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In order to explore a mobile virtual reality railway traffic congestion prediction algorithm based on convolutional neural network, an expanded causal convolution neural network (DCFCN) was proposed, which introduced the expanded convolution to increase the size of the receptive field and obtain the long-term memory of the sequence. At the same time, causal convolution is introduced to solve the problem of information leakage. DCFCN is made up of 6 convolutional layers, each layer achieves causal convolution through padding, and the expansion coefficient increases exponentially layer by layer. Experimental results show that LSTM and GRU can obtain the time sequence relationship of mobile virtual reality traffic flow sequence, and the prediction effect is better than simple method and traditional ARIMA model, but still inferior to DCFCN. The RMSE of DCFCN decreased by 0.38 compared with single-layer LSTM, 0.52 compared with double-layer LSTM, and 0.38 compared with single-layer and double-layer GRU. It shows that TCN model can indeed do better than RNN in sequence modeling. It is proved that the proposed DCFCN is superior to other comparison models in mobile virtual reality traffic flow prediction, and the computational efficiency on GPU is significantly improved.

In this paper, we provide partial solutions to a problem raised by Vaisala on local properties of free quasiconformal mappings. In particular, we show that a locally free quasiconformal mapping is a globally free quasiconformal mapping under the condition of locally relative quasisymmetry.

The main aim of this paper is to investigate the invariant properties of uniform domains under flattening and sphericalization in nonlocally compact complete metric spaces. Moreover, we show that quasi-Möbius maps preserve uniform domains in nonlocally compact spaces as well.

The aim of this paper is to investigate the equivalence conditions for uniform perfectness of quasimetric spaces. We also obtain the invariance property of uniform perfectness under quasim?bius mappings in quasimetric spaces. In the end, two applications are given.

By introducing new deformations on symbolic Cantor sets and ultrametric spaces, we prove that doubling ultrametric spaces admit bilipschitz embedding into Cantor sets. If in addition the spaces are uniformly perfect, we show that they are quasisymmetrically equivalent to Cantor sets. As an application, we provide a new proof for a recent work of Heer (Anal. Geom. Metr. Spaces, 2017) regarding quasim\"obius uniformization of Cantor set., Comment: 8 pages, BMMS revised

In this paper, we introduce a class of mappings related to the distance ratio metric and study its connection to the freely quasiconformal mapping in Banach spaces. Moreover, we investigate the invariance of Gromov hyperbolic, natural and uniform domains under this class of mappings.

The main purpose of this paper is to investigate the properties of a mapping which is required to be roughly bilipschitz with respect to the Apollonian metric (roughly Apollonian bilipschit...

The main purpose of this paper is to study the characterizations of John spaces. We obtain five equivalent characterizations for length John spaces. As an application, we establish a dimension-free quasisymmetric invariance of length John spaces. This result is new also in the case of the Euclidean space.

We show that every quasihyperbolic geodesic in a John space admitting a roughly starlike Gromov hyperbolic quasihyperbolization is a cone arc. This result provides a new approach to the elementary metric geometry question, formulated in \cite[Question 2]{Hei89}, which has been studied by Gehring, Hag, Martio and Heinonen. As an application, we obtain a simple geometric condition connecting uniformity of the space with the existence of Gromov hyperbolic quasihyperbolization.

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that $G\subset X$ and $G'\subset Y$ are domains, and that $f: G\to G'$ is a homeomorphism. In this paper, we first give some basic properties of short arcs, and then we show that: if $f$ is a weakly quasisymmetric mapping and $G'$ is a quasiconvex domain, then the image $f(D)$ of every uniform subdomain $D$ in $G$ is uniform. As an application, we get that if $f$ is a weakly quasisymmetric mapping and $G'$ is an uniform domain, then the images of the short arcs in $G$ under $f$ are uniform arcs in the sense of diameter., Comment: Comput. Methods Funct. Theory;2018. arXiv admin note: substantial text overlap with arXiv:1501.07375

As some steel enterprises are at a disadvantage in the choice of the mode of transportation, this paper made further studies of the characteristics of the iron ore logistics, taking comprehensive consideration of optimizing the waiting time under the conditions with limited loading capacity and setting up a procedural model of the iron ore logistics system with minimum cost of transportation, storage, loading, unloading, and transportation equipment waiting. Finally, taking the iron ore transport system of one steel enterprise as example, the solution and the validity of the model were analyzed and verified in this paper.

The main aim of this paper is to give a complete solution to one of the open problems, raised by Heinonen from 1989, concerning the subinvariance of John domains under quasiconformal mappings in $${\mathbb {R}}^n$$ . As application, the quasisymmetry of quasiconformal mappings is discussed.

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain $$G \subset {\mathbb R}^n$$ . In the sequel, we investigate a class of domains, so called $$\varphi $$ -uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism $$\varphi $$ from $$[0,\infty )$$ to itself. Finally, we discuss a number of stability properties of $$\varphi $$ -uniform domains. In particular, we show that the class of $$\varphi $$ -uniform domains is stable in the sense that removal of a geometric sequence of points from a $$\varphi $$ -uniform domain yields a $$\varphi _1$$ -uniform domain.

Suppose that E and E′ denote real Banach spaces with dimension at least 2, that D $$ \subseteq $$ E and D′ $$ \subseteq $$ E′ are domains, that f: D → D′ is an (M,C)-CQH homeomorphism, and that D is uniform. The aim of this paper is to prove that D′ is a uniform domain if and only if f extends to a homeomorphism $$\overline f :\overline D \to {\overline D ^\prime }$$ and $$\overline f $$ is η-QM relative to ∂D. This result shows that the answer to one of the open problems raised by Vaisala from 1991 is affirmative.

Let E be real normed vector spaces with the dimension at least 2. In this paper we study the following questions: When is the union of two John domains in E a John domain and when is the union of two uniform domains in E a uniform domain? 1. Introduction and main results Throughout the paper, we always assume that E denotes a real normed vector space with dimE ‚ 2 and that D is a proper subdomain in E. The norm of a vector z in E is written as jzj, and for any two points z1;z2 in E, the distance between them is denoted by jz1 i z2j, and the closed line segment with endpoints z1 and z2 by (z1;z2). For x 2 E and r > 0, we let B(x;r) denote the open ball in E with center x and radius r. For real numbers r and s, we use the notation: r ^ s = minfr;sg. John domains in Euclidean spaces R n were introduced by John (1) in connection with his work on elasticity. The term is due to Martio and Sarvas (3). Roughly speaking, a domain is a John domain if it is possible to travel from one point of the domain to another without going too close to the boundary. The precise definition is as follows. Definition 1.1. D is called a c-John domain if for every pair of points x1;x2 2 D there is a rectifiable arc ∞ joining them with '(∞(x1;x)) ^ '(∞(x2;x)) • c d(x) for all x 2 ∞, where c is a positive constant, ∞(xj;x) denotes the closed subarc of ∞ with endpoints xj and x (j = 1;2), '(∞(xj;x)) the arclength of ∞(xj;x). ∞ is called a c-John arc joining x1 and x2. See (4) for several characterizations of John domains. In the study of John do

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least $2$ and that $D\subset E$ and $D'\subset E'$ are domains. In this paper, we establish, in terms of the $j_D$ metric, a necessary and sufficient condition for the homeomorphism $f: E \to E'$ to be FQC. Moreover, we give, in terms of the $j_D$ metric, a sufficient condition for the homeomorphism $f: D\to D'$ to be FQC. On the other hand, we show that this condition is not necessary., 10 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1110.6269

We study the stability of John domains in Banach spaces under removal of a countable set of points. In particular, we prove that the class of John domains is stable in the sense that removing a certain type of closed countable set from the domain yields a new domain which also is a John domain. We apply this result to prove the stability of the inner uniform domains. Finally, we consider a wider class of domains, so called $\psi$-John domains and prove a similar result for this class., Comment: 22pages

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\not=E$ and $D'\not=E'$ are bounded domains with connected boundaries, that $f: D\to D'$ is an $M$-QH homeomorphism, and that $D'$ is uniform. The main aim of this paper is to prove that $f$ extends to a homeomorphism $\bar \bar{D}\to \bar{D}'$ and $\bar{f}|\partial D$ is bilipschitz if and only if $f$ is bilipschitz in $\bar{D}$. The answer to some open problem of V\"ais\"al\"a is affirmative under an natural additional condition., Comment: arXiv admin note: substantial text overlap with arXiv:1105.4684

Let E be a real Banach space with dimension at least 2. In this paper, we prove that if D ⊂ E is a John domain which is homeomorphic to an inner uniform domain via a CQH map, then each neargeodesic in D is a cone arc.