The purpose of this paper is to describe string topology from the viewpoint of Chen’s iterated integrals. Let M be a compact closed oriented d-manifold and LM be the free loop space of M , the set of unbased smooth maps from S to M . Let H∗(LM) be the homology of the free loop space shifted by the dimension of the manifold i.e. H∗(LM) = H∗+d(LM). Chas and Sullivan found the product on H∗(LM) wh...

Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M . In [2] Chas and Sullivan defined a product on the homology H∗(LM) of degree −d. They then investigated other structure that this product induces, including a Lie algebra structure on H∗(LM), and an induced product on the S 1 equivariant homology, H S ∗ (LM). These algebraic structures, as well as oth...

the aim of this paper is to study the high order difference scheme for the solution of a fractional partial differential equation (pde) in the electroanalytical chemistry. the space fractional derivative is described in the riemann-liouville sense. in the proposed scheme we discretize the space derivative with a fourth-order compact scheme and use the grunwald- letnikov discretization of the ri...

the sequential $p$-convergence in a fuzzy metric space, in the sense of george and veeramani, was introduced by d. mihet as a weaker concept than convergence. here we introduce a stronger concept called $s$-convergence, and we characterize those fuzzy metric spaces in which convergent sequences are $s$-convergent. in such a case $m$ is called an $s$-fuzzy metric. if $(n_m,ast)$ is a fuzzy metri...

In this paper we introduce strong $I^K$-convergence of functions which is common generalization of strong $I^*$-convergence of functions in probabilistic metric spaces. We also define and study strong $I^{K}$-limit points of functions in same space.

where M is a closed, compact, oriented Riemannian manifold, LM is the space of continuous maps from the circle into M–the free loop space of M–and C∗(LM) is the complex of singular chains on LM . Inspired by an idea in [6], chain-level string topology operations were constructed in [5] using a subspace of our space of string diagrams. This subspace • has a different homotopy type from the space...

In this paper, we prove the Spanne-Guliyev type boundedness of the generalized fractional integral operator Iρ from the vanishing generalized local Morrey spaces V LM {x0} p,φ1 to V LM {x0} q,φ2 , 1 < p < q < ∞, and from the space V LM {x0} 1,φ1 to the weak space VWLM {x0} q,φ2 , 1 < q < ∞. We also prove the Adams-Guliyev type boundedness of the operator Iρ from the vanishing generalized Morrey...

This paper proposes a new learning method for process neural networks (PNNs) based on the Gaussian mixture functions and particle swarm optimization (PSO), called PSO-LM. First, the weight functions of the PNNs are specified as the generalized Gaussian mixture functions (GGMFs). Second, a PSO algorithm is used to optimize the parameters, such as the order of GGMFs, the number of hidden neurons,...

We develop a theory of stratified LM -filters which generalizes the theory of stratified L-filters. Our stratification condition explicitly depends on a suitable mapping between the lattices L and M . If L and M are identical and the mapping is the identity mapping, then we obtain the theory of stratified L-filters. Based on the stratified LM -filters, a general theory of lattice-valued converg...

We apply Preuss' concept of $mbbe$-connectedness to the categories of lattice-valued uniform convergence spaces and of lattice-valued uniform spaces. A space is uniformly $mbbe$-connected if the only uniformly continuous mappings from the space to a space in the class $mbbe$ are the constant mappings. We develop the basic theory for $mbbe$-connected sets, including the product theorem. Furtherm...