characterization of oriented graphs of rank 2 abstract let d be an oriented graph i e a digraph in which all arcs are not symmetric with vertex set 1 n the adjacency matrix a a i j of d is the n n matrix defined as a i j 1 if ij is an arc of d and a i j 0 if ij is not an arc of d the rank of d is defined to be the rank of its adjacency matrix in this paper oriented graphs of rank 2 are characterized

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

fast adaptive 2d vortex methods we present a new approach to vortex methods for the 2d euler equations we obtain long time high order accuracy at almost optimal cost by using three tools fast adaptive quadrature rules a free lagrangian formulation and a useful new analysis of the consistency error our error analysis halves the order of differentiability required of the flow and suggests an efficient new balance of smoothing parameters which works well with fast summation schemes numerical experiments with our methods confirm our theoretical predictions and display excellent long time accuracy

directed strongly regular graphs obtained from coherent algebras abstract the notion of a directed strongly regular graph was introduced by a duval in 1988 as one of the possible generalizations of classical strongly regular graphs to the directed case we investigate this generalization with the aid of coherent algebras in the sense of d g higman we show that the coherent algebra of a mixed directed strongly regular graph is a non commutative algebra of rank at least 6 with this in mind we examine the group algebras of dihedral groups the flag algebras of a steiner 2 designs in search of directed strongly regular graphs as a result a few new infinite series of directed strongly regular graphs are constructed in particular this provides a positive answer to a question of duval on the existence of a graph with certain parameter set having 20 vertices one more open case with 14 vertices listed in duval s paper is ruled out while new interpretations in terms of coherent algebras are given for many of duval s results

linear syzygies of curves with prescribed gonality abstract we prove two statements concerning the linear strand of the minimal free resolution of a k gonal curve c of genus g firstly we show that a general curve c of genus g of non maximal gonality k g 1 2 satisfies schreyer s conjecture that is b g k 1 c c g k this statement goes beyond green s conjecture and predicts that all highest order linear syzygies in the canonical embedding of c are determined by the syzygies of the k 1 dimensional scroll containing c secondly we prove an optimal effective version of the gonality conjecture for general k gonal curves which makes more precise the asymptotic gonality conjecture proved by ein lazarsfeld and improves results of rathmann

geometry of curves with exceptional secant planes linear series along the general curve we study linear series on a general curve of genus g whose images are exceptional with regard to their secant planes working in the framework of an extension of brill noether theory to pairs of linear series we prove that a general curve has no linear series with exceptional secant planes in a very precise sense whenever the total number of series is finite next we partially solve the problem of computing the number of linear series with exceptional secant planes in a one parameter family in terms of tautological classes associated with the family by evaluating our hypothetical formula along judiciously chosen test families as an application we compute the number of linear series with exceptional secant planes on a general curve equipped with a one dimensional family of linear series we pay special attention to the extremal case of d secant d 2 planes to 2d 1 dimensional series which appears in the study of hilbert schemes of points on surfaces in that case our formula may be rewritten in terms of hypergeometric series which allows us both to prove that it is nonzero and to deduce its asymptotics in d

accuracy of the random vortex method for a problem with non smooth initial conditions abstract the accuracy of the random vortex method in approximating the solution of the navier stokes equation is investigated using the model problem of a circular vortex as suggested by milinazzo and saffman 13 the method consists of partitioning the vorticity into vortex blobs these blobs are moved via two actions first a blob is deterministically moved under the action of the velocity field associated with the other blobs then to simulate viscosity a random component is added to the position of the blob for this model problem the nonlinear terms of the navier stokes equation vanish thus the major error inherent in the deterministic component of the method vanishes consequently for this model problem concentration is on the interaction of the deterministic and random components of the method results show that the accuracy of the method depends heavily on the initial distribution and strength of the computational elements i e the vortex blobs with the right choice of initial conditions it is found that e1 t l t a t a t is o r 1 2 n 1 2 where l t and a t are respectively the exact and computed angular moment of vorticity distribution at time t for n vortex blobs at a reynolds number r

fast triangulated vortex methods for the 2d euler equations abstract vortex methods for inviscid incompressible two dimensional fluid flow are usually based on blob approximations this paper presents a vortex method in which the vorticity is approximated by a piecewise polynomial interpolant on a delaunay triangulation of the vortices an efficient reconstruction of the delaunay triangulation at each step makes the method accurate for long times the vertices of the triangulation move with the fluid velocity which is reconstructed from the vorticity via a simplified fast multipole method for the biot savart law with a continuous source distribution the initial distribution of vortices is constructed from the initial vorticity field by an adaptive approximation method which produces good accuracy even for discontinuous initial data numerical results show that the method is highly accurate over long time intervals experiments with single and multiple circular and elliptical rotating patches of both piecewise constant and smooth vorticity indicate that the method produces much smaller errors than blob methods with the same number of degrees of freedom at little additional cost generalizations to domains with boundaries viscous flow and three space dimensions are discussed

a characterization of graphs with rank 5 abstract the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph g with rank 5 or equivalently what is the structure of a connected n vertex graph g whose adjacency matrix has nullity n 5 in this paper we completely characterize connected graphs g whose adjacency matrix has rank 5

on the nullity of bipartite graphs abstract the nullity of a graph is defined to be the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph in this paper we obtain the nullity set of bipartite graphs of order n and characterize the bipartite graphs with nullity n 4 and the regular bipartite graphs with nullity n 6

a note on nikulin surfaces and their moduli spaces we study a number of natural linear systems carried by any polarized nikulin surface of genus g determine their positivity and establish their brill noether theory as an application we compute the classes of some natural effective divisors on the moduli space of nikulin surfaces relying upon recent work of farkas and rimanyi

deterministic high order vortex methods for the 2d navier stokes equation with rezoning in this paper we extend the use of automatic rezoning toviscousflow in two dimensions in a previous paper we tested this technique on inviscid flow with very good results to simulate viscosity we follow fishelov s idea of explicitly taking the laplacian of the cutoff function but unlike fishelov we use a moving grid this eliminates the need to approximate the gradient of the vorticity but rezoning needs to be used to keep the discretization error low we first test the method on a radially symmetric problem where the exact vorticity is known for all time using both an eighth order cutoff function and an infinite order cutoff function we obtain low errors and high rates of convergence then we calculate the evolution of two circular vortex patches and of a square vorticity patch the exact solution for the last two problems is not known in all test problems we use a viscosity coefficient of 0 0005

g circulant solutions to the 0 1 matrix equation am jn this paper discusses the structure of g circulant solutions to a m jn where a is an unknown 0 1 matrix and jn is a matrix of order n with all entries equal to 1 wang has made a conjecture on the form of all such solutions partially verifying his conjecture we discover a close relationship among the hall polynomial a x the shifting parameter g and the order n of any 0 1 g circulant solution a to a m jn as a consequence all the g circulant solutions to a m jn are completely determined in the case that n is a prime power moreover in the case that the constant line sum r of a is square free all g circulant solutions to a m jn are proved to be permutation similar to the adjacency matrix of the de bruijn digraph b r m motivated by the current status of this subject we identify all 0 1 g circulant solutions to a m jn whose hall polynomials have some specific properties and we further determine the possible values that the shifting parameter g of such solutions may take the uniqueness of these solutions up to isomorphism is also investigated our paper is concluded with some open problems in particular we give the concept of standard factorization and conjecture that all factorizations of x n 1 x 1 into a product of 0 1 polynomials must be standard and thus point out the close similarity between wang s conjecture and a conjecture appearing in the study of perfect graph 2002 elsevier science inc all rights reserved

the generic green lazarsfeld secant conjecture using lattice theory on special k3 surfaces calculations on moduli stacks of pointed curves and voisin s proof of green s conjecture on syzygies of canonical curves we prove the prym green conjecture on the naturality of the resolution of a general prym canonical curve of odd genus as well as many cases of the green lazarsfeld secant conjecture on syzygies of non special line bundles on general curves

further results on the minimum rank of regular classes of 0 1 matrices abstract let b n k be the set of all 0 1 matrices of order n with constant line sum k and let n k be the minimum rank over b n k it is known that n k n k n k n k k where n k is the rank of a recursively defined matrix a b n k brualdi manber and ross showed that n k n k if and only if k n in this paper we show that n k n k k if and only if n k satisfies one of the following three relations i n 1 mod k k 2 or 3 ii n k 1 k 2 iii n 4 q 3 k 4 and q 1 moreover we obtain the exact values of n 4 for all n 4 and determine all the possible ranks of regular 0 1 matrices in b n 4 we also present some positive integer pairs n k such that n k n k n k k which gives a positive answer to a question posed by pullman and stanford

skew rank of an oriented graph in terms of matching number abstract an oriented graph g is a digraph without loops and multiple arcs where g is called the underlying graph of g let s g denote the skew adjacency matrix of g the rank of s g is called the skew rank of g denoted by sr g which is even since s g is skew symmetric li and yu 2015 12 proved that the skew rank of an oriented unicyclic graph g is either 2 m g 2 or 2 m g where m g denotes the matching number of g in this paper we extend this result to general cases it is proved that the skew rank of an oriented connected graph g is an even integer satisfying 2 m g 2 g sr g 2 m g where g e g v g 1 is the number of fundamental cycles also called the first betti number besides the oriented graphs satisfying sr g 2 m g 2 g are characterized definitely

on the nef cone of symmetric products of a generic curve the nef cone of the symmetric product c of a curve c with general moduli is easily determined when k is at least equal to the gonality gon c of the curve in this paper we describe the nef cone for k gon c 1 when the genus of c is even in this case the boundary of the nef cone has rational slope and is determined using curves on c associated to the g s of c 1 introduction let c be a smooth irreducible curve of genus g 1 for any k 2 denote by c its mi symmetric product that parametrizes effective degree k divisors z x x2 on c let ni c k resp nl c k be the set of r linear combinations of 1 cycles resp divisors on k modulo numerical equivalence nx k is called the n ron severi group of c and denoted ns c k n k and nl k are finite dimensional vector spaces and the bilinear form given by the intersection pairing is nondegenerate if c has general moduli n c and nl c have rank two an interesting problem is to try to determine inside n c the convex cone of classes of effective 1 cycles denoted by ne c and called the cone of curves or mori cone dually one can study inside nl c the nef cone nef k i e the closure of the convex cone of classes of ample divisors by kleiman s criterion of ampleness

the geometry of the moduli space of odd spin curves the spin moduli spacesg is the parameter space of theta characteristics spin structures on stable curves of genus g it has two connected components s g ands g depending on the parity of the spin structure we establish a complete birational classication by kodaira dimension of the odd components g of the spin moduli space we show thats g is uniruled for g 12 and even unirational for g 8 in this range introducing the concept of cluster for the mukai variety whose one dimensional linear sections are general canonical curves of genus g we construct new birational models ofs g these we then use to explicitly describe the birational structure of s g for instance s 8 is birational to a locally trivial p 7 bundle over the moduli space of elliptic curves with seven pairs of marked points for g 12 we prove thats g is a variety of general type in genus 12 this requires the construction of a counterexample to the slope conjecture on eective divisors on the moduli space of stable curves of genus 12

on the relationship between the skew rank of an oriented graph and the rank of its underlying graph abstract an oriented graph g is a digraph without loops and multiple arcs where g is the underlying graph of g let s g denote the skew adjacency matrix of g and a g be the adjacency matrix of g the rank resp skew rank of g resp g written as r g resp s r g refers to the rank of a g resp s g it is natural and interesting to study the relationship between s r g and r g wong ma and tian european j combin 54 2016 76 86 30 determined the sharp upper bound on s r g r g as a continuance of it in this paper a sharp lower bound on s r g r g is determined as well a sharp upper bound on s r g r g is determined all the corresponding extremal oriented graphs g are characterized respectively

irreducibility brill noether loci and vojta s inequality this paper deals with generalizations of hilbert s irreducibility theorem the classical hilbert irreducibility theorem states that for any cover f of the projective line defined over a number field k there exist infinitely many k rational points on the projective line such that the fiber of f over p is irreducible over k in this paper we consider similar statements about algebraic points of higher degree on curves of any genus we prove that hilbert s irreducibility theorem admits a natural generalization to rational points on an elliptic curve and thus via a theorem of abramovich and harris to points of degree 3 or less on any curve we also present examples that show that this generalization does not hold for points of degree 4 or more these examples come from an earlier geometric construction of debarre and fahlaoui some additional necessary facts about this construction can be found in the appendix provided by debarre we exhibit a connection between these irreducibility questions and the sharpness of vojta s inequality for algebraic points on curves in particular we show that vojta s inequality is not sharp for the algebraic points arising in our examples

minimal resolutions chow forms and ulrich bundles on k3 surfaces the minimal resolution conjecture mrc for points on a projective varietyx p predicts that the minimal graded free resolution of a general set x of points is as simple as the geometry of x allows originally the most studied case has been that when x p see epsw the general form of the mrc for subvarieties x p was formulated in mus and fmp the betti diagram of a large enough set x consisting of general points is obtained from the betti diagram of x by adding two rows indexed by u 1 and u where u is an integer depending on all differences bi 1 u 1 bi u are known and depend on the hilbert polynomial px and i u and see fmp the minimal resolution conjecture for general points on x predicts that

the rank of a signed graph in terms of the rank of its underlying graph abstract let g be a signed graph and a be its adjacency matrix where g is the underlying graph of the rank r of is the rank of a we know that for a signed graph g is balanced if and only if g g that is when is balanced then r r g where r g is the rank of the underlying graph g of a natural problem is that how about the relations between the rank of an unbalanced signed graph and the rank of its underlying graph in this paper we first prove that r g 2 d g r r g 2 d g for an unbalanced signed graph with d g 1 where d g e g v g g is the dimension of cycle spaces of g g is the number of connected components of g as an application we also prove that 1 d g r r g 1 d g for an unbalanced signed graph with d g 1 all corresponding extremal graphs are characterized

equiangular tight frames from complex seidel matrices containing cube roots of unity we derive easily verifiable conditions which characterize when complex seidel matrices containing cube roots of unity have exactly two eigenvalues the existence of such matrices is equivalent to the existence of equiangular tight frames for which the inner product between any two frame vectors is always a common multiple of the cube roots of unity we also exhibit a relationship between these equiangular tight frames complex seidel matrices and highly regular directed graphs we construct examples of such frames with arbitrarily many vectors

a characterization of graphs with rank 4 the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph with rank 4 this question has not yet been fully answered in the literature and only some partial results are known in this paper we resolve this question by completely characterizing graphs g whose adjacency matrix has rank 4

most odd degree hyperelliptic curves have only one rational point consider the smooth projective models c of curves y 2 f x with f x 2 z x monic and separable of degree 2g 1 we prove that for g 3 a positive fraction of these have only one rational point the point at innity we prove a lower bound on this fraction that tends to 1 as g 1 finally we show that c q can be algorithmically computed for such a fraction of the curves the method can be summarized as follows using p adic analysis and an idea of mccallum we develop a reformulation of chabauty s method that shows that certain computable conditions imply c q 1 on the other hand using further p adic analysis the theory of arithmetic surfaces a new result on torsion points on hyperelliptic curves and crucially the bhargava gross theorems on the average number and equidistribution of nonzero 2 selmer group elements we prove that these conditions are often satised for p 2

moduli of theta characteristics via nikulin surfaces we study moduli spaces of k3 surfaces endowed with a nikulin involution and their image in the moduli space r g of prym curves of genus g we observe a striking analogy with mukai s well known work on ordinary k3 surfaces many of mukai s results have a very precise prym nikulin analogue for instance a general prym curve from r g is a section of a nikulin surface if and only if g 7 and g 6 furthermore r 7 has the structure of a fibre space over the corresponding moduli space of polarized nikulin surfaces we then use these results to study the geometry of the moduli space s g of even spin curves with special emphasis on the transition case of s 8 which is a 21 dimensional calabi yau variety

syzygies of canonical curves and special linear series 0 introduction 105 1 line bundles on scrolls 109 2 scrolls and pencils 113 3 syzygies of a subvariety of a scroll 115 4 canonical curves 118 5 on the equation of rational surfaces 121 6 d gonal curves for d 5 126 7 proof of the main result 131 references 136

projectivity and birational geometry of bridgeland moduli spaces we construct a family of nef divisor classes on every moduli space of stable complexes in the sense of bridgeland this divisor class varies naturally with the bridgeland stability condition for a generic stability condition on a k3 surface we prove that this class is ample thereby generalizing a result of minamide yanagida and yoshioka our result also gives a systematic explanation of the relation between wall crossing for bridgeland stability and the minimal model program for the moduli space we give three applications of our method for classical moduli spaces of sheaves on a k3 surface 1 we obtain a region in the ample cone in the moduli space of gieseker stable sheaves only depending on the lattice of the k3 2 we determine the nef cone of the hilbert scheme of n points on a k3 surface of picard rank one when n is large compared to the genus 3 we verify the hassett tschinkel huybrechts sawon conjecture on the existence of a birational lagrangian fibration for the hilbert scheme in a new family of cases

singularities of moduli spaces of sheaves on k3 surfaces and nakajima quiver varieties abstract the aim of this paper is to study the singularities of certain moduli spaces of sheaves on k3 surfaces by means of nakajima quiver varieties the singularities in question arise from the choice of a non generic polarization with respect to which we consider stability and admit natural symplectic resolutions corresponding to choices of general polarizations for sheaves that are pure of dimension one we show that these moduli spaces are locally around a singular point isomorphic to a quiver variety and that via this isomorphism the natural symplectic resolutions correspond to variations of git quotients of the quiver variety

moduli of roots of line bundles on curves we treat the problem of completing the moduli space for roots of line bundles on curves special attention is devoted to higher spin curves within the universal picard scheme two new different constructions both using line bundles on nodal curves as boundary points are carried out and compared with pre existing ones

rank of adjacency matrices of directed strongly regular graphs for a positive integer r we consider the set br of all values of kn for which there exists an n n matrix with entries 0 and 1 such that each row and each column has exactly k 1 s and the matrix has rank r we prove that the set br is finite for every r if there exists a k regular directed graph on n vertices such that its adjacency matrix has rank r then kn br we use this to exclude existence of directed strongly regular graphs for infinitely many feasible parameter sets

on the characterization of digraphs with given rank abstract the rank of a digraph is defined as the rank of its adjacency matrix in this paper we show how to find all digraphs of rank k from the reduced bipartite graphs of rank 2k in particular since the bipartite graphs of rank 2 4 and 6 are known then we characterize digraphs of rank 1 2 and 3 the results are based on rank preserving operations on digraphs such as splitting and fusion of vertices or twin deletion and vertex multiplication

directed strongly regular graphs with rank 5 abstract from the parameters n k t of a directed strongly regular graph dsrg a duval 1988 4 showed how to compute the eigenvalues and multiplicities of the adjacency matrix and thus the rank of the adjacency matrix for every rational number q where 1 5 q 7 10 there is a feasible i e satisfying duval s conditions parameter set for a dsrg with rank 5 and with k n q in this paper we show that there exist a dsrg with such a feasible parameter set only if k n is 1 5 1 3 2 5 1 2 3 5 or 2 3 every dsrg with rank 5 therefore has parameters of a known graph the proof is based on an enumeration of 5 5 matrices with entries in 0 1

mixed graphs with h rank 3 abstract in the article mohar 2016 9 mohar determined all mixed graphs with h rank 2 and used it to classify cospectral graphs with respect to their hermitian adjacency matrix constructing a class of graphs which can not be determined by their h spectrum in the present paper we investigate the h rank of mixed graphs further determining the h ranks of those mixed graphs with trees cycles and complete bipartite graphs as underlying graphs respectively moreover we characterize all mixed graphs with h rank 3 and show that all connected mixed graphs with h rank 3 can be determined by their h spectrum

syzygies of torsion bundles and the geometry of the level l modular variety over overline mathcal m _ g we formulate and in some cases prove three statements concerning the purity or more generally the naturality of the resolution of various modules one can attach to a generic curve of genus g and a torsion point of l in its jacobian these statements can be viewed an analogues of green s conjecture and we verify them computationally for bounded genus we then compute the cohomology class of the corresponding non vanishing locus in the moduli space mathcal r g ell of twisted level l curves of genus g and use this to derive results about the birational geometry of mathcal r g ell for instance we prove that mathcal r g 3 is a variety of general type when g 11 and the kodaira dimension of mathcal r 11 3 is greater than or equal to 19 in the last section we explain probabilistically the unexpected failure of the prym green conjecture in genus 8 and level 2

efficient and highly accurate computation of a class of radially symmetric solutions of the navier stokes equation and the heat equation in two dimensions in this paper we test several different formulas for the computation of the exact vorticity and angular velocity in certain radially symmetric solutions of the two dimensional navier stokes equation in vorticity stream function form the class of initial conditions for the vorticity considered here has often been used by many authors in the study of vortex methods however only in the case of zero viscosity has it been possible to efficiently compute the exact vorticity and velocity at later times the expressions for the vorticity and angular velocity given in this paper enable us to compute these quantities both efficiently and highly accurately for nonzero viscosity this makes it feasible to obtain reliable error measurements in the study of vortex methods for the navier stokes equation

level structures on abelian varieties kodaira dimensions and lang s conjecture abstract assuming lang s conjecture we prove that for a prime p number field k and positive integer g there is an integer r such that no principally polarized abelian variety a k has full level p r structure to this end we use a result of zuo to prove that for each closed subvariety x in the moduli space a g of principally polarized abelian varieties of dimension g there exists a level m x such that the irreducible components of the preimage of x in a g m are of general type for m m x

nikulin involutions on k3 surfaces we study the maps induced on cohomology by a nikulin i e a symplectic involution on a k3 surface we parametrize the 11 dimensional irreducible components of the moduli space of algebraic k3 surfaces with a nikulin involution and we give examples of the general k3 surface in various components we conclude with some remarks on morrison nikulin involutions these are nikulin involutions which interchange two copies of e 8 1 in the neron severi group