some results on almost square banach spaces abstract we study almost square banach spaces under a topological point of view indeed we prove that the class of banach spaces which admits an equivalent norm to be asq is that of those banach spaces which contain an isomorphic copy of c 0 we also prove that the symmetric projective tensor products of an almost square banach space have the strong diameter two property

geometric aspects of the daugavet property let x be a closed subspace of a banach space y and j be the inclusion map we say that the pair x y has the daugavet property if for every rank one bounded linear operator t from x to y the following equality kj tk 1 ktk 1 holds a new characterization of the daugavet property in terms of weak open sets is given it is shown that the operators not xing

emerging notions of norm attainment for lipschitz maps between banach spaces abstract we classify several notions of norm attaining lipschitz maps which were introduced previously and present the relations among them in order to verify proper inclusions we also analyze some results for the sets of lipschitz maps satisfying each of these properties to be dense or not in lip 0 x y for instance we characterize a banach space y with the radon nikod m property in terms of the denseness of norm attaining lipschitz maps with values in y further we introduce a property called the local directional bishop phelps bollobas property for lipschitz compact maps which extends the one studied previously for scalar valued functions and provide some new positive results

shape classification via image based multiscale description we introduce a new multiscale fourier based object description in 2 d space using a low pass gaussian filter lpgf and a high pass gaussian filter hpgf separately using the lpgf at different scales standard deviation represents the inner and central part of an object more than the boundary on the other hand using the hpgf at different scales represents the boundary and exterior parts of an object more than the central part our algorithms are also organized to achieve size translation and rotation invariance evaluation indicates that representing the boundary and exterior parts more than the central part using the hpgf performs better than the lpgf based multiscale representation and in comparison to zernike moments and elliptic fourier descriptors with respect to increasing noise multiscale description using hpgf in 2 d also outperforms wavelet transform based multiscale contour fourier descriptors and performs similar to the perimeter descriptors without any noise

on strongly norm attaining lipschitz maps abstract we study the set sna m y of those lipschitz maps from a complete pointed metric space m to a banach space y which strongly attain their lipschitz norm i e the supremum defining the lipschitz norm is a maximum extending previous results we prove that this set is not norm dense when m is a length space or local or when m is a closed subset of r with positive lebesgue measure providing new examples which have very different topological properties than the previously known ones on the other hand we study the linear properties which are sufficient to get lindenstrauss property a for the lipschitz free space f m over m and show that all of them actually provide the norm density of sna m y in the space of all lipschitz maps from m to any banach space y next we prove that sna m r is weakly sequentially dense in the space of all lipschitz functions for all metric spaces m finally we show that the norm of the bidual space of f m is octahedral provided the metric space m is discrete but not uniformly discrete or m is infinite

algebroid functions wirsing s theorem and their relations in this paper we first point out a relationship between the second main theorem for algebriod functions in nevanlinna theory and wirsing s theorem in diophantine approximation this motivates a unified proof for both theorems the second part of this paper deals with moving targets problem for holomorphic maps to riemann surfaces its counterpart in diophantine approximation follows from a recent work of thomas j tucker in this paper we point out tucker s result in the special case of the approximation by rational points could be obtained by doing a translation and applying the corresponding result with fixed target however we could not completely recover tucker s result concerning the approximation by algebraic points in the last part of this paper cases in higher dimensions are studied some partial results in higher dimensions are obtained and some conjectures are raised

linearity measure for curve segments in this paper we consider linearity measure for a bounded length curves first we define a new linearity measure for open curve segments and then extend method to closed curves contours the derived measures for both open curve segments and closed curves are invariant with respect to similarity transformations the linearity measure for open curve segments picks the value 1 if and only if the measured open line segment is a perfect straight line segment while the established linearity measures for closed curves never reach 1 as preferred

a new convexity measure for polygons convexity estimators are commonly used in the analysis of shape in this paper we define and evaluate a new convexity measure for planar regions bounded by polygons the new convexity measure can be understood as a boundary based measure and in accordance with this it is more sensitive to measured boundary defects than the so called area based convexity measures when compared with the convexity measure defined as the ratio between the euclidean perimeter of the convex hull of the measured shape and the euclidean perimeter of the measured shape then the new convexity measure also shows some advantages particularly for shapes with holes the new convexity measure has the following desirable properties 1 the estimated convexity is always a number from 0 1 2 the estimated convexity is i if and only if the measured shape is convex 3 there are shapes whose estimated convexity is arbitrarily close to 0 4 the new convexity measure is invariant under similarity transformations and 5 there is a simple and fast procedure for computing the new convexity measure

pattern recognition by affine moment invariants abstract the paper deals with moment invariants which are invariant under general affine transformation and may be used for recognition of affine deformed objects our approach is based on the theory of algebraic invariants the invariants from second and third order moments are derived and shown to be complete the paper is a significant extension and generalization of recent works several numerical experiments dealing with pattern recognition by means of the affine moment invariants as the features are described

octahedral norms and convex combination of slices in banach spaces abstract we study the relation between octahedral norms daugavet property and the size of convex combinations of slices in banach spaces we prove that the norm of an arbitrary banach space is octahedral if and only if every convex combination of w slices in the dual unit ball has diameter 2 which answers an open question as a consequence we get that the banach spaces with the daugavet property and its dual spaces have octahedral norms also we show that for every separable banach space containing l 1 and for every e 0 there is an equivalent norm so that every convex combination of w slices in the dual unit ball has diameter at least 2 e

a new convexity measure based on a probabilistic interpretation of images in this paper we present a novel convexity measure for object shape analysis the proposed method is based on the idea of generating pairs of points from a set and measuring the probability that a point dividing the corresponding line segments belongs to the same set the measure is directly applicable to image functions representing shapes and also to gray scale images which approximate image binarizations the approach introduced gives rise to a variety of convexity measures which make it possible to obtain more information about the object shape the proposed measure turns out to be easy to implement using the fast fourier transform and we would consider this in detail finally we illustrate the behavior of our measure in different situations and compare it to other similar ones

arithmetic distance functions and height functions in diophantine geometry in virtually all areas of diophantine geometry the theory of height functions plays a crucial role this theory associates to each cartier divisor d on a projective variety v a function ho mapping the group of rational points of v to the real numbers this function which is defined up to a bounded function on v has very nice functorial properties for example it is linear in d and depends only on the linear equivalence class of d as a consequence any relation between divisor classes such as the theorem of the square for abelian varieties will yield a corresponding relation for height functions further it is possible to write the height function ho as a sum of local height functions 20 v where v ranges over the distinct absolute values of the given field these local heights are also known as logarithmic distance functions or weil functions each 2 o is defined away from the support of d and gives a measure of the v adic distance from the given point of v to the divisor d in this paper we propose to define local height functions 2 x for every closed subscheme x of a projective variety v as above 2x v will give a measure of the v adic distance from a point of v to x these functions will have many nice functorial properties for example 2xny will equal the minimum of 2x and hr in this way relations between closed subschemes or equivalently between ideal sheaves will yield relations between local height functions we will thus be able to convert geometric statements into arithmetic statements relatively painlessly more generally we will deal with the case that v is merely quasi projective by defining a function 20v v which will measure the v adic distance to the boundary of v we will then be able to assign to each closed subscheme x of v a local height function 2x which will be well defined up to addition of a multiple of 2 v this extra generality is useful for example when one has a complete family of varieties and wishes to discard the bad fibers as one application of the machinery that we have developed we prove a quantitative version of the inverse function theorem thus given a finite map we use local height functions to describe how far away from the ramification locus

notes on shape based tools for treating the objects ellipticity issues we establish the average shape ellipticity measure as an average value of the estimated similarities between the shape considered and all the ellipses whose axes length ratio belongs to a certain predefined interval we derive the maximal shape ellipticity measure as the maximal similarity estimate between a given shape and all the ellipses in this paper we put under the same umbrella several well known results and establish several new results related to the shape based tools which treat the objects ellipticity issues we start with a derivation of an explicit and closed formula for the computation of the ellipticity measures from an infinite family introduced recently the new formula enables a fast computation of such measures since it does not require any optimizing procedure for the computation as it was the case before in addition the established formula enables an easy theoretical manipulation as a result we have discovered new shape features as they are i the average shape ellipticity measure which might be interpreted as an average value of the estimated similarities between the shape considered and all the ellipses whose axes length ratio belongs to a certain predefined interval ii the maximal shape ellipticity measure which might be understood as the maximal possible similarity estimate between a given shape and an ellipses some of the other results obtained relate strongly to the well known measures and methods broadly used in shape based object analysis tasks

simple and fast computation of moments abstract in this paper we address the problem of efficient computation of moments from the boundary of a digital area boundary based computation is superior to usual region based approaches as the data dimension of boundary representations is substantially smaller than that of region representations we investigate the inter order relationship of moments one of our results is that moments of higher order can be computed from moments of lower order based on this relationship a simple iterative algorithm is proposed for the computation of moments from a polygonal approximation of the boundary in comparison with a direct computation method our algorithm is simpler to program the memory requirement is minimum simulation results show that a speed up of factor 8 can be achieved using our algorithm a special version of the algorithm can be utilized to compute moments from the run length chain code of the boundary our algorithm can be applied to compute the most popular geometric moments as well as other types of moments like legendre zernike rotational and complex moments

a generalization of theorems of faltings and thue siegel roth wirsing in 1929 siegel proved a celebrated theorem on finiteness for integral solutions of certain diophantine equations this theorem applies to systems of polynomial equations which either a describe an irreducible curve whose projective closure has positive genus or b describe an irreducible curve of genus zero with at least three points at infinity for such systems siegel s theorem says that there are only finitely many solutions in the ring of integers of any given number field siegel s proof used the method of diophantine approximations as pioneered by thue in 1909 t to give an example if x y is a large integral solution of the equation x3 2y3 1 then

relatively weakly open sets in closed balls of banach spaces and real jb triples of finite rank we prove that given a real jb triple x there exists a nonempty relatively weakly open subset of the closed unit ball of x with diameter less than 2 if and only if the banach space of x is isomorphic to a hilbert space moreover we give the structure of real jb triples whose banach spaces are isomorphic to hilbert spaces such real jb triples are also characterized in two different purely algebraic ways

the daugavet property of c algebras jb triples and of their isometric preduals a banach space x is said to have the daugavet property if every rank one operator t x x satisfies id t 1 t we give geometric characterizations of this property in the settings of c algebras jb triples and their isometric preduals we also show that in these settings the daugavet property passes to ultrapowers and thus it is equivalent to an stronger property called the uniform daugavet property

measuring linearity of planar point sets our goal is to design algorithms that give a linearity measure for planar point sets there is no explicit discussion on linearity in literature although some existing shape measures may be adapted we are interested in linearity measures which are invariant to rotation scaling and translation these linearity measures should also be calculated very quickly and be resistant to protrusions in the data set the measures of eccentricity and contour smoothness were adapted from literature the other five being triangle heights triangle perimeters rotation correlation average orientations and ellipse axis ratio the algorithms are tested on 30 sample curves and the results are compared against the linear classifications of these curves by human subjects it is found that humans and computers typically easily identify sets of points that are clearly linear and sets of points that are clearly not linear they have trouble measuring sets of points which are in the gray area in between although they appear to be conceptually very different approaches we prove theoretically and experimentally that eccentricity and rotation correlation yield exactly the same linearity measurements they however provide results which are furthest from human measurements the average orientations method provides the closest results to human perception while the other algorithms proved themselves to be very competitive

orientation and anisotropy of multi component shapes from boundary information we define a method for computing the orientation of compound shapes based on boundary information the orientation of a given compound shape s is taken as the direction a that maximises the integral of the squared length of projections of all the straight line segments whose end points belong to particular boundaries of components of s to a line that has the slope a just as the concept of orientation can be extended from single component shapes to multiple components elongation can also be applied to multiple components and we will see that it effectively produces a measure of anisotropy since it is maximised when all components are aligned in the same direction the presented method enables a closed formula for an easy computation of both orientation and anisotropy

separate weak continuity of the triple product in dual real jb triples we prove that if e is a real jb triple having a predual e then e is the unique predual of e and the triple product on e is separately sigma e e continuous

sur la non densite des points entiers we give non density results for integral points on affine varieties inthe spirit of lang vojta conjecture in particular let x be a projective variety of dimension d 2 over a number field k resp over c let h be the sum of 2d properly intersecting ample divisors on x we show that any set of quasi integral points resp any integral curve on x h is not zariski dense resume on donne des resultats de non densite pour les points entiers sur des varietes affines dans l esprit de la conjecture de lang vojta en particulier soit x une variete pro jective de dimension d 2 sur un corps de nombres k resp sur c soit h la somme de 2d diviseurs amples sur x qui se coupent proprement on montre que tout ensemble de points quasi entiers resp toute courbe entiere sur x h est non zariski dense

complete metric spaces with property z are length spaces abstract a metric space m is length when every two points have an approximate midpoint while m has property z when for every x y m there exists z such that d x z d z y d x y is arbitrarily smaller than d x z and d z y answering a problem posed by garcia lirola prochazka and rueda zoca we prove that every complete metric space with property z is length as a consequence we get that for the lipschitz free space of a metric space a number of geometric properties are all equivalent like the daugavet property the diameter 2 property the absence of strongly exposed points or the absence of preserved extreme points

fractal dimension of time indexed paths abstract measuring animal track or other trajectories for example the track of a flowing particle by divider method to obtain the fractal dimension without knowing the chronology of the points of the path one can obtain erroneous results the importance of time indexing of the track s point will be demonstrated by a simply geometrical example neglecting these indices the error can be high enough to mask relevant differences between various movement paths

characterisation of the weak star symmetric strong diameter 2 property in lipschitz spaces abstract we give a characterisation of the weak symmetric strong diameter 2 property for lipschitz function spaces in terms of a property of the corresponding metric space using this characterisation we show that the weak symmetric strong diameter 2 property is different from the weak strong diameter 2 property in lipschitz spaces thereby answering a question posed in a recent paper by haller langemets lima and nadel

the measurement of circularity and ellipticity on a digital grid abstract a new method is presented based on the discrete fourier transform of an ordered sequence of boundary points for measuring the circularity and ellipticity of object boundaries together with a parametric description of their best fitting circles or ellipses techniques for rapid computation are discussed an analysis of quantization error with verification by computer experiment is presented

holomorphic curves into algebraic varieties this paper establishes a defect relation for algebraically nondegenerate holomorphic mappings into an arbitrary nonsingular complex projective variety v rather than just the projective space intersecting possible nonlinear hypersurfaces extending the result of h cartan

geometric moment invariants this paper derives moment invariants in an intuitive way by multiple integrals of invariant geometric primitives like distance area and volume many existing 2 d and 3 d moment invariants are re expressed to prove the correctness of this method furthermore this construction method can be easily extended to higher dimensional space and get higher order moment invariants explicit expressions of 3 d moment invariants are generated automatically with the help of symbolic computation software and can be regarded as a kind of shape descriptor for the representation of solid objects

integral invariants for shape matching for shapes represented as closed planar contours we introduce a class of functionals which are invariant with respect to the euclidean group and which are obtained by performing integral operations while such integral invariants enjoy some of the desirable properties of their differential counterparts such as locality of computation which allows matching under occlusions and uniqueness of representation asymptotically they do not exhibit the noise sensitivity associated with differential quantities and therefore do not require presmoothing of the input shape our formulation allows the analysis of shapes at multiple scales based on integral invariants we define a notion of distance between shapes the proposed distance measure can be computed efficiently and allows warping the shape boundaries onto each other its computation results in optimal point correspondence as an intermediate step numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts missing parts and noise as a quantitative analysis we report matching scores for shape retrieval from a database

on a general thue s equation we analyze the integral points on varieties defined by one equation of the form f 1 f r g where the f i g are polynomials in n variables with algebraic coefficients and g has small degree we shall use a method that we recently introduced in the context of siegel s theorem for integral points on curves classical very particular instances of our equations arise e g in a well known corollary of roth s theorem the case n 2 f i linear forms deg g r 2 and with the norm form equations treated by w m schmidt here we shall prove thm 1 that the integral points are not zariski dense provided deg f i n max deg f i deg g and provided the f i g satisfy certain mild assumptions which are generically verified our conclusions also cover certain complete intersection subvarieties of our hypersurface thm 2 finally we shall prove thm 3 an analogue of the schmidt s subspace theorem for arbitrary polynomials in place of linear forms

generalizations of siegel s and picard s theorems we prove new theorems that are higher dimensional generalizations of the classical theorems of siegel on integral points on affine curves and of picard on holomorphic maps from c to affine curves these include results on integral points over varying number fields of bounded degree and results on kobayashi hyperbolicity we give a number of new conjectures describing from our point of view how we expect siegel s and picard s theorems to optimally generalize to higher dimensions in this article we prove new theorems that are higher dimensional generaliza tions of the classical theorems of siegel on integral points on affine curves and of picard on holomorphic maps from c to affine curves in section 2 we will give the statements of siegel s and picard s theorems and we will recall how these two theorems from such seemingly different areas of mathematics are related we will then proceed to give a number of new conjectures describing from our point of view how we expect siegel s and picard s theorems to optimally generalize to higher dimensions these include conjectures on integral points over varying number fields of bounded degree and conjectures addressing hyperbolic questions these conjectures appear to be fundamentally new we will then summarize our progress on these conjectures we have been able to get results in all dimensions with best possible results in many cases for surfaces

holomorphic curves into algebraic varieties intersecting divisors in subgeneral position recently there are many developments on the second main theorem for holomorphic curves into algebraic varieties intersecting divisors in general position or subgeneral position in this paper we refine the concept of subgeneral position by introducing the notion of the index of subgeneral position with this new notion we give some surprising improvement of the previous known second main theorem type results moreover via the analogue between nevanlinna theory and diophantine approximation the corresponding schmidt s subspace type theorems are also established in the final section

almost square banach spaces abstract we single out and study a natural class of banach spaces almost square banach spaces in an almost square space we can find given a finite set x 1 x 2 x n in the unit sphere a unit vector y such that x i y is almost one these spaces have duals that are octahedral and finite convex combinations of slices of the unit ball of an almost square space have diameter 2 we provide several examples and characterizations of almost square spaces we prove that non reflexive spaces which are m ideals in their biduals are almost square we show that every separable space containing a copy of c 0 can be renormed to be almost square a local and a weak version of almost square spaces are also studied

a characterisation of the daugavet property in spaces of lipschitz functions abstract we study the daugavet property in the space of lipschitz functions lip 0 m on a complete metric space m namely we show that lip 0 m has the daugavet property if and only if m is a length metric space this condition also characterises the daugavet property in the lipschitz free space f m moreover when m is compact we show that either f m has the daugavet property or its unit ball has a strongly exposed point if m is an infinite compact subset of a strictly convex banach space then the daugavet property of lip 0 m is equivalent to the convexity of m

notes on shape orientation where the standard method does not work in this paper we consider some questions related to the orientation of shapes with particular attention to the situation where the standard method does not work there are irregular and non symmetric shapes whose orientation cannot be computed in a standard way but in the literature the most studied situations are those where the shape under consideration has more than two axes of symmetry or where it is an n fold rotationally symmetric shape with n 2 the basic reference for our work is w h tsai s l chou detection of generalized principal in rotationally symmetric shapes pattern recognition 24 1991 95 104 we give a very simple proof of the main result from w h tsai s l chou detection of generalized principal in rotationally symmetric shapes pattern recognition 24 1991 95 104 and suggest a modification of the proposal on how the principal axes of rotationally symmetric shapes should be computed we show some desirable property in defining the orientation of such shapes if the modified approach is applied also we give some comments on the problems that arise when computing shape elongation

points of differentiability of the norm in lipschitz free spaces abstract we consider convex series of molecules in lipschitz free spaces i e elements of the form n n x n y n d x n y n such that n n we characterise these elements in terms of geometric conditions on the points x n y n of the underlying metric space and determine when they are points of g teaux differentiability of the norm in particular we show that g teaux and frechet differentiability are equivalent for finitely supported elements of lipschitz free spaces over uniformly discrete and bounded metric spaces and that their tensor products with g teaux resp frechet differentiable elements of a banach space are g teaux resp frechet differentiable in the corresponding projective tensor product

a hu moment invariant as a shape circularity measure in this paper we propose a new circularity measure which defines the degree to which a shape differs from a perfect circle the new measure is easy to compute and being area based is robust e g with respect to noise or narrow intrusions also it satisfies the following desirable properties it ranges over 0 1 and gives the measured circularity equal to 1 if and only if the measured shape is a circle it is invariant with respect to translations rotations and scaling compared with the most standard circularity measure which considers the relation between the shape area and the shape perimeter the new measure performs better in the case of shapes with boundary defects which lead to a large increase in perimeter and in the case of compound shapes in contrast to the standard circularity measure the new measure depends on the mutual position of the components inside a compound shape also the new measure performs consistently in the case of shapes with very small i e close to zero measured circularity it turns out that such a property enables the new measure to measure the linearity of shapes in addition we propose a generalisation of the new measure so that shape circularity can be computed while controlling the impact of the relative position of points inside the shape an additional advantage of the generalised measure is that it can be used for detecting small irregularities in nearly circular shapes damaged by noise or during an extraction process in a particular image processing task

octahedrality in lipschitz free banach spaces the aim of this note is to study octahedrality in vector valued lipschitz free banach spaces on a metric space under topological hypotheses on it by analysing the weak star strong diameter 2 property in lipschitz function spaces also we show an example that proves that our results are optimal and that octahedrality in vector valued lipschitz free banach spaces actually relies on the underlying metric space as well as on the banach one

octahedral norms in duals and biduals of lipschitz free spaces abstract we continue with the study of octahedral norms in the context of spaces of lipschitz functions and in their duals first we prove that the norm of f m is octahedral as soon as m is unbounded or is not uniformly discrete further we prove that a concrete sequence of uniformly discrete and bounded metric spaces k m satisfies that the norm of f k m is octahedral for every m finally we prove that if x is an arbitrary banach space and the norm of lip 0 m is octahedral then the norm of lip 0 m x is octahedral these results solve several open problems from the literature

shape ellipticity from hu moment invariants in this paper we derive an explicit formula which uses the first two hu moment invariants to compute a shape ellipticity measure i e to evaluate how much a planar shape differs from an ellipse the ellipticity measure computed by this formula is invariant with respect to translation rotation and scaling transformations also the highest possible value is obtained if and only if the shape considered is an ellipse several experiments are also provided to confirm the theoretical observations a by product of the derivations made in this paper is an implicit interpretation of geometric shape meaning of the second hu moment invariant a formula which connects the shape ellipticity and the first hu moment invariant both having a well understood behavior and the second hu moment invariant is derived

on cartan s theorem and cartan s conjecture we give a mild generalization of cartan s theorem on value distribution for a holomorphic curve in projective space relative to hyperplanes this generalization is used to complete the proof of the following theorem claimed in an earlier paper by the author given hyperplanes in projective space in general position there exists a finite union of proper linear subspaces such that all holomorphic curves not contained in that union even linearly degenerate curves satisfy the inequality of cartan s theorem except for the ramification term in addition it is shown how these methods can lead to a shorter proof of nochka s theorem on cartan s conjecture and in the number field case how nochka s theorem gives a short proof of wirsing s theorem on approximation of algebraic numbers by algebraic numbers of bounded degree