jet schemes of locally complete intersection canonical singularities let x be a variety defined over an algebraically closed field k of characteristic zero the mth jet scheme xm of x is a scheme whose closed points over x x are morphisms ox x k t tm 1 when x is a smooth variety this is an affine bundle over x of dimension m 1 dim x the space of arcs x of x is the projective limit x proj limm xm our main result is a proof of the following theorem which was conjectured by eisenbud and frenkel

a remark on a standard and linear vector network equilibrium problem with capacity constraints in this paper weak vector equilibrium principle with capacity constraints is introduced a necessary condition that a vector minimum cost flow is a vector equilibrium flow with capacity constraints is obtained when the number of paths connecting with each pair of source and sink is less than or equal to 2 a sufficient condition for a vector minimum cost flow to be a vector equilibrium flow is also obtained a generalized weak vector equilibrium principle is also introduced without any additional assumption a necessary and sufficient condition for a weak vector minimum cost flow to be a generalized weak vector equilibrium flow is obtained

on multiplicities of graded sequences of ideals abstract we generalize a result from l ein et al math ag 0202303 proving that for an arbitrary graded sequence of zero dimensional ideals the multiplicity of the sequence is equal to its volume this is done using a deformation to monomial ideals as a consequence of our result we obtain a formula which computes the multiplicity of an ideal i in terms of the multiplicities of the initial monomial ideals of the powers i m we use this to give a new proof of the inequality between multiplicity and the log canonical threshold from de fernex et al j alg geom to appear

absolute optimal solution for a compact and convex game abstract this paper investigates the existence of absolute optimal solutions for a partition p in continuous and quasiconcave games we show that the p consistency property introduced in the paper together with the quasiconcavity and continuity of payoffs permits the existence of p absolute optimal solutions in games with compact and convex strategy spaces the p consistency property is a general condition that cannot be dispensed with for the existence of p absolute optimal solutions we also characterize the existence of p absolute optimal solutions by providing necessary and sufficient conditions moreover we suggest an algorithm for efficiently computing p absolute optimal solutions

vanishing of principal value integrals on surfaces principal value integrals are associated to multi valued rational differential forms with normal crossings support on a non singular algebraic variety we prove their vanishing on rational surfaces in the context of a conjecture of denef jacobs as an appli cation we obtain a strong vanishing result for candidate poles of p adic and motivic igusa zeta functions

towards the singular locus of the space of arcs we give a theory on the space of arcs x infty of a singular variety x based on the finiteness property of the stable points of x infty we also show how computations can be made in this theory as a consequence we obtain a process of determining invariants of x from its space of arcs x infty and from its spaces of wedges arcs in the space of arcs more precisely we give two concepts of regularity for the stable points of x infty and we show that we have regularity in codimension 1 and for all stable points of the space of arcs of a curve as another application we show concrete examples obtained from the spaces of arcs of noetherian 1 dimensional local rings which are analytically ramified

a multicriteria pareto optimal path algorithm abstract in this paper we present an algorithm for finding in a multicriteria network pareto optimal paths one each for each efficient objective vector validation of the algorithm and two illustrative examples are included computational experience is also reported this algorithm is a generalization of an earlier one for the bicriterion case

effective uniform bounding in partial differential fields abstract motivated by the effective bounds found in 12 for ordinary differential equations we prove an effective version of uniform bounding for fields with several commuting derivations more precisely we provide an upper bound for the size of finite solution sets of partial differential polynomial equations in terms of data explicitly given in the equations and independent of parameters our methods also produce an upper bound for the degree of the zariski closure of solution sets whether they are finite or not

log canonical thresholds in positive characteristic in this paper we study the singularities of pairs in arbitrary characteristic via jet schemes for a smooth variety x in characteristic 0 ein lazarsfeld and musta showed that there is a correspondence between irreducible closed cylinders and divisorial valuations on x via this correspondence one can relate the codimension of a cylinder to the log discrepancy of the corresponding divisorial valuation we now extend this result to positive characteristic in particular we prove musta s log canonical threshold formula avoiding the use of log resolutions making the formula available also in positive characteristic as a consequence we get a comparison theorem via reduction modulo p and a version of inversion of adjunction in positive characteristic

potential value and consistency let p be a real valued function defined on the space of cooperative games with transferable utility satisfying the following condition in every game the marginal contributions of all players are efficient it is proved that there exists just one such function p called the potential and moreover that the resulting payoff vector coincides with the shapley value

vector equilibrium problem and vector optimization this paper examines the vector equilibrium model based on a vector cost consideration this is a generalization of the well known wardrop traffic equilibrium principle where road users choose paths based on just a single cost the concept of parametric equilibria is introduced and used to establish relations with parametric complementarity and variational inequality problems relations with some vector optimization problems via scalarization techniques are given under appropriate conditions some solution methods for solving vector equilibrium problems are also discussed

elliptic genera of singular varieties the notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced and the relation between them is studied the elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of calabi yau hypersurfaces in fano gorenstein toric varieties introduced earlier the orbifold elliptic genus is given in terms of the fixed point sets of the action we show that the generating function for the orbifold elliptic genus ellorb x 6n p for symmetric groups 6n acting on n fold products coincides with the one proposed by r dijkgraaf g moore e verlinde and h verlinde the two notions of elliptic genera are conjectured to coincide

the pairwise egalitarian solution abstract the main aim of this paper is to introduce an ad hoc solution concept for transportation games this solution is related to the structure of the underlying transportation problem in the game in particular to the optimal solutions for it this is significantly called the pairwise egalitarian solution we compare the new solution with the core and the shapley value of the game finally we study the properties satisfied by the pairwise egalitarian solution and provide two axiomatic characterizations of it

the arc space of the grassmannian abstract we study the arc space of the grassmannian from the point of view of the singularities of schubert varieties our main tool is a decomposition of the arc space of the grassmannian that resembles the schubert cell decomposition of the grassmannian itself just as the combinatorics of schubert cells is controlled by partitions the combinatorics in the arc space is controlled by plane partitions sometimes also called 3d partitions a combination of a geometric analysis of the pieces in the decomposition and a combinatorial analysis of plane partitions leads to invariants of the singularities as an application we reduce the computation of log canonical thresholds of pairs involving schubert varieties to an easy linear programming problem we also study the nash problem for schubert varieties showing that the nash map is always bijective in this case

arcs on determinantal varieties we study arc spaces and jet schemes of generic determinantal varieties using the natural group action we decompose the arc spaces into orbits and analyze their structure this allows us to compute the number of irreducible components of jet schemes log canonical thresholds and topological zeta functions

asymptotic equivalence of bayes cross validation and widely applicable information criterion in singular learning theory in regular statistical models the leave one out cross validation is asymptotically equivalent to the akaike information criterion however since many learning machines are singular statistical models the asymptotic behavior of the cross validation remains unknown in previous studies we established the singular learning theory and proposed a widely applicable information criterion the expectation value of which is asymptotically equal to the average bayes generalization loss in the present paper we theoretically compare the bayes cross validation loss and the widely applicable information criterion and prove two theorems first the bayes cross validation loss is asymptotically equivalent to the widely applicable information criterion as a random variable therefore model selection and hyperparameter optimization using these two values are asymptotically equivalent second the sum of the bayes generalization error and the bayes cross validation error is asymptotically equal to 2 n where is the real log canonical threshold and n is the number of training samples therefore the relation between the cross validation error and the generalization error is determined by the algebraic geometrical structure of a learning machine we also clarify that the deviance information criteria are different from the bayes cross validation and the widely applicable information criterion

the nash problem on arc families of singularities nash 21 proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that appears on every resolution he asked if the converse also holds does every such exceptional divisor correspond to an arc family we prove that the converse holds for toric singularities but fails in general

jet closures and the local isomorphism problem abstract if a morphism of germs of schemes induces isomorphisms of all local jet schemes does it follow that the morphism is an isomorphism this problem is called the local isomorphism problem in this paper we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem this approach leads to a partial solution of the local isomorphism problem which is shown to have a negative answer in general and a positive one in several situations of geometric interest

dimensions of jet schemes of log singularities we characterize kawamata log terminal singularities and log canonical singularities by dimensions of jet schemes our main result is theorem 2 4

germs of arcs on singular algebraic varieties and motivic integration we study the scheme of formal arcs on a singular algebraic variety and its images under truncations we prove a rationality result for the poincare series of these images which is an analogue of the rationality of the poincare series associated to p adic points on a p adic variety the main tools which are used are semi algebraic geometry in spaces of power series and motivic integration a notion introduced by m kontsevich in particular we develop the theory of motivic integration for semi algebraic sets of formal arcs on singular algebraic varieties we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of oesterle

differential and sensitivity properties of gap functions for minty vector variational inequalities the purpose of this paper is to investigate differential properties of a class of set valued maps and gap functions involving minty vector variational inequalities relationships between their contingent derivatives are discussed an explicit expression of the contingent derivative for the class of set valued maps is established optimality conditions of solutions for minty vector variational inequalities are obtained

the arc space of a toric variety abstract the nash problem on arc families is affirmatively answered for a toric variety by ishii and kollar s paper which also shows the negative answer for general case the nash problem is one of questions about the relation between arc families and valuations in this paper the relation is described clearly for a toric variety the arc space of a toric variety admits an action of the group scheme determined by the torus each orbit on the arc space corresponds to a lattice point in the cone and therefore corresponds to a toric valuation the dominant relation among the orbits is described in terms of the lattice points as a corollary we obtain the answer to the embedded version of the nash problem for an invariant ideal on a toric variety

birationally rigid hypersurfaces we prove that for n 4 all smooth hypersurfaces of degree n in n are birationally superrigid first discovered in the case n 4 by iskovskikh and manin in a work that started this whole direction of research this property was later conjectured to hold in general by pukhlikov the proof relies on the method of maximal singularities in combination with a formula on restrictions of multiplier ideals

analytic efficient solution set for multi criteria quadratic programs abstract we present active set methods to evaluate the exact analytic efficient solution set for multi criteria convex quadratic programming problems mcqp subject to linear constraints the idea is based on the observations that a strictly convex programming problem admits a unique solution and that the efficient solution set for a multi criteria strictly convex quadratic programming problem with linear equality constraints can be parameterized the case of bi criteria quadratic programming bcqp is first discussed since many of the underlying ideas can be explained much more clearly in the case of two objectives in particular we note that the efficient solution set of a bcqp problem is a curve on the surface of a polytope the extension to problems with more than two objectives is straightforward albeit some slightly more complicated notation two numerical examples are given to illustrate the proposed methods

iterative methods for variational and complementarity problems in this paper we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems included among such methods are the newton and several successive overrelaxation algorithms for the most part the study is concerned with the family of linear approximation methods these are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches the paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria finally several convergence results are obtained for some nonlinear approximation methods

totally balanced games arising from controlled programming problems a cooperative game in characteristic function form is obtained by allowing a number of individuals to esercise partial control over the constraints of a generally nonlinear mathematical programming problem either directly or through committee voting conditions are imposed on the functions defining the programming problem and the control system which suffice to make the game totally balanced this assures a nonempty core and hence a stable allocation of the full value of the programming problem among the controlling palyers in the linear case the core is closely related to the solutions of the dual problem applications are made to a variety of economic models including the transferable utility trading economies of shapley and shubik and a multishipper one commodity transshipment model with convex cost functions and concave revenue functions dropping the assumption of transferable utility leads to a class of controlled multiobjective or pareto programming problems which again yield totally balanced games

pairwise solutions and the core of transportation situations the main aim of this paper is to study the relationship between the so called pairwise solutions and the core of transportation situations or games with this objective in mind we show that every core element of a transportation situation is contained in a pairwise solution with a specific weight system therefore every core element is in fact a particular pairwise distribution furthermore given a core element we outline a procedure in three steps to obtain the corresponding pairwise distribution with weight system

bounds for p adic exponential sums and log canonical thresholds we propose a conjecture for exponential sums which generalizes both a conjecture by igusa and a local variant by denef and sperber in particular it is without the homogeneity condition on the polynomial in the phase and with new predicted uniform behavior the exponential sums have summation sets consisting of integers modulo p m lying p adically close to y and the proposed bounds are uniform in p y and m we give evidence for the conjecture by showing uniform bounds in p y and in some values for m on the way we prove new bounds for log canonical thresholds which are closely related to the bounds predicted by the conjecture

holomorphic arcs on singularities the study of formal arcs was initiated by nash in a 1967 preprint arc spaces of smooth varieties have a rather transparent structure but difficult problems arise for arcs passing through singularities the nash conjecture on the irreducible components of such arc spaces was proved for surfaces by fernandez de bobadilla and pe pereira and for toric singularities by ishii and kollar but counter examples were found in higher dimensions by ishii and kollar de fernex and johnson and kollar here we start the study of holomorphic arcs these are holomorphic maps from the closed unit disk to a complex analytic space as one expects there is not much conceptual difference between the set of formal arcs and the set of holomorphic arcs since every formal arc can be approximated by holomorphic arcs however a formal deformation of an arc is a much more local object than a holomorphic deformation thus in many cases the space of holomorphic arcs has infinitely many connected components while the space of formal arcs always has only finitely many for a complex analytic space we define two variants the space of arcs and the space of short arcs and we study their connected components for short arcs we obtain complete answers for surface singularities and for isolated quotient singularities in all dimensions

totally balanced games and games of flow a class of characteristic function games arising from maximum flow problems is introduced and is shown to coincide with the class of totally balanced games the proof relies on the max flow min cut theorem of ford and fulkerson and on the observation that the class of totally balanced games is the span of the additive games with the minimum operation

arcs valuations and the nash map this paper gives a map from the set of families of arcs on a variety to the set of valuations on the rational function field of the variety we characterize a family of arcs which corresponds to a divisorial valuation by this map we can see that both the nash map and a certain mckay correspondence are the restrictions of this map this paper also gives the affirmative answer to the nash problem for a non normal variety in a certain category as a corollary we get the affirmative answer for a non normal toric variety

on f pure thresholds using the frobenius map we introduce a new invariant for a pair r a of a ring r and an ideal a r which we call the f pure threshold c a of a and study its properties we see that the f pure threshold characterizes several ring theoretic properties by virtue of hara and yoshida s result trans amer math soc 355 2003 3143 the f pure threshold c a in characteristic zero corresponds to the log canonical threshold lc a which is an important invariant in birational geometry using the f pure threshold we prove some ring theoretic properties of three dimensional terminal singularities of characteristic zero also in fixed prime characteristic we establish several properties of the f pure threshold similar to those of the log canonical threshold with quite simple proofs

zeta functions for analytic mappings log principalization of ideals and newton polyhedra in this paper we provide a geometric description of the possi ble poles of the igusa local zeta function z s f associated to an analytic mapping f f1 fl u k n k l and a locally constant function with support in u in terms of a log principalizaton of the k x ideal if f1 fl typically our new method provides a much shorter list of possible poles compared with the previous methods we determine the largest real part of the poles of the igusa zeta function and then as a corollary we ob tain an asymptotic estimation for the number of solutions of an arbitrary sys tem of polynomial congruences in terms of the log canonical threshold of the subscheme given by if we associate to an analytic mapping f f1 fl a newton polyhedron f and a new notion of non degeneracy with respect to f the novelty of this notion resides in the fact that it depends on one newton polyhedron and khovanskii s non degeneracy notion depends on the newton polyhedra of f1 fl by constructing a log principalization we give an explicit list for the possible poles of z s f l 1 in the case in which f is non degenerate with respect to f

traveling salesman games in this paper we discuss the problem of how to divide the total cost of a round trip along several institutes among the institutes visited we introduce two types of cooperative games fixed route traveling salesman games and traveling salesman games as a tool to attack this problem under very mild conditions we prove that fixed route traveling salesman games have non empty cores if the fixed route is a solution of the classical traveling salesman problem core elements provide us with fair cost allocations a traveling salesman game may have an empty core even if the cost matrix satisfies the triangle inequality in this paper we introduce a class of matrices defining ts games with non empty cores

learning coefficient of generalization error in bayesian estimation and vandermonde matrix type singularity the term algebraic statistics arises from the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry sturmfels 2009 the purpose of our study is to consider the generalization error and stochastic complexity in learning theory by using the log canonical threshold in algebraic geometry such thresholds correspond to the main term of the generalization error in bayesian estimation which is called a learning coefficient watanabe 2001a 2001b the learning coefficient serves to measure the learning efficiencies in hierarchical learning models in this letter we consider learning coefficients for vandermonde matrix type singularities by using a new approach focusing on the generators of the ideal which defines singularities we give tight new bound values of learning coefficients for the vandermonde matrix type singularities and the explicit values with certain conditions by applying our results we can show the learning coefficients of three layered neural networks and normal mixture models

on existence of vector equilibrium flows with capacity constraints of arcs in this paper we propose a weak vector equilibrium principle with capacity constraints of arcs by proving the existence of solutions for the weighted variational inequality we establish the existence results of weak vector traffic equilibrium flows with capacity constraints of arcs