On $$Z$$ -Sets in the Space of Idempotent Probability Measures

Mathematical Notes - It is shown that the spaces of probability measures and of idempotent probability measures on the same metrizable compact space are homeomorphic. An example is given which...     

Fibration of Idempotent Measures

Ukrainian Mathematical Journal - We prove that the idempotent barycenter map restricted to the points with nontrivial fibers is a trivial fibration with Hilbert cube fibers whenever it is open.     

A Compact Space Is Homotopy Equivalent to a CW-Complex If and Only If It Is an Absolute Neighborhood h-Retract

It is proved that a compact space is homotopy equivalent to a CW-complex if and only if it is an absolute neighborhood h-retract.     

Generalized Control Systems in the Space of Probability Measures

In this paper we formulate a time-optimal control problem in the space of probability measures. The main motivation is to face situations in finite-dimensional control systems evolving deterministically where the initial position of the controlled particle is not exactly known, but can be expressed by a probability measure on \(\mathbb {R}^{d}\). We propose for this problem a generalized version of some concepts from classical control theory in finite dimensional systems (namely, target set, dynamic, minimum time function...) and formulate an Hamilton-Jacobi-Bellman equation in the space of probability measures solved by the generalized minimum time function, by extending a concept of approximate viscosity sub/superdifferential in the space of probability measures, originally introduced by Cardaliaguet-Quincampoix in Cardaliaguet and Quincampoix (Int. Game Theor. Rev. 10, 1–16, 2008). We prove also some representation results linking the classical concept to the corresponding generalized ones. The main tool used is a superposition principle, proved by Ambrosio, Gigli and Savaré in Ambrosio et al. [3], which provides a probabilistic representation of the solution of the continuity equation as a weighted superposition of absolutely continuous solutions of the characteristic system.     

Moore空间P~n(8)及其同伦群

令G是一Abel群,m≥2是一整数．一个型为(G,m)的Moore空间是一单连通的CW-复形X,使得■_i(X)=G(i=m),0(i≠m)．这里J_i为X的第i个整系数的约化同调群．众所周知,Moore空间存在,且任何两个型为(G,n)的Moore空间有相同伦型．取G=Z_k(模k的剩余类加群)．p~n(k)=S~(n-1)∪_(kl_(n-1))e~n为型为(Z_k,n-1)的Moore空间．特别地,考虑k=8,决定了Moore空间p~n(8)的一些同伦群．主要证明工具是Toda引进的复合工具-Toda积,Gray的关于从p~n(8)到n维球面S~n的pinchin映射的同伦纤维的胞腔结构,以及关于亚稳定相对同伦群π_k(X,A)的同伦切割定理,其中A为维数小于n-1的有限CW-复形,X=A∪e~n．     

Moore空间Pn(8)及其同伦群

令G是一Abel群,m≥2是一整数.一个型为(G,m)的Moore空间是一单连通的CW-复形X,使得Hi(X)=G(i=m),0(i≠m).这里Hi为X的第i个整系数的约化同调群.众所周知, Moore空间存在,且任何两个型为(G,n)的Moore空间有相同伦型.取G=Zk(模k的剩余类加群).Pn(k)=Sn-1∪kln-1en为型为(Zk,n-1)的Moore空间.特别地,考虑k=8,决定了Moore空间Pn(8)的一些同伦群.主要证明工具是Toda引进的复合工具-Toda积,Gray的关于从Pn(8)到n维球面Sn的pinchin映射的同伦纤维的胞腔结构,以及关于亚稳定相对同伦群π(X,A)的同伦切割定理,其中A为维数小于n-1的有限CW-复形,X=A∪en.     

概率逻辑伪度量空间及其性质

论证有限多个公式的概率分布与生成它的原子公式集的概率分布之间的关系,然后把计量逻辑学与概率逻辑学相结合,提出了概率真度、概率逻辑伪度量空间(F(n),ρP);指出当取均匀概率分布时,概率真度就转化为计量逻辑学中的真度,同时两公式间的概率逻辑伪距离Pρ就转化为计量逻辑学中的伪距离ρ.从而在有限理论中建立了一种更具一般性的概率逻辑伪度量空间理论。     

Optimisation of Measures on a Hyperfinite Adapted Probability Space

The minimisation problem for a functional is considered, where is an ℝ ⁿ -valued stochastic process, defined on some filtered probability space , and P is an admissible probability measure in the sense that it obeys (1) some uniform equivalence condition with respect to the given measure ℙ on Γ, and (2) a finite number (possibly zero) of arbitrarily given other conditions that require the expectation (with respect to P) of some continuous bounded function φ of , for t ₁,…,t _k ∈[0,1], to lie within some closed set. We assume that u can be formulated through finite compositions of conditional expectations and bounded continuous functions. Under the assumption of |φ| being uniformly bounded from below and some condition on the dimension of , the existence of a solution on hyperfinite adapted probability spaces, as well as its minimality among admissible measures on any other adapted probability space, is proven. Also, a coarseness result for the Loeb operation is established. The main result of this paper, however, is a “standard result”: It does not include any reference to nonstandard analysis and can be perfectly understood without any familiarity with nonstandard analysis.      

Chaotic Properties of Mappings on a Probability Space

Sensitive dependence on initial conditions is widely understood as being the central idea of chaos. We first give sufficient conditions (both topological and ergodic) on an endomorphism to ensure the sensitivity property. Then, a strong sensitivity concept is introduced. Sufficient conditions on a transformation implying strong sensitivity are given. We also provide bounds for the strong sensitivity constant.     

On Weak Convergence of Probability Measures in a Banach Space

Necessary and sufficient conditions are obtained for weak convergence of Borel probability distributions in a separable strictly normed Banach space with a continuous modulus of convexity. These conditions are given in terms of simultaneous convergence of the distribution functions of the norm and of all linear continuous functionals with respect to the sequence of distributions under consideration. Bibliography: 9 titles.     

三幂等矩阵的一些性质

重点探索了三幂等矩阵的性质.主要从矩阵乘积、线性变换和矩阵的秩等角度出发,将幂等矩阵的性质向三幂等矩阵推广,对三幂等矩阵的性质进行探究,得到了15个相关结论,并给出部分性质的详细推导过程.     

向量值鞅空间pHST(X)与pHσr(X)的共轭

证明了B值鞅空间,pH^Sr(X)和pH^σr(X)的共轭分别是qK^Sr′(X^*)与qK^σr′(X^*)，此外还讨论了pK^Sr(X)和pKr(X)，pK^σr(X)和pK^ r(X)的相互嵌入关系与Banach空间的P一致光滑性和q一致凸性之间的密切联系．     

（m，l）幂等矩阵的代数等价与正交的一些性质

应用数域上（m,l）幂等矩阵与m幂等矩阵的关系,得到了数域上（m,l）幂等矩阵的l次方幂的代数等价、相似和特征多项式相等是互为确定的结论,由此推广改进了数域上m幂等矩阵的代数等价与正交性的相应结果．     

幂等矩阵的性质及其推广

首先对幂等矩阵的简单性质进行了归纳总结,接着论证了幂等矩阵的等价条件及其特征值的取值范围,并讨论了幂等矩阵与实对称矩阵的关系、幂等矩阵与其伴随矩阵的特征值和特征向量的对应关系及幂等矩阵在群逆中的一个性质.最后讨论了幂等矩阵的两种分解形式.     

Absolute Continuity and Singularity of Two Probability Measures on a Filtered Space

Let μ and ν be fixed probability measures on a filtered space \((\varOmega, \mathcal{F}, \allowbreak(\mathcal{F}_{t} )_{t\in \mathbf{R}^{+} } )\). Denote by μ _T and ν _T (respectively, μ _T? and ν _T?) the restrictions of the measures μ and ν on \(\mathcal{F}_{T} \) (respectively, on \(\mathcal{F}_{T-} \)) for a stopping time T. We find the Hahn decomposition of μ _T and ν _T using the Hahn decomposition of the measures μ, ν and the Hellinger process h _t in the strict sense of order \(\frac{1}{2}\). The norm of the absolutely continuous component of μ _T? with respect to ν _T? is computed in terms of density processes and Hellinger integrals.     

Some Properties of the Idempotent Graph of a Ring

The idempotent graph of a ring R, denoted by I(R), is a graph whose vertices are all nontrivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = yx = 0. In this paper, we show that diam\({(I(M_n(D))) = 4}\), for all natural number \({n \geq 4}\) and diam\({(I(M_3(D))) = 5}\), where D is a division ring. We also provide some classes of rings whose idempotent graphs are connected. Moreover, the regularity, clique number and chromatic number of idempotent graphs are studied.     

A Homotopy 2-Groupoid of a Hausdorff Space

If X is a Hausdorff space we construct a 2-groupoid G ₂ X with the following properties. The underlying category of G ₂ X is the `path groupoid" of X whose objects are the points of X and whose morphisms are equivalence classes f, g of paths f, g in X under a relation of thin relative homotopy. The groupoid of 2-morphisms of G ₂ X is a quotient groupoid X / N X, where X is the groupoid whose objects are paths and whose morphisms are relative homotopy classes of homotopies between paths. N X is a normal subgroupoid of X determined by the thin relative homotopies. There is an isomorphism G ₂ X(f,f) ₂(X, f(0)) between the 2-endomorphism group of f and the second homotopy group of X based at the initial point of the path f. The 2-groupoids of function spaces yield a 2-groupoid enrichment of a (convenient) category of pointed spaces.We show how the 2-morphisms may be regarded as 2-tracks. We make precise how cubical diagrams inhabited by 2-tracks can be pasted.     

Space $$ \mathcal{L} $$( X) of local dynamical systems and the Filippov space Aceu( X)

In the present paper, we establish a relationship between continuous local dynamical systems and spaces of the class A _ceu(X) of the Filippov theory. We suggest a construction method for a space of the class A _ceu(X) on the basis of a locally given dynamical system and conversely, a dynamical system is constructed locally in a specific way on the basis of a given space of the class A _ceu(X). The suggested construction method provides a homeomorphism between the space of all local dynamical systems on a locally compact metric space X and the space A _ceu(X). The obtained results generalize the Filippov theory to locally dynamical systems.