resolutions of fat point ideals involving eight general points of p2 abstract the main result theorem 1 provides an algorithm for determining the minimal free resolution of fat point subschemes of p 2 involving up to eight general points of arbitrary multiplicities the resolutions obtained hold for any algebraically closed field independent of the characteristic the algorithm works by giving a formula in nice cases and a reduction to the nice cases otherwise the algorithm which does not involve grobner bases is very fast partial information is also obtained in certain cases with n 8

qualitative analysis of a recurrent neural network for nonlinear continuously differentiable convex minimization over a nonempty closed convex subset we investigate the qualitative properties of a recurrent neural network rnn for minimizing a nonlinear continuously differentiable and convex objective function over any given nonempty closed and convex subset which may be bounded or unbounded by exploiting some key inequalities in mathematical programming the global existence and boundedness of the solution of the rnn are proved when the objective function is convex and has a nonempty constrained minimum set under the same assumption the rnn is shown to be globally convergent in the sense that every trajectory of the rnn converges to some equilibrium point of the rnn if the objective function itself is uniformly convex and its gradient vector is a locally lipschitz continuous mapping then the rnn is globally exponentially convergent in the sense that every trajectory of the rnn converges to the unique equilibrium point of the rnn exponentially these qualitative properties of the rnn render the network model well suitable for solving the convex minimization over any given nonempty closed and convex subset no matter whether the given constrained subset is bounded or not

on the cohen macaulay type of s lines in an 1 abstract let a be the coordinate ring of s lines through the origin in a n 1 k we discuss what it means for these lines to be in sufficiently general position and then with this restriction on the lines we attempt to verify our belief that the cohen macaulay type of a depends only on s and n we characterize those s and n for which a is a gorenstein ring i e of c m type 1 and explicitly calculate the cohen macaulay type in several other cases we then extend some of our results to the case of an arbitrary reduced curve whose tangent cone consists of lines in sufficiently general position finally we calculate the hubert samuel polynomials of the curves we have been considering

linear systems of plane curves through fixed fat points of p2 abstract given any s points p1 ps in p 2 and s positive integers m1 ms let sn be the linear system of plane curves of degree n through pi with multiplicity at least mi 1 i s we give numerical bounds for the regularity of sn in the following cases a the points pi are non singular points of an integral curve of degree d b the pi s are in general position c the pi s are in uniform position d the pi s are generic points of p 2 we also study the sharpness of such bounds

regularity of linear systems of plane curves the subject of this paper is the study of linear systems of plane curves which are given by assigning a set of fixed multiple points this study not only has an interest per se but is connected to other subjects for example results on rational surfaces and curves on them can be found in ro a and gil a linear system of plane curves of projective dimension r is a subvector space i s

a posteriori error bounds for the linearly constrained varitional inequality problem in this paper we derive a posteriori error bound of an approximate solution to a linearly constrained variational inequality problem three measures of error are used with each depending on a particular characterization of an exact solution and related to a certain class of iterative methods

on nagata s conjecture abstract this paper gives an improved lower bound on the degrees d such that for general points p 1 p n p 2 and m 0 there is a plane curve of degree d vanishing at each p i with multiplicity at least m

on some integral inequalities similar to hardy s inequality abstract the aim of the present paper is to establish some new integral inequalities which claim their origin in the well known hardy s integral inequality the method by which we obtain our results is quite elementary and based on the idea used by levinson to obtain the generalizations of hardy s integral inequality

some hardy type inequalities in this paper some new hardy type inequalities involving many functions are obtained these on the one hand generalize and on the other hand improve some existing results by isumi and isumi levinson and pachpatte on this famous type of inequalities

the geometric interpretation of froberg iarrobino conjectures on infinitesimal neighbourhoods of points in projective space abstract the study of infinitesimal deformations of a variety embedded in projective space requires at ground level that of deformation of a collection of points as specified by a zero dimensional scheme further basic problems in infinitesimal interpolation correspond directly to the analysis of such schemes an optimal hilbert function of a collection of infinitesimal neighbourhoods of points in projective space is suggested by algebraic conjectures of r froberg and a iarrobino we discuss these conjectures from a geometric point of view they give for each such collection a function based on dimension number of points and order of each neighbourhood which should serve as an upper bound to its hilbert function weak conjecture the strong conjecture predicts when the upper bound is sharp in the case of equal order throughout in general we refer to the equality of the hilbert function of a collection of infinitesimal neighbourhoods with that of the corresponding conjectural function as the strong hypothesis we interpret these conjectures and hypotheses as accounting for the infinitesimal neighbourhoods of projective subspaces naturally occurring in the base locus of a linear system with prescribed singularities at fixed points we develop techniques and insight toward the conjectures verification and refinement the main result gives an upper bound on the hilbert function of a collection of infinitesimal neighbourhoods in p n based on hilbert functions of certain such subschemes of p n 1 further equality occurs exactly when the scheme has only the expected linear obstructions to the linear system at hand it follows that an infinitesimal neighbourhood scheme obeys the weak conjecture provided that the schemes identified in codimension one satisfy the strong hypothesis this observation is then applied to show that the weak conjecture does hold valid in p n for n 3 the main feature here is that the result is obtained although the strong hypothesis is not known to hold generally in p 2 and further p 2 presents special exceptional cases consequences of the main result in higher dimension are then examined we note then that the full weight of the strong conjecture and validity of the strong hypothesis are not necessary toward using the main theorem in the next dimension we end with the observation of how our viewpoint on the strong hypothesis pertains to extra algebraic information namely on the structure of the minimal free resolution of an ideal generated by linear forms

a recurrent neural network for nonlinear optimization with a continuously differentiable objective function and bound constraints this paper presents a continuous time recurrent neural network model for nonlinear optimization with any continuously differentiable objective function and bound constraints quadratic optimization with bound constraints is a special problem which can be solved by the recurrent neural network the proposed recurrent neural network has the following characteristics 1 it is regular in the sense that any optimum of the objective function with bound constraints is also an equilibrium point of the neural network if the objective function to be minimized is convex then the recurrent neural network is complete in the sense that the set of optima of the function with bound constraints coincides with the set of equilibria of the neural network 2 the recurrent neural network is primal and quasiconvergent in the sense that its trajectory cannot escape from the feasible region and will converge to the set of equilibria of the neural network for any initial point in the feasible bound region 3 the recurrent neural network has an attractivity property in the sense that its trajectory will eventually converge to the feasible region for any initial states even at outside of the bounded feasible region 4 for minimizing any strictly convex quadratic objective function subject to bound constraints the recurrent neural network is globally exponentially stable for almost any positive network parameters simulation results are given to demonstrate the convergence and performance of the proposed recurrent neural network for nonlinear optimization with bound constraints

linear systems of plane curves with base points of equal multiplicity in this article we address the problem of computing the dimenlsion of the space of plane curves of degree d with n general points of multiplicity m a conjecture of harbourne and hirschowitz implies that when d 3m the dimension is equal to the expected dimension given by the riemann roch theorem also systems for which the dimension is larger than expected should have a fixed part containing a multiple 1 curve we reformulate this conjecture by explicitly listing those systems which have unexpected dimension then we use a degeneration technique developed to show that the conjecture holds for all m 12

on the analysis of a recurrent neural network for solving nonlinear monotone variational inequality problems we investigate the qualitative properties of a recurrent neural network rnn for solving the general monotone variational inequality problems vips defined over a nonempty closed convex subset which are assumed to have a nonempty solution set but need not be symmetric the equilibrium equation of the rnn system simply coincides with the nonlinear projection equation of the vip to be solved we prove that the rnn system has a global and bounded solution trajectory starting at any given initial point in the above closed convex subset which is positive invariant for the rnn system for general monotone vips we show by an example that the trajectory of the rnn system can converge to a limit cycle rather than an equilibrium in the case that the monotone vips are not symmetric contrary to this for the strictly monotone vips it is shown that every solution trajectory of the rnn system starting from the above closed convex subset converges to the unique equilibrium which is also locally asymptotically stable in the sense of lyapunov no matter whether the vips are symmetric or nonsymmetric for the uniformly monotone vips the aforementioned solution trajectory of the rnn system converges to the unique equilibrium exponentially

a recurrent neural network for nonlinear continuously differentiable optimization over a compact convex subset we propose a general recurrent neural network rnn model for nonlinear optimization over a nonempty compact convex subset which includes the bound subset and spheroid subset as special cases it is shown that the compact convex subset is a positive invariant and attractive set of the rnn system and that all the network trajectories starting from the compact convex subset converge to the equilibrium set of the rnn system the above equilibrium set of the rnn system coincides with the optimum set of the minimization problem over the compact convex subset when the objective function is convex the analysis of these qualitative properties for the rnn model is conducted by employing the properties of the projection operator of euclidean space onto the general nonempty closed convex subset a numerical simulation example is also given to illustrate the qualitative properties of the proposed general rnn model for solving an optimization problem over various compact convex subsets

plane curves of minimal degree with prescribed singularities we prove that there exists a positive such that for any integer d 3 and any topological types s1 sn of plane curve singularities satisfying

on a new class of hardy type inequalities in this paper we establish a new class of integral inequalities which originate from the well known hardy s inequality the analysis used in the proofs is quite elementary and is based on the idea used by levinson to obtain generalisations of hardy s inequality

geometry of families of nodal curves on the blown up projective plane let p2r be the projective plane blown up at r generic points denote by e0 e1 er the strict transform of a generic straight line on p2 and the exceptional divisors of the blown up points on p2r respectively we consider the variety virr of all irreducible curves c with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness irreducibility and non emptiness moreover we extend our conditions for the smoothness and the irreducibility on families of reducible curves for r 9 we give the complete answer concerning the existence of nodal curves in virr

nonsingular curves passing through points of p2 in generic position i abstract the least integer is looked for such that there exists a nonsingular curve of degree less than or equal to passing through any assigned set of n points of p 2 in generic position here generic takes different meanings and the generic n tuples of points describe nonempty open subsets u of p 2 n indeed we find i for u 1 n tuples in general position u 1 n 2 ii if n d 2 2 h 0 h d 1 for u2 n tuples in n generic position iii with the notations of ii for u3 n tuples in uniform position in any case a nonempty open subset u u3 is found in which u d 1

on some generalizations of hardy s integral inequality in this paper we establish some new generalizations of the hardy s inequality by using a fairly elementary analysis the inequalities given here contain in the special cases some of the recent generalizations of hardy s integral inequality appearing in the literature

linear systems on the blow up of p1 n abstract in this note we study linear systems on the blow up of p 1 n at r points in very general position we prove that the fibers of the projections p 1 n p 1 s 1 s n 1 can give contribution to the speciality of the linear system this allows us to give a new definition of expected dimension of a linear system in p 1 n which we call fiber dimension finally we state a conjecture about linear systems on p 1 3

resolutions of ideals of quasiuniform fat point subschemes of the notion of a quasiuniform fat point subscheme z p 2 is introduced and conjectures for the hilbert function and minimal free resolution of the ideal i defining z are put forward in a large range of cases it is shown that the hilbert function conjecture implies the resolution conjecture in addition the main result gives the first determination of the resolution of the mth symbolic power i m n of an ideal defining n general points of p 2 when both m and n are large in particular for infinitely many m for each of infinitely many n and for infinitely many n for every m 2 resolutions in other cases such as fat points with tails are also given except where an explicit exception is made all results hold for an arbitrary algebraically closed field k as an incidental result a bound for the regularity of i m n is given which is often a significant improvement on previously known bounds

analytic order of singular and critical points we deal with the following closely related problems i for a germ of a reduced plane analytic curve what is the minimal degree of an algebraic curve with a singular point analytically equivalent isomorphic to the given one ii for a germ of a holomorphic function in two variables with an isolated critical point what is the minimal degree of a polynomial equivalent to the given function up to a local holomorphic coordinate change classically known estimates for such a degree d in these questions are 1 d 1 where is the milnor number our result in both the problems is d a with an absolute constant a as a corollary we obtain asymptotically proper sufficient conditions for the existence of algebraic curves with prescribed singularities on smooth algebraic surfaces

on some variants of hardy s inequality abstract the present paper deals with some new generalizations and extensions of a certain variant of hardy s inequality given by izumi and izumi the method employed in our analysis is quite elementary and the results established in this paper provide new estimates on this type of integral inequalities