rational curves and rational singularities we study rational curves on algebraic varieties especially on normal affine varieties endowed with a action for varieties with an isolated singularity covered by a family of rational curves with a general member not passing through the singular point we show that this singularity is rational in particular this provides an explanation of classical results due to h a schwartz and g h halphen on polynomial solutions of the generalized fermat equation

thresholding in a generalized model for translation invariant systems abstract in a previous paper we introduced a generalized model for translation invariant ti operators we considered the space of all maps from an abelian group g to called lg fuzzy sets where is a complete lattice ordered group and we defined ti operators on this space in this paper thresholding is considered in the same general framework and in this regard positive ti operators are studied the threshold of an lg fuzzy set a at level t is defined to be the set g g a g t also a ti operator is defined to be positive if its kernel consists of elements which are positive except possibly at some points which are as the main result of this note it is proved that a positive and isotone ti operator commutes with thresholding if and only if it has a crisp kernel or base also the theorem is used to derive a similar result for an operator whose kernel is bounded from below

reconstruction in a generalized model for translation invariant systems abstract we consider translation invariant ti operators on the set of maps from an abelian group g to called lg fuzzy sets where 0 is a complete lattice ordered group by defining minkowski and morphological operations on and considering order preserving operators we prove a reconstruction theorem this theorem which is called the strong reconstruction theorem srt is similar to the convolution theorem in the theory of linear and shift invariant systems and states that for an order preserving ti operator y one can explicitly compute y a for any a from a specific subset of called the base of y the introduced framework is a general model for the theory of translation invariant systems and srt shows the consistency of it for the special cases when g r z srt implies the results of maragos and schafer

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

regularity on abelian varieties i we introduce the notion of mukai regularity m regularity for coherent sheaves on abelian varieties the definition is based on the fourier mukai transform and in a special case depending on the choice of a polarization it parallels and strengthens the usual castelnuovo mumford regularity mukai regularity has a large number of applications ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties to higher dimensional type statements and to a study of special classes of vector bundles some of these applications are explained here while others are the subject of upcoming sequels

simplicial differential forms with poles shadow forms consider a simplicial complex k whitney associated to each simplicial cochain 4 a differential form v 4 on k whitney s map v from simplicial cochains to differential forms is a map of cochain complexes which induces the derham isomorphism we call the forms v 4 whitney forms they have been useful in several contexts most notably integral geometry and rational homotopy theory in this paper we introduce a generalization of whitney forms called shadow forms we associate to certain simplicial chains g with respect to the first derived subdivision k of k a shadow form w on k let us illustrate the construction of shadow forms when k is a single n simplex which we now denote by a in this case w g is defined for k simplices g of ax which do not lie in the boundary of a or for linear combinations of these for example if a is the 2 simplex then either of the 1 simplices or 2 pictured below has an associated shadow form

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

chattering avoidance by second order sliding mode control relying on the possibility of generating a second order sliding motion by using as control the first derivative of the control signal instead of the actual control a new solution to the problem of chattering elimination in variable structure control vsc is presented such a solution inspired by the classical bang bang optimal control strategy is first depicted and expressed in terms of a control algorithm by introducing a suitable auxiliary problem involving a second order uncertain system with unavailable velocity then the applicability of the algorithm is extended via suitable modifications to the case of nonlinear systems with uncertainties of more general types the proposed algorithm does not require the use of observers and differential inequalities and can be applied in practice by exploiting such commercial components as peak detectors or other approximated methods to evaluate the change of the sign of the derivative of the quantity accounting for the distance to the sliding manifold

on the geography of threefolds of general type abstract let x be a complex nonsingular projective 3 fold of general type we show that there are positive constants c c and m 1 such that x c vol x and p m x c m 3 vol x for all m m 1

morphological representation of discrete and binary images a general theory for the morphological representation of discrete and binary images is presented the basis of this theory relies on the generation of a set of nonoverlapping segments of an image via repeated erosions and set transformations which in turn produces a decomposition that guarantees exact reconstruction the relationship between the proposed representation and some existing shape analysis tools e g discrete size transform pattern spectrum skeletons is investigated thus introducing the representation as the basis of a unified theory for geometrical image analysis particular cases of the general representation scheme are shown to yield a number of useful image decompositions which are directly related to various forms of morphological skeletons the relationship between the representation and the various forms of morphological skeletons is studied as a result of this study a unified theory is developed for the mathematical description of the morphological skeleton decomposition of discrete and binary images

design of continuous twisting algorithm abstract for the double integrator with matched lipschitz disturbances we propose a continuous homogeneous controller providing finite time stability of the origin the disturbance is compensated exactly in finite time using a discontinuous function through an integral action since the controller is dynamic the closed loop is a third order system that achieves a third order sliding mode in the steady state the stability and robustness properties of the controller are proven using a smooth and homogeneous strict lyapunov function lf in a first stage the gains of the controller and the lf are designed using a method based on polya s theorem in a second stage the controller s gains are adjusted through a sum of squares representation of the lf

relative monodromy weight filtrations in this paper we establish the existence of relative weight filtrations of cones of monodromy logarithms and identities between relative weight filtrations in a context more general than m kashiwara s infinitesimal mixed hodge modules this generalizes a theorem of m kashiwara and adds conceptual clarity to the problem our approach to working with relative weight filtrations w n w is to split the weight filtrations w with canonical splittings introduced by p deligne these splittings agree with the splittings that occur in the mathfrak sl 2 theory of variations of pure hodge structure of w schmid and of e cattani a kaplan and w schmid but are more general as they don t require filtrations w to be monodromy weight filtrations

discrete differentiators based on sliding modes abstract sliding mode based differentiation of the input f t yields exact estimations of the derivatives f f n provided an upper bound l t of f n 1 t is available in real time in practice it involves discrete sampling and numerical integration of the internal variables between the measurements accuracy asymptotics of different discretization schemes are calculated for discrete noisy sampling whereas sampling and integration steps are independently variable or constant proposed discrete differentiators restore the optimal accuracy asymptotics of their continuous time counterparts event triggered sampling is considered extensive numeric experiments are presented and analyzed

sliding order and sliding accuracy in sliding mode control the synthesis of a control algorithm that stirs a nonlinear system to a given manifold and keeps it within this constraint is considered usually what is called sliding mode is employed in such synthesis this sliding mode is characterized in practice by a high frequency switching of the control it turns out that the deviation of the system from its prescribed constraints sliding accuracy is proportional to the switching time delay a new class of sliding modes and algorithms is presented and the concept of sliding mode order is introduced these algorithms feature a bounded control continuously depending on time with discontinuities only in the control derivative it is also shown that the sliding accuracy is proportional to the square of the switching time delay

morphological filters part i their set theoretic analysis and relations to linear shift invariant filters this paper examines the set theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations which involve supremum infimum operations and additions binary signals are classified as sets and multilevel signals as functions two set theoretic representations of signals are reviewed filters are classified as set processing sp or function processing fp conditions are provided for certain fp filters that pass binary signals to commute with signal thresholding because then they can be analyzed and implemented as sp filters the basic morphological operations of set erosion dilation opening and closing are related to minkowski set operations and are used to construct fp morphological filters emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions the latter case allows the definition of morphological convolutions and correlations toward this goal various properties of fp morphological filters are also examined linear shift invariant filters due to their translation invariance are uniquely characterized by their kernel which is a special collection of input signals increasing linear filters are represented as the supremum of erosions by their kernel functions if the filters are also discrete and have a finite extent impulse response they can be represented as the supremum of erosions only by their minimal with respect to a signal ordering kernel functions stable linear filters can be represented as the sum of at most two weighted suprema of erosions these results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal shaping strategies

adaptive continuous higher order sliding mode control this paper is concerned with the development of an adaptation structure which can be applied to conventional super twisting and higher order sliding mode schemes the objective is to alter the modulation gains associated with these schemes in such a way that they are as small as possible to mitigate chattering effects but large enough to ensure that sliding can be maintained in the presence of bounded and derivative bounded uncertainties in all the proposed schemes the equivalent control is used to drive the adaptive mechanism the approach is based on a novel dual layer nested adaptive methodology which is quite different to the existing schemes proposed in the sliding mode literature the new adaptive schemes do not require knowledge of the minimum and maximum allowed values of the adaptive gain and in their most general form do not require information about the bound on the disturbances and their derivatives

a kollar type vanishing theorem let f x rightarrow y be a smooth fibration between two complex manifolds x and y and let l be a pseudo effective line bundle on x we obtain a sufficient condition for r q f k x y otimes l to be reflexive and hence derive a kollar type vanishing theorem

the decomposition theorem perverse sheaves and the topology of algebraic maps we give a motivated introduction to the theory of perverse sheaves culminating in the decomposition theorem of beilinson bernstein deligne and gabber a goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties while most proofs are omitted we discuss several approaches to the decomposition theorem indicate some important applications and examples

polarized mixed hodge structures and the local monodromy of a variation of hodge structure a variation of polarized hodge structures on the complement of a divisor with normal crossings gives rise locally on the divisor to a commuting set ni of nilpotent endomorphisms of the vector space underlying the variation the logarithms of the unipotent parts of the various picard lefschetz transformations these reflect properties of the singularities of the period map associated to the variation cf 5 7 9 and sect 3 for example a variation depending on a single parameter defines asymptotically a mixed hodge structure whose weight filtration is the monodromy weight filtration of the corresponding endomorphism n 9 11 this paper is mainly concerned with the properties of the set n arising in the several parameters case in particular theorem 3 3 asserts that all the elements in the open polyhedral cone c spanned over r by the n s determine the same monodromy weight filtration it also describes the relationship between this common filtration and those associated to the faces of the cone this statement was conjectured by deligne based on his analogous result 4 1 9 2 for the q cone associated to those variations which arise from families of polarized non singular algebraic varieties in w 1 we discuss some general properties of commuting sets of nilpotent endomorphisms and their associated filtrations section2 focuses on the case when each element in such set defines in the sense of 2 4 a polarized mixed hodge structure with a fixed hodge filtration f as is the case for the monodromy cone c of a variation these conditions imply cf 2 16 a generic uniqueness for the monodromy weight filtration we also consider those pola

quantum k theory of grassmannians we show that equivariant k theoretic 3 point gromov witten invariants of genus zero on a grassmann variety are equal to triple intersec tions computed in the equivariant k theory of a two step flag manifold thus generalizing an earlier result of buch kresch and tamvakis in the process we show that the gromov witten variety of curves passing through 3 general points is irreducible and rational our applications include pieri and giambelli formulas for the quantum k theory ring of a grassmannian which determine the multiplication in this ring we also compute the dual schubert basis for this ring and show that its structure constants satisfy s3 symmetry our for mula for gromov witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of chaput manivel and perrin

a sharp lower bound for the canonical volume of 3 folds of general type let v be a smooth projective 3 fold of general type denote by k 3 a rational number the self intersection of the canonical sheaf of any minimal model of v one defines k 3 as the canonical volume of v assume p g v 2 we show that k 3 ge frac 1 3 which is a sharp lower bound then we classify those v with small volume we also give some new examples with p g 2 which have maximal canonical stability index finally we give an application to 4 folds of general type

homogeneity approach to high order sliding mode design it is shown that a general uncertain single input single output regulation problem is solvable only by means of discontinuous control laws giving rise to the so called high order sliding modes the homogeneity properties of the corresponding controllers yield a number of practically important features in particular finite time convergence is proved and asymptotic accuracy is calculated in a very general way in the presence of input noises discrete measurements and switching delays a robust homogeneous differentiator is included in the control structure thus yielding robust output feedback controllers with finite time convergence it is demonstrated that homogeneity features significantly simplify the design and investigation of a new family of high order sliding mode controllers finally simulation results are presented

simple homogeneous sliding mode controller high order sliding mode hosm control is known to provide for finite time exact output regulation of uncertain systems with known relative degrees yet the corresponding universal hosm controllers are typically constructed by special recursive procedures and have complicated form we propose two new families of homogeneous hosm controllers of a very simple form lyapunov functions are provided for a significant part of the first family controllers the second family consists of quasi continuous controllers which can be done arbitrarily smooth everywhere outside of the hosm manifold a regularization procedure ensures high accuracy output regulation by means of control with required smoothness level output feedback controllers are constructed controllers of the orders 3 5 are demonstrated

deformation theory of singular symplectic n folds by a symplectic manifold or a symplectic n fold we mean a compact kaehler manifold of even dimension n with a non degenerate holomorphic 2form i e is a nowhere vanishing n form this notion is generalized to a variety with singularities we call x a projective symplectic variety if x is a normal projective variety with rational gorenstein singularities and if the regular locus u of x admits a non degenerate holomorphic 2 form a symplectic variety will play an important role together with a singular calabi yau variety in the generalized bogomolov decomposition conjecture now that essentially a few examples of symplectic manifolds are discovered it seems an important task to seek new symplectic manifolds by deforming symplectic varieties in this paper we shall study a projective symplectic variety from a view point of deformation theory if x has a resolution x x such that x is a symplectic manifold we say that x has a symplectic resolution our first results are concerned with a birational contraction map of a symplectic manifold

pluricanonical maps of varieties of maximal albanese dimension let x be a smooth complex projective algebraic variety of maximal albanese dimension we give a characterization of kappa x in terms of the set v 0 x omega x p in mbox rm pic 0 x h 0 x omega x otimes p ne 0 an immediate consequence of this is that the kodaira dimension kappa x is invariant under smooth deformations we then study the pluricanonical maps varphi m x dashrightarrow mathbb p h 0 x mk x we prove that if x is of general type varphi m is generically finite for m geq 5 and birational for m geq 5 mbox rm dim x 1 more generally we show that for m geq 6 the image of varphi m is of dimension equal to kappa x and for m geq 6 kappa x 2 varphi m is the stable canonical map

continuous terminal sliding mode controller for uncertain systems with relative degree two a continuous homogeneous sliding mode control algorithm is proposed this algorithm ensures finite time convergence to the third order sliding set using only information about the output and its first derivative we prove the convergence of the proposed algorithm via a homogeneous continuously differentiable and strict lyapunov function

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

higher order super twisting for perturbed chains of integrators in this paper we present a generalization of the super twisting algorithm for perturbed chains of integrators of arbitrary order this higher order super twisting host controller is homogeneous with respect to a family of dilations and is continuous it is built as a dynamic controller with respect to the state variable of the chain of integrators and the convergence analysis is performed by the use of a homogeneous strict lyapunov function which is explicitly constructed the effectiveness of the controller is finally illustrated with simulations for a chain of integrators of order four first pure then perturbed where we compare the performances of two host controllers

frobenius modules and hodge asymptotics we exhibit a direct correspondence between the potential defining the h1 1 small quantum module structure on the cohomology of a calabi yau manifold and the asymptotic data of the a model variation of hodge structure this is done in the abstract context of polarized variations of hodge structure and frobenius modules

l2 cohomology of arithmetic varieties abstract the l2 cohomology of arithmetic quotients of bounded symmetric domains is studied we establish the conjecture of zucker equating the l2 cohomology of these spaces to the intersection cohomology of their baily borel compactifications

on the cohomology groups of local systems over hilbert modular varieties via higgs bundles let x be a hilbert modular variety and v a non trivial local system over x in this paper we study deligne saito s mixed hodge structure mhs on the cohomology groups h k x v using the method of higgs bundles among other results we give a dimension formula for the hodge numbers and show that the mixed hodge structure is split over r these results are analogous to 18 in the cocompact case and complement the results in 8 for constant coefficients

algebraic cycles and mumford griffiths invariants let x be a projective algebraic manifold and let ch r x be the chow group of algebraic cycles of codimension r on x modulo rational equivalence working with a candidate bloch beilinson filtration f 0 on ch r x q due to the second author we construct a space of arithmetic hodge theoretic invariants j r x and corresponding map r x gr f ch r x q j r x and determine conditions on x for which the kernel and image of r x are uncountably large

morphological filters part ii their relations to median order statistic and stack filters this paper extends the theory of median order statistic os and stack filters by using mathematical morphology to analyze them and by relating them to those morphological erosions dilations openings closings and open closings that commute with thresholding the max min representation of os filters is introduced by showing that any median or other os filter is equal to a maximum of erosions moving local minima and also to a minimum of dilations moving local maxima thus os filters can be computed by a closed formula that involves a max min on prespecified sets of numbers and no sorting stack filters are established as the class of filters that are composed exactly of a finite number of max min operations the kernels of median os and stack filters are collections of input signals that uniquely represent these filters due to their translation invariance the max min functional definitions of these nonlinear iliters is shown to be equivalent to a maximum of erosions by minimal with respect to a signal ordering kernel elements and also to a minimum of dilations by minimal kernel elements of dual filters the representation of stack filters based on their minimal kernel elements is proven to be equivalent to their representation based on irreducible sum of products expressions of boolean functions it is also shown that median filtering and its iterations of any signal by convex 1 d windows is bounded below by openings and above by closings a signal is a root fixed point of the median iff it is a root of both an opening and a closing the open closing and clos opening yield median roots in one pass suppress impulse noise similarly to the median can discriminate between positive and negative noise impulses and are computationally less complex than the median some similar results are obtained for 2 d median filtering

generic vanishing in characteristic p 0 and the characterization of ordinary abelian varieties we prove a generic vanishing type statement in positive characteristic and apply it to prove positive characteristic versions of kawamata s theorems a characterization of smooth varieties birational to ordinary abelian varieties and the surjectivity of the albanese map when the frobenius stable kodaira dimension is zero

duality in a generalized model for translation invariant systems abstract in a previous paper we introduced a generalized model for translation invariant ti operators in this model we considered the space of all maps from an abelian group g to u called lg fuzzy sets where is a complete lattice ordered group and we defined ti operators on this space also in that paper we proved strong reconstruction theorem to show the consistency of this model this theorem states that for an order preserving ti operator y one can explicitly compute y a for any a from a specific subset of called the base of y in this paper duality is considered in the same general framework and in this regard continuous ti operators are studied this kind of operators are characterized in terms of critical points of their kernels a critical point is defined to be a point at which there exists a net which converges to when is the value of a nontrivial limit function of the kernel at that point a critical point is called to be of first type if for every such net there exists some other point at which this net converges to and it is called to be of second type otherwise as the main result of this paper we prove that a welldefined isotone ti operator is continuous if and only if it has no critical point of second type moreover in this case a ti operator has a continuous extension by duality

residuated semigroups and morphological aspects of translation invariant systems abstract the main goal of this paper is to verify classical properties of morphological operators within the general model of translation invariant ti systems in this model ti operators are defined on the space of lg fuzzy sets i e a g in which g is an abelian group and is a complete lattice ordered group a ti operator y is an operator on which is invariant under translation on g and as groups we consider the generalization of minkowski addition d on and we emphasize that is an involutive residuated topological monoid we verify all properties of traditional set theoretic morphological operators as well as classical representations matheron 1967 for openings closures and granulometries in this general setting we also study spectral skeleton decompositions in this model using the same techniques as in the crisp case goutsias and schonfeld 1991 this formal approach not only clarifies the role of morphological operators but also it gives rise to simpler and clearer proofs using standard results of the theory of residuated semigroups and categories

a generalized fuzzy mathematical morphology and its application in robust 2 d and 3 d object representation in this paper the generalized fuzzy mathematical morphology gfmm is proposed based on a novel definition of the fuzzy inclusion indicator fii fii is a fuzzy set used as a measure of the inclusion of a fuzzy set into another that is proposed to be a fuzzy set it is proven that the fii obeys a set of axioms which are proposed to be extensions of the known axioms that any inclusion indicator should obey and which correspond to the desirable properties of any mathematical morphology operation the gfmm provides a very powerful and flexible tool for morphological operations the binary and grayscale mathematical morphologies can be considered as special cases of the proposed gfmm an application for robust skeletonization and shape decomposition of two dimensional 2 d and three dimensional 3 d objects is presented simulation examples show that the object reconstruction from their skeletal subsets that can be achieved by using the gfmm is better than by using the binary mathematical morphology in most cases furthermore the use of the gfmm for skeletonization and shape decomposition preserves the shape and the location of the skeletal subsets and spines

complex analytic neron models for arbitrary families of intermediate jacobians given a family of intermediate jacobians for a polarizable variation of integral hodge structure of odd weight on a zariski open subset of a complex manifold we construct an analytic space that naturally extends the family its two main properties are a the horizontal and holomorphic sections are precisely the admissible normal functions without singularities b the graph of any admissible normal function has an analytic closure inside our space as a consequence we obtain a new proof for the zero locus conjecture of m green and p griffiths the construction uses filtered mathcal d modules and m saito s theory of mixed hodge modules it is functorial and does not require normal crossing or unipotent monodromy assumptions

hirzebruch s conjecture on cusp singularities hirzebruch conjectured in h1 that the signature defect of a hilbert modular cusp should coincide with the special value of the shimizu l function corresponding to the cusp atiyah et al lads1 ads2 and miiller mu proved the conjecture in this paper we prove a relation between some geometric invariants associated with a cusp which together with a recent result of ishida i gives an algebro geometric proof of the hirzebruch conjecture we treat more general cusps called tsuchihashi cusps including hilbert modular cusps for this cusp singularity v p we define two invariants namely the contribution of the cusp x p to the arithmetic genus and the signature defect a p of the cusp the signature defect was defined by hirzebruch h1 which coincides with ours see sect 1 in the case of hilbert modular cusps we can also find other generalizations by morita mo and looijenga l the invariant z p of a hilbert modular cusp is called the q invariant by ehlers e satake fsa defined z p in a more general context showing that it coincides with the contribution of the cusp to the dimension formula of the space of cusp forms as calculated by means of the riemann roch theorem

higher order sliding mode control using discontinuous integral action in this article we propose a novel family of higher order sliding mode hosm controllers with discontinuous integral term for a class of single input single output siso nonlinear systems they belong to a recently introduced class of controllers named continuous hosm since they are able to compensate exactly and in finite time perturbations uncertainties using a continuous control signal they are obtained by introducing a discontinuous integral action at the input of the plant here continuously differentiable lyapunov functions are used to rigorously prove the stability of the closed loop system we also investigate an approximation of this strategy where a continuous integral term replaces the discontinuous one the robustness properties of the closed loop controller with continuous and discontinuous integral term are studied

foundations of fuzzy sets abstract this paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice it then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy and connective within these categories weak forms of subobject representations provide an internal second order logic an approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal second order theory