weak disorder and strong magnetic field the energy spectrum consists of bands of extended states separated

a new parameterization of digital straight lines a 1 1 correspondence is established between digital straight lines which start at a fixed point and a simple set of quadruples of integer parameters such a representation by parameters is useful for enumeration first the authors show a 1 1 correspondence between point pairs in a planar set of points and the linear dichotomies of this set then from the equivalence between digital lines and linear dichotomies of points on the digitization grid they prove a 1 1 and onto correspondence between digital straight lines starting at a fixed point and a well defined set of pairs of grid points it follows that four parameters uniquely represent any given digital line with a fixed starting point an o n algorithm is given for determining the parameters from the digital line as well as o log n algorithms for transforming between these parameters and the parameters suggested by l dorst and a w m smeulders 1984

fluctuation estimates for sub quadratic gradient field actions in this article we estimate fluctuations of the scalar field for a special class of sub quadratic actions which grow like 2 0 1 in particular if 1 2 we show that in three dimensions e 0 is bounded for small for each edge jk we introduce an auxiliary field tjk r to express the action as a superposition of gaussian free fields the effective action which arises from integrating over the gaussian field is shown to be convex in t the brascamp lieb inequality is then applied to obtain the desired estimates on a nonuniformly elliptic green s function

a lie algebraic approach to non hermitian hamiltonians with real spectra an algebraic technique useful in studying non hermitian hamiltonians with real spectra is presented the method is illustrated by explicit application to a family of one dimensional potentials

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

faithful polygonal representation of the convex and concave parts of a digital curve from results about digital convexity we define a reversible polygon that faithfully represents the maximal convex and concave parts of a digital curve such a polygon always exists and is unique in the general case it is computed from a given digital curve in linear time using well known routines adding a point at the front of a digital straight segment and removing a point from the back of a digital straight segment it may helps to extract perceptually meaningful parts of shape outlines or lines

factorization and superpotential of the pt symmetric hamiltonian we study the factorization of the pt symmetric hamiltonian the general expression for the superpotential corresponding to the pt symmetric potential is obtained and explicit examples are presented

three dimensional digital line segments digital arcs in 3 d digital pictures are defined the digital image of an arc is also defined a digital arc is defined to be a digital line segment if it is the digital image of a line segment it is shown that a digital line segment may be characterized by the chord property holding for its projections onto the coordinate planes it is also shown that a digital line segment may not be characterized by its own chord property a linear time algorithm is presented that determines whether or not a digital arc is a digital line segment

digital topology and geometry in medical imaging a survey digital topology and geometry refers to the use of topologic and geometric properties and features for images defined in digital grids such methods have been widely used in many medical imaging applications including image segmentation visualization manipulation interpolation registration surface tracking object representation correction quantitative morphometry etc digital topology and geometry play important roles in medical imaging research by enriching the scope of target outcomes and by adding strong theoretical foundations with enhanced stability fidelity and efficiency this paper presents a comprehensive yet compact survey on results principles and insights of methods related to digital topology and geometry with strong emphasis on understanding their roles in various medical imaging applications specifically this paper reviews methods related to distance analysis and path propagation connectivity surface tracking image segmentation boundary and centerline detection topology preservation and local topological properties skeletonization and object representation correction and quantitative morphometry a common thread among the topics reviewed in this paper is that their theory and algorithms use the principle of digital path connectivity path propagation and neighborhood analysis

digital convexity straightness and convex polygons new schemes for digitizing regions and arcs are introduced it is then shown that under these schemes sklansky s definition of digital convexity is equivalent to other definitions digital convex polygons of n vertices are defined and characterized in terms of geometric properties of digital line segments also a linear time algorithm is presented that given a digital convex region determines the smallest integer n such that the region is a digital convex n gon

computing distance transformations in convex and non convex domains abstract it is shown that the well known two pass sequential local transformation algorithm for computing a distance transformation in rectangular domains may fail in some convex integer domains but that a four pass algorithm is sufficient in all two dimensional convex domains for non convex domains the number of passes necessary is shown to be generally greater two propagation algorithms for computing the distance transformation are described and shown theoretically and experimentally to be computationally more efficient than the sequential local transformation algorithm in non convex domains of complex shape the relationship of the distance transformation in non convex domains to some more general transformations is explored

digital parallelism perpendicularity and rectangles the slope of digital line segments is defined and an algorithm to evaluate it is presented parallelism and perpendicularity of two digital line segments are also defined finally rectangular digital regions are defined and characterized and an algorithm that determines whether or not a given digital region is a digital rectangle is presented

three dimensional digital planes definitions of 3 d digital surface and plane are introduced many geometric properties of these objects are examined in particular it is shown that digital convexity is neither a necessary nor a sufficient condition for a digital surface element to be a convex digital plane element but it is both necessary and sufficient for a digital surface to be a digital plane also algorithms are presented to determine whether or not a finite set of digital points is a convex digital plane element

systematic search for symmetric potentials with real energy spectra changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions here we relax the standard hermiticity requirement and find an innovative construction which leads to unusual complex potentials their energy spectrum is shown to stay real after a weakening of the hermiticity of the schr dinger equation to its mere invariance under the combined parity and time reversal symmetry this ultimately results in richer bound state spectra some of our new exactly solvable potentials generalize the current textbook models details are given for constructions based on the hypergeometric and confluent hypergeometric special functions

separating point sets by circles and the recognition of digital disks given two sets a and b of points in the plane we give a polynomial time algorithm to find the set s of points which are centers of circles that contain all the members of a and none of b a consequence of the algorithm is that s is polygonal another application is an o n2 time and o n2 space algorithm for the recognition of digital disks

on recursive o n partitioning of a digitized curve into digital straight segments a simple online algorithm for partitioning of a digital curve into digital straight line segments of maximal length is given the algorithm requires o n time and o 1 space and is therefore optimal efficient representations of the digital segments are obtained as byproducts the algorithm also solves a number theoretical problem concerning nonhomogeneous spectra of numbers

convex digital solids a definition of convexity of digital solids is introduced then it is proved that a digital solid is convex if and only if it has the chordal triangle property other geometric properties which characterize convex digital regions are shown to be only necessary but not sufficient conditions for a digital solid to be convex an efficient algorithm that determines whether or not a digital solid is convex is presented

structure of the space of ground states in systems with non amenable symmetries we investigate classical spin systems in d 1 dimensions whose transfer operator commutes with the action of a nonamenable unitary representation of a symmetry group here so 1 n these systems may alternatively be interpreted as systems of interacting quantum mechanical particles moving on hyperbolic spaces in sharp contrast to the analogous situation with a compact symmetry group the following results are found and proven i spontaneous symmetry breaking already takes place for finite spatial volume finitely many particles and even in dimensions d 1 2 the tuning of a coupling temperature parameter cannot prevent the symmetry breaking ii the systems have infinitely many non invariant and non normalizable generalized ground states iii the linear space spanned by these ground states carries a distinguished unitary representation of so 1 n the limit of the spherical principal series iv the properties i iii hold universally irrespective of the details of the interaction

a paradigm for class identification problems the following problem arises in many applications involving classification identification and inference there is a set of objects x and a particular x spl isin x is chosen unknown to us based on information obtained about x in a sequential manner one wishes to decide whether x belongs to one class of objects a sub 0 or a different class of objects a sub 1 the authors study a general paradigm applicable to a broad range of problems of this type which they refer to as problems of class identification or discernibility they consider various types of information sequences and various success criteria including discernibility in the limit discernibility with a stopping criterion uniform discernibility and discernibility in the cesaro sense they consider decision rules both with and without memory necessary and sufficient conditions for discernibility are provided for each case in terms of separability conditions on the sets a sub 0 and a sub 1 they then show that for any sets a sub 0 and a sub 1 various types of separability can be achieved by allowing failure on appropriate sets of small measure applications to problems in language identification system identification and discrete geometry are discussed

the discrete green theorem and some applications in discrete geometry the discrete version of green s theorem and bivariate difference calculus provide a general and unifying framework for the description and generation of incremental algorithms it may be used to compute various statistics about regions bounded by a finite and closed polygonal path more specifically we illustrate its use for designing algorithms computing many statistics about polyominoes regions whose boundary is encoded by four letter words area coordinates of the center of gravity moment of inertia set characteristic function the intersection with a given set of pixels hook lengths higher order moments and also q statistics for projections

curvedness of a line picture abstract the shape analysis of a binary picture is of great importance for pictorial pattern recognition in this paper we propose a useful geometric feature parameter called curvedness curvedness represents which lines are dominant in a binary picture straight or curved our algorithm for measuring curvedness is based on the relationship between the distance to a boundary point on each black point along each quantized direction and the mean width of a binary picture we investigate the fundamental property of curvedness the experimental results for japanese hiragana and kanji characters show the validity of curvedness

symmetric effective mass schrodinger equations we outline a general method of obtaining exact solutions of symmetric schrodinger equations with a position dependent effective mass using this method exact solutions of some symmetric potentials have been obtained we have also discussed the construction of isospectral symmetric potentials

some results on fuzzy digital convexity abstract the definition of fuzzy convexity is reviewed and some results on projections of convex and fuzzy convex sets are established digital fuzzy convexity is defined and relationships among alternative definitions are investigated

digital planarity of rectangular surface segments we generalize the concept of evenness which has been developed for digital straight lines evenness is a necessary and sufficient condition for a digital arc segment to be a digital straight line segment we prove that evenness is also a necessary and sufficient condition for a rectangular surface segment to be a digital plane segment this is not true for surface segments of arbitrary shape to clarify the relation between shape and evenness we introduce the notion of a regular shape and of an arbitrarily extendable even set

the number of digital straight lines on an n n grid the number of digital straight lines on an n n grid is shown a digital straight line is equivalent to a linear dichotomy of points on a square grid the result is obtained by determining a way of counting the number of linearly separable dichotomies of points on the plane that are not necessarily in general position the analysis is easily modified to provide a simple solution to a similar problem considered by c berenstein and d lavine 1988 on the number of digital straight lines from a fixed starting point

geometric algorithms for digitized pictures on a mesh connected computer although mesh connected computers are used almost exclusively for low level local image processing they are also suitable for higher level image processing tasks we illustrate this by presenting new optimal in the o notational sense algorithms for computing several geometric properties of figures for example given a black white picture stored one pixel per processing element in an n n mesh connected computer we give n time algorithms for determining the extreme points of the convex hull of each component for deciding if the convex hull of each component contains pixels that are not members of the component for deciding if two sets of processors are linearly separable for deciding if each component is convex for determining the distance to the nearest neighboring component of each component for determining internal distances in each component for counting and marking minimal internal paths in each component for computing the external diameter of each component for solving the largest empty circle problem for determining internal diameters of components without holes and for solving the all points farthest point problem previous mesh connected computer algorithms for these problems were either nonexistent or had worst case times of n2 since any serial computer has a best case time of n2 when processing an n n image our algorithms show that the mesh connected computer provides significantly better solutions to these problems

digital representation schemes for 3d curves digital representation of three dimensional 3 d curves is considered two dimensional representation schemes are first reviewed then methods applicable to curves in three and higher dimensional spaces are described desirable properties of curve representation schemes are identified and used as a basis for qualitative and quantitative comparison of the various methods that have been suggested conclusions concerning the selection of a digital 3 d curve representation scheme for shape analysis and coding are drawn it is shown that grid intersect quantization is a poor choice for curve representation in 3 d space and that cube quantization that leads to 6 connected chain codes meets all the identified requirements and should be preferred

on properties of discretized convex curves the connection between a continuous convex curve and its discrete image is investigated using an appropriate definition of discrete convexity it is shown that the discrete image of continuous convex curve may not be convex however its deviation from convexity in the discrete definition is bounded by a small constant the actual pixel patterns that are obtained by discretizing convex curves are studied certain constraints on the context of discrete images of continuous convex curves were discovered

digital straight lines and convexity of digital regions it is shown that a digital region is convex if and only if every pair of points in the region is connected by a digital straight line segment contained in the region the midpoint property is shown to be a necessary but not a sufficient condition for the convexity of digital regions however it is shown that a digital region is convex if and only if it has the median point property

a compact chord property for digital arcs abstract rosenfeld ieee trans comput c 23 12 1264 1269 1974 defined the chord property and proved that a digital arc is a digital straight segment if and only if it satisfies the chord property a new property is defined which we call the compact chord property and the two properties are proved to be equivalent the compact chord property offers a useful alternative for testing a digital arc for straightness by exploiting the notion of visibility in computational geometry

on the straightness of digital arcs the straightness of a digital arc can simply be determined by the absence of unevenness in its chain code the absence of unevenness is a necessary and sufficient condition for a digital arc to have the chord property and it offers a much simpler alternative for examining whether a digital arc has the chord property or not this condition is also the most comprehensive and precise expression of the third criterion about digital straight lines given by freeman the relations among the chord property freeman s three criteria and the hierarchical structure that a digital straight line supposedly have become much more clear a procedure to detect the uneven segments and algorithms for determining the straightness of digital arcs is also included

a new class of non hermitian hamiltonians with real spectra we construct a new class of non hermitian hamiltonians with real spectra the hamiltonians possess one explicitly known eigenfunction