rational points on k3 surfaces a new canonical height a fundamental tenet of diophantine geometry is that the geometric properties of an algebraic variety should determine its basic arithmetic properties this is certainly true for curves where the sign of the euler characteristic of c determines whether the set of rational points on c is finite x c 0 for higher dimensional varieties there are some precise conjectures due to bombieri lang and vojta 15 which predict when the rational points on a variety should be finite or degenerate i e not zariski dense and some conjectures of manin et al 2 5 on the distribution of rational points in those cases when they are zariski dense but except for abelian varieties their subvarieties and some fano varieties varieties for which the anticanonical bundle is ample there are very few general theorems in this paper we will study the rational points on a certain class of k 3 surfaces defined over a number field k the moduli space of marked algebraic k3 surfaces is a countable union of 19 dimensional quasi projective varieties we are going to look at the 18 dimensional family studied by wehler 1 17 wehler s family consists of k3 surfaces s whose automorphism group aut s contains a subgroup d isomorphic to the free product 7 2 2 of two cyclic groups of order 2 we will use the geometric information provided by this infinite automorphism group to study the k rat ional points on s for any point pes we can look at the orbit of p under the action of d

smoothness of equisingular families of curves francesco severi 1921 showed that equisingular families of plane nodal curves are t smooth i e smooth of the expected dimension whenever they are non empty for families with more complicated singularities this is no longer true given a divisor d on a smooth projective surface it thus makes sense to look for conditions which ensure that the family v irr d s 1 s r of irreducible curves in the linear system d l with precisely r singular points of types s 1 s r is t smooth considering different surfaces including the projective plane general surfaces in p 3 c products of curves and geometrically ruled surfaces we produce a sufficient condition of the type e t i 1 s i d k 2 where is some invariant of the singularity type and is some constant this generalises the results of greuel lossen and shustin 2001 for the plane case combining their methods and the method of bogomolov instability for many singularity types the invariant leads to essentially better conditions than the invariants used by greuel lossen and shustin 1997 and for most classes of geometrically ruled surfaces our results are the first known for t smoothness at all

mapping tori of free group automorphisms are coherent the mapping torus of an endomorphism of a group g is the hnnextensiong g with bonding maps the identity and we show that a mapping torus of an injective free group endomorphism has the property that its flnitely generated subgroups are flnitely presented and moreover these subgroups are of flnite type

algebraic cycles and higher k theory here g x is the grothendieck group of coherent sheaves on x 13 gr refers to the graded group defined by the y filtration on g x cf kratzer 14 soult 20 and ch x is the chow group of codimension i algebraic cycles defined by fulton 9 the left hand isomorphism is a formal consequence of the existence of a i structure on g x while the existence of r is the central theme of the b f m rr theorem the main purpose of this paper is to define a theory of higher chow groups ch x n n 2 0 so as to obtain isomorphisms

curves on threefolds and a conjecture of griffiths harris we prove that any arithmetically gorenstein curve on a smooth general hypersurface x subset mathbb p 4 of degree at least 6 is a complete intersection this gives a characterisation of complete intersection curves on general type hypersurfaces in mathbb p 4 we also verify that certain 1 cycles on a general quintic hypersurface are non trivial elements of the griffiths group

moishezon s theorem and degeneration abstract let l be a tensor product of two very ample line bundles on the smooth projective complex threefold x under the hypothesis that h 0 x k x l 0 we show that the restriction map r pic x pic y is an isomorphism for very general y in the linear system l for such l this result recovers the noether lefschetz theorem of moishezon who extended the original topological and hodge theoretic arguments of lefschetz we give a degeneration argument which is almost completely algebraic in nature

the fractional fourier transform and time frequency representations the functional fourier transform frft which is a generalization of the classical fourier transform was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community to which it may however be potentially useful the frft depends on a parameter spl alpha and can be interpreted as a rotation by an angle spl alpha in the time frequency plane an frft with spl alpha spl pi 2 corresponds to the classical fourier transform and an frft with spl alpha 0 corresponds to the identity operator on the other hand the angles of successively performed frfts simply add up as do the angles of successive rotations the frft of a signal can also be interpreted as a decomposition of the signal in terms of chirps the authors briefly introduce the frft and a number of its properties and then present some new results the interpretation as a rotation in the time frequency plane and the frft s relationships with time frequency representations such as the wigner distribution the ambiguity function the short time fourier transform and the spectrogram these relationships have a very simple and natural form and support the frft s interpretation as a rotation operator examples of frfts of some simple signals are given an example of the application of the frft is also given

modular forms and special cubic fourfolds abstract we study the degrees of special cubic divisors on moduli space of cubic fourfolds with at worst ade singularities in this paper we show that the generating series of the degrees of such divisors is a level three modular form

a simple proof that rational curves on k3 are nodal the picard group of s is generated by two effective divisors c and f with c 2 f 2 0 and c f 1 it can be realized as an elliptic fibration over p with a unique section c fibers f and 2 here c1 is the first chern class of the hodge bundle of the fibration s p see h m it is a standard result that the number of nodal fibers of an elliptic fibration are given by 12 h m p 158 so there are exactly 24 rational nodal curves in the linear series f for s general this is the same special k3 surface used by bryan and leung in their counting of curves on k3 surfaces b l it is actually the attempt to understand their method that leads us to our proof we will call a k3 surface with picard lattice 1 1 a bl k3 surface a bl k3 surface s lies on the boundary of the moduli space of k3 surfaces of genus g with c gf as the corresponding primitive divisor every curve in the linear series os c gf is totally reducible i e it consists of the 2 curve c and g elliptic tails attached to c a curve d os c gf is the image of a stable rational map only if d c m1f1 m2f2 m24f24 where f1 f2 f24 are 24

discrete fractional fourier transforms based on closed form hermite gaussian like dft eigenvectors in this paper we construct discrete fractional fourier transforms dfrft using recently introduced closed form hermite gaussian like hgl eigenvectors with respect to such eigenvectors we discuss the convergence of their components to samples of the corresponding continuous hermite gaussian functions and propose solutions to deal with some restrictions related to their construction this allows us to give new procedures for obtaining orthonormal bases of hgl eigenvectors which are used to fractionalize the discrete fourier transform we illustrate the application of the resulting dfrft in the scenario of filtering in the fractional domain and compare the results with existing dfrft approaches

spectral analysis and reconstruction for periodic nonuniformly sampled signals in fractional fourier domain nonuniform sampling occurs in many applications due to imperfect timebase or random events periodic nonuniform sampling is a special type of nonuniform sampling the properties and applications of periodic nonuniform sampling signals in traditional fourier domain have been extensively studied but so far none of the research papers covering the spectral analysis and reconstruction of periodic nonuniformly sampled signals in fractional fourier domain have been published this correspondence is to explore the spectral properties of periodic nonuniformly sampled signals associated with the fractional fourier transform first based on the uniform fractional fourier transform kernel the general spectral representation of periodic nonuniformly sampled signals has been derived second detailed analysis of periodic nonuniformly sampled chirp signals in the fractional fourier domain have been performed the results can be used to estimate the chirp rate and the sampling offsets finally a more simple relationship between the discrete fractional spectrum of periodic nonuniformly sampled signals and the continuous spectrum has been obtained based on this relationship the original continuous spectrum can be reconstructed from periodic nonuniformly sampled signals in fractional fourier domain in addition the simulations are carried out to verify the correctness of the results

relatively hyperbolic groups are c simple we characterize relatively hyperbolic groups whose reduced c algebra is simple as those which have no non trivial finite normal subgroups

on the noether lefschetz theorem and some remarks on codimension two cycles here of general moduli means that there is a countable union v of subvarieties of the space pn of surfaces of degree d in p3 such that the statement pie s z holds for s p n v noether it would seem stated this theorem but never completely proved it instead he gave a plausibility argument based on the following construction 1 let g be the hilbert scheme of curves of degree n and genus 9 in p3 l gr3 d l ip n the space of surfaces of degree d in p3 and zc t the incidence correspondence s o a c s c z s

group rings of proper ascending hnn extensions of countably infinite free groups are primitive abstract we prove that every group ring of a proper ascending hnn extension of a free group with at most countably infinite rank is primitive

polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms we prove that every mapping torus of any free group endomorphism is residually finite we show how to use a not yet published result of e hrushovski to extend our result to arbitrary linear groups the proof uses algebraic self maps of affine spaces over finite fields in particular we prove that when such a map is dominant the set of its fixed closed scheme points is zariski dense in the affine space

optimal filtering in fractional fourier domains for time invariant degradation models and stationary signals and noise the classical fourier domain wiener filter which can be implemented in o n log n time gives the minimum mean square error estimate of the original undistorted signal for time varying degradations and nonstationary processes however the optimal linear estimate requires o n sup 2 time for implementation we consider filtering in fractional fourier domains which enables significant reduction of the error compared with ordinary fourier domain filtering for certain types of degradation and noise especially of chirped nature while requiring only o n log n implementation time thus improved performance is achieved at no additional cost expressions for the optimal filter functions in fractional domains are derived and several illustrative examples are given in which significant reduction of the error by a factor of 50 is obtained

maximal components of noether lefschetz locus for beilinson hodge cycles we discuss the noether lefschetz locus for beilinson s hodge cycles on the complement of two or three hyperplane sections in a smooth projective surface in p 3 the main theorem gives an explicit description of maximal components of the noether lefschetz locus

density of rational curves on k3 surfaces using the dynamics of self rational maps of elliptic k3 surfaces together with deformation theory we prove that the union of rational curves is dense on a very general k3 surface and that the union of elliptic curves is dense in the 1st jet space of a very general k3 surface both in the strong topology

the gonality of complete intersection curves abstract the purpose of this paper is to show that for a complete intersection curve c in projective space other than a few exceptions stated below any morphism f c p r satisfying deg f o p r 1 deg c is obtained by projection from a linear space in particular we obtain bounds on the gonality of such curves and compute the gonality of general complete intersection curves we also prove a special case of one of the well known cayley bacharach conjectures posed by eisenbud green and harris

fractional spectrum analysis for nonuniform sampling in the presence of clock jitter and timing offset this paper focuses on analyzing the fractional spectrum of nonuniform sampling which is affected by clock jitter and timing offset both cases of single channel and recurrent samplings with limited bandwidth in the fractional fourier domain are considered for which we study two types of discrete time fractional fourier transforms dtfrfts for the first type dtfrfts we perform their statistical analyses and meanwhile establish relationships between the fractional spectra of the input and output signals of non ideal single multi channel analog to digital converter such relationships indicate that the statistical mean of each fractional spectrum contains periodic replicas of the spectrum for the input signal and these replicas are particularly modulated by the characteristic function of the perturbations introduced by the clock jitter moreover we interpret the first type dtfrfts from the perspective of linear frequency modulation signal decomposition for nonuniform sampling the second type dtfrfts serve as approximate dtfrfts and they especially deal with the general situation when the nonuniform sampling instants are unknown to this end we study modified forms of the first type dtfrfts with known inputs in kernels based on which the statistical mean of the approximate fractional spectra for both cases are derived then we compensate the fractional spectrum bias with high accuracy by developing optimal filters that minimize the mean square error between the original and the compensated fractional spectra for both cases different from single channel sampling the compensation for recurrent sampling involves an additional step before optimal filtering which gives better accuracy simulation results show that the proposed methods outperform existing methods

tight geodesics in the curve complex the curve graph mathcal g associated to a compact surface is the 1 skeleton of the curve complex defined by harvey masur and minsky showed that this graph is hyperbolic and defined the notion of a tight geodesic therein we prove some finiteness results for such geodesics for example we show that a slice of the union of tight geodesics between any pair of points has cardinality bounded purely in terms of the topological type of we deduce some consequences for the action of the mapping class group on mathcal g in particular we show that it satisfies an acylindricity condition and that the stable lengths of pseudoanosov elements are rational with bounded denominator

arithmetically cohen macaulay bundles on complete intersection varieties of sufficiently high multidegree recently it has been proved that any arithmetically cohen macaulay acm bundle of rank two on a general smooth hypersurface of degree at least three and dimension at least four is a sum of line bundles when the dimension of the hypersurface is three a similar result is true provided the degree of the hypersurface is at least six we extend these results to complete intersection subvarieties by proving that any acm bundle of rank two on a general smooth complete intersection subvariety of sufficiently high multi degree and dimension at least four splits we also obtain partial results in the case of threefolds

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

spectral analysis of randomly sampled signals suppression of aliasing and sampler jitter nonuniform sampling can facilitate digital alias free signal processing dasp i e digital signal processing that is not affected by aliasing this paper presents two dasp approaches for spectrum estimation of continuous time signals the proposed algorithms named the weighted sample ws and weighted probability wp density functions respectively utilize random sampling to suppress aliasing both methods produce unbiased estimators of the signal spectrum to achieve this effect the computational procedure for each method has been suitably matched with the probability density function characterising the pseudorandom generators of the sampling instants both proposed methods are analyzed and the qualities of the estimators they produce have been compared with each other although none of the proposed spectrum estimators is universally better than the other one it has been shown that in practical cases the wp estimator produces generally smaller errors than those obtained from ws estimation a practical limitation of the approaches caused by the sampling instant jitter is also studied it has been proven that in the presence of jitter the theoretically infinite bandwidths of ws and wp signal analyses are limited the maximum frequency up to which these analyses can be performed is inversely proportional to the size of the jitter

filterbank reconstruction of bandlimited signals from nonuniform and generalized samples this paper introduces a filterbank interpretation of various sampling strategies which leads to efficient interpolation and reconstruction methods an identity which is referred to as the interpolation identity is developed and is used to obtain particularly efficient discrete time systems for interpolation of generalized samples as well as a class of nonuniform samples to uniform nyquist samples either for further processing in that form or for conversion to continuous time the interpolation identity also leads to new sampling strategies including an extension of papoulis 1977 generalized sampling expansion

noether lefschetz locus for surfaces we generalize m green s explicit noether lefschetz theorem to the family of smooth complete intersection surfaces in the higher dimensional projective spaces moreover we give a new proof of the density theorem due to c ciliberto j harris and r miranda 5

property p_ naive p naive for acylindrically hyperbolic groups we prove that every acylindrically hyperbolic group that has no non trivial finite normal subgroup satisfies a strong ping pong property the p naive property for any finite collection of elements h 1 dots h k there exists another element gamma ne 1 such that for all i langle h i gamma rangle langle h i rangle langle gamma rangle we also show that if a collection of subgroups h 1 dots h k is a hyperbolically embedded collection then there is gamma ne 1 such that for all i langle h i gamma rangle h i langle gamma rangle

unified fractional fourier transform and sampling theorem the fractional fourier transform frt is an extension of the ordinary fourier transform ft applying the language of the unified ft we develop frt expressions for discrete and continuous signals introducing a particular form of periodicity chirp periodicity the frt sampling theorem is derived as an extension of its ordinary counterpart

rank 3 acm bundles on general hypersurfaces in p5 abstract we prove that a general hypersurface in p 5 of degree d 3 does not support an indecomposable rank 3 arithmetically cohen macaulay acm bundle this settles the base case of a generic version of a conjecture of buchweitz greuel and schreyer

primitivity of group rings of non elementary torsion free hyperbolic groups abstract we use a recent result of alexander and nishinaka to show that if g is a non elementary torsion free hyperbolic group and r is a countable domain then the group ring rg is primitive this implies that the group ring kg of any non elementary torsion free hyperbolic group g over a field k is primitive

integral motives and special values of zeta functions for each field k we define a category of rationally decomposed mixed motives with z coefficients when k is finite we show that the category is tannakian and we prove formulas relating the behaviour of zeta functions near integers to certain ext groups

non noetherian groups and primitivity of their group algebras abstract we prove that the group algebra kg of a group g over a field k is primitive provided that g has a non abelian free subgroup with the same cardinality as g and that g satisfies the following condition for each subset m of g consisting of a finite number of elements not equal to 1 and for any positive integer m there exist distinct a b and c in g so that if x 1 1 g 1 x 1 x m 1 g m x m 1 where g i is in m and x i is equal to a b or c for all i between 1 and m then x i x i 1 for some i this generalizes results of 1 9 17 and 18 and proves that for every countably infinite group g satisfying kg is primitive for any field k we use this result to determine the primitivity of group algebras of one relator groups with torsion

on the integral hodge conjecture for real varieties i we formulate the real integral hodge conjecture a version of the integral hodge conjecture for real varieties and raise the question of its validity for cycles of dimension 1 on uniruled and calabi yau threefolds and on rationally connected varieties we relate it to the problem of determining the image of the borel haefliger cycle class map for 1 cycles with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of computing the torsion of the chow group of 1 cycles of real threefolds new results about these problems are obtained along the way

on locally indicable groups a group is said to be locally indicable if each of its non trivial finitely generated subgroups has the infinite cyclic group as a homomorphic image such groups were studied by higman 9 in connection with the zero divisor and unit problems for group rings more recently they have arisen i 2 12 in the study of equations over groups in 12 i proved a freiheitsatz for locally indicable groups this has been proved independently by brodskii 2 and short 22 the present paper arises from an investigation of a question put to me by s j pride whether torsionfree 1 relator groups are locally indicable the question was raised originally by baumslag i 1 problem 19 and an affirmative solution has recently been announced by brodski 2 as a consequence the group algebra rg of a torsion free 1 relator group g over an integral domain r has no non trivial zero divisors and no non trivial units using higman s results 9 the first fact was also proved by lewin and lewin 16 by embedding rg in a division ring the second appears to be new as was pointed out to me by k a brown a second consequence is that no 1 relator group has a non trivial finitely generated perfect subgroup which answers 1 problem 7 and 10 question 1 in fact using the freiheitsatz and the tower method described in 12 it is possible to prove the following general version of brodskii s theorem

sparse discrete fractional fourier transform and its applications the discrete fractional fourier transform is a powerful signal processing tool with broad applications for nonstationary signals in this paper we propose a sparse discrete fractional fourier transform sdfrft algorithm to reduce the computational complexity when dealing with large data sets that are sparsely represented in the fractional fourier domain the proposed technique achieves multicomponent resolution in addition to its low computational complexity and robustness against noise in addition we apply the sdfrft to the synchronization of high dynamic direct sequence spread spectrum signals furthermore a sparse fractional cross ambiguity function sfrcaf is developed and the application of sfrcaf to a passive coherent location system is presented the experiment results confirm that the proposed approach can substantially reduce the computation complexity without degrading the precision

the noether lefschetz conjecture and generalizations we prove the noether lefschetz conjecture on the moduli space of quasi polarized k3 surfaces this is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles i e sub arithmetic manifolds of the same type for compact manifolds this was proved in 3 here we extend the results of 3 to non compact manifolds this allows us to apply our results to the moduli spaces of quasi polarized k3 surfaces

discrete fractional fourier transform based on orthogonal projections the continuous fractional fourier transform frft performs a spectrum rotation of signal in the time frequency plane and it becomes an important tool for time varying signal analysis a discrete fractional fourier transform has been developed by santhanam and mcclellan see ibid vol 42 p 994 98 1996 but its results do not match those of the corresponding continuous fractional fourier transforms we propose a new discrete fractional fourier transform dfrft the new dfrft has dft hermite eigenvectors and retains the eigenvalue eigenfunction relation as a continous frft to obtain dft hermite eigenvectors two orthogonal projection methods are introduced thus the new dfrft will provide similar transform and rotational properties as those of continuous fractional fourier transforms moreover the relationship between frft and the proposed dfrft has been established in the same way as the conventional dft to continuous fourier transform

on sampling of band limited signals associated with the linear canonical transform sampling is one of the fundamental topics in the signal processing community theorems proposed under this topic form the bridge between the continuous time signals and discrete time signals several sampling theorems which aid in the reconstruction of signals in the linear canonical transform lct domain have been proposed in the literature however two main practical issues associated with the sampling of the lct still remain unresolved the first one relates to the reconstruction of the original signal from nonuniform samples and the other issue relates to the fact that only a finite number of samples are available practically focusing on these issues this paper seeks to address the above from the lct point of view first we extend several previously developed theorems for signals band limited in the fourier domain to signals band limited in the lct domain followed by the derivation of the reconstruction formulas for finite uniform or recurrent nonuniform sampling points associated with the lct simulation results and the potential applications of the theorem are also proposed

noether lefschetz locus for beilinson hodge cycles i we discuss the noether lefschetz locus for beilinson s hodge cycles here beilinson s hodge cycles mean the cohomology class of bloch s higher chow cycles on smooth varieties the main result is to give an estimate of the dimension of the maximal component of the noether lefschetz locus for the universal family of open complete intersections

rational curves on k3 surfaces we show that projective k3 surfaces with odd picard rank contain infinitely many rational curves our proof extends the bogomolov hassett tschinkel approach i e uses moduli spaces of stable maps and reduction to positive characteristic

sampling and sampling rate conversion of band limited signals in the fractional fourier transform domain the fractional fourier transform frft has become a very active area in signal processing community in recent years with many applications in radar communication information security etc this study carefully investigates the sampling of a continuous time band limited signal to obtain its discrete time version as well as sampling rate conversion for the frft firstly based on product theorem for the frft the sampling theorems and reconstruction formulas are derived which explain how to sample a continuous time signal to obtain its discrete time version for band limited signals in the fractional fourier domain secondly the formulas and significance of decimation and interpolation are studied in the fractional fourier domain using the results the sampling rate conversion theory for the frft with a rational fraction as conversion factor is deduced which illustrates how to sample the discrete time version without aliasing the theorems proposed in this study are the generalizations of the conventional versions for the fourier transform finally the theory introduced in this paper is validated by simulations

sampling and reconstruction of signals in function spaces associated with the linear canonical transform the linear canonical transform lct has been shown to be useful and powerful in signal processing optics etc many results of this transform are already known including sampling theory most existing sampling theories of the lct consider the class of bandlimited signals however in the real world many analog signals arising in engineering applications are non bandlimited in this correspondence we propose a sampling and reconstruction strategy for a class of function spaces associated with the lct which can provide a suitable and realistic model for real applications first we introduce definitions of semi and fully discrete convolutions for the lct then we derive necessary and sufficient conditions pertaining to the lct under which integer shifts of a chirp modulated function generate a riesz basis for the function spaces by applying the results we present a more comprehensive sampling theory for the lct in the function spaces and further a sampling theorem which recovers a signal from its own samples in the function spaces is established moreover some sampling theorems for shift invariant spaces and some existing sampling theories for bandlimited signals associated with the fourier transform ft the fractional ft or the lct are noted as special cases of the derived results finally some potential applications of the derived theory are presented