castelnuovo mumford regularity bounds for singular surfaces we prove the regularity conjecture namely eisenbud goto conjecture for a normal surface with rational gorenstein elliptic and log canonical singularities along the way we bound the regularity for a dimension zero scheme by its loewy length and for a curve allowing embedded or isolated point components by its arithmetic degree

a vanishing theorem for log canonical pairs using inversion of adjunction we deduce from nadel s theorem a vanishing property for ideals sheaves on projective varieties a special case of which recovers a result due to bertram ein lazarsfeld this enables us to generalize to a large class of projective schemes certain bounds on castelnuovo mumford regularity previously obtained by bertram ein lazarsfeld in the smooth case and by chardin ulrich for locally complete intersection varieties with rational singularities our results are tested on several examples

a distortion theorem for biholomorphic mappings in c 2 the classical distortion theorems for families of univalent functions have been studied not later than 1907 when kbe discovered his verzerrangsatz

linear invariance of locally biholomorphic mappings in the unit ball of a jb triple abstract we study the notion of linear invariance on the unit ball of a j b triple x and we obtain some connection between the norm order of a linear invariant family and the starlikeness of order 1 2 also we give some result concerning the radius of univalence of some linear invariant families finally if the dimension of x is finite and if the norm order of a linear invariant family is finite then we prove the normality of the linear invariant family and we also obtain upper bounds on the distortion and the growth of mappings in a linear invariant family with specified norm order in particular our results are valid for the classical cartan domains and the unit polydisc

on the irreducibility of secant cones and an application to linear normality given a smooth subvariety of dimension greater than 2 3 r 1 in mathbb p sp r we show that the double locus upstairs of its generic projection to mathbb p sp r 1 is irreducible this implies a version of zak s linear normality theorem

soft learning vector quantization and clustering algorithms based on non euclidean norms single norm algorithms this paper presents the development of soft clustering and learning vector quantization lvq algorithms that rely on a weighted norm to measure the distance between the feature vectors and their prototypes the development of lvq and clustering algorithms is based on the minimization of a reformulation function under the constraint that the generalized mean of the norm weights be constant according to the proposed formulation the norm weights can be computed from the data in an iterative fashion together with the prototypes an error analysis provides some guidelines for selecting the parameter involved in the definition of the generalized mean in terms of the feature variances the algorithms produced from this formulation are easy to implement and they are almost as fast as clustering algorithms relying on the euclidean norm an experimental evaluation on four data sets indicates that the proposed algorithms outperform consistently clustering algorithms relying on the euclidean norm and they are strong competitors to non euclidean algorithms which are computationally more demanding

suboptimal minimum cluster volume cover based method for measuring fractal dimension a new method for calculating fractal dimension is developed in this paper the method is based on the box dimension concept however it involves direct estimation of a suboptimal covering of the data set of interest by finding a suboptimal cover this method is better able to estimate the required number of covering elements for a given cover size than is the standard box counting algorithm moreover any decrease in the error of the covering element count directly increases the accuracy of the fractal dimension estimation in general our method represents a mathematical dual to the standard box counting algorithm by not solving for the number of boxes used to cover a data set given the size of the box instead the method chooses the number of covering elements and then proceeds to find the placement of smallest hyperellipsoids that fully covers the data set this method involves a variant of the fuzzy c means clustering algorithm as well as the use of the minimum cluster volume clustering algorithm a variety of fractal dimension estimators using this suboptimal covering method are discussed finally these methods are compared to the standard box counting algorithm and wavelet decomposition methods for calculating fractal dimension by using one dimensional cantor dust sets and a set of standard brownian random fractal images

constrained clustering as an optimization method a deterministic annealing approach to clustering is derived on the basis of the principle of maximum entropy this approach is independent of the initial state and produces natural hierarchical clustering solutions by going through a sequence of phase transitions it is modified for a larger class of optimization problems by adding constraints to the free energy the concept of constrained clustering is explained and three examples are are given in which it is used to introduce deterministic annealing the previous clustering method is improved by adding cluster mass variables and a total mass constraint the traveling salesman problem is reformulated as constrained clustering yielding the elastic net en approach to the problem more insight is gained by identifying a second lagrange multiplier that is related to the tour length and can also be used to control the annealing process the open path constraint formulation is shown to relate to dimensionality reduction by self organization in unsupervised learning a similar annealing procedure is applicable in this case as well

generic projections the equations defining projective varieties and castelnuovo regularity for a reduced irreducible projective variety x of degree d and codimension e in mathbb p n the castelnuovo mumford regularity rm reg x is defined as the least k such that x is k regular i e h i mathbb p n mathcal i x k i 0 for i ge1 where mathcal i x subset cal o mathbb p n is the sheaf of ideals of x there is a long standing conjecture about k regularity see 5 rm reg x le d e 1 here we show that rm reg x le d e 1 10 for any smooth fivefold and rm reg x le d e 1 20 for any smooth sixfold by extending methods used in 10 furthermore we give a bound for the regularity of a reduced connected and equidimensional locally cohen macaulay curve or surface in terms of degree d codimension e and an arithmetic genus rho a see theorem 4 1

a time domain constrained fuzzy clustering method and its application to signal analysis this paper introduces a new fuzzy clustering method with time domain constraints which is used to signal analysis proposed method makes it possible to include natural constraints for signal analysis using fuzzy clustering that is the neighboring samples of signal belong to the same cluster this method can be called time domain constrained fuzzy clustering this paper introduces two approaches to include the above kind of constraints the first approach leads to the time domain constrained fuzzy c regression models method the second approach leads to the insensitive version of the above method which results in additional robustness for outliers and non gaussian noise finally simulations on synthetic as well as real life signals are realized to evaluate the performance of the time domain constrained fuzzy clustering methods a comparison with the traditional fuzzy c regression models is also made large scale simulations demonstrate the competitiveness of the proposed methods for signal analysis with respect to the traditional fuzzy clustering methods

on the precise growth covering and distortion theorems for normalized biholomorphic mappings abstract in this paper we consider a normalized biholomorphic mapping f x defined on the unit ball in a complex banach space where the origin 0 is a zero of order k 1 of f x x the precise growth and covering theorem for f x is obtained when f x is a starlike mapping or a starlike mapping of order especially the precise growth and covering theorem for f x is also established when f x is a quasi convex mapping moreover the precise distortion theorem for f x is given when f x is a convex mapping our result includes many known results

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

trace order and a distortion theorem for linearly invariant families on the unit ball of a finite dimensional jb triple abstract we give a distortion theorem for linearly invariant families on the unit ball b of a finite dimensional jb triple x by using the trace order the exponents in the distortion bounds depend on the bergman metric at 0 further we introduce a new definition for the trace order of a linearly invariant family on b based on a jacobian argument we also construct an example of a linearly invariant family on b which has minimum trace order and is not a subset of the normalized convex mappings of b for dim x 2 finally we prove a regularity theorem for linearly invariant families on b all four types of classical cartan domains are the open unit balls of jb triples and the same holds for any finite product of these domains thus the unit balls of jb triples are natural generalizations of the unit disc in c and we have a setting in which a large number of bounded symmetric homogeneous domains may be studied simultaneously

clustering algorithms based on volume criteria clustering algorithms such as the k means algorithm and the fuzzy c means algorithm are based on the minimization of the trace of the fuzzy within fluster scatter matrix in this paper we explore the use of determinant volume criteria for clustering we derive an algorithm called the minimum scatter volume msv algorithm that minimizes the scatter volume and another algorithm called the minimum cluster volume mcv that minimizes the sum of the volumes of the individual clusters the behavior of msv is shown to be similar to that of k means whereas mcv is more versatile

fuzzy clustering with volume prototypes and adaptive cluster merging two extensions to objective function based fuzzy clustering are proposed first the point prototypes are extended to hypervolumes whose size can be fixed or can be determined automatically from the data being clustered it is shown that clustering with hypervolume prototypes can be formulated as the minimization of an objective function second a heuristic cluster merging step is introduced where the similarity among the clusters is assessed during optimization starting with an overestimation of the number of clusters in the data similar clusters are merged in order to obtain a suitable partitioning an adaptive threshold for merging is proposed the extensions proposed are applied to gustafson kessel and fuzzy c means algorithms and the resulting extended algorithm is given the properties of the new algorithm are illustrated by various examples

distortion theorems for biholomorphic convex mappings in cn abstract in the paper the problem of sharp lower estimation for df z in the class of normalized biholomorphic mappings f between the open unit ball b n and convex domains in c n has been considered

generic initial ideals of singular curves in graded lexicographic order abstract in this paper we are interested in the generic initial ideals of singular projective curves with respect to the graded lexicographic order let c be a singular irreducible projective curve of degree d 5 with the arithmetic genus a c in p r where r 3 if m i c is the regularity of the lexicographic generic initial ideal of i c in a polynomial ring k x 0 x r then we prove that m i c is 1 d 1 2 a c which is obtained from the monomial x r 3 x r 1 d 1 2 a c provided that dim tan p c 2 for every singular point p c this number is equal to one plus the number of secant lines through the center of general projection into p 2 our result generalizes the work of j ahn 2008 1 for smooth projective curves and that of a conca and j sidman 2005 9 for smooth complete intersection curves in p 3 the case of singular curves was motivated by a conca and j sidman 2005 9 example 4 3 we also provide some illuminating examples of our results via calculations done with macaulay 2 and singular decker et al 2011 10 grayson and stillman 16

relations among starlikeness convexity and k fold symmetry of locally biholomorphic mappings in cn abstract the subject of the paper concerns the existence of different inclusions between families s t s c of c n biholomorphic starlike or convex mappings and a family s k k 2 of c n locally biholomorphic mappings in the definition of s k we used a combination of a kikuchi matsuno suffridge s starlikeness condition and a property of j k symmetrical mappings in c n this definition generalizes a notion of planar sakaguchi s functions onto n dimensional case a motivation for the family s k comes from the paper 11 of the second author and the paper 6 of hamada and kohr the family s k is a superset of a family which is connected with a problem posed in the first paper and solved in the second one

a castelnuovo mumford regularity bound for scrolls abstract let x p r be a scroll of codimension e and degree d over a smooth projective curve of genus g the purpose of this paper is to prove a linear castelnuovo mumford regularity bound that reg x d e 1 g e 1 this bound works over an algebraically closed field of arbitrary characteristic

robust clustering methods a unified view clustering methods need to be robust if they are to be useful in practice in this paper we analyze several popular robust clustering methods and show that they have much in common we also establish a connection between fuzzy set theory and robust statistics and point out the similarities between robust clustering methods and statistical methods such as the weighted least squares technique the m estimator the minimum volume ellipsoid algorithm cooperative robust estimation minimization of probability of randomness and the epsilon contamination model by gleaning the common principles upon which the methods proposed in the literature are based we arrive at a unified view of robust clustering methods we define several general concepts that are useful in robust clustering state the robust clustering problem in terms of the defined concepts and propose generic algorithms and guidelines for clustering noisy data we also discuss why the generalized hough transform is a suboptimal solution to the robust clustering problem

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

roper suffridge extension operator and the lower bound for the distortion abstract liczberski starkov gave a sharp lower bound for d n f z near the origin where n is the roper suffridge extension operator and f is a normalized convex mapping on the unit disk in c they gave a conjecture that the sharp lower bound holds on the euclidean unit ball b n in c n in this paper we will give a sharp lower bound on b n for a more general extension operator and for normalized univalent mappings f or normalized convex mappings f we will give a lower bound for mappings f in a linear invariant family we will also give a similar sharp lower bound on bounded convex complete reinhardt domains in c n

a note on the gustafson kessel and adaptive fuzzy clustering algorithms in this letter we show that the gustafson kessel g k algorithm 1979 and the original adaptive fuzzy clustering afc algorithm can be thought of as special cases of a more general algorithm our analysis shows that the g k algorithm is better suited for ellipsoidal clusters of equal volume whereas the original afc algorithm is better suited for linear clusters we also discuss a new variation of these algorithms which can be used to improve the results of the g k and afc algorithms in some cases

liaison equivalence classes we will work in ip n 2 over an algebraically closed field the notion of liaison among subvarieties of ip was introduced in 3 we will consider here liaison in codimension two generalizing results in 4 recall therefore that two subschemes v 1 v 2 of ip that are locally cohenmacaulay and equidimensional of codimension two are said to be linked algebraically via the complete intersection x of two hypersurfaces denoted v l v z via x if d e v j x v r o v x t j they are said to be linked geometrically if i 1 and v2 in the above definition have no components in common according to 4 theorem 1 7 the two notions of linkage generate the same equivalence relation among those subschemes v which are also generically locally complete intersections we will hence consider only algebraic linkage

vanishing theorems a theorem of severi and the equations defining projective varieties in 1903 severi s proved that if x x c p are smooth surfaces whose union is the complete intersection of r 2 hypersurfaces of degrees di dr 2 then hypersurfaces of degree k z di r cut out a complete linear series on x cf sr xiii 9 8 the purpose of this paper is first of all to show that elementary arguments using the kodaira vanishing theorem lead to a simple variant of severi s statement theorem 7 below which extends it in serval directions more importantly we hope to convince the reader that this result has a surprising number of applications to questions involving the equations defining projective varieties consider to begin with a smooth complex projective variety x c pr of dimension n and codimension e r n in this setting our theorem asserts the vanishing of the higher cohomology of suitable twists of the ath power of the ideal sheaf yx of x in apr proposition 1 assume that x is cut out scheme theoretically in p by hypersurfaces of degrees d1 d2 d then h p r k 0 for i 1 provided k ad d2 de r note that only the degrees of the first e codim x pr defining equations come into play here when n 2 and a 1 this is a consequence of severi s

distortion of locally biholomorphic bloch mappings on bounded symmetric domains abstract we generalize bonk s distortion theorem on the unit disc in the complex plane to locally biholomorphic mappings on finite dimensional bounded symmetric domains as an application we obtain a lower bound for the bloch constant for various classes of locally biholomorphic bloch mappings

an application from partial sums of e z to a problem in several complex variables abstract let b n r z c n z where is the euclidean norm and for x c n let h x denote the closed convex hull of x in c n in 1990 graham showed that if f is a normalized holomorphic map from bn1 into c n and if f is either an open map or a polynomial map then there is a sharp uniform constant a a given by ae1 a 1 such that h f bn1 bna graham posed the question to find for normalized polynomial maps f of degree m the best constant am so that h f bn1 bnam we answer this question and obtain for each m the sharp constant a m a a ln m 2 1 a m a ln 2 1 a a 1 a m 1 m m we also note that this solution extends an old result of polya and szeg

growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation abstract let b be the unit ball in c n with respect to an arbitrary norm and let f z t be a g loewner chain such that e t f z t z has a zero of order k 1 at z 0 in this paper we obtain growth and covering theorems for f 0 moreover we consider coefficient bounds and examples of mappings in s g k 1 0 b

a sharp bound for the castelnuovo mumford regularity of subspace arrangements abstract we show that the ideal of an arrangement of d linear subspaces of projective space is d regular in the sense of castelnuovo and mumford answering a question of b sturmfels in particular this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d

two point distortion bounds for biholomorphic mappings of the ball in cn abstract lower and upper two point distortion bounds for families f of biholomorphic mappings on the unit ball b of c n are given in terms of the trace order of the linear invariant family generated by f bounds on ratios involving the derivative and jacobian of the mappings in f and the caratheodory distance on b by two point distortion bounds we mean estimates on f b f a for a b b and f f this immediately results in growth bounds for such mappings a contrast is drawn between these bounds and two point distortion bounds in terms of the norm order of the generated linear invariant family as part of our work we develop a lower distortion bound for automorphisms of b that may be of independent interest

distortion theorems for convex mappings on homogeneous balls abstract let b be the open unit ball of a complex banach space x and let b be homogeneous we prove distortion results for normalized convex mappings f b x which generalize various finite dimensional distortion theorems and improve some infinite dimensional ones in particular our results are valid for the open unit balls of complex hilbert spaces and the cartan domains

on the castelnuovo mumford regularity of connected curves in this paper we prove that the regularity of a connected curve is bounded by its degree minus its codimension plus 1 we also investigate the structure of connected curves for which this bound is optimal in particular we construct connected curves of arbitrarily high degree in p 4 having maximal regularity but no extremal secants we also show that any connected curve in p 3 of degree at least 5 with maximal regularity and no linear components has an extremal secant

pluriharmonic mappings in cn and complex banach spaces abstract in this paper we obtain a sufficient condition for pluriharmonic mappings on the euclidean unit ball b n to be univalent sense preserving quasiconformal and bi lipschitz diffeomorphisms on b n and to have linearly connected images also we give a sufficient condition for pluriharmonic mappings on b n to have quasiconformal extensions to c n next we generalize the harmonic schwarz lemma to pluriharmonic mappings of the unit ball b x of a complex banach space x into the unit ball b n of c n with respect to an arbitrary norm further we obtain a generalization of the harmonic schwarz pick lemma to the case of pluriharmonic mappings of the homogeneous unit ball b x of a complex banach space x into the unit ball b n we also obtain a version of the holomorphic schwarz pick lemma for the jacobian determinant on the euclidean unit ball b n to the case of pluriharmonic mappings of the homogeneous unit ball b x into b n in the case that b x is an open subset of c n finally we obtain the landau and the bloch theorems for pluriharmonic or holomorphic mappings on finite dimensional homogeneous unit balls

generalised kernel weighted fuzzy c means clustering algorithm with local information abstract to improve the performance of segmentation for the images corrupted by noise many variants of standard fuzzy c means fcm clustering algorithm have been proposed that incorporate the local spatial neighbourhood information to perform image segmentation among them the kernel weighted fuzzy local information c means kwflicm algorithm gives robust to noise image segmentation results by using local spatial image neighbourhood information it is limited to one dimensional input data i e image intensity in this paper we propose a generalisation of kwflicm gkwflicm that is applicable to m dimensional input data sets the proposed algorithm incorporates neighbourhood information among the m dimensional data which mitigates the disadvantages of the standard fcm clustering algorithm sensitive to noise and outliers poor performance for differently sized clusters and for different density clusters and greatly improves the clustering performance experiments have been performed on several noisy and non noisy data sets as well as natural and real world images to demonstrate the effectiveness efficiency and robustness to noise of the gkwflicm algorithm by comparing it to kernel fuzzy c means kfcm kernel possibilistic fuzzy c means kpfcm fuzzy local information c means flicm and kwflicm

bloch functions on bounded symmetric domains we introduce and characterize bloch functions on bounded symmetric domains which may be infinite dimensional by extending several well known equivalent conditions for bloch functions on the open unit disc u in cc we also generalize a number of results concerning bloch functions on u to bounded symmetric domains given a holomorphic mapping between bounded symmetric domains bxbx and byby we derive criteria for boundedness and compactness of the composition operator c c between the bloch spaces b by b by and b bx b bx extending several known results for finite dimensional domains

growth and distortion theorems for linearly invariant families on homogeneous unit balls in cn abstract let b be a homogeneous unit ball in x c n in this paper we obtain growth and distortion theorems for linearly invariant families f of locally biholomorphic mappings on the unit ball b with finite norm order ord e 1 f we use the euclidean norm for the target space instead of the norm of x because we are able to obtain lower bounds in the two point distortion theorems for linearly invariant families on any homogeneous unit ball in c n we also obtain similar results for affine and linearly invariant families a l i f s of pluriharmonic mappings of the unit ball b into c n again in most of these results we use the euclidean norm for the target space to obtain lower bounds in the two point distortion theorems for a l i f s on b these results are generalizations to homogeneous unit balls of recent results due to graham kohr and pfaltzgraff the authors of this paper and duren hamada and kohr in the last section we consider two point distortion theorems for l i f s and a l i f s on the unit polydisc u n in c n

three themes of syzygies we present three exciting themes of syzygies where major progress was made recently boij soderberg theory stillman s question and syzygies over complete intersections

fuzzy c varieties elliptotypes clustering in reproducing kernel hilbert space abstract fuzzy clustering algorithms are successfully applied to a wide variety of problems such as pattern recognition image analysis modeling and so on the fuzzy c means fcm method is one of the most popular clustering methods based on the minimization of a criterion function however the performance of the fcm method is good only when a data set contains clusters that are approximately the same size and shape in this paper a simple idea will be used to overcome this problem the original input data space will be mapped into the high possibly infinite dimensional feature space f through some nonlinear mapping in this space the data structures will be modeled by the linear varieties or elliptotypes this method is called kernel fuzzy c varieties elliptotypes clustering algorithm performance of the new clustering algorithm is experimentally compared with fcm and fuzzy c varieties elliptotypes methods using synthetic datasets and real life datasets

a bound for castelnuovo mumford regularity by double point divisors abstract let x p r be a non degenerate smooth projective variety of dimension n codimension e and degree d defined over an algebraically closed field of characteristic zero in this paper we first show that reg o x d e and classify the extremal and the next to extremal cases our result reduces the eisenbud goto regularity conjecture for the smooth case to the problem finding a castelnuovo type bound for normality it is worth noting that mccullough peeva recently constructed counterexamples to the regularity conjecture by showing that reg o x is not even bounded above by any polynomial function of d when x is not smooth for a normality bound in the smooth case we establish that reg x n d 2 1 which improves previous results obtained by mumford bertram ein lazarsfeld and noma finally by generalizing mumford s method on double point divisors we prove that reg x d 1 m where m is an invariant arising from double point divisors associated to outer general projections using double point divisors associated to inner projections we also obtain a slightly better bound for reg x under suitable assumptions

isometries between jb triples let z and w be jb triples and let t be a linear isometry from z into w for any z z with z 1 we show that