uniform q series asymptotics for staircase polygons we present a uniform asymptotic expansion for the area perimeter generating function of staircase polygons by calculating the asymptotic behaviour of the alternating q series 1 phi 1 0 y q x as q to 1 from a new integral representation this leads to a direct calculation of the scaling function for this model

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

a chain code for representing 3d curves abstract a chain code for representing three dimensional 3d curves is defined any 3d continuous curve can be digitalized and represented as a 3d discrete curve this 3d discrete curve is composed of constant straight line segments thus the chain elements represent the orthogonal direction changes of the constant straight line segments of the discrete curve the proposed chain code only considers relative direction changes which allows us to have a curve descriptor invariant under translation and rotation also this curve descriptor may be starting point normalized for open and closed curves and invariant under mirroring transformation in the content of this work the main characteristics of this chain code are presented this chain code is inspired by the work of guzman mcc technical report number aca 254 87 july 13 1987 for representing 3d stick bodies finally we present some results of this chain code to represent and process 3d discrete curves as linear features over the terrain by means of digital elevation model dem data also we use this chain code for representing solids composed of voxels thus each solid represents a dem which is described by only one chain

q linear approximants scaling functions for polygon models the perimeter and area generating functions of exactly solvable polygon models satisfy q functional equations where q is the area variable the behaviour in the vicinity of the point where the perimeter generating function diverges can often be described by a scaling function we develop the method of q linear approximants in order to extract the approximate scaling behaviour of polygon models when an exact solution is not known we test the validity of our method by approximating exactly solvable q linear polygon models this leads to scaling functions for a number of q linear polygon models notably generalized rectangles ferrers diagrams and stacks

quantum groups via hall algebras of complexes we describe quantum enveloping algebras of symmetric kac moody lie algebras via a nite eld hall algebra construction involving z2 graded complexes of quiver representations

exact scaling functions for self avoiding loops and branched polymers it is shown that a recently conjectured form for the critical scaling function for planar self avoiding polygons weighted by their perimeter and area also follows from an exact renormalization group flow into the branched polymer problem combined with the dimensional reduction arguments of parisi and sourlas the result is generalized to higher order multicritical points yielding exact values for all their critical exponents and exact forms for the associated scaling functions

the bridgeland s ringel hall algebra associated to an algebra with global dimension at most two abstract for any finite dimensional associative algebra with global dimension 2 we show that there is an embedding from the twisted ringel hall algebra to the bridgeland s ringel hall algebra in particular this result is true for tilted algebras and canonical algebras

triangulated categories and kac moody algebras by using the ringel hall algebra approach we find a lie algebra arising in each triangulated category with t 2 1 where t is the translation functor in particular the generic form of the lie algebras determined by the root categories the 2 period orbit categories of the derived categories of finite dimensional hereditary associative algebras gives a realization of all symmetrizable kac moody lie algebras

natural representations for straight lines and the hough transform on discrete arrays the grid of discrete sampled data points in a digital image supports a limited set of lines at angles and displacements natural to that grid the effect of this implicit line quantization on the parametrization of the hough transform is presented the function describing the discrete hough transforms line detection sensitivity is derived expressions for the orientation frequency and popularity of lines in the natural set are given the results obtained are of importance to data arrays of small size the distribution of lines in the natural set is also important as it determines the precision and reliability with which straight lines can be measured on a discrete imaging array from the natural line set concept a general a d slope offset straight line parametrization is developed for which the hough transform is compact and fast to compute and which is as easy to interpret as the class p theta parametrization

directed compact percolation cluster size and hyperscaling exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice the corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained the work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice a duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve the moments of both distributions have critical exponents which satisfy a constant gap hypothesis with gap exponents nu 2 and delta 3 corresponding to the scaling length and scaling size respectively the usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced by delta nu d nu perpendicular to where nu perpendicular to is the exponent corresponding to the cluster width and d is the number of transverse dimensions 1 for the square lattice

an optimal parallel algorithm for digital curve segmentation first we give an optimal erew pram algorithm that finds an unknown discrete monotone function with domain and range of size n in o log n time using o n independent threshold queries of kind x y here independent means that simultaneous queries always refer to mutually disjoint values x and y this is used for solving within the same resources a certain segmentation problem for words over semigroups the classical problem of partitioning a digital curve into a minimum number of digital line segments which is of interest in digital image processsing turns out to be a special case of this and can therefore be solved in o log n time using o n work on an erew pram this strengthens and generalizes all known algorithmic results about digital curve segmentation as a further prerequisite we use the dorst smeulders parametrization of digital line segments

exact solution of the partially directed compact lattice animal model the authors calculate exactly the number generating function for partially directed compact lattice animals on the square lattice for the fully directed model they report results of new numerical studies

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

meaningful scales detection along digital contours for unsupervised local noise estimation the automatic detection of noisy or damaged parts along digital contours is a difficult problem since it is hard to distinguish between information and perturbation without further a priori hypotheses however solving this issue has a great impact on numerous applications including image segmentation geometric estimators contour reconstruction shape matching or image edition we propose an original strategy to detect what the relevant scales are at which each point of the digital contours should be considered it relies on theoretical results of asymptotic discrete geometry a direct consequence is the automatic detection of the noisy or damaged parts of the contour together with its quantitative evaluation or noise level apart from a given maximal observation scale the proposed approach does not require any parameter tuning and is easy to implement we demonstrate its effectiveness on several datasets we present different direct applications of this local measure to contour smoothing and geometric estimators whose algorithms initially required a noise scale parameter to tune they show the pertinence of the proposed measure for digital shape analysis and reconstruction

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

border and surfacetracing theoretical foundations in this paper we define and study digital manifolds of arbitrary dimension and provide in particular a general theoretical basis for curve or surface tracing in picture analysis the studies involve properties such as the one dimensionality of digital curves and n 1 dimensionality of digital hypersurfaces that makes them discrete analogs of corresponding notions in continuous topology the presented approach is fully based on the concept of adjacency relation and complements the concept of dimension as common in combinatorial topology this work appears to be the first one on digital manifolds based on a graph theoretical definition of dimension in particular in the n dimensional digital space a digital curve is a one dimensional object and a digital hypersurface is an n 1 dimensional object as it is in the case of curves and hypersurfaces in the euclidean space relying on the obtained properties of digital hypersurfaces we propose a uniform approach for studying good pairs defined by separations and obtain a classification of good pairs in arbitrary dimension we also discuss possible applications of the presented definitions and results

efficiency of characterizing ellipses and ellipsoids by discrete moments in this paper our studies are focused on ellipses and problems related to their representation and reconstruction from the data resulting from their digitization the main result of the paper is that a finite number of discrete moments corresponded to digital ellipses is in one to one correspondence with digital ellipses which enables coding of digital ellipses with an asymptotically optimal amount of memory in addition the problem of reconstruction based on the same parameters is considered since the digitization of real shapes causes an inherent loss of information about the original objects the precision of the original shape estimation from the corresponding digital data is limited we derive a sharp upper bound for the errors in reconstruction of the center position and half axes of the ellipse in function of the applied picture resolution i e the number of pixels per unit an extension of these results to the 3d case is also given

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

exact enumeration of parallel walks on directed lattices mutually avoiding pairs of parallel walks on a number of d dimensional lattices are mapped onto undirected random walks which return to the origin on a projected d 1 dimensional lattice generating functions for the number of such pairs of a given length are thus expressed in terms of standard green s functions directed lattices considered include the directed hypercubic and body centred hypercubic the generating function for a directed triangular lattice is also obtained this work is a generalization of known results for the square lattice

root categories and simple lie algebras abstract by using the t 2 orbit category of the derived category of a hereditary algebra which is proved to be a triangulated category too we give a complete realization of a simple lie algebra this is a global version of ringel s work for the positive parts of simple lie algebras in this realization the grothendieck group provides the cartan subalgebra the indecomposable objects correspond to the chevalley basis and the structure constants are given by the evaluations of hall polynomials

directed compact percolation near a wall iii exact results for the mean length and number of contacts existing exact results for the percolation probability and mean cluster size for compact percolation near a dry wall are extended to the mean cluster length and the mean number of contacts with the wall the results are derived from our previous work on vesicles near an attractive wall and involve elliptic integrals as opposed to the simple rational forms found for the percolation probability and cluster size below the results for the cluster length satisfy previously conjectured differential equations a closed expression is conjectured for the mean size above in terms of a hypergeometric function

discrete representation of straight lines if a continuous straight line segment is digitized on a regular grid obviously a loss of information occurs as a result the discrete representation obtained e g a chaincode string can be coded more conveniently than the continuous line segment but measurements of properties such as line length performed on the representation have an intrinsic inaccuracy due to the digitization process in this paper two fundamental properties of the quantization of straight line segments are treated 1 it is proved that every straight chaincode string can be represented by a set of four unique integer parameters definitions of these parameters are given 2 a mathematical expression is derived for the set of all continuous line segments which could have generated a given chaincode string the relation with the chord property is briefly discussed

bridgeland s hall algebras of tilted algebras and lusztig s symmetries abstract for each finite dimensional algebra a of finite global dimension over a finite field bridgeland considered the ringel hall algebra of 2 cyclic complexes of projective a modules and achieved an algebra called bridgeland s hall algebra by taking some localization and reduction in this paper we show that there exist homomorphisms of algebras from quantum groups to bridgeland s hall algebras of tilted algebras in particular for apr tilting this homomorphism provides a realization of lusztig s symmetries of quantum groups via bridgeland s hall algebras

drinfeld double and ringel green theory of hall algebras abstract as a continuation of the work of ringel and green on hall algebras the hopf algebra structure of a extended twisted hall algebra is presented here by introducing the ringel pairing of hall algebras the drinfeld double of a hall algebra is constructed the reduced form of this drinfeld double is applicable to the generic composition algebra which gives a complete realization of quantum group this ringel green theory makes it possible to give the formulae for the computation of commutators in quantum groups in terms of hall algebras

enumeration of directed compact site animals in two dimensions the authors study the problem of directed compact site animals in two dimensions using exact enumeration methods the numerical analysis of the animal number series strongly indicates that the critical exponent theta equals zero they also estimate the non universal growth parameter lambda 2 66185 or 0 00050

estimating shortest paths and minimal distances on digitized three dimensional surfaces abstract the estimation of minimal distances and shortest paths between points on a continuous three dimensional 3d surface given in digitized form is considered the suggested approach combines recently developed length estimators for digitized 3d curves with well known algorithms for computing shortest paths in sparse graphs to estimate the minimal distances and the corresponding shortest paths in a systematic computationally efficient way in comparison to differential geometry techniques for solving the continuous geodesic problem and to computational geometry algorithms for solving the discrete geodesic probelm the suggested hybrid continuous discrete approach is well suited to applications in which continuous surfaces are given in digitized form computation time must be relatively short and an approximate solution is acceptable in particular the suggested method can be used to efficiently obtain a good initial approximation as an input to other algorithms thus improving their speed of convergence and reducing their tendency to converge to insignificant local minima some confusion has recently arisen with respect to the relation between the design of length estimators and the optimization of local distances for distance transformations since 3d length estimation is a central concept in the suggested approach an effort is made to resolve discrepancies and to highlight the similarities and differences between these problems it is shown that while in two dimensions the design of length estimators and the design of local weights for distance transformations are rather similar in three dimensions digital geometry dictates that these problems are different the contexts in which each of the two approaches should be preferred are identified

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

self avoiding polygons on the square lattice we have developed an improved algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 90 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 529 27 1 biased and the critical exponent 0 500 0005 10 unbiased the critical point is indistinguishable from a root of the polynomial 581x4 7x2 13 0 an asymptotic expansion for the coefficients is given for all n there is strong evidence for the absence of any non analytic correction to scaling exponent

critical exponents from nonlinear functional equations for partially directed cluster models we present a method for the derivation of the generating function and computation of critical exponents for several cluster models staircase bar graph and directed column convex polygons as well as partially directed self avoiding walks starting with nonlinear functional equations for the generating function by linearizing these equations we first give a derivation of the generating functions the nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz alternatively taking the continuum limit leads to nonlinear differential equations from which one can extract the scaling function we find that all the above models are in the same universality class with exponents y 1 2 7 1 3 and b 2 3 all models have as their scaling function the logarithmic derivative of the airy function

chain code probabilities and optimal length estimators for digitized three dimensional curves abstract length estimation of three dimensional digitized curves is considered three dimensional 26 directional chain code representation is assumed recent results on three dimensional euclidean distance maps are inapplicable to optimal three dimensional length estimation to support optimal design of three dimensional length estimators chain code probabilities in three dimensional chain encoded digital straight lines are derived optimal three dimensional length estimators are determined and their performance is evaluated analytically and experimentally

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

surface to bulk crossover in directed compact percolation for directed compact percolation in two dimensions the authors conjecture the exact analytical expressions for the percolation probability near a non conducting wall the initial wet site is at a varying distance from the wall which is along the oriented time direction their results allow an explicit evaluation of the surface to bulk crossover in the percolation probability

estimation of the original length of a straight line segment from its digitization in three dimensions abstract estimation of different properties of a continuous predigitized curve from its digitized contour is an important problem with applications in various disciplines of computer science in this context the problem of estimating the original length of a line segment in three dimensions given its quantization in a suitable form is considered four types of characterization schemes for the digitized contour are presented along with a set of estimators the average errors of these estimators are found out experimentally the domains of characterization are formulated for the first three schemes and the theoretical average errors are calculated for the first two the theoretical and the experimental errors are found to tally within computational accuracy

on an exactly solvable compact cluster growth model on a square lattice we consider a cluster growth model on a square lattice that is in the compact directed percolation universality class the model is exactly solvable as regards its static cluster properties and expressions are given for various quantities of interest such as the mean perimeter length and the mean cluster size the results also provide new information on area perimeter generating functions for a class of self avoiding polygons on square lattices

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

directed compact percolation near a wall i biased growth the directed compact percolation cluster model of domany and kinzel 1984 is considered in the presence of a wall which is parallel to the growth direction and hence restricts the lateral growth of the cluster in one direction the critical exponents are found to depend on whether the wall is wet or dry in the former case the model is solved exactly for all the standard percolation functions and the critical behaviour is found to be the same as that for cluster growth with no wall present with this boundary condition the cluster is completely attached to the wall and the model may also be viewed as one of symmetric compact cluster growth in the case of a dry wall the cluster may repeatedly leave and return to the wail as it grows and in this case the percolation probability has been derived exactly by lin 1992 and found to have a critical exponent different from that of the bulk lin s result is rederived and an exact formula for the percolation probability is found for a more general model in which the cluster growth is biased either towards or away from the wall it is found that the unbiased case is special in that any bias away from the wall recovers the bulk critical exponent and a bias towards the wall produces a problem in the same class as the wet wall model

elliptic lie algebras and tubular algebras abstract this article is to study relations between tubular algebras of ringel and elliptic lie algebras in the sense of saito yoshii using the explicit structure of the derived categories of tubular algebras given by happel ringel we prove that the elliptic lie algebra of type d 4 1 1 e 6 1 1 e 7 1 1 or e 8 1 1 is isomorphic to the ringel hall lie algebra of the root category of the tubular algebra with the same type as a by product of our proof we obtain a chevalley basis of the elliptic lie algebra following indecomposable objects of the root category of the corresponding tubular algebra this can be viewed as an analogue of the frenkel malkin vybornov theorem in which they described a chevalley basis for each untwisted affine kac moody lie algebra by using indecomposable representations of the corresponding affine quiver

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

hall algebras hereditary algebras and quantum groups let r be an associative hereditary algebra over a finite field k and let r fin be the full subcategory of r mod whose objects are those left r modules x which are finite as sets ix oc assume also that r isfinitary in c ringel s sense i e that iext l s s c for all simple s s in r fin this condition is met for example if r is finitely generated as k algebra 4 pp 435 436 let be the set of all isomorphism classes in r fin if 2 e then u will denote an r module in class 2 the class of all zero left r modules is denoted 0 let i c be the set of all isomorphism classes of simple modules in r fin thus u i e i is a complete set of simple finite left r modules we identify the grothendieck group k0 r fin with the free abelian group e1 v i v e 7z having i as free basis so that if x e r fin then the corresponding element of k0 r fin is the dimension vector dimx y i vii where for each i e vi is the multiplicity of the simple module u in any composition series of x clearly d imx lies in the subsemigroup

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

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

gift wrapping based preimage computation algorithm based on a classical convex hull algorithm called gift wrapping the purpose of the paper is to provide a new algorithm for computing the vertices of a polytope called preimage roughly the set of naive digital planes containing a finite subset s of z 3 the vertices of the upper hemisphere the ones of the lower hemisphere and at last the equatorial vertices are computed independently the principle of the algorithm is based on duality and especially on the fact that the vertices of the preimage correspond to faces of the input set s or of its chords set email protected email protected 0 0 1 it allows to go from one vertex to another by gift wrapping until the whole region of interest has been explored

scaling function and universal amplitude combinations for self avoiding polygons we analyse new data for self avoiding polygons saps on the square and triangular lattices enumerated by both perimeter and area providing evidence that the scaling function is the logarithm of an airy function the results imply universal amplitude combinations for all area moments and suggest that rooted saps may satisfy a q algebraic functional equation

double hall algebras and derived equivalences abstract we show that the reduced drinfeld double of the ringel hall algebra of a hereditary category is invariant under derived equivalences by associating an explicit isomorphism to a given derived equivalence we also extend the results of burban and schiffmann 5 6 sevenhant and van den bergh 1999 25 and xiao and yang 2001 31

subpixel edge location in binary images using dithering this paper concerns the problem of obtaining subpixel estimates of the locations of straight edges in binary digital images using dithering by adding uniformly distributed independent random noise it is shown that estimation bias may be removed and that the estimation variance is inversely proportional to the length of the line segment the sensitivity to incorrect dither amplitude is calculated and implementation is discussed

iterated floor function algebraic numbers discrete chaos beatty subsequences semigroups for a real number a the floor function laj is the integer part of a the sequence lmaj m 1 2 3 is the beatty sequence of ac identities are proved which express the sum of the iterated floor functional ai for 1 i n operating on a nonzero algebraic number a of degree n in terms of only ai lmcaj m and a bounded term applications include discrete chaos discrete dynamical systems explicit construction of infinite nonchaotic subsequences of chaotic sequences discrete order identities explicit construction of nontrivial beatty subsequences and certain arithmetical semigroups beatty sequences have a large literature in combinatorics they have also been used in nonperiodic tilings quasicrystallography periodic scheduling computer vision digital lines and formal language theory

compact directed percolation with modified boundary rules a forest fire model a modified version of compact directed percolation on a square lattice is examined in the context of a model problem for the spread of forest fires the modification relates to conditioning the extent of the fire spread along the forest boundaries exact expressions are given for the mean perimeter length and the mean size of the damaged forest under such conditioning

directed compact percolation near a wall ii cluster length and size for paper i see ibid vol 27 p 3743 50 1994 the mean cluster size and length for the unbiassed growth of compact clusters near a dry wall are considered in the case of the cluster size below pc an exact expression is obtained for a seed of arbitrary width and distance from the surface it is found that the critical exponent gamma 1 for any finite distance from the surface crossover to the bulk value gamma 2 as the distance from the surface tends to infinity is observed this extends an existing result for the exponent beta of the percolation probability which changes from a value of 2 in the presence of a surface to 1 in the bulk limit the value delta 3 of the scaling size exponent is unchanged by the introduction of the surface the cluster size above pc and the mean cluster length are investigated using differential approximants from which we conjecture that these functions satisfy second order differential equations accepting this conjecture gives a mean size exponent the same as below pc and a logarithmic divergence of the mean length from both sides of the critical point the latter result together with scaling theory predicts that the exponent v sub has the value 2 the same as for the bulk problem

derived categories tilted algebras and drinfel d doubles abstract given a finite dimensional hereditary algebra over a finite field k in the derived category db we obtain some formulae on hall numbers associated to triangles using them we prove that for any tilted algebra of there is an embedding of the twisted ringel hall algebra of into the drinfel d double d of the twisted ringel hall algebra of furthermore if is also hereditary we show that this embedding can be extended as an isomorphism between the two corresponding drinfel d doubles these formulae are also used to construct an automorphism of d directly associated to auslander reiten translation and this automorphism follows by b sevenhant m van den bergh s construction in the case of quivers

algebraic languages and polyominoes enumeration abstract in this paper the use of algebraic languages theory in solving an open problem in combinatorics is shown by constructing a bijection between convex polyominoes and words of an algebraic language and by solving the corresponding algebraic system we prove that the number of convex polyominoes with perimeter 2n 8 is 2n 11 4 n 4 2n 1 2n n

reconstruction of a digital circle abstract since digitization always causes some loss of information reconstruction of the original figure from a given digitization is a challenging task reconstruction of digital circles has already been addressed in the literature however an in depth analysis of an obq image of a continuous circle as well as a solution to its domain construction problem is still lacking in this paper a detailed analysis of digital circles has been carried out a modified i r method is formulated to numerically compute the domain of each digital quarter circle for a given radius several properties of the obq image of a circle reveal that in many cases it is possible to split a digital circle into four digital quarter circles such that the domains of the individual quarter circles can be combined to obtain the domain of the full circle moreover the domain of a quarter circle is geometrically characterized

a note on bridgeland s hall algebra of two periodic complexes we show that the hall algebra of two periodic complexes which is recently introduced by t bridgeland coincides with the drinfeld double of the ordinary hall bialgebra

hall polynomials for the representation finite hereditary algebras let k be a field let r be a finite dimensional k algebra with centre k which is representation finite and hereditary thus r is morita equivalent to the tensor algebra of a k species with underlying graph a a disjoint union of dynkin diagrams and the set of isomorphism classes of indecomposable r modules corresponds bijectively to the set of positive roots of the corresponding semisimple complex lie algebra g see g and drl consequently the grothendieck group k r mod of all finitely generated r modules modulo split exact sequences is the free abelian group with basis indexed by let h be a cartan subalgebra of g and g n h 0 n the corresponding triangular decomposition note that n is the direct sum of one dimensional complex vectorspaces indexed by the elements of so we may identify k r mod q and n as vectorspaces and we deal with the problem of how to recover the lie multiplication of n on k r mod we have shown in r2 that the grothendieck group k r mod may be considered in a natural way as a lie algebra by using as structure constants the evaluations of hall polynomials at 1 the aim of this paper is to show that this lie algebra k r mod can be identified with a chevalley h form of n in particular k r mod and n are isomorphic as lie algebras we are going to determine all possible polynomials which occur as hall polynomials p where x y z e there are precisely 16 different polynomials including the zero polynomial cpo and the absolute value of their evaluations at 1 is bounded by 3 one easily observes that q x 0 in case y z x thus let us assume y z x in this case precisely one of the two polynomials cp and cpzz is non zero the non zero polynomials cp can be written in the form irqi where c i t with 1 d r 6 3 and pi is one of the following 12 integral polynomials

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

exact solution of the staircase and row convex polygon perimeter and area generating function an explicit expression is obtained for the perimeter and area generating function g y z sigma n or 2 sigma m or 1 cn mynzm where cn m is the number of row convex polygons with area m and perimeter n a similar expression is obtained for the area perimeter generating function for staircase polygons both expressions contain q series

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

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

staircase polygons scaling functions and asymmetric compact directed percolation the scaling function for compact directed percolation on a square lattice is investigated for the asymmetric case where two parameters control the critical behaviour a simple representation for the area perimeter generating function for staircase polygons is found which can be recast as a non linear functional equation from this the exact scaling function is extracted in the process the most concise derivations to date are given for the exact low order cluster moments

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

hall algebras associated to triangulated categories by counting triangles and applying the octahedral axiom we find a direct way to prove the formula of toen in t section 5 1 for a triangulated category satisfying the left homological finiteness condition

derived hall algebras the purpose of this work is to define a derived hall algebra mathcal dh t associated to any dg category t under some finiteness conditions our main theorem states that mathcal dh t is associative and unital it is shown that mathcal dh t contains the usual hall algebra mathcal h t when t is an abelian category we will also prove an explicit formula for the derived hall numbers purely in terms of invariants of the triangulated category associated to t as an example we describe the derived hall algebra of an hereditary abelian category

digitized circular arcs characterization and parameter estimation the digitization of a circular arc causes an inherent loss of geometrical information arcs with slightly different local curvature or position may lead to exactly the same digital pattern in this paper the authors give a characterization of all centers and radii of circular arcs yielding the same digitization pattern the radius of the arcs varies over the set however only one curvature or radius estimate can be assigned to the digital pattern the authors derive an optimal estimator and give expressions for the bound on the precision of estimation this bound due to digitization is the deterministic equivalent of the cramer rao bound known from parameter estimation theory consider the estimation of the local curvature and local radius of a smooth object typically such parameters are estimated by moving a window along the digital boundary methods in literature show a poor precision in estimating curvature values relative errors of over 40 are often found from the definition of curvature it follows that locally the curve can be considered a circular arc and hence the method presented in this paper can be applied to the pattern in the window giving estimates with optimal precision and a measure for the remaining error on the practical side the authors present examples of the residual error due to the discrete grid the estimation of the radius or curvature of a circular arc at random position with an estimation window containing 10 points coded with nine freemancodes has a relative deviation exceeding 2 for a full disk the deviation is below 1 when the radius r exceeds four grid units the presented method is particularly useful for problems where some prior knowledge on the distribution of radii is known and where there is a noise free sampling