pullback and pushout constructions in c algebra theory abstract a systematic study of pullback and pushout diagrams is conducted in order to understand restricted direct sums and amalgamated free products of c algebras particular emphasis is given to the relations with tensor products both with the minimal and the maximal c tensor norm thus it is shown that pullback and pushout diagrams are stable under tensoring with a fixed algebra and stable under crossed products with a fixed group general tensor products between diagrams are also investigated the relations between the theory of extensions and pullback and pushout diagrams are explored in some detail the crowning result is that if three short exact sequences of c algebras are given with appropriate morphisms between the sequences allowing for pullback or pushout constructions at the levels of ideals algebras and quotients then the three new c algebras will again form a short exact sequence under some mild extra conditions as a generalization of a theorem of t a loring it is shown that each morphism between a pair of c algebras combined with its extension to the stabilized algebras gives rise to a pushout diagram this result has applications to corona extendibility and conditional projectivity finally the pullback and pushout constructions are applied to the class of noncommutative cw complexes defined by s eilers t a loring and g k pedersen j reine angew math 1998 499 101 143 to show that this category is stable under tensor products and under restricted direct sums

linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

regularizing effect of absorption terms in singular problems abstract we prove existence of solutions to problems whose model is p u u q f u in u 0 in u 0 on where is an open bounded subset of r n n 2 p u is the p laplacian operator for 1 p n q 0 0 and f is a nonnegative function in l m for some m 1 in particular we analyze the regularizing effect produced by the absorption term in order to infer the existence of finite energy solutions in case 1 we also study uniqueness of these solutions as well as examples which show the optimality of the results finally we find local w 1 p solutions in case 1

on the energy spectrum and parameter spaces of classicalcp n models it is known that a large class of smooth solutions ofcpn models can be constructed starting from holomorphic maps of an algebraic curve into complex projective spaces here we apply results from algebraic geometry to describe the energy spectrum and the parameter spaces for such models

limits as p of p laplacian concave convex problems abstract we study the behavior as p of the sequel of positive weak solutions of the concave convex problem p div u p 2 u u q p u r p in u 0 in u 0 on where r n is a bounded domain 0 and the exponents q r satisfy lim p q p p 1 q lim p r p p 1 r with 0 q 1 r we characterize any positive uniform limit of a sequence of weak solutions of p as a viscosity solution of min u max u q u r u 0 in notice that the limit process decouples the nonlinearity we obtain existence non existence and global multiplicity of positive viscosity solutions of the limit problem in terms of the parameter

on the profile of solutions with two sharp layers to a singularly perturbed semilinear dirichlet problem we discuss the existence of positive solutions of some singularity perturbed elliptic equations on convex domains with nonlinearity changing sign in particular we obtain solutions with both a boundary layer and a sharp interior peak

a hybridizable discontinuous galerkin method for the p laplacian we propose the first hybridizable discontinuous galerkin method for the p laplacian equation when using polynomials of degree k geq 0 for the approximation spaces of u boldsymbol nabla u and boldsymbol nabla u p 2 boldsymbol nabla u the method exhibits optimal k 1 order of convergence for all variables in l 1 and l p norms in our numerical experiments for k geq 1 an elementwise computation allows us to obtain a new approximation u h that converges to u with order k 2 we rewrite the scheme as discrete minimization problems in order to solve them with nonlinear minimization algorithms the unknown of the first problem is the approximation of u on the skeleton of the mesh but requires solving nonlinear local problems the second problem has the approximation on the elements as an additional unknown but it only requires solving linear local problems we present numerical results displaying the convergence properties of the methods demonstrating the utility of

anzellotti s pairing theory and the gauss green theorem abstract in this paper we obtain a very general gauss green formula for weakly differentiable functions and sets of finite perimeter this result is obtained by revisiting anzellotti s pairing theory and by characterizing the measure pairing a d u when a is a bounded divergence measure vector field and u is a bounded function of bounded variation

numerical solution of boundary value problems for the eikonal equation in an anisotropic medium abstract a dirichlet problem is considered for the eikonal equation in an anisotropic medium the nonlinear boundary value problem bvp formulated in the present work is the limit of the diffusion reaction problem with a diffusion parameter tending to zero to solve numerically the singularly perturbed diffusion reaction problem monotone approximations are employed numerical examples are presented for a two dimensional bvp for the eikonal equation in an anisotropic medium the standard lagrangian finite element approximation in space is used in computations

absolute continuity of the best sobolev constant abstract let q inf u l p p u l q p u w 0 1 p 0 be the best sobolev constant of the immersion w 0 1 p l q where is a bounded and smooth domain of r n 1 p n and 1 q p n p n p we prove that the function q 1 p q is absolutely continuous

on the solutions to 1 laplacian equation with l1 data in the present paper we study the behaviour as p goes to 1 of the renormalized solutions to the problems 0 1 div up p 2 up fin up 0on where p 1 is a bounded open set of rn n 2 with lipschitz boundary and f belongs to l1 we prove that these renormalized solutions pointwise converge up to subsequences to a function u with a suitable definition of solution we also prove that u is a solution to a limit problem moreover we analyze the situation occurring when more regular data f are considered

a refined brill noether theory over hurwitz spaces let f c rightarrow mathbb p 1 be a degree k genus g cover the stratification of line bundles l in mathrm pic d c by the splitting type of f l is a refinement of the stratification by brill noether loci w r d c we prove that for general degree k covers these strata are smooth of the expected dimension in particular this determines the dimensions of all irreducible components of w r d c for a general k gonal curve there are often components of different dimensions extending results of pflueger adv math 312 46 63 2017 and jensen and ranganathan brill noether theory for curves of a fixed gonality arxiv 1701 06579 2017 the results here apply over any algebraically closed field

overdetermined problems with possibly degenerate ellipticity a geometric approach given an open bounded connected subset of n we consider the overdetermined boundary value problem obtained by adding both zero dirichlet and constant neumann boundary data to the elliptic equation div a u u 1 in we prove that if this problem admits a solution in a suitable weak sense then is a ball this is obtained under fairly general assumptions on and a in particular a may be degenerate and no growth condition is required our method of proof is quite simple it relies on a maximum principle for a suitable p function combined with some geometric arguments involving the mean curvature of

classification of bifurcation diagrams of a p laplacian dirichlet problem with examples abstract we study bifurcation diagrams of positive solutions of the p laplacian dirichlet problem p u x f u x 0 1 x 1 u 1 u 1 0 where p y y p 2 y p u is the one dimensional p laplacian and p 1 and 0 are two bifurcation parameters assume that f u g u h u where g h c 0 c 2 0 satisfy hypotheses h1 h5 presented herein for different values p with 1 p 2 and with p 2 we give a classification of totally six different bifurcation diagrams we prove that on the u plane each possible bifurcation diagram consists of exactly one curve with exactly one turning point where the curve turns to the right hence we are able to determine the exact multiplicity of positive solutions in addition for 1 p 2 and for p 2 we give interesting examples f u k u p 1 u q u r satisfying r q p 1 and k 0 and show complete evolution of bifurcation diagrams as evolution parameter k varies from 0 to

on the shape of the nonnegative solutions to a singularly perturbed quasilinear dirichlet problem abstract we study the shape of the nontrivial nonnegative solutions for a singularly perturbed quasilinear dirichlet problem with non lipschitz nonlinearity we find that there exists at least one nontrivial nonnegative solution of the problem which consists of some bubbles the profile of each bubble is discussed

a neumann problem for the p x laplacian with p 1 in a subdomain abstract in this paper we study a neumann problem with non homogeneous boundary conditions for the p x laplacian in particular we assume that p is a step function defined in a domain and equals to 1 in a subdomain 1 and 2 in its complementary 2 by considering a suitable sequence p k of variable exponents such that p k p and replacing p with p k in the original problem we prove the existence of a solution u k for each of those intermediate ones we also show that under a hypothesis concerning the boundary data g the limit of the sequence u k is a function u which belongs to the space of functions of bounded variation and is a solution to the original p problem

limit linear series basic theory in this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves curves of compact type the technically much simpler special case of 1 dimensional series was developed by beauville 2 knudsen 21 23 harris and mumford 17 in the guise of admissible covers it has proved very useful for studying the moduli space of curves the above papers and harris 16 and the simplest sorts of weierstrass points diaz 4 with our extended tools we are able to prove for example that in this paper we present the basic theory of limit linear series necessary for proving these results the results themselves will be taken up in our forthcoming papers 8 12 simpler applications not requiring the tools developed in this paper but perhaps clarified by them have already been given in our papers 5 7

solutions of the cheeger problem via torsion functions abstract the cheeger problem for a bounded domain r n n 1 consists in minimizing the quotients e e among all smooth subdomains e and the cheeger constant h is the minimum of these quotients let p c 1 be the p torsion function that is the solution of torsional creep problem p p 1 in p 0 on where p u div u p 2 u is the p laplacian operator p 1 the paper emphasizes the connection between these problems we prove that lim p 1 p l 1 p h lim p 1 p l 1 1 p moreover we deduce the relation lim p 1 p l 1 c n lim p 1 p l where c n is a constant depending only of n and h explicitely given in the paper an eigenfunction u bv l of the dirichlet 1 laplacian is obtained as the strong l 1 limit as p 1 of a subsequence of the family p p l 1 p 1 almost all t level sets e t of u are cheeger sets and our estimates of u on the cheeger set e 0 yield b 1 h b 1 n e 0 h n where b 1 is the unit ball in r n for convex we obtain u e 0 1 e 0

relaxation of hamilton jacobi equations we study the relaxation of hamilton jacobi equations the relaxation in our terminology is the following phenomenon the pointwise supremum over a certain collection of subsolutions in the almost everywhere sense of a hamilton jacobi equation yields a viscosity solution of the convexified hamilton jacobi equation this phenomenon has recently been observed in 13 in eikonal equations we show in this paper that this relaxation is a common phenomenon for a wide range of hamilton jacobi equations

spike layered solutions with compact support to some singularly perturbed quasilinear elliptic problems in general smooth domains in this paper we study the structure of nonnegative nontrivial solutions of the following problem e pu f u in u 0 in u 0 on as e 0 where pu div du p 2du with p 1 e 0 is a small parameter and is a bounded smooth domain in rn n 1 f is a class of logistic type nonlinearities satisfying f 0 f z1 f z2 0 with 0 0 in z1 z2 and limu 0 f u up 1 by virtue of the sub and supersolution method we prove that there are many nonnegative nontrivial solutions and they are spike layered solutions moreover the measure of each spike layer is estimated as e 0

chern schwartz macpherson classes and the euler characteristic of degeneracy loci and special divisors this concept overlaps a large family of interesting varieties for example the varieties of special divisors studied in section 3 several authors have worked out explicit formulas for the euler characteristic of dr 9 in terms of different cohomological and numerical invariants under the assumption that x is nonsingular and v is appropriately general for instance if 0 is a section of a vector bundle then the formulas for the euler characteristic x do fp were given by hirzebruch h and navarro aznar n if dr is a curve or a surface in a nonsingular x some explicit formulas for x dr v were given by harris and tu h t in terms of the chern classes of e f and x but under the extra assumption dr 1 v 0 which implies that dr q is nonsingular in loc cit the authors also posed the problem of finding a general formula for x dr o if such exists under the assumption dr 1 0 or even stronger without this assumption the first problem was solved positively by the second named author prl proposition 5 7 by the use of polynomials universally supported on degeneracy

the element free galerkin method for the nonlinear p laplacian equation abstract the element free galerkin efg method is developed in this paper for solving the nonlinear p laplacian equation the moving least squares approximation is used to generate meshless shape functions the penalty approach is adopted to enforce the dirichlet boundary condition the galerkin weak form is employed to obtain the system of discrete equations and two iterative procedures are developed to deal with the strong nonlinearity then the computational formulas of the efg method for the p laplacian equation are established numerical results are finally given to verify the convergence and high computational precision of the method

the convective eigenvalues of the one dimensional p laplacian as p 1 abstract this paper studies the limit behavior as p 1 of the eigenvalue problem u x p 2 u x x c u x p 2 u x u p 2 u 0 x 1 u 0 u 1 0 we point out that explicit expressions for both the eigenvalues n and associated eigenfunctions are not available see 16 in spite of this hindrance we obtain the precise values of the limits lim p 1 n in addition a complete description of the limit profiles of the eigenfunctions is accomplished moreover the formal limit problem as p 1 is also addressed the results extend known features for the special case c 0 6 28

anisotropic p q laplacian equations when p goes to 1 abstract in this paper we prove a stability result for an anisotropic elliptic problem more precisely we consider the dirichlet problem for an anisotropic equation which is as the p laplacian equation with respect to a group of variables and as the q laplacian equation with respect to the other variables 1 p q with datum f belonging to a suitable lebesgue space for this problem we study the behaviour of the solutions as p goes to 1 showing that they converge to a function u which is almost everywhere finite regardless of the size of the datum f moreover we prove that this u is the unique solution of a limit problem having the 1 laplacian operator with respect to the first group of variables furthermore the regularity of the solutions to the limit problem is studied and explicit examples are shown

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

an extension of the pairing theory between divergence measure fields and bv functions abstract in this paper we introduce a nonlinear version of the notion of anzellotti s pairing between divergence measure vector fields and functions of bounded variation motivated by possible applications to evolutionary quasilinear problems as a consequence of our analysis we prove a generalized gauss green formula

uniqueness of the cheeger set of a convex body abstract we prove that if c r n is an open bounded convex set then there is only one cheeger set inside c and it is convex a cheeger set of c is a set which minimizes the ratio perimeter over volume among all subsets of c

linked symplectic forms and limit linear series in rank 2 with special determinant abstract we generalize the prior linked symplectic grassmannian construction applying it to prove smoothing results for rank 2 limit linear series with fixed special determinant on chains of curves we apply this general machinery to prove new results on nonemptiness and dimension of rank 2 brill noether loci in a range of degrees

free products of completely positive maps and spectral sets let f be the free group on n generators and for each word w e f let 1 wi be its length haagerup lo showed that for each 0 r 1 the function h wp r is a positive definite function on f consider the positive definite function q n rinl on h then the free product function cpi cpr on ff z h coincides with h and the result was extended this way by de michele and fig talamanca 7 and by boiejko 4 51 in s it is proved that the free product of the unital positive defined functions ui gi z z is still positive defined on the free product group gi the correspondence between the positive defined functions on a discrete group g and the completely positive maps on the full c algebra c g and the isomorphism c g g2 n c g z c g suggested we consider amalgamated products of unital linear maps on the amalgamated product of a family of unital c algebras over a c subalgebra in section 3 we prove that the amalgamated product of a family of unital completely positive b bimodule maps qi a c is completely positive on the biggest amalgamated free product i a i as an application of our main result in section 2 we obtain some results concerning the dilation of noncommutative families of operators

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

limit linear series for curves not of compact type we introduce a notion of limit linear series for nodal curves which are not of compact type we give a construction of a moduli space of limit linear series which works also in smoothing families and we prove a corresponding specialization result for a more restricted class of curves which simultaneously generalizes two component curves and curves of compact type we give an equivalent definition of limit linear series which is visibly a generalization of the eisenbud harris definition finally for the same class of curves we prove a smoothing theorem which constitutes an improvement over known results even in the compact type case

a hk mortar spectral element method for the p laplacian equation abstract we present a constrained vertex variant of the mortar spectral element method to solve the p laplacian equation to show reliability of the method first we investigate convergence rate of h version and k version for a sufficiently smooth solution then for solutions with limited regularity we use a h k strategy in which the size of elements is geometrically reduced towards the singularity while the polynomial degree is linearly reduced in fact we numerically study convergence rate of the h k mortar spectral element method in the l 2 norm and h 1 norm using geometrical meshes with elements of non uniform polynomial degree in this regard we consider several benchmarks with various choices of the parameter p as well as geometric grading factors to deal with possible degeneracy in the p laplacian operator we add artificial diffusion to the original equation and then study the effects of this extra diffusion on convergence rate to find the solution of highly nonlinear system arising from discretization we use an optimization technique based on the trust region method in which an analytical jacobian matrix is utilized

on the projective normality of line bundles of extremal degree we propose a new method using deformation theory to study the maximal rank conjecture for line bundles of extremal degree which can be viewed as the first case to test the conjecture we prove that maximal rank conjecture holds by our new method

estimates on the support of the solutions of some nonlinear elliptic and parabolic problems we obtain existence and uniqueness of solutions with compact support for some nonlinear elliptic and parabolic problems including the equations of one dimensional motion of a non newtonian fluid precise estimates for the support of these solutions are obtained and the optimality of our hypotheses is discussed

decay rate estimates for the quasi linear timoshenko system with nonlinear control and damping terms in this work we consider a class of quasi linear timoshenko systems with nonlinear control and damping terms we investigate the stabilization of weak solutions to the initial boundary value problem and obtain explicit decay rate estimation depending on control and damping terms

on the existence of certain families of curves introduction 25 w 1 generalities on families of curves in ip 28 w the map 0 d 33 w 3 the number of moduli of a family 40 w 4 plane curves existence of families with the expected number of moduli 43 w 5 smoothing certain reducible curves in ip r r 3 47 w 6 curves in ip r 3 existence of families with the expected number of moduli 52 w curves in ip3 a more general existence theorem 55

weak solutions of quasilinear problems with nonlinear boundary condition the growing attention for the study of the p laplacian operator p in the last few decades is motivated by the fact that it arises in various applications for instance in fluid mechanics the shear stress and the velocity gradient pu of certain uids obey a relation of the form x a x pu x where pu u p 2 u here p 1 is an arbitrary real number and the case p 2 respectively pi 2 p 2 corresponds to a newtonian respectively pseudoplastic dilatant uid the resulting equations of motion then involve div a pu which reduces to a pu a div pu provided that a is constant the p laplacian also appears in the study of torsional creep elastic for p 2 plastic as p see 7 ow through porous media p 2 see 12 or glacial sliding p 1 3 see 9 let rn be an unbounded domain with possible noncompact smooth boundary and n is the unit outward normal on we consider the nonlinear elliptic boundary value problem div a x u p 2 u 1 x 1 u p 2u 1 x 2 u q 2u in

on a family of torsional creep problems involving rapidly growing operators in divergence form let n n 2 be a bounded domain with smooth boundary and p n be a sequence of real numbers converging to as n for each integer n 1 we define the function varphi n t p n vert t vert p n 2 te vert t vert p n for all t and we prove the existence of a unique nonnegative variational solution for the problem div n u x u x u x n 1 when x subject to the homogeneous dirichlet boundary condition next we establish the uniform convergence in of the sequence of solutions for the above family of equations to the distance function to the boundary of our result complements the earlier developments on the topic obtained by payne and philippin 26 kawohl 21 bhattacharya dibenedetto and manfredi 2 perez llanos and rossi 27 and bocea and mih ilescu 4

elliptic equations involving the 1 laplacian and a subcritical source term abstract in this paper we deal with a dirichlet problem for an elliptic equation involving the 1 laplacian operator and a source term we prove that when the growth of the source is subcritical there exist two bounded nontrivial solutions to our problem moreover a pohozaev type identity is proved which holds even when the growth is supercritical we also show explicit examples of our results

principal eigenvalue of a very badly degenerate operator and applications abstract in this paper we define and investigate the properties of the principal eigenvalue of the singular infinity laplace operator u d 2 u d u d u d u d u this operator arises from the optimal lipschitz extension problem and it plays the same fundamental role in the calculus of variations of l functionals as the usual laplacian does in the calculus of variations of l 2 functionals our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of berestycki nirenberg and varadhan h berestycki l nirenberg s r s varadhan the principal eigenvalue and maximum principle for second order elliptic operators in general domains comm pure appl math 47 1 1994 47 92 in the case of uniformly elliptic linear operators as an application we obtain existence and uniqueness results for certain related non homogeneous problems and decay estimates for the solutions of the evolution problem associated to the infinity laplacian

divisors on general curves and cuspidal rational curves in t roduc t ion 371 0 pre l iminar ies 377 1 a pl i icker formula 378 2 d imens iona l t ransversa l i ty 380 3 d i m e n s i o n of families of r a t iona l curves with fixed degree and ass igned cusps in a project ive space 383 4 l inear series on curves with universa l s ingular i t ies 386 5 l inear series and project ive embedd ings of general curves 391 6 embedd ings of general cuspidal curves 400 7 degenera t ion of schuber t in tersect ions some special cases 403 8 degene ra t ion of schuber t in tersect ions ma in results 407 9 g c is reduced for general cuspidal r a t iona l c 414

computing the first eigenpair of the p laplacian in annuli abstract we propose a method for computing the first eigenpair of the dirichlet p laplacian p 1 in the annulus a b x r n a x b n 1 for each t a b we use an inverse iteration method to solve two radial eigenvalue problems one in the annulus a t with the corresponding eigenvalue t and boundary conditions u a 0 u t and the other in the annulus t b with the corresponding eigenvalue t and boundary conditions u t 0 u b next we adjust the parameter t using a matching procedure to make t coincide with t thereby obtaining the first eigenvalue p hence by a simple splicing argument we obtain the positive l normalized radial first eigenfunction u p the matching parameter is the maximum point of u p in order to apply this method we derive estimates for t and t and we prove that these functions are monotone and locally lipschitz continuous moreover we derive upper and lower estimates for the maximum point which we use in the matching procedure and we also present a direct proof that u p converges to the l normalized distance function to the boundary as p we also present some numerical results obtained using this method

limit linear series and ranks of multiplication maps we develop a new technique for studying ranks of multiplication maps for linear series via limit linear series and degenerations to chains of genus 1 curves we use this approach to prove a purely elementary criterion for proving cases of the maximal rank conjecture and then apply the criterion to several ranges of cases giving a new proof of the case of quadrics and also treating several families in the case of cubics our proofs do not require restrictions on direction of approach so we recover new information on the locus in the moduli space of curves on which the maximal rank condition fails

minimizing within convex bodies using a convex hull method we present numerical methods to solve optimization problems on the space of convex functions or among convex bodies hence convexity is a constraint on the admissible objects whereas the functionals are not required to be convex to deal with this our method mixes geometrical and numerical algorithms we give several applications arising from classical problems in geometry and analysis alexandrov s problem of finding a convex body of prescribed surface function cheeger s problem of a subdomain minimizing the ratio surface area on volume newton s problem of the body of minimal resistance in particular for the latter application the minimizers are still unknown except in some particular classes we give approximate solutions better than the theoretical known ones hence demonstrating that the minimizers do not belong to these classes