a unified approach to some recurrence sequences via faa di bruno s formula abstract the well known faa di bruno formula for higher derivatives of a composite function plays an important role in combinatorics in this paper we obtain a unified approach to some recurrence sequences for the complete bell polynomials by using faa di bruno s formula furthermore we obtain an inverse relation of lah numbers of two types and by using this relation we obtain some new and interesting combinatorial identities

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

brill noether theory of curves on enriques surfaces i the positive cone and gonality we study the existence of linear series on curves lying on an enriques surface and general in their complete linear system using a method that works also below the bogomolov reider range we compute in all cases the gonality of such curves we also give a new result about the positive cone of line bundles on an enriques surface and we show how this relates to the gonality

the sum of like powers of the zeros of the riemann zeta function on analyse une methode de calcul de la somme r p r ou r est un entier superieur a 1 la somme etant prise sur tous les zeros complexes de la fonction zeta de riemann

lmi based criteria for global robust stability of bidirectional associative memory networks with time delay abstract in this paper several new sufficient conditions are given to ensure existence uniqueness and globally exponential robust stability of the equilibrium point for bidirectional associative memory bam networks with delays this novel approach based on the linear matrix inequality lmi technique removes some existing restrictions on the system s parameters and the derived conditions are easy to verify via the lmi toolbox in addition two examples are given to show the effectiveness and the advantage of the proposed results

on the eigenvalues of the electrostatic integral operator abstract a fundamental result in scattering and potential theory in r 3 states that the spectrum of the electrostatic integral operator lies in the interval 1 1 in the case of a sphere and a prolate spheroid it is known that the spectrum of the operator lies in the interval 1 0 a fundamental question arises whether the spectrum of the operator always lies in this interval or whether there exists a smooth surface for which the electrostatic integral operator has a positive eigenvalue in this paper this question is answered a surface is produced whereby the underlying integral operator has a positive eigenvalue

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

on the convergence of interior reflective newton methods for nonlinear minimization subject to bounds we consider a new algorithm an interior reflective newton approach for the problem of minimizing a smooth nonlinear function of many variables subject to upper and or lower bounds on some of the variables this approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths reflection paths the interior reflective approach does not require identification of an activity set in this paper we establish that the interior reflective newton approach is globally and quadratically convergent moreover we develop a specific example of interior reflective newton methods which can be used for large scale and sparse problems

on the magnetostatic integral operator for ellipsoids the magnetostatic integral operator in r3is of considerable interest in both scattering and potential theory as well as for its scalar pendant the electrostatic integral operator its eigenvalues and its eigenfunctions play a fundamental role in the theory of permanent magnetization while the eigenvalues of both operators are closely related explicit expressions for the eigenfunctions of the magnetostatic operator are known only in the case when the underlying surface is a sphere for the cases of ellipsoids and spheroids we present explicit expressions for the eigenvalues in terms of lame functions and associated legendre functions and expressions for eigenfunctions in terms of surface ellipsoidal and surface spheroidal harmonics respectively as an application the permanent magnetization of ellipsoids is discussed

divided differences and a general explicit formula for sequences of mansour mulay shattuck divided differences is a very important tool often used both in theory and in practice for the approximation of functions in this paper we employ divided differences with repetitions to find a general explicit formula for the sequences satisfying mansour mulay shattuck s two term recurrence with arbitrary b i i 0

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

normal generation and clifford index on algebraic curves for a smooth curve c it is known that a very ample line bundle mathcal l on c is normally generated if cliff mathcal l cliff c and there exist extremal line bundles mathcal l non normally generated very ample line bundle with cliff mathcal l cliff c with h 1 mathcal l le 1 however it has been unknown whether there exists an extremal line bundle mathcal l with h 1 mathcal l ge 2 in this paper we prove that for any positive integers g c with g 2c 5 and c equiv 0 mod 2 there exists a smooth curve of genus g and clifford index c carrying an extremal line bundle mathcal l with h 1 mathcal l 2 in fact a smooth quadric hypersurface section c of a general projective k3 surface always has an extremal line bundle mathcal l with h 1 mathcal l 2 more generally if c has a line bundle mathcal m computing the clifford index c of c with 3c 2 3 deg mathcal m leq g 1 then c has such an extremal line bundle mathcal l

a hyper kahler compactification of the intermediate jacobian fibration associated with a cubic 4 fold let x be a general cubic fourfold it was observed by donagi and markman that the relative intermediate jacobian fibration j u with u p5 x associated to the smooth hyperplane sections of x carries a natural holomorphic symplectic form making the fibration lagrangian in this paper we obtain a smooth projective compactification j of j with the property that the holomorphic symplectic form on j extends to a holomorphic symplectic form on j in particular j is a 10 dimensional compact hyper kahler manifold which we show to be deformation equivalent to the exceptional example of o grady this proves a conjecture by markushevich

letter to the editor stability analysis on delayed neural networks based on an improved delay partitioning approach in this paper the asymptotical stability is investigated for a class of delayed neural networks dnns in which one improved delay partitioning idea is employed by choosing an augmented lyapunov krasovskii functional and utilizing general convex combination method two novel conditions are obtained in terms of linear matrix inequalities lmis and the conservatism can be greatly reduced by thinning the partitioning of delay intervals moreover the lmi based criteria heavily depend on both the upper and lower bounds on time delay and its derivative which is different from the existent ones though the results are not presented via standard lmis they still can be easily checked by resorting to matlab lmi toolbox finally three numerical examples are given to demonstrate that our results can be less conservative than the present ones

crandall s computation of the incomplete gamma function and the hurwitz zeta function with applications to dirichlet l series this paper extends tools developed by crandall 2012 16 to provide robust high precision methods for computation of the incomplete gamma function and the lerch transcendent we then apply these to the corresponding computation of the hurwitz zeta function and so of dirichlet l series and character polylogarithms

electric field and potential calculation for a bioelectric current dipole in an ellipsoid computer based modelling in ellipsoidal geometry can be useful in electro encephalography and gastrography due to the resemblance of the human brain and stomach to an ellipsoid both theoretically and computationally the bioelectric current dipole model is important for the study of electrical activity in these two organs computing the electric potential and electric field e due to a current dipole located in an ellipsoid requires truncated series expansions involving ellipsoidal harmonics which are the products of lame functions a theoretical model for this has appeared in the literature only for expansions of order 2 in however this may be insufficient in forward electrogastrography while an analogous situation is encountered in some geodetic problems where similar expansions are required in this paper we propose a generalized model for computing and e numerically using harmonic expansions of arbitrary order and degree the implementation of such a procedure involves finding roots of lame polynomials of degree 5 or higher using an optimization technique for solving nonlinear systems of equations this process can allow one to make a knowledgeable decision concerning the optimal expansion size required for the physical problem under investigation without compromising the overall accuracy of the computation

two chain rules for divided differences and faa di bruno s formula in this paper we derive two formulas for divided differences of a function of a function both formulas lead to other divided difference formulas such as reciprocal and quotient rules the two formulas can also be used to derive faa di bruno s formula and other formulas for higher derivatives of composite functions we also derive a divided difference version of faa di bruno s determinant formula

generalized lazarsfeld mukai bundles and a conjecture of donagi and morrison abstract let s be a k3 surface and assume for simplicity that it does not contain any 2 curve using coherent systems we express every non simple lazarsfeld mukai bundle on s as an extension of two sheaves of some special type that we refer to as generalized lazarsfeld mukai bundles this has interesting consequences concerning the brill noether theory of curves c lying on s from now on let g denote the genus of c and a be a complete linear series of type g d r on c such that d g 1 and the corresponding brill noether number is negative first we focus on the cases where a computes the clifford index if r 1 and with only some completely classified exceptions we show that a coincides with the restriction to c of a line bundle on s this is a refinement of green and lazarsfeld s result on the constancy of the clifford index of curves moving in the same linear system then we study a conjecture of donagi and morrison predicting that under no hypothesis on its clifford index a is contained in a g e s which is cut out from a line bundle on s and satisfies e g 1 we provide counterexamples to the last inequality already for r 2 a slight modification of the conjecture which holds for r 1 2 is proved under some hypotheses on the pair c a and its deformations we show that the result is optimal in the sense that our hypotheses cannot be avoided by exhibiting in appendix a some counterexamples obtained jointly with andreas leopold knutsen

on the vanishing of higher syzygies of curves ii we verify green s conjecture for a generic k gonal curve of genus g for g geq k k 1 2

magnetoencephalography in ellipsoidal geometry an exact analytic solution for the forward problem in the theory of biomagnetics of the human brain is known only for the 1d case of a sphere and the 2d case of a spheroid where the excitation field is due to an electric dipole within the corresponding homogeneous conductor in the present work the corresponding problem for the more realistic ellipsoidal brain model is solved and the leading quadrupole approximation for the exterior magnetic field is obtained in a form that exhibits the anisotropic character of the ellipsoidal geometry the results are obtained in a straightforward manner through the evaluation of the interior electric potential and a subsequent calculation of the surface integral over the ellipsoid using lame functions and ellipsoidal harmonics the basic formulas are expressed in terms of the standard elliptic integrals that enter the expressions for the exterior lame functions the laborious task of reducing the results to the spherical geometry is also included

global convergence of delayed dynamical systems we discuss some delayed dynamical systems investigating their stability and convergence in a critical case to ensure the stability the coefficients of the dynamical system must satisfy some inequalities in most existing literatures the restrictions on the coefficients are strict inequalities the tough question is what will happen in the case critical case the strict inequalities are replaced by nonstrict inequalities i e is replaced by spl les the purpose of the paper is to discuss this critical case and give an affirmative answer in the case that the activation functions are hyperbolic tangent

a sum property of the eigenvalues of the electrostatic integral operator abstract the double layer potential integral operator and the electrostatic integral operator in r 3 occur while using integral equation methods in scattering and potential theory if the underlying surface is either a sphere a spheroid or a triaxial ellipsoid explicit expressions of the eigenvalues and eigenfunctions of order n n 0 are known the main result of this paper is the following the sum of all eigenvalues of order n n 0 each counted with respect to its multiplicity is 1 this is trivial for the sphere in the case of spheroids a proof is given for n n 0 for the triaxial ellipsoid this sum property is verified for n 0 1 2 3 whereas for n 3 some numerical results are provided

convergence for a class of delayed recurrent neural networks without m matrix condition in this paper we investigate the asymptotic behavior of solutions to a class of recurrent neural network model with delays without assuming m matrix condition it is shown that every solution of the network tends to an equilibrium point as t our results improve and extend some corresponding ones already known

green s conjecture for curves on rational surfaces with an anticanonical pencil green s conjecture is proved for smooth curves c lying on a rational surface s with an anticanonical pencil under some mild hypotheses on the line bundle l mathcal o s c constancy of clifford dimension clifford index and gonality of curves in the linear system vert l vert is also obtained

electroencephalography in ellipsoidal geometry the human brain is shaped in the form of an ellipsoid with average semiaxes equal to 6 6 5 and 9 cm this is a genuine 3 d shape that reflects the anisotropic characteristics of the brain as a conductive body the direct electroencephalography problem in such anisotropic geometry is studied in the present work the results which are obtained through successively solving an interior and an exterior boundary value problem are expressed in terms of elliptic integrals and ellipsoidal harmonics both in jacobian as well as in cartesian form reduction of our results to spheroidal as well as to spherical geometry is included in contrast to the spherical case where the boundary does not appear in the solution the boundary of the realistic conductive brain enters explicitly in the relative expressions for the electric field moreover the results in all three geometrical models reveal that to some extend the strength of the electric source is more important than its location

combinatorial formulae for the derivatives of lame functions lame functions play a central role in the theory of ellipsoidal harmonics and have many varied applications in mathematical physics in this article a generalized approach to the computation of lame function derivatives of arbitrary order is derived and it is demonstrated how these derivatives can be expressed recursively in terms of the original lame functions by making use of combinatorial formulae discovered by di bruno girard and waring

the stieltjes function definition and properties definition et proprietes d une fonction liee au developpement en serie de laurent de la fonction zeta de riemann au voisinage du point singulier s 1 fonction de stieltjes

exotic g_2 g2 manifolds we exhibit the first examples of closed 7 dimensional riemannian manifolds with holonomy g 2 that are homeomorphic but not diffeomorphic these are also the first examples of closed ricci flat manifolds that are homeomorphic but not diffeomorphic the examples are generated by applying the twisted connected sum construction to fano 3 folds of picard rank 1 and 2 the smooth structures are distinguished by the generalised eells kuiper invariant introduced by the authors in a previous paper

computation and theory of extended mordell tornheim witten sums we consider some fundamental generalized mordell tornheim witten mtw zeta function values along with their derivatives and explore connections with multiple zeta values mzvs to achieve this we make use of symbolic integration high precision numerical integration and some interesting combinatorics and special function theory our original motivation was to represent unresolved constructs such as eulerian log gamma integrals we are able to resolve all such integrals in terms of a mtw basis we also present for a substantial subset of mtw values explicit closed form expressions in the process we signicantly extend methods for high precision numerical computation of polylogarithms and their derivatives with respect to order

convergence for pseudo monotone semiflows on product ordered topological spaces in this paper we consider a class of pseudo monotone semiflows which only enjoy some weak monotonicity properties and are defined on product ordered topological spaces under certain conditions several convergence principles are established for each precompact orbit of such a class of semiflows to tend to an equilibrium which improve and extend some corresponding results already known some applications to delay differential equations are presented

multidimensional bell polynomials of higher order we develop some extensions of the classical bell polynomials previously obtained by introducing a further class of these polynomials called multidimensional bell polynomials of higher order they arise considering the derivatives of functions f in several variables f i i 1 2 m where f i are composite functions of different orders i e f i t i 1 i 2 i r i t i 1 2 m we show that these new polynomials are always expressible in terms of the ordinary bell polynomials by means of suitable recurrence relations or formal multinomial expansions moreover we give a recurrence relation for their computation

the riemann zeta function and its derivatives formulas for higher derivatives of the riemann zeta function are developed from ramanujan s theory of the constant of series by using the euler maclaurin summation methods formulas for zeta n s zeta n 1 s and zeta n 0 are obtained additional formulas involving the stieltjes constants are also derived analytical expression for error bounds is given in each case the formulas permit accurate derivative evaluation and the error bounds are shown to be realistic a table of zeta prime s is presented to 20 significant figures for s 20 0 1 20 for rational arguments zeta 1 k zeta prime 1 k are given for k 10 1 10 the first ten zeros of zeta prime s are also tabulated because the stieltjes constants appear in many formulas the constants were evaluated freshly for this work formulas for the gamma n are derived with new error bounds and a tabulation of the constants is given from n 0 to 100

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

the mercat conjecture for stable rank 2 vector bundles on generic curves we prove the mercat conjecture for rank 2 vector bundles on generic curves of every genus for odd genus we identify the effective divisor in mg where the mercat conjecture fails and compute its slope the clifford index of a smooth curve c defined as the non negative quantity cliff c min deg l 2r l l pic c h c l 2 deg l g 1 stratifies the moduli space mg of curves of genus g with the smallest stratum being that consisting of hyperelliptic curves which have clifford index zero while the definition is obviously inspired by the classical clifford theorem the notion came to real prominence in the works of green lazarsfeld g gl voisin v2 and many others in the context of syzygies of canonical curves for a general curve c of genus g one has cliff c bg 1 2 c the locus of curves c mg with cliff c bg 1 2 c is a subvariety of codimension 1 respectively 2 when g is odd respectively even much studied by harris and mumford hm it has been a long standing problem to find an adequate definition of the clifford index for higher rank vector bundles on curves a higher rank clifford index should not only capture the behavior of the generic curve from the point of view of special higher rank vector bundles but also provide a geometrically meaningful stratification ofmg an interesting definition has been put forward by lange and newstead ln1 for a semistable vector bundle e of rank 2 and slope e on a curve c of genus g one defines its clifford index as cliff e e h c e 2 0 the rank 2 clifford index of c is then defined as the quantity cliff2 c min cliff e e uc 2 d d 2g 2 h c e 4 observe that if l is a line bundle on c with deg l 2h0 c l 2 cliff c that is l computes the clifford index of c then cliff l l cliff c in particular the inequality cliff2 c cliff c holds for every curve c of genus g the main achievement of this paper is the proof of the following result theorem 1 for a general curve c of genus g the following equality holds cliff2 c cliff c g 1 2 equivalently if e is a semistable rank 2 vector bundle on c contributing to cliff2 c then h c e e 2 cliff c mercat me conjectured the equality cliff2 c cliff c for every smooth curve c mg counterexamples to this expectation have been found using noether lefschetz special k3 surfaces in fo1 fo2 and ln2 however these examples are special in moduli and theorem 1 proves mercat s conjecture for generic curves of every genus

the gonality and the clifford index of curves on a toric surface abstract we determine the gonality and the clifford index for curves on a compact smooth toric surface moreover it is shown that their gonality is computed by pencils on the ambient surface from the geometrical viewpoint this means that the gonality can be read off from the lattice polygon associated to the curve

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

a spectral property of the electrostatic integral operator for an ellipsoid the eigenfunctions of the electrostatic integral operator and its adjoint are governed by the lame polynomials it is shown that the arithmetic mean of the eigenvalues corresponding to the maximum number of linearly independent eigenfunctions of equal order is the same for the ellipsoid as for the sphere

formulas for higher derivatives of the riemann zeta function the functional equation for c s is used to obtain formulas for all derivatives t k s a closed form evaluation of t k 0 is given and numerical values are computed to

a multivariate faa di bruno formula with applications a multivariate faa di bruno formula for computing arbitrary partial derivatives of a function composition is presented it is shown by way of a general identity how such derivatives can also be expressed in the form of an infinite series applications to stochastic processes and multivariate cumulants are then delineated

on the splitting of lazarsfeld mukai bundles on k3 surfaces abstract let x be a k3 surface let c be a smooth curve on x and let z be a base point free pencil on c then the lazarsfeld mukai bundle e c z of rank 2 associated with c and z is given by an extension of the torsion free sheaf j z o x c by o x where j z is the ideal sheaf of z in x we can see that if c is very ample as a divisor on x e c z is an acm bundle with respect to o x c in this paper by using this fact we will characterize a necessary condition for e c z to be given by an extension of two line bundles on x by acm line bundles with respect to o x c

some extensions of faa di bruno s formula with divided differences the well known formula of faa di bruno s for higher derivatives of a composite function has played an important role in combinatorics in this paper we generalize the divided difference form of faa di bruno s formula and give an explicit formula for the n th divided difference of a multicomposite function more generally we establish the relationships of the bell polynomials with respect to multicomposite functions applying these to multicomposite functions we obtain some extensions of faa di bruno s formula

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

new spectral results for the electrostatic integral operator abstract in this paper two new fundamental results are established concerning the spectra of both the double layer potential and the electrostatic integral operators in r 3 while it is known that 0 lies in the spectrum of each operator its specific nature is not well understood for some geometries e g a sphere and a prolate spheroid it is known that 0 lies in their continuous spectra in this paper however specific examples are produced where 0 lies in the point spectrum of each operator thus it is established that the spectral nature of 0 depends upon the geometry of the underlying surface the second new result given is an important generalization of a recent paper concerning the possible refinement of a classical result of plemelj

an extension of the bell polynomials abstract after recalling the most important properties and applications of the bell polynomials we introduce an extension of this special class of functions more precisely we consider the case of multicomposite functions and we show connections with the ordinary bell polynomials

some identities involving exponential functions and stirling numbers and applications guo and qi 2013 posed a problem asking to determine the coefficients a k i 1 for 1 i k such that 1 1 e t k 1 i 1 k a k i 1 1 e t 1 i 1 the authors answer this question alternatively by faa di bruno s formula unify the eight identities due to guo and qi to two identities involving two parameters and apply them to obtain an explicit expression for the apostol bernoulli numbers and the fubini numbers respectively