stochastic production planning with production constraints this paper considers an infinite horizon stochastic production planning problem with the constraint that production rate must be nonnegative it is shown that an optimal feedback solution exists for the problem moreover this solution is characterized and is then compared with the solution of the unconstrained problem also obtained by using a policy iteration procedure are computational solutions to the related problems with upper bounds on the production rate which determines production rates over time to minimize an integral representing a discounted quadratic loss function the loss function is defined in terms of the deviations of production and inventory levels from their rated or factory optimal values the model is solved both with and without nonnegative production constraints a stochastic extension of this paper involving a white noise process 1 is analyzed by sethi and thompson 9 10 closed form solutions for optimal feedback production policy for both finite and infinite horizon versions of the model without production constraints are obtained in particular the model must allow negative production rates or disposals in this paper we consider the stochastic production planning problem with the constraint that production rates must be nonnegative only the infinite horizon problem is treated it is shown that an optimal feedback solution exists for the problem this solution is characterized also obtained by using a policy iteration procedure are computational solutions to the related problems with upper bounds on the production rate

splittable ideals and the resolutions of monomial ideals abstract we provide a new combinatorial approach to study the minimal free resolutions of edge ideals that is quadratic square free monomial ideals with this method we can recover most of the known results on resolutions of edge ideals with fuller generality and at the same time obtain new results past investigations on the resolutions of edge ideals usually reduced the problem to computing the dimensions of reduced homology or koszul homology groups our approach circumvents the highly nontrivial problem of computing the dimensions of these groups and turns the problem into combinatorial questions about the associated simple graph we also show that our technique extends successfully to the study of graded betti numbers of arbitrary square free monomial ideals viewed as facet ideals of simplicial complexes

the core of zero dimensional monomial ideals the core of an ideal is the intersection of all its reductions we describe the core of a zero dimensional monomial ideal i as the largest monomial ideal contained in a general reduction of i this provides a new interpretation of the core in the monomial case as well as an efficient algorithm for computing it we relate the core to adjoints and first coefficient ideals and in dimension two and three we give explicit formulas

on the ideal theory of graphs we study algebras defined by finite sets g m1 mq of monomials of a polynomial ring r there are two basic algebras i k g k m1 mq the k subalgebra of r spanned by the mi and ii the quotient ring r i g where i g m1 mq they come together in the construction of the rees algebra r i g of the ideal i g the emphasis is almost entirely on sets of squarefree monomials of degree two and their attached graphs the main results are assertions about the cohen macaulay behaviour of the koszul homology of i g and how normality or cohen macaulayness of one of the algebras can be read off the properties of the graph or in the other algebra

the h vector of a ladder determinantal ring cogenerated by 2 2 minors is log concave we show that the h vector of a ladder determinantal ring cogenerated by m u1 v1 is log concave thus we prove an instance of a conjecture of stanley respectively conca and herzog

extensions of the multiplicity conjecture the multiplicity conjecture of herzog huneke and srinivasan states an upper bound for the multiplicity of any graded k algebra as well as a lower bound for cohen macaulay algebras in this note we extend this conjecture in several directions we discuss when these bounds are sharp find a sharp lower bound in the case of not necessarily arithmetically cohen macaulay one dimensional schemes of 3 space and propose an upper bound for finitely generated graded torsion modules we establish this bound for torsion modules whose codimension is at most two

cellular resolutions of monomial modules we construct a canonical free resolution for arbitrary monomial modules and lattice ideals this includes monomial ideals and defining ideals of toric varieties and it generalizes our joint results with irena peeva for generic ideals bps ps introduction given a field k we consider the laurent polynomial ring t k x 1 1 x 1 n as a module over the polynomial ring s k x1 xn the module structure comes from the natural inclusion of semigroup algebras s k n k z t a monomial module is an s submodule of t which is generated by monomials x x1 1 xan n a z of special interest are the two cases when m has a minimal monomial generating set which is either finite or forms a group under multiplication in the first case m is isomorphic to a monomial ideal in s in the second case m coincides with the lattice module ml s x a l k x b n l t for some sublattice l z whose intersection with n is the origin 0 0 0 we shall derive free resolutions of m from regular cell complexes whose vertices are the generators of m and whose faces are labeled by the least common multiples of their vertices the basic theory of such cellular resolutions is developed in section 1 our main result is the construction of the hull resolution in section 2 we rescale the exponents of the monomials in m so that their convex hull in r is a polyhedron pt whose bounded faces support a free resolution of m this resolution is new and interesting even for monomial ideals it need not be minimal but unlike minimal resolutions it respects symmetry and is free from arbitrary choices in section 3 we relate the lattice module ml to the z l graded lattice ideal il x x a b l s this class of ideals includes ideals defining toric varieties we express the cyclic smodule s il as the quotient of the infinitely generated s module ml by the action of l in fact we like to think of ml as the universal cover of il many questions about il can thus be reduced to questions about ml in particular we obtain the hull resolution of a lattice ideal il by taking the hull resolution of ml modulo l this paper is inspired by the work of barany howe and scarf bhs who introduced the polyhedron pt in the context of integer programming the hull resolution generalizes results in bps for generic monomial ideals and in ps for generic lattice ideals in these generic cases the hull resolution is minimal

on deterministic control problems an approximation procedure for the optimal cost i the stationary problem we study deterministic optimal control problems having stopping time continuous and impulse controls in each strategy we obtain the optimal cost considered as the maximum element of a suitable set of subsolutions of the associated hamilton jacobi equation using an approximation method a particular derivative discretization scheme is employed convergence of approximate solutions is shown taking advantage of a discrete maximum principle which is also proved for the numerical solutions of approximate problems we use a method of relaxation type the algorithm is very simple it can be run on computers with small central memory in part i we study the stationary case in part ii siam j control optim 23 1985 pp 267 285 we study the nonstationary case

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

minimizing a quadratic payoff with monotone controls many important problems in economic dynamics can be formulated as optimal control problems in which the controls must be monotone functions of time in this paper we will completely solve a problem with quadratic payoff involving the control and the derivative of the control by constructing optimal monotone controls several generalizations and examples are also given

simplicial ideals 2 linear ideals and arithmetical rank abstract in the first part of this paper we study scrollers and linearly joined varieties scrollers were introduced in barile and morales 2004 bm4 linearly joined varieties are an extension of scrollers and were defined in eisenbud et al 2005 eghp there they proved that scrollers are defined by homogeneous ideals having a 2 linear resolution a particular class of varieties of important interest in classical geometry are cohen macaulay varieties of minimal degree they were classified geometrically by the successive contribution of del pezzo 1885 dp bertini 1907 b and xambo 1981 x and algebraically in barile and morales 2000 bm2 they appear naturally studying the fiber cone of a codimension two toric ideals morales 1995 m gimenez et al 1993 1999 gms1 gms2 barile and morales 1998 bm1 ha 2006 h ha and morales 2009 hm let s be a polynomial ring and i s a homogeneous ideal defining a sequence of linearly joined varieties we compute depth s i we prove that c v depth s i 1 where c v is the connectedness dimension of the algebraic set defined by i we characterize sets of generators of i and give an effective algorithm to find equations as an application we prove that ara i projdim s i in the case where v is a union of linear spaces in particular this applies to any square free monomial ideal having a 2 linear resolution in the case where v is a union of linear spaces the ideal i can be characterized by a tableau which is an extension of a ferrer or young tableau all these results are new and extend results in barile and morales 2004 bm4 eisenbud et al 2005 eghp

straightening law and powers of determinantal ideals of hankel matrices in this paper we establish a standard monomial theory for generic hankel matrices by a generic hankel matrix we mean a matrix y yij with yij x i j 1 where the x i are indeterminates over a field k we use this structure to determine the symbolic powers and the primary decomposition of the powers of the determinantal ideals it of y further we prove that the symbolic and ordinary rees algebras of it are cohen macaulay normal domains the first standard monomial theory was developed by hodge h to study the homogeneous coordinate ring of the grassmannian variety later standard monomial theories were established for generic matrices by doubilet rota and stein drs and for generic symmetric and generic skew symmetric matrices by de concini and procesi dp these are all examples of algebras with straightening law asl for short over a poset or over a doset the abstract notion of asl was introduced and developed by eisenbud e1 and by de concini eisenbud and procesi dep2 see also bv these structures turned out to be an extremely powerful tool in studying determinantal rings and ideals arising from the above mentioned generic matrices let now x1 xn be indeterminates over an arbitrary field k for j 1 n we denote by xj the j n 1 j hankel matrix with entries x1 xn that is

a formula for the core of certain strongly stable ideals abstract the core of an ideal is the intersection of all of its reductions the core is a natural object to study from the point of view of blow up algebras and integral closure and it also has geometric significance coming for example from its connection to adjoint and multiplier ideals in general the core is difficult to describe explicitly in this paper we investigate a particular family of strongly stable ideals we prove that ideals in this family satisfy an artin nagata property yet fail to satisfy other stronger standard depth conditions we then show that there is a surprisingly simple explicit formula for the core of these ideals

many toric ideals generated by quadratic binomials possess no quadratic grobner bases abstract let g be a finite connected simple graph and i g the toric ideal of the edge ring k g of g in the present paper we study finite graphs g with the property that i g is generated by quadratic binomials and i g possesses no quadratic grobner basis first we give a nontrivial infinite series of finite graphs with the above property second we implement a combinatorial characterization for i g to be generated by quadratic binomials and by means of the computer search we classify the finite graphs g with the above property up to 8 vertices

monomial and toric ideals associated to ferrers graphs each partition a 1 2 an determines a so called ferrers tableau or equivalently a ferrers bipartite graph its edge ideal dubbed a ferrers ideal is a squarefree monomial ideal that is generated by quadrics we show that such an ideal has a 2 linear minimal free resolution i e it defines a small subscheme in fact we prove that this property characterizes ferrers graphs among bipartite graphs furthermore using a method of bayer and sturmfels we provide an explicit description of the maps in its minimal free resolution this is obtained by associating a suitable polyhedral cell complex to the ideal graph along the way we also determine the irredundant primary decomposition of any ferrers ideal we conclude our analysis by studying several features of toric rings of ferrers graphs in particular we recover establish formulae for the hilbert series the castelnuovo mumford regularity and the multiplicity of these rings while most of the previous works in this highly investigated area of research involve path counting arguments we offer here a new and self contained approach based on results from gorenstein liaison theory

small schemes and varieties of minimal degree we prove that if x p r is any 2 regular scheme in the sense of castelnuovo mumford then x is small this means that if l is a linear space and y l x is finite then y is linearly independent in the sense that the dimension of the linear span of y is deg y 1 the converse is true and well known for finite schemes but false in general the main result of this paper is that the converse small implies 2 regular is also true for reduced schemes algebraic sets this is proven by means of a delicate geometric analysis leading to a complete classification we show that the components of a small algebraic set are varieties of minimal degree meeting in a particularly simple way from the classification one can show that if x p r is 2 regular then so is xred and so also is the projection of x from any point of x our results extend the del pezzo bertini classification of varieties of minimal degree the charac terization of these as the varieties of regularity 2 by eisenbud goto and the construction of 2 regular square free monomial ideals by froberg 0 introduction throughout this paper we will work with projective schemes x p r over an algebraically closed field k the castelnuovo mumford regularity of x p r is a basic homological measure of the complexity of x and its embedding in p r that gives a bound for the degrees of the generators of the defining ideal ix of x and for many other invariants the only schemes of regular ity 1 are the linear spaces but no classification is known for projective schemes of regularity 2 in this paper we prove a structure theorem for reduced 2 regular schemes showing that their irreducible components are varieties of minimal degree and characterizing how these components can meet we also show that the reduced structure on any 2 regular scheme is 2 regular and thus we obtain a complete description of the reduced structures on 2 regular schemes since a high veronese re embedding of any zero dimensional scheme is 2 regular one cannot hope to characterize the isomorphism types of all 2 regular nonreduced schemes before stating our results we review some basic notions for any subscheme x p r we write span x for the smallest linear subspace of p r containing x recall that every variety reduced irreducible scheme x p r satisfies the condition deg x 1 codim x span x

an optimization approach for well targeted transcranial direct current stimulation transcranial direct current stimulation tdcs is a noninvasive brain stimulation technique which modifies neural excitability by providing weak currents through scalp electrodes the aim of this study is to introduce and analyze a novel optimization method for safe and well targeted multiarray tdcs for optimization we consider an optimal control problem for a laplace equation with neumann boundary conditions with control and pointwise gradient state constraints we prove well posedness results for the proposed methods and provide computer simulation results in a highly realistic six compartment geometry adapted hexahedral head model for discretization of the proposed minimization problem the finite element method is employed and the existence of at least one minimizer to the discretized optimization problem is shown for numerical solution of the corresponding discretized problem we employ the alternating direction method of multipliers and comprehensively examine the cortical current flow field with regard to focality target intensity and orientation the numerical results reveal that the optimized current flow fields show significantly higher focality and in most cases higher directional agreement to the target vector in comparison to standard bipolar electrode montages