Image of the Nash map in terms of wedges

M. Lejeune-Jalabert (Lecture Notes in Math., vol. 777, Springer-Verlag, 1980, pp. 303–336) proposed the following ‘problem of wedges’: let X be a surface over an algebraically closed field k of characteristic zero. Given a wedge $\text{φ}\text{:}\text{Spec}\text{k[[ξ,t]]→X}$, whose special arc lifts to the minimal resolution Y of X in an arc transversal to an irreducible component of the exceptional locus in a general point, does φ lift to Y? The main result in this Note is to extend this problem to a problem of wedges in a variety X of any dimension and to prove that, if the wedge problem is true for X, then the Nash problem is true for X. From this, necessary and sufficient conditions are given for an essential divisor to belong to the image of the Nash map, and we conclude that the Nash problem is true for sandwiched surface singularities. To cite this article: A.J. Reguera, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On existence of log minimal models and weak Zariski decompositions

We first introduce a weak type of Zariski decomposition in higher dimensions: an -Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective -Cartier divisor. We then prove that there is a very basic relation between Zariski decompositions and log minimal models which has long been expected: we prove that assuming the log minimal model program in dimension d − 1, a lc pair (X/Z, B) of dimension d has a log minimal model (in our sense) if and only if K _X  + B has a weak Zariski decomposition/Z.     

Uncoupled automata and pure Nash equilibria

We study the problem of reaching a pure Nash equilibrium in multi-person games that are repeatedly played, under the assumption of uncoupledness: EVERY player knows only his own payoff function. We consider strategies that can be implemented by finite-state automata, and characterize the minimal number of states needed in order to guarantee that a pure Nash equilibrium is reached in every game where such an equilibrium exists.     

Arithmetically Buchsbaum divisors on varieties of minimal degree

In this paper we consider integral arithmetically Buchsbaum subschemes of projective space. First we show that arithmetical Buchsbaum varieties of sufficiently large degree have maximal Castelnuovo-Mumford regularity if and only if they are divisors on a variety of minimal degree. Second we determine all varieties of minimal degree and their divisor classes which contain an integral arithmetically Buchsbaum subscheme. Third we investigate these varieties. In particular, we compute their Hilbert function, cohomology modules and (often) their graded Betti numbers and obtain an existence result for smooth arithmetically Buchsbaum varieties.

     

A Class of Solvable Lie Algebras with Triangular Decompositions

So far there has been elementary proof for Frobenius's theorem only in special cases: if the complement is solvable, see e.g. [3], if the complement is of even order, see e.g. [6]. In the first section we consider the case, when the order of the complement is odd. We define a graph the vertices of which are the set K^# of elements of our Frobenius group with 0 fixed points. Two vertices are connected with an edge if and only if the corresponding elements commute. We prove with elementary methods that K is a normal subgroup in G if and only if there exists an element x in K^# such that all elements of K^# belonging to the connected component C of K^# containing x are at most distance 2 from c and N^G(C) is not a -group, where is the set of prime divisors of the Frobenius complement of G. In the second section we generalize the case when the order of the complement is even, proving that the Frobenius kernel is a normal subgroup, if a fixed element a of the complement, the order of which is a minimal prime divisor of the order of the complement, generates a solvable subgroup together with any ofits conjugates. In the third section we prove a generalization of the Glauberman-Thompson normal p-complement theorem, and using this wegive another sufficient condition for the Frobenius kernel to be a normal subgroup for |G| odd, namely we prove this under the conditionthat all the Sylow normalizers in G intersect some of the complements     

Approximations and Well-Posedness in Multicriteria Games

First, sufficient conditions of minimal character are given which guarantee the sequential closedness of the set-valued function defined by the parametric weak-multicriteria Nash equilibria of a parametric multicriteria game, that is to say: a convergent sequence of parametric weak-multicriteria Nash equilibria, corresponding to an approximate value of the parameter x_n, converges to a weak-multicriteria Nash equilibrium corresponding to the limit value x of the sequence (x_n)_n. Then, approximating sequences and parametrically well-posedness for a multicriteria game are introduced and investigated.     

Hierarchic Control for the Wave Equation

This paper deals with the hierarchical control of the wave equation. We use Stackelberg–Nash strategies. As usual, we consider one leader and two followers. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we look for a leader that solves an exact controllability problem. We consider linear and semilinear equations.     

Poincaré series of divisorial filtration corresponding to a curve with one place at infinity

The exceptional divisor component of the projective plane modified by a sequence of blow-ups determines filtration on the ring of polynomials in two variables. The set of such components determines the multi-index filtration on this ring. The Poincaré series of this filtration is calculated for some sets of components provided that the modification under study is the minimal resolution of a plane algebraic curve with one place at infinity.     

Nondominated Nash points: application of biobjective mixed integer programming

We study the connection between biobjective mixed integer linear programming and normal form games with two players. We first investigate computing Nash equilibria of normal form games with two players using single-objective mixed integer linear programming. Then, we define the concept of efficient (Pareto optimal) Nash equilibria. This concept is precisely equivalent to the concept of efficient solutions in multi-objective optimization, where the solutions are Nash equilibria. We prove that the set of all points in the payoff (or objective) space of a normal form game with two players corresponding to the utilities of players in an efficient Nash equilibrium, the so-called nondominated Nash points, is finite. We demonstrate that biobjective mixed integer linear programming, where the utility of each player is an objective function, can be used to compute the set of nondominated Nash points. Finally, we illustrate how the nondominated Nash points can be used to determine the disagreement point of a bargaining problem.     

Feedback Nash equilibria in a problem of optimal fishery management

The paper deals with a problem of optimal management of a common-property fishery, modelled as a two-player differential game. Under nonclassical assumptions on harvest rates and utilities, a feedback Nash equilibrium is determined, using a bionomic equilibrium concept. Later on, this assumption is relaxed and a feedback Nash equilibrium is established under minimal hypotheses.     

Trivial Ring Extensions Defined by Arithmetical-Like Properties

In this article we investigate the transfer of the notions of elementary divisor ring, Hermite ring, Bezout ring, and arithmetical ring to trivial ring extensions of commutative rings by modules. Namely, we prove that the trivial ring extension R: = A ? B defined by extension of integral domains is an elementary divisor ring if and only if A is an elementary divisor ring and B = qf(A); and R is an Hermite ring if and only if R is a Bezout ring if and only if A is a Bezout domain and qf(A) = B. We provide necessary and sufficient conditions for R = A ? E to be an arithmetical ring when E is a nontorsion or a finitely generated A ? module. As an immediate consequences, we show that A ? A is an arithmetical ring if and only if A is a von Neumann regular ring, and A ? Q(A) is an arithmetical ring if and only if A is a semihereditary ring.     

Lower semicontinuity for approximate social Nash equilibria

We introduce a suitable concept of approximate social Nash equilibrium and we determine sufficient conditions of minimal character which guarantee, for a parametric social Nash equilibrium problem, the lower semicontinuity of the set-valued function defined by these approximate solutions. Received: December 2001     

Multilateral non-cooperative bargaining in a general utility space

We consider an n-player bargaining problem where the utility possibility set is compact, convex, and stricly comprehensive. We show that a stationary subgame perfect Nash equilibrium exists, and that, if the Pareto surface is differentiable, all such equilibria converge to the Nash bargaining solution as the length of a time period between offers goes to zero. Without the differentiability assumption, convergence need not hold.     

有限个互素因子链上幂GCD矩阵与幂LCM矩阵的行列式的整除性

设S={x1,x2,...,xn}是由n个不同的正整数组成的集合,并设a为正整数.如果一个n阶矩阵的第i行j列元素是S中元素xi和xj的最大公因子的a次幂(xi,xj)a,则称该矩阵为定义在S上的a次幂最大公因子(GCD)矩阵,用(Sa)表示;类似定义a次幂LCM矩阵[Sa].如果存在{1,2,...,n}上的一个置换σ使得xσ(1)|xσ(2)|···|xσ(n),则称S为一个因子链.如果存在正整数k,使得S=S1∪S2∪···∪Sk,其中每一个Si(1ik)均为一个因子链,并且对所有的1i=jk,Si中的每个元素与Sj中的每个元素互素,则称S由有限个互素因子链构成.本文中,设S由有限个互素的因子链构成,并且1∈S.我们首先给出幂GCD矩阵与幂LCM矩阵的行列式的公式,然后证明:如果a|b,则det(Sa)|det(Sb),det[Sa]|det[Sb],det(Sa)|det[Sb].最后我们指出:如果构成S的有限个因子链不互素,则此结论一般不成立.     

Algebraicity of Global Real Analytic Hypersurfaces

Let X be an algebraic manifold without compact component and let V be a compact coherent analytic hypersurface in X, with finite singular set. We prove that V is diffeotopic (in X) to an algebraic hypersurface in X if and only if the homology class represented by V is algebraic and singularities are locally analytically equivalent to Nash singularities. This allows us to construct algebraic hypersurfaces in X with prescribed Nash singularities.     

A general theorem of existence of quasi absolutely minimal Lipschitz extensions

In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain other mild conditions, a quasi absolutely minimal Lipschitz extension must exist as well. Here we use the qualifier “quasi” to indicate that the extending function in question nearly satisfies the conditions of being an absolutely minimal Lipschitz extension, up to several factors that can be made arbitrarily small.     

Jacobians with a vanishing theta-null in genus 4

In this paper we prove a conjecture of Hershel Farkas [11] that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the Hessian of the theta function at the corresponding 2-torsion point is degenerate, the abelian variety is a Jacobian. We also discuss possible generalizations to higher genera, and an interpretation of this condition as an infinitesimal version of Andreotti and Mayer’s local characterization of Jacobians by the dimension of the singular locus of the theta divisor.     

An algorithm for road coloring

A coloring of edges of a finite directed graph turns the graph into a finite-state automaton. A synchronizing word of a deterministic automaton is a word in the alphabet of colors of its edges (regarded as letters) which maps the automaton to a single state. A coloring of edges of a directed graph of uniform outdegree (constant outdegree of any vertex) is synchronizing if the corresponding deterministic finite automaton possesses a synchronizing word.The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph of uniform outdegree if the greatest common divisor of lengths of all its cycles is one. Posed in 1970, it has evoked noticeable interest among the specialists in the theory of graphs, automata, codes, symbolic dynamics, and well beyond these areas.We present an algorithm for the road coloring of cubic worst-case time complexity and quadratic time complexity in the majority of studied cases. It is based on the recent positive solution of the road coloring problem. The algorithm was implemented in the freeware package TESTAS.     

On the generalized Ritt problem as a computational problem

The Ritt problem asks if there is an algorithm that decides whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing whether a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the proof of this equivalence yields algorithms for computing a canonical decomposition of a radical differential ideal into prime components and a canonical generating set of a radical differential ideal. Both proposed representations of a radical differential ideal are independent of the given set of generators and can be made independent of the ranking.