proof of wahl s conjecture on surjectivity of the gaussian map for flag varieties introduction let x be a smooth projective variety over c with ample line bundles t and 2 on x we can form their external tensor product to get a line bundle t fxi 2 on x x x let 5d be the ideal sheaf of the diagonal d c x x x wahl defines a map which he calls as the gaussian map i21 2 h x x x d 0 1 2 h x lc 4 0 l c 0 2 which is induced from the canonical projection id d d by identifying the cxxxkkd y od module d k supported in d with the sheaf of 1 forms ql on d x wahl has extensively studied this map in w1 w2 w3 and w4 and he made the following conjecture in w41

deformations of quasi homogeneous surface singularities let a a be a finitely generated normal graded domain over e so that i 0 v spec a has an isolated singularity we say v is a quasi homogeneous singularity or admits a good ie action or is defined by weighted homogeneous polynomials we are concerned with two questions about the deformations of v there are two natural notions of equisingular deformation of v first one has deformations of weight 0 that is one perturbs the defining equations of vby terms of weight at least as big to make a good notion out of this dock sang rim observed one is really deforming a and the weight filtration ij where ij ai

koszul divisors on moduli spaces of curves given a moduli space how can one construct the best in the sense of higher dimensional algebraic geometry effective divisor on it we show that at least in the case of the moduli space of curves the answer is provided by the koszul divisor defined in terms of the syzygies of the parameterized objects in this paper we find a formula for the slopes of all koszul divisors on overline cal m g in particular we obtain the first infinite series of counterexamples to the harris morrison slope conjecture and we prove the maximal rank conjecture in the case when the brill noether number of the corresponding linear series equals 0 we also find shorter proofs for the formulas of the class of the brill noether and gieseker petri divisors finally we improve most of logan s results on the kodaira dimension of the moduli spaces overline cal m g n of pointed stable curves

solutions of fractional logistic equations by euler s numbers abstract in this paper we solve in the convergence set the fractional logistic equation making use of euler s numbers to our knowledge the answer is still an open question the key point is that the coefficients can be connected with euler s numbers and then they can be explicitly given the constrained of our approach is that the formula is not valid outside the convergence set the idea of the proof consists to explore some analogies with logistic function and euler s numbers and then to generalize them in the fractional case

generalized luneburg canonical varieties and vector fields on quasicaustics some aspects of a particular class of bifurcation varieties which are provided by simple and unimodal boundary singularities are studied their correspondence to diffraction theory is established the generic caustics by diffraction on apertures are derived and their generating families for the corresponding lagrangian varieties are calculated it is proved that the quasicaustics associated to simple singularities are smooth hypersurfaces or whitney s cross caps the procedure for calculating the modules of logarithmic vector fields is given and the minimal sets of the corresponding generators are explicitly calculated the general boundary singularities are constructed and the structure of quasicaustics defined by parabolic singularities is investigated

unsupervised segmentation and classification of cervical cell images the pap smear test is a manual screening procedure that is used to detect precancerous changes in cervical cells based on color and shape properties of their nuclei and cytoplasms automating this procedure is still an open problem due to the complexities of cell structures in this paper we propose an unsupervised approach for the segmentation and classification of cervical cells the segmentation process involves automatic thresholding to separate the cell regions from the background a multi scale hierarchical segmentation algorithm to partition these regions based on homogeneity and circularity and a binary classifier to finalize the separation of nuclei from cytoplasm within the cell regions classification is posed as a grouping problem by ranking the cells based on their feature characteristics modeling abnormality degrees the proposed procedure constructs a tree using hierarchical clustering and then arranges the cells in a linear order by using an optimal leaf ordering algorithm that maximizes the similarity of adjacent leaves without any requirement for training examples or parameter adjustment performance evaluation using two data sets show the effectiveness of the proposed approach in images having inconsistent staining poor contrast and overlapping cells

koregulare und aquidimensionale darstellungen zusammenfassung the aim of this note is to give a classification of all irreducible representations v of a connected semisimple algebraic group g such that either the quotient v g is smooth or the quotient map v v g has equidimensional fibres it follows by the classification that if the quotient map is equidimensional then the quotient v g is smooth

homological properties of determinantal arrangements abstract we explore a natural extension of braid arrangements in the context of determinantal arrangements we show that these determinantal arrangements are free divisors additionally we prove that free determinantal arrangements defined by the minors of 2 n matrices satisfy nice combinatorial properties we also study the topology of the complements of these determinantal arrangements and prove that their higher homotopy groups are isomorphic to those of s 3 furthermore we find that the complements of arrangements satisfying those same combinatorial properties above have poincare polynomials that factor nicely

differential forms on free and almost free divisors we introduce a variant of the usual kahler forms on singular free divisors and show that it enjoys the same depth properties as kahler forms on isolated hypersurface singularities using these forms it is possible to describe analytically the vanishing cohomology and the gauss manin connection in families of free divisors in precise analogy with the classical description for the milnor fibration of an isolated complete intersection singularity due to brieskorn and greuel this applies in particular to the family formula of discriminants of a versal deformation formula of a singularity of a mapping

on families of square matrices in this paper we classify families of square matrices up to the following natural equivalence thinking of these families as germs of smooth mappings from a manifold to the space of square matrices we allow arbitrary smooth changes of co ordinates in the source and pre and post multiply our family of matrices by generally distinct families of invertible matrices all dependent on the same variables we obtain a list of all the corresponding simple mappings that is those that do not involve adjacent moduli this is a non linear generalisation of the classical notion of linear systems of matrices we also make a start on an understanding of the associated geometry 2000 mathematics subject classification 58k40 58k50 58k60 32s25

a note on the fractional logistic equation abstract in this short note we show that the real function recently proposed by west 2015 is not an exact solution for the fractional logistic equation

can fractional calculus help improve tumor growth models abstract ode based population models remain viable tools to investigate tumor growth and support clinical evidence by following a fractional approach this study derives analytical solutions for five of these models whose parameters are best fitted against extant clinical data in terms of tumor growth prediction results show that fractional models not only have better performance which is mostly wanted for decision making in oncology but also reveal interesting characteristics to be further explored

volume preserving subgroups of a and k and singularities in unimodular geometry for a germ of a smooth map f from mathbb k n to mathbb k p and a subgroup g omega q of any of the mather groups g for which the source or target diffeomorphisms preserve some given volume form q in mathbb k q q n or p we study the g omega q moduli space of f that parameterizes the g omega q orbits inside the g orbit of f we find for example that this moduli space vanishes for g omega q mathcal a omega p and mathcal a stable maps f and for g omega q mathcal k omega n and mathcal k simple maps f on the other hand there are mathcal a stable maps f with infinite dimensional mathcal a omega n moduli space

cytoplasm and nucleus segmentation in cervical smear images using radiating gvf snake a radiating gradient vector flow rgvf snake aiming at accurate extraction of both the nucleus and cytoplasm from a single cell cervical smear image is proposed after preprocessing the areas in the image are roughly clustered into nucleus cytoplasm and the background by a spatial k means clustering algorithm after initial contours are extracted the image is segmented using rgvf rgvf involves a new edge map computation method and a stack based refinement and is thus robust to contaminations and can effectively locate the obscure boundaries the boundaries can also be correctly traced even if there are interferences near the cytoplasm and nucleus regions experiments performed on the herlev dataset which contains 917 images show the effectiveness of the proposed algorithm

growth motion and 1 parameter families of symmetry sets associated to every plane curve there is the locus of centres of circles bitangent to that curve the so called symmetry set of the curve we can view this set as the spine of our curve which can be recovered by taking the envelope of circles of varying radii along this spine varying the symmetry set in some isotopy while keeping the radius function fixed may be viewed as crudely modelling motion of the original curve viewed as a biological object fixing the symmetry set and varying the radius function can be considered to model growth crudely in this paper we shall describe the generic changes in the curves which take place in the process of growth and motion and outline the corresponding results for spheres centred on a space curve we also use the idea of a stratified morse function to describe the generic changes which occur in one parameter families of full bifurcation sets in the plane applying this to the bifurcation set of distance squared functions we find all the transitions of a symmetry set and evolute which occur in a generic isotopy of a plane curve

free divisors in prehomogeneous vector spaces we study linear free divisors that is free divisors arising as discriminants in prehomogeneous vector spaces and in particular in quiver representation spaces we give a characterization of the prehomogeneous vector spaces containing such linear free divisors for reductive linear free divisors we prove that the numbers of geometric and representation theoretic irreducible components coincide as a consequence we find that a quiver can only give rise to a linear free divisor if it has no oriented or unoriented cycles we also deduce that the linear free divisors which appear in sato and kimura s list of irreducible prehomogeneous vector spaces are the only irreducible reductive linear free divisors furthermore we show that all quiver linear free divisors are strongly euler homogeneous that they are locally weakly quasihomogeneous at points whose corresponding representation is not regular and that all tame quiver linear free divisors are locally weakly quasihomogeneous in particular the latter satisfy the logarithmic comparison theorem

modified fractional logistic equation abstract in the article west 2015 the author has obtained a function as the solution to fractional logistic equation fle as demonstrated later in area et al 2016 this function west function is not the solution to fle but nevertheless as shown by west it is in good agreement with the numerical solution to fle the west function indicates a compelling feature in which the exponentials are substituted by mittag leffler functions in this paper a modified fractional logistic equation mfle is introduced to which the west function is a solution the proposed fractional integro differential equation possesses a nonlinear additive term related to the solution of the logistic equation le the method utilized in this article may be applied to the analysis of solutions to nonlinear fractional differential equations of mathematical physics

generic structure of two dimensional dimages under gaussian blurring for medical imaging problems computer scientists pizer and eberly introduced the core as the analogue for grayscale images of the medial axis it is defined for two dimensional images as the ridge of a medial function defined on 2 1 dimensional scale space in this paper we extend methods from singularity theory to analyze the generic properties of ridges and cores for two dimensional images including the generic transitions which occur for sequences of images varying with a parameter such as time we do so by placing the ridge construction into a larger relative critical set structure which captures properties undetected by the disjoint ridge core segments the genericity properties are established using transversality conditions for associated mappings we introduce a method for establishing transversality conditions for medial functions obtained by gaussian blurring by using a basis of local solutions to the heat equation the method applies to many other properties besides those we consid

toy model of fractional transport of cancer cells due to self entrapping a simple mathematical model is proposed to study the influence of cell fission on transport the model describes fractional in time tumor development which is a one dimensional continuous time random walk the model is relevant for consideration of both solid and diffusive cancers

mathematical modeling of tumor immune competitive system considering the role of time delay abstract in this paper we consider a three dimensional nonlinear delay differential system tumor cells cytotoxic t lymphocytes t helper cells with single interaction delay we perform linear stability of the equilibria and the existence of hopf bifurcation in which the discrete time delay is used as a bifurcation parameter we estimate the length of delay to preserve the stability of period 1 limit cycle we also investigate the direction period and the stability of bifurcated periodic solutions by applying normal form method and center manifold theory we observe that the discrete time delay plays an important role in stability switching numerical simulations are presented to illustrate the rich dynamical behavior of the model with different values for the time delay including the existence of periodic oscillations which demonstrate the phenomena of long term tumor relapse

on a riemann liouville generalized taylor s formula in this paper a generalized taylor s formula of the kindfx j 0najx a j 1 1 tnx whereaj r x a 0 1 is established such expression is precisely the classical taylor s formula in the particular case 1 in addition detailed expressions fortn x andaj involving the riemann liouville fractional derivative and some applications are also given

cohomology of the complement of a free divisor we prove that if d is a strongly quasihomogeneous free divisor in the stein manifold x and u is its complement then the de rham cohomology of u can be computed as the cohomology of the complex of meromorphic differential forms on x with logarithmic poles along d with exterior derivative the class of strongly quasihomogeneous free divisors introduced here includes free hyperplane arrangements and the discriminants of stable mappings in mather s nice dimensions and in particular the discriminants of coxeter groups

obstructions and hypersurface sections minimally elliptic singularities we study the obstruction space t 2 for minimally elliptic surface singularities we apply the main lemma of our previous paper 3 which relates t 2 to deformations of hypersurface sections to use this we classify general hypersurface sections of minimally elliptic singularities as in the rational singularity case there is a simple formula for the minimal number of generators for t 2 as a module over the local ring this number is in many cases e g for cusps of hilbert modular surfaces equal to the vector space dimension of t 2

on recent developments in the theory of abstract differential equations with fractional derivatives abstract this note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives we show that the existence results in agarwal et al 2009 1 belmekki and benchohra 2010 2 darwish et al 2009 3 hu et al 2009 4 mophou and n guerekata 2009 6 7 mophou 2010 8 9 muslim 2009 10 pandey et al 2009 11 rashid and el qaderi 2009 12 and tai and wang 2009 13 are incorrect since the considered variation of constant formulas is not appropriate in this note we also consider a different approach to treat a general class of abstract fractional differential equations

on the legacy of free divisors discriminants and morse type singularities we investigate when the freeness of a divisor v 0 is inherited by the discriminants for the versal deformations of nonlinear sections of v we introduce morse type singularities for sections and give a criterion for freeness of the discriminant in terms of v 0 generically having morse type singularities this criterion is applied to determine when the bifurcation sets of mappings and smoothings of space curves and complete intersections are free it also explains the failure of freeness for discriminantal arrangements of hyperplanes

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

the number of critical points of a product of powers of linear functions be the complement of a given a complex n vector 2 21 2n e cn consider the multivalued holomorphic function defined on m by studying bethe vectors in statistical mechanics a varchenko 18 conjectured that for generic 2 all critical points of qs are nondegenerate and the number of critical points is equal to the absolute value of the euler characteristic of m m he proved the conjecture for complexified real arrangements in 18 a similar result was known to k aomoto 1 example 1 who stated without proof that for positive 2 the number of critical points of b equals the number of bounded components of the complement of the real arrangement see 2 theorem 4 4 1 1 for details in this paper we prove varchenko s conjecture for all arrangements more generally we give a formula for the number of critical points of the multivalued holomorphic function

a new look at the fractionalization of the logistic equation abstract the fractional version of the logistic equation will be studied in this paper motivated by unsuccessful previous papers we showed how to obtain the correct solution the algorithm is very simple its numerical implementation will be studied and exemplified using a pade approximation

accurate cervical cell segmentation from overlapping clumps in pap smear images accurate segmentation of cervical cells in pap smear images is an important step in automatic pre cancer identification in the uterine cervix one of the major segmentation challenges is overlapping of cytoplasm which has not been well addressed in previous studies to tackle the overlapping issue this paper proposes a learning based method with robust shape priors to segment individual cell in pap smear images to support automatic monitoring of changes in cells which is a vital prerequisite of early detection of cervical cancer we define this splitting problem as a discrete labeling task for multiple cells with a suitable cost function the labeling results are then fed into our dynamic multi template deformation model for further boundary refinement multi scale deep convolutional networks are adopted to learn the diverse cell appearance features we also incorporated high level shape information to guide segmentation where cell boundary might be weak or lost due to cell overlapping an evaluation carried out using two different datasets demonstrates the superiority of our proposed method over the state of the art methods in terms of segmentation accuracy

projective degenerations of k3 surfaces gaussian maps and fano threefolds in this article we exhibit certain projective degenerations of smoothk3 surfaces of degree 2g 2 in g whose picard group is generated by the hyperplane class to a union of two rational normal scrolls and also to a union of planes as a consequence we prove that the general hyperplane section of suchk3 surfaces has a corank one gaussian map ifg 11 org 13 we also prove that the general such hyperplane section lies on a uniquek3 surface up to projectivities finally we present a new approach to the classification of prime fano threefolds of index one which does not rely on the existence of a line

transversality in families of mappings by a classical result of thom 20 if n and p are smooth manifolds and q is a smooth submanifold of p the set of maps f n p transverse to q is residual in c n p if q is a closed subset of p the set of transverse maps is also open however if u is a further smooth manifold and fu a family of smooth mappings we cannot expect to be able to deform the family to make each fu transverse to q this situation was considered by bruce 2 who showed that for a residual set of maps f n p u we have a for each u u the set of points in n at which fu is not transverse to q is discrete and b at each such point the class of the contact of fu n with q has finite codimension this is the result he established the formal statement given was more explicit but weaker this result is not entirely satisfactory we may hope also for example that each contact class presented is universally unfolded in the family we address this problem and show that this hope is justified under a hypothesis of nice dimensions after recalling mather s theory of topological stability we establish a result valid in all dimensions with topologically versal unfoldings since the proofs of these results involve numerous translations between equivalent versality conditions we begin with a section developing a notation and approach to facilitate such proofs we then consider the corresponding question with the submanifold q replaced by a stratified subset of p this requires an extensive study of theories of contact equivalence relative to a subset of the target good algebraic conditions and criteria are only known under an analyticity hypothesis for some results this needs to be strengthened to a holonomic condition with these preliminaries we obtain a parallel to the theory of the earlier section however a full account of a topological analogue remains beyond our reach

automorphism groups for quasi homogeneous gorenstein surface singularities abstract let x 0 be a non log canonical quasi homogeneous surface singularity germ and letg aut x 0 be the maximal reductive subgroup in this paper we bound the order ofg c by p p a purely topological invariant hurwitz s theorem comes out as a corollary we use the standard approach of taking the quotient space of x 0 by the action of a finite group of automorphisms acting freely onx 0

a and the vanishing topology of discriminants suppose thatf n 0 p 0 is finitely a determined withn p we define a milnor fiber for the discriminant off it is the discriminant of a stabilization off we prove that this discriminant milnor fiber has the homotopy type of a wedge of spheres of dimensionp 1 whose number we denote by f one of the main theorems of the paper is a type result if n p is in the range of nice dimensions in the sense of mather then mu delta f geqq a e codium with equality iff is weighted homogeneous outside the nice dimensions we obtain analogous formulae with correction terms measuring the presence of unstable but topologically stable germs in the stabilization these results are further extended to nonlinear sections of free divisors

a note on property of the mittag leffler function abstract recently the authors have found in some publications that the following property 0 1 of mittag leffler function is taken for granted and used to derive other properties 0 1 e a t s e a t e a s t s 0 where a is a real constant and 0 in this note it is proved that the above property is unavailable unless 1 or a 0 moreover a new equality on e a t is developed whose limit state as 1 is just the property 0 1

an improved joint optimization of multiple level set functions for the segmentation of overlapping cervical cells in this paper we present an improved algorithm for the segmentation of cytoplasm and nuclei from clumps of overlapping cervical cells this problem is notoriously difficult because of the degree of overlap among cells the poor contrast of cell cytoplasm and the presence of mucus blood and inflammatory cells our methodology addresses these issues by utilizing a joint optimization of multiple level set functions where each function represents a cell within a clump that have both unary intracell and pairwise intercell constraints the unary constraints are based on contour length edge strength and cell shape while the pairwise constraint is computed based on the area of the overlapping regions in this way our methodology enables the analysis of nuclei and cytoplasm from both free lying and overlapping cells we provide a systematic evaluation of our methodology using a database of over 900 images generated by synthetically overlapping images of free lying cervical cells where the number of cells within a clump is varied from 2 to 10 and the overlap coefficient between pairs of cells from 0 1 to 0 5 this quantitative assessment demonstrates that our methodology can successfully segment clumps of up to 10 cells provided the overlap between pairs of cells is sim 0 5 we also evaluate our approach quantitatively and qualitatively on a set of 16 extended depth of field images where we are able to segment a total of 645 cells of which only sim 10 are free lying finally we demonstrate that our method of cell nuclei segmentation is competitive when compared with the current state of the art

a review of definitions of fractional derivatives and other operators abstract given the increasing number of proposals and definitions of operators in the scope of fractional calculus it is important to introduce a systematic classification nonetheless many of the definitions that emerged in the literature can not be considered as fractional derivatives we analyze a list of expressions to have a general overview of the concept of fractional integrals derivatives moreover some formulae that do not involve the term fractional are also included due to their particular interest in the area

generalized exponents of a free arrangement of hyperplanes and shepherd todd brieskorn formula we define an n arrangement as a finite family of hyperplanes through the origin in c 1 in 11 and 12 we studied the free arrangement and defined its structure sequence their definitions will be given again in sect 2 in this article we say the generalized exponents instead of the structure sequence let d o d 1 d be the generalized exponents of a free n arrangement x let xj u h our main result is hex

exact solution to fractional logistic equation abstract the logistic equation is one of the most familiar nonlinear differential equations in the biological and social sciences herein we provide an exact solution to an extension of this equation to incorporate memory through the use of fractional derivatives in time the solution to the fractional logistic equation fle is obtained using the carleman embedding technique that allows the nonlinear equation to be replaced by an infinite order set of linear equations which we then solve exactly the formal series expansion for the initial value solution of the fle is shown to be expressed in terms of a series of weighted mittag leffler functions that reduces to the well known analytic solution in the limit where the fractional index for the derivative approaches unity the numerical integration to the fle provides an excellent fit to the analytic solution we propose this approach as a general technique for solving a class of nonlinear fractional differential equations

moduli of curves on enriques surfaces abstract we compute the number of moduli of all irreducible components of the moduli space of smooth curves on enriques surfaces in most cases the moduli maps to the moduli space of prym curves are generically injective or dominant exceptional behavior is related to existence of enriques fano threefolds and to curves with nodal prym canonical model

the jacobian module the milnor fiber and the d module generated by f s for a germ f on a complex manifold x we introduce a complex derived from the liouville form acting on logarithmic differential forms and give an exactness criterion we use this liouville complex to connect properties of the d module generated by f s to homological data of the jacobian ideal specifically we show that for a large class of germs the annihilator of f s is generated by derivations through local cohomology we connect the cohomology of the milnor fiber to the jacobian module via logarithmic differentials in particular we consider not necessarily reduced hyperplane arrangements we prove a conjecture of terao on the annihilator of 1 f we confirm in many cases a corresponding conjecture on the annihilator of f s but we disprove it in general we show that the bernstein sato polynomial of an arrangement is not determined by its intersection lattice we prove that arrangements for which the annihilator of f s is generated by derivations fulfill the strong monodromy conjecture and that this includes as very special cases all arrangements of coxeter and of crystallographic type and all multi arrangements in dimension 3

modular properties of nodal curves on k3 surfaces in this article we are going to address the following issues 1 the first is a rigidity property for pairs s c consisting of a general projective k3 surface s and a curve c obtained as the normalization of a nodal hyperplane section hat c hookrightarrow s we prove that a non trivial deformation of a pair s c induces a non trivial deformation of c 2 the second question concerns the wahl map of curves c obtained as above we prove that the wahl map of the normalization of a nodal curve contained in a general projective k3 surface is non surjective in both cases we impose upper bounds on the number of nodes of hat c

multiscale gradient watersheds of color images we present a new framework for the hierarchical segmentation of color images the proposed scheme comprises a nonlinear scale space with vector valued gradient watersheds our aim is to produce a meaningful hierarchy among the objects in the image using three image components of distinct perceptual significance for a human observer namely strong edges smooth segments and detailed segments the scale space is based on a vector valued diffusion that uses the additive operator splitting numerical scheme furthermore we introduce the principle of the dynamics of contours in scale space that combines scale and contrast information the performance of the proposed segmentation scheme is presented via experimental results obtained with a wide range of images including natural and artificial scenes

on universal severi varieties of low genus k3 surfaces we prove the irreducibility of universal severi varieties parametrizing irreducible reduced nodal hyperplane sections of primitive k3 surfaces of genus g with 3 g 11 g 10

classification of varieties with canonical curve section via gaussian maps on canonical curves the purpose of this article is to develop further a method to classify varieties x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i having canonical curve section using gaussian map computations in a previous article we applied these techniques to classify prime fano threefolds that is fano threefolds whose picard group is generated by the hyperplane bundle in this article we extend this method and classify fano threefolds of higher index and mukai varieties i e varieties of dimension four or more with canonical curve sections first we determine when the hilbert scheme inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i of such varieties x is nonempty moreover in the case of picard number one we prove that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i is irreducible and that the examples of fano iskovskih and mukai form a dense open subset of smooth points of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i

on mittag leffler type functions in fractional evolution processes abstract we review a variety of fractional evolution processes so defined being governed by equations of fractional order whose solutions turn out to be related to mittag leffler type functions the chosen equations are the simplest of the fractional calculus and include the abel integral equations of the second kind which are relevant in typical inverse problems and the fractional differential equations which govern generalized relaxation and oscillation phenomena

the gaussian map for rational ruled surfaces in this paper the gaussian map 2 h 0 c k h 0 c 3k of a smooth curve c lying on a minimal rational ruled surface is computed it is shown that the corank of is determined for almost all such curves by the rational surface in which it lies hence expect for some special cases a curve cannot lie on two nonisomorphic minimal rational ruled surfaces

milnor number and tjurina number of complete intersections two basic invariants of a complex analytic isolated complete intersection singularity x x are its milnor number g x x and its tjurina number z x x the former is the n th betti number minus one if n 0 of a no nsingular nearby fibre whereas the latter is the dimension of the base space of a miniversal deformation of x x it is conjectured that for n 1 we always have p x x z x x so far this has been verified in the following cases 1 if x x is a hypersurface singularity 2 if n 1 3 if the link of x in x is a rational homology sphere 4 if x x is quasi homogeneous then we actually have equality of p x x and z x x 5 if n 2 the first case is rather trivial the next three are all due to greuel 3 and the fifth case is due to the first author 4 subsequently j wahl expressed for n 2 the difference of milnor number and tjurina number as a sum of non negative nvariants of the germ x x 7 the purpose of this note is to prove the conjecture n general in fact we obtain a higher dimensional generalization of wahl s result