Metric distortion in the geometric Schottky problem

Science China Mathematics - The classical Schottky problem is concerned with characterization of Jacobian varieties of compact Riemann surfaces among all abelian varieties, or the identification of...     

Shape distortion of space structures,postulates and controversy

An attempt to resolve the controversy connected with the definition of shape distortion of space structures is proposed by generalization of the different approaches. A generalization of the criterion for shape control robustness of space structures is also proposed.On leave from Department of Aerospace Engineering, Technion, Haifa, Israel.     

Degenerate Poisson structures in dimension 3

Formal normal forms of degenerate Poisson structures in dimension 3 are described. The main tool of the study is a spectral sequence previously introduced by the author. In particular, this method allows one to obtain a new proof of the linearizability of Poisson structures with semisimple linear part. However, there are nonlinearizable Poisson structures in dimension 3 as well.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 579–592, April, 1998.The author wishes to thank the referee for pointing out reference [3] and for other useful remarks.     

Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings

We investigate how the integrability of the derivatives of Orlicz-Sobolev mappings defined on open subsets of Rn affect the sizes of the images of sets of Hausdorff dimension less than n. We measure the sizes of the image sets in terms of generalized Hausdorff measures.     

Metric dimension of symplectic dual polar graphs and symmetric bilinear forms graphs

In this paper, by constructing resolving sets of symplectic dual polar graphs and symmetric bilinear forms graphs, we obtain upper bounds on their metric dimension.     

Fractal dimension for fractal structures: A Hausdorff approach revisited

In this paper, we use fractal structures to study a new approach to the Hausdorff dimension from both continuous and discrete points of view. We show that it is possible to generalize the Hausdorff dimension in the context of Euclidean spaces equipped with their natural fractal structure. To do this, we provide three definitions of fractal dimension for a fractal structure and study their relationships and mathematical properties.     

Beliefs and engagement structures: behind the affective dimension of mathematical learning

Beliefs influencing students’ mathematical learning and problem solving are structured and intertwined with larger affective and cognitive structures. This theoretical article explores a psychological concept we term an engagement structure, with which beliefs are intertwined. Engagement structures are idealized, hypothetical constructs, analogous in many ways to cognitive structures. They describe complex “in the moment” affective and social interactions as students work on conceptually challenging mathematics. We present engagement structures in a self-contained way, paying special attention to their theoretical justification and relation to other constructs. We suggest how beliefs are characteristically woven into their fabric and influence their activation. The research is based on continuing studies of middle school students in inner-city classrooms in the USA.     

Resistance dimension,random walk dimension and fractal dimension

In this paper, we generalize a theorem due to Telcs concerning random walks on infinite graphs, which describes the relation of random walk dimension, fractal dimension and resistance dimension. Moreover, we obtain a reasonable upper bound and lower bound on the hitting time in terms of resistance for some nice graphs. In fact, the conditions given in this paper are weaker than those obtained by A. Telcs.Partly supported by National Natural Science Foundation and State Educational Committee of China.     

Counterexamples in theory of fractal dimension for fractal structures

Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. For this purpose, it is always used the box dimension, since it is easy to calculate, though the Hausdorff dimension, which is the oldest and also the most accurate fractal dimension, presents the best analytical properties. Additionally, fractal structures provide an appropriate topological context where new models of fractal dimension for a fractal structure could be developed in order to generalize the classical models of fractal dimension. In this survey, we gather different definitions and counterexamples regarding these new models of fractal dimension in order to show the reader how they behave mathematically with respect to the classical models, and also to point out which features of such models can be exploited to powerful effect in applications.     

The Representive of Metric Pro jection on the Finite Co dimension Subspace in Banach Space

In the paper we introduce the notions of the separation factor κ and give a representive of metric projection on an n-codimension subspace(or an affine set)under certain conditions in Banach space. Further, we obtain the distance formula from any point x to a finite n-codimension subspace. Results extend and improve the corresponding results in Hilbert space.     

Homogeneous almost complex structures in dimension 6 with semi-simple isotropy

We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homogeneous types or a left-invariant structure on a Lie group. For integrable invariant almost complex structures we classify all compatible invariant Hermitian structures on these homogeneous manifolds, indicate their integrability properties (Kähler, SNK, SKT) and mark the other interesting geometric properties (including the Gray-Hervella type).     

Mass transportation on sub-Riemannian structures of rank two in dimension four

This paper is concerned with the study of the Monge optimal transport problem in sub-Riemannian manifolds where the cost is given by the square of the sub-Riemannian distance. Our aim is to extend previous results on existence and uniqueness of optimal transport maps to cases of sub-Riemannian structures which admit many singular minimizing geodesics. We treat here the case of sub-Riemannian structures of rank two in dimension four.     

Metric regularity of composition set-valued mappings: Metric setting and coderivative conditions

The paper concerns a new method to obtain a proof of the openness at linear rate/metric regularity of composite set-valued maps on metric spaces by the unification and refinement of several methods developed somehow separately in several works of the authors. In fact, this work is a synthesis and a precise specialization to a general situation of some techniques explored in the last years in the literature. In turn, these techniques are based on several important concepts (like error bounds, lower semicontinuous envelope of a set-valued map, local composition stability of multifunctions) and allow us to obtain two new proofs of a recent result having deep roots in the topic of regularity of mappings. Moreover, we make clear the idea that it is possible to use (co)derivative conditions as tools of proof for openness results in very general situations.     

Metric Inequality, Subdifferential Calculus and Applications

In this paper we establish characterizations of Asplund spaces in terms of conditions ensuring the metric inequality and intersection formulae. Then we establish chain rules for the limiting Fréchet subdifferentials. Necessary conditions for constrained optimization problems with non-Lipschitz data are derived.     

Metric fluctuations,entropy, and the Wheeler-deWitt equation

Quantization using quantum potentials is described in terms of momentum fluctuations, and it is related to Fisher information and entropy. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 32–44, July, 2007.