Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities.     

Decomposition of Log Crepant Birational Morphisms between Log Terminal Surfaces

We prove that every log crepant birational morphism between log terminal surfaces can be decomposed into log flopping type divisorial contraction morphisms and log blowdowns. Repeating these two kinds of contractions, we reach a minimal log minimal surface from any log minimal surface.     

The set of toric minimal log discrepancies

We describe the set of minimal log discrepancies of toric log varieties, and study its accumulation points. This work is supported by a 21st Century COE Kyoto Mathematics Fellowship, and a JSPS Grant-in-Aid No 17740011.     

On the Weak Semi-continuity of Vector Functions and Minimum Problems

Lower semi-continuity from above or upper semi-continuity from below has been used by many authors in recent papers. In this paper, we first study the weak semi-continuity for vector functions having particular form as that of Browder in ordered normed vector spaces; we obtain several new results on the lower semi-continuity from above or upper semi-continuity from below for these vector functions. Our results generalize some well-known results of Browder in scalar case. Secondly, we study the minimum or maximum problems for vector functions satisfying lower semi-continuous from above or upper semi-continuous from below conditions; several new results on the existence of minimal points or maximal points are obtained. We also use these results to study vector equilibrium problems and von Neumann’s minimax principle in ordered normed vector spaces.     

Semi-continuity properties of weighted log canonical thresholds of toric plurisubharmonic functions

In this note, we prove a semi-continuity theorem for certain weighted log canonical thresholds of toric plurisubharmonic functions.     

Letters of a Bi-rationalist. VII Ordered termination

To construct a resulting model in the LMMP, it is sufficient to prove the existence of log flips and their termination for some sequences. We prove that the LMMP in dimension d − 1 and the termination of terminal log flips in dimension d imply, for any log pair of dimension d, the existence of a resulting model: a strictly log minimal model or a strictly log terminal Mori log fibration, and imply the existence of log flips in dimension d + 1. As a consequence, we prove the existence of a resulting model of 4-fold log pairs, the existence of log flips in dimension 5, and Geography of log models in dimension 4. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 184–208. To V.A. Iskovskikh, who has greatly shaped my vision of mathematics     

Notes on minimal compactifications of the affine plane

In this paper, we consider the minimal compactifications X of the complex affine plane with at most log canonical singular points. We classify the surfaces X in the case X has at least one non-log terminal singular point.      

Singularities with the highest Mather minimal log discrepancy

This paper characterizes singularities with Mather minimal log discrepancies in the highest unit interval, i.e., the interval between $d-1$ and $d$ , where $d$ is the dimension of the scheme. The class of these singularities coincides with one of the classes of (1) compound Du Val singularities, (2) normal crossing double singularities, (3) pinch points, and (4) pairs of non-singular varieties and boundaries with multiplicities less than or equal to 1 at the point. As a corollary, we also obtain one implication of an equivalence conjectured by Shokurov for the usual minimal log discrepancies.     

Continuity of the value and optimal strategies when common priors change

We show that the value of a zero-sum Bayesian game is a Lipschitz continuous function of the players?? common prior belief with respect to the total variation metric on beliefs. This is unlike the case of general Bayesian games where lower semi-continuity of Bayesian equilibrium (BE) payoffs rests on the ??almost uniform?? convergence of conditional beliefs. We also show upper semi-continuity (USC) and approximate lower semi-continuity (ALSC) of the optimal strategy correspondence, and discuss ALSC of the BE correspondence in the context of zero-sum games. In particular, the interim BE correspondence is shown to be ALSC for some classes of information structures with highly non-uniform convergence of beliefs, that would not give rise to ALSC of BE in non-zero-sum games.     

Inversion of adjunction for non-degenerate hypersurfaces

 We prove a precise inversion of adjunction formula for the log variety (ℂ ^{d +1},X), where X is a non-degenerate hypersurface. As a corollary, the minimal log discrepancies of non-degenerate normal hypersurface singularities are bounded by dimension. Received: 17 September 2002 / Revised version: 22 November 2002 Published online: 14 February 2003 Current address: DPMMS, CMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England. e-mail: f.ambro@dpmms.cam.ac.uk Mathematics Subject Classification (2000): Primary 14B05; Secondary 14M25, 52B20     

Genericity and Hölder Stability in Semi-Algebraic Variational Inequalities

The aim of this paper is twofold. We first present generic properties of semi-algebraic variational inequalities: “typical” semi-algebraic variational inequalities have finitely many solutions, around each of which they admit a unique “active manifold” and such solutions are nondegenerate. Second, based on these results, we offer Hölder stability, upper semi-continuity, and lower semi-continuity properties of the solution map of parameterized variational inequalities.     

Long time behavior of stochastic {MHD} equations perturbed by multiplicative noises

In this paper, 2-dimensional (2D) magnetohydrodynamics (MHD) equations perturbed by multiplicative noises in both the velocity and the magnetic field is studied. We first considered the stability, or the upper semi-continuity, for equivalent random dynamical systems (RDS), and then applying the abstract result we established the existence and the upper semi-continuity of tempered random attractors for the stochastic MHD equations. This result shows that the asymptotic behavior of MHD equations is stable under stochastic perturbations.     

Minimal discrepancies of toric singularities

The main purpose of this paper is to prove that minimal discrepancies ofn-dimensional toric singularities can accumulate only from above and only to minimal discrepancies of toric singularities of dimension less thann. I also prove that some lower-dimensional minimal discrepancies do appear as such limit.     

On the number of minimal models of a log smooth threefold

We give a topological bound on the number of minimal models of a class of three-dimensional log smooth pairs of log general type.     

The stability of the solution sets for set optimization problems via improvement sets

ABSTRACT

The aim of this paper is to investigate the stability of the solution sets for set optimization problems via improvement sets. Firstly, we consider the relations among the solution sets for optimization problem with set optimization criterion. Then, the closeness and the convexity of solution sets are discussed. Furthermore, the upper semi-continuity, Hausdorff upper semi-continuity and lower semi-continuity of solution mappings to parametric set optimization problems via improvement sets are established under some suitable conditions. These results extend and develop some recent works in this field.     

An overview of semi-continuity results on the spectral radius and positivity

Our aim is to provide a novelly comprehensive and unifying approach to showing the continuous dependence of the spectral radius of compact linear operators defined on Banach spaces (with specific attention to positive operators defined on normal Banach spaces) and emphasizing that the upper semi-continuity generally holds unlike the lower semi-continuity.     

Strong proximinality and renormings

We characterize finite-dimensional normed linear spaces as strongly proximinal subspaces in all their superspaces. A connection between upper Hausdorff semi-continuity of metric projection and finite dimensionality of subspace is given.

     

Attractors for Multi-valued Non-autonomous Dynamical Systems: Relationship,Characterization and Robustness

In this paper we study cocycle attractors, pullback attractors and uniform attractors for multi-valued non-autonomous dynamical systems. We first consider the relationship between the three attractors and find that, under suitable conditions, they imply each other. Then, for generalized dynamical systems, we find that these attractors can be characterized by complete trajectories, which implies that the uniform attractor is lifted invariant, though it has no standard invariance by definition. Finally, we study both upper and lower semi-continuity of these attractors. A weak equi-attraction method is introduced to study the lower semi-continuity, and we show with an example the advantages of this method. A reaction-diffusion system and a scalar ordinary differential inclusion are studied as applications.     

Global Stability Results for the Weak Vector Variational Inequality

In this paper, we consider the global stability of solutions of a Weak Vector Variational Inequality in a finite-dimensional Euclidean space. Upper semi-continuity of the solution set mapping is established. And by a scalarization method, we derive a sufficient condition that guarantees the lower semi-continuity of the solution set mapping for the Weak Vector Variational Inequality     

On the weak limit of mappings with finite distortion

We give a new proof that the limit of a weakly convergent sequence of mappings with finite distortion also has finite distortion. The result has been recently proved by Gehring and Iwaniec using the biting convergence of Jacobians. We present a different proof using simply the lower semi-continuity of quasiconvex functionals.