On Convergence Approach Spaces

The extension of two axioms due to Cook and Fischer to the category CAP of convergence approach spaces leads to the study of non-Archimedean approach spaces as well as two versions of regularity appropriate to CAP and related categories.     

Approach Spaces, Limit Tower Spaces, and Probabilistic Convergence Spaces

The category LTS of limit tower spaces is defined and shown to be isomorphic to the category CAP of convergence approach spaces. The full subcategory of LTS determined by the objects satisfying a diagonal axiom due to Cook and Fischer is shown to be isomorphic to the category AP of approach spaces. A family of isomorphisms is also obtained between LTS and certain full subcategories of the category PCS of probabilistic convergence spaces.     

Complexity of Homogeneous Spaces and Growth of Multiplicities

The complexity of a homogeneous space G/H under a reductive group G is by definition the codimension of general orbits in G/H of a Borel subgroup B\subseteq G. We give a representation-theoretic interpretation of this number as the exponent of growth for multiplicities of simple G-modules in the spaces of sections of homogeneous line bundles on G/H. For this, we show that these multiplicities are bounded from above by the dimensions of certain Demazure modules. This estimate for multiplicities is uniform, i.e., it depends not on G/H, but only on its complexity.     

Min-Max Spaces and Complexity Reduction in Min-Max Expansions

Idempotent methods have been found to be extremely helpful in the numerical solution of certain classes of nonlinear control problems. In those methods, one uses the fact that the value function lies in the space of semiconvex functions (in the case of maximizing controllers), and approximates this value using a truncated max-plus basis expansion. In some classes, the value function is actually convex, and then one specifically approximates with suprema (i.e., max-plus sums) of affine functions. Note that the space of convex functions is a max-plus linear space, or moduloid. In extending those concepts to game problems, one finds a different function space, and different algebra, to be appropriate. Here we consider functions which may be represented using infima (i.e., min-max sums) of max-plus affine functions. It is natural to refer to the class of functions so represented as the min-max linear space (or moduloid) of max-plus hypo-convex functions. We examine this space, the associated notion of duality and min-max basis expansions. In using these methods for solution of control problems, and now games, a critical step is complexity-reduction. In particular, one needs to find reduced-complexity expansions which approximate the function as well as possible. We obtain a solution to this complexity-reduction problem in the case of min-max expansions.     

Kleisli Monoids Describing Approach Spaces

We study the functional ideal monad \(\mathbb {I} = (\mathsf {I}, m, e)\) on S e t and show that this monad is power-enriched. This leads us to the category \(\mathbb {I}\)- M o n of all \(\mathbb {I}\)-monoids with structure preserving maps. We show that this category is isomorphic to A p p, the category of approach spaces with contractions as morphisms. Through the concrete isomorphism, an \(\mathbb {I}\)-monoid (X,ν) corresponds to an approach space \((X, \mathfrak {A}),\) described in terms of its bounded local approach system. When I is extended to R e l using the Kleisli extension \(\check {\mathsf {I}},\) from the fact that \(\mathbb {I}\)- M o n and \((\mathbb {I},2)\)- C a t are isomorphic, we obtain the result that A p p can be isomorphically described in terms of convergence of functional ideals, based on the two axioms of relational algebras, reflexivity and transitivity. We compare these axioms to the ones put forward in Lowen (2015). Considering the submonad \(\mathbb {B}\) of all prime functional ideals, we show that it is both sup-dense and interpolating in \(\mathbb {I}\), from which we get that \((\mathbb {I},2)\)- C a t and \((\mathbb {B},2)\)- C a t are isomorphic. We present some simple axioms describing A p p in terms of prime functional ideal convergence.     

Convergence Approach Spaces : Actions

Properties of continuous actions and pseudoquotients are studied in the category of convergence approach spaces. Invariance properties of continuous actions on convergence approach spaces are given. It is shown that the formation of pseudoquotient spaces is idempotent. Function space actions are also investigated.     

An Elementary Approach to Exponential Spaces

In 1970, Day and Kelly characterized exponential spaces by a condition (C). Eight years later, Hofmann and Lawson pointed out that this is equivalent to quasi-local compactness, i.e. every neighborhood V of a point contains a smaller one W such that any open cover of V admits a finite subcover of W. These characterizations work with topologies on topologies and may be felt to be not really elementary. This note instead offers an elementary approach which applies to quotient-reflective subcategories as well and includes a natural generalization of the compact-open topology on function spaces.     

Functional Approach of Large Deviations in General Spaces

Let X be a topological space, ( ) a net of Borel probability measures on X, and (t) a net in ]0,[ converging to 0. Let be a set of continuous functions such that for all x X that can be suitably distinguished by some continuous functions from any closed set not containing contains such a distinguishing function. Assuming that exists for all , we give a sufficient condition in order that ( ₎ satisfies a large deviation principle with powers (t₎ and not necessary tight rate function. When X is completely regular (not necessary Hausdorff), this condition is also necessary, and so strictly weaker than exponential tightness; this allows us to strengthen Brycs theorem in various ways. We give the general form of a rate function in terms of . A Prohorov-type theorem with a weaker notion than exponential tightness is obtained, which improves known results.     

First-Order Complexity of Subgraph Isomorphism via Kneser Graphs

Mathematical Notes - The problem of finding the least number of variables of a first-order formula expressing the statement that an input-graph contains a subgraph isomorphic to a given...     

一种基于复杂性理论的装备作战仿真方法

采用什么样的计算机仿真方法来隐喻真实的作战系统是装备作战仿真研究的关键问题.从复杂性科学的研究角度,引入了复杂适应系统(CAS)理论及其技术体系,提炼了基于Agents/space的建模与仿真框架,说明了框架实现的关键技术——可计算模型、复杂性解决方案和仿真实现平台.进而利用该方法进行了典型装备作战仿真问题研究,包括:利用神经网络、三维连续空间可计算模型,并选用Mason平台实现了装甲装备战损规律仿真;利用三层元胞自动机、产生式系统可计算模型,并选用Repast平台实现了装备群对抗仿真.为基于复杂性理论开展装备作战仿真或具有类似特征问题的研究提供了一种新的试验途径.     

Complexity in Complex Analysis

We show that the classical kernel and domain functions associated to an n-connected domain in the plane are all given by rational combinations of three or fewer holomorphic functions of one complex variable. We characterize those domains for which the classical functions are given by rational combinations of only two or fewer functions of one complex variable. Such domains turn out to have the property that their classical domain functions all extend to be meromorphic functions on a compact Riemann surface, and this condition will be shown to be equivalent to the condition that an Ahlfors map and its derivative are algebraically dependent. We also show how many of these results can be generalized to finite Riemann surfaces.     

Characterizations of Function Spaces via Boundedness of Commutators

In this paper, we give some creative characterizations of Campanato spaces via the boundedness of commutators associated with the Calder ′on-Zygmund singular integral operator, fractional integrals and Hardy type operators. Furthermore, we put forward a few problems on the characterizations of Campanato type spaces via the boundedness of commutators.     

Tilting Theoretical Approach to Moduli Spaces Over Preprojective Algebras

We apply tilting theory over preprojective algebras Λ to the study of moduli spaces of Λ-modules. We define the categories of semistable modules and give equivalences, so-called reflection functors, between them by using tilting modules over Λ. Moreover we prove that the equivalence induces an isomorphism of K-schemes between moduli spaces. In particular, we study the case when the moduli spaces are related to Kleinian singularities, and generalize some results of Crawley-Boevey (Am J Math 122:1027–1037, 2000).     

Characterizations of the BMO and Lipschitz Spaces via Commutators on Weak Lebesgue and Morrey Spaces

Acta Mathematicae Applicatae Sinica, English Series - We prove that the weak Morrey space WM is contained in the Morrey space $$M_{{q_1}}^p$$ for 1 ≤ q1 < q ≤ p < ∞....     

Interpolation,the Rudimentary Geometry of Spaces of Lipschitz Functions,and Geometric Complexity

Foundations of Computational Mathematics - We consider seriously the analogy between interpolation of nonlinear functions and manifold learning from samples, and examine the results of transferring...     

Extension Theorems for Large Scale Spaces via Coarse Neighbourhoods

We introduce the notion of (hybrid) large scale normal space and prove coarse geometric analogues of Urysohn’s Lemma and the Tietze Extension Theorem for these spaces, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from which we obtain both the classical topological results and the (hybrid) large scale results as corollaries. We prove that all metric spaces are hybrid large scale normal, and characterize those locally compact abelian groups which (as hybrid large scale spaces) are hybrid large scale normal. Finally, we look at some properties of the Higson compactifications and coronas of hybrid large scale normal spaces.     

Fractal Characteristics of Measures; An Approach Via Function Spaces

Let be a positive Radon measure in R ⁿ with compact support . Let Q _jm be cubes with side-length 2^-j+1 originating from the canonical tiling of R ⁿ where j\in N ₀ and m\in Z ⁿ. If \in R, 0 < p \le , 0 < q \le , then _pq is the mixed _q – _p -quasi-norm of the sequence 2 ^j (Q _jm ). Quantities of this type are considered in fractal geometry (multifractal formalism) and in the theory of the function spaces B ^s _pq (R ⁿ) and F ^s _pq (R ⁿ). In Theorem 1 we deal with the question when _pq is an equivalent quasi-norm in some of these spaces (-property). If || = 0, then _S consists of those points (t,s) in the t–s-diagram in Figure 1 for which belongs to B ^s _p (R ⁿ) with pt = 1. Theorem 2 deals with the interrelation of S and _pq . Some applications to truncated Riesz potentials, Bessel potentials and Fourier transforms of are given.