Every N-Dimensional Separable Metric Space Contains a Totally Disconnected (n-1)-Dimensional Subset with no True Quasi-Components

Владимир Тодоров, Петър Стоев - Тази бележка съдържа елементарна конструкция на множество с указаните в заглавието свойства. Да отбележим в допълнение, че така полученото множество остава напълно несвързано дори и след като се допълни с краен брой елементи.

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.

Optimal trajectory planning for path convergence in three-dimensional space

This article addresses the problem of determining the shortest path that connects a given initial configuration (position, heading angle, and flight path angle) to a given rectilinear or a circular path in three-dimensional space for a constant speed and turn-rate constrained aerial vehicle. The final path is assumed to be located relatively far from the starting point. Due to its simplicity and low computational requirements the algorithm can be implemented on a fixed-wing type unmanned air vehicle in real time in missions where the final path may change dynamically. As wind has a very significant effect on the flight of small aerial vehicles, the method of optimal path planning is extended to meet the same objective in the presence of wind comparable to the speed of the aerial vehicles. But, if the path to be followed is closer to the initial point, an off-line method based on multiple shooting, in combination with a direct transcription technique, is used to obtain the optimal solution. Optimal paths are generated for a variety of cases to show the efficiency of the algorithm. Simulations are presented to demonstrate tracking results using a 6-degrees-of-freedom model of an unmanned air vehicle.

Iris recognition algorithm based on point covering of high-dimensional space and neural network

In this paper, we constructed a Iris recognition algorithm based on point covering of high-dimensional space and Multi-weighted neuron of point covering of high-dimensional space, and proposed a new method for iris recognition based on point covering theory of high-dimensional space. In this method, irises are trained as "cognition" one class by one class, and it doesn't influence the original recognition knowledge for samples of the new added class. The results of experiments show the rejection rate is 98.9%, the correct cognition rate and the error rate are 95.71% and 3.5% respectively. The experimental results demonstrate that the rejection rate of test samples excluded in the training samples class is very high. It proves the proposed method for iris recognition is effective.

Blind image restoration based on the theory of high-dimensional space geometry

In practical situations, the causes of image blurring are often undiscovered or difficult to get known. However, traditional methods usually assume the knowledge of the blur has been known prior to the restoring process, which are not practicable for blind image restoration. A new method proposed in this paper aims exactly at blind image restoration. The restoration process is transformed into a problem of point distribution analysis in high-dimensional space. Experiments have proved that the restoration could be achieved using this method without re-knowledge of the image blur. In addition, the algorithm guarantees to be convergent and has simple computation.

High-Dimensional Space Geometrical Informatics and its applications to image restoration

With a view to solve the problems in modern information science, we put forward a new subject named High-Dimensional Space Geometrical Informatics (HDSGI). It builds a bridge between information science and point distribution analysis in high-dimensional space. A good many experimental results certified the correctness and availability of the theory of HDSGI. The proposed method for image restoration is an instance of its application in signal processing. Using an iterative "further blurring-debluring-further blurring" algorithm, the deblured image could be obtained.

A novel image restoration algorithm based on high-dimensional space geometry

A novel image restoration approach based on high-dimensional space geometry is proposed, which is quite different from the existing traditional image restoration techniques. It is based on the homeomorphisms and "Principle of Homology Continuity" (PHC), an image is mapped to a point in high-dimensional space. Begin with the original blurred image, we get two further blurred images, then the restored image can be obtained through the regressive curve derived from the three points which are mapped form the images. Experiments have proved the availability of this "blurred-blurred-restored" algorithm, and the comparison with the classical Wiener Filter approach is presented in final.

A novel image restoration approach based on point location in high-dimensional space geometry

The goal of image restoration is to restore the original clear image from the existing blurred image without distortion as possible. A novel approach based on point location in high-dimensional space geometry method is proposed, which is quite different from the thought ways of existing traditional image restoration approaches. It is based on the high-dimensional space geometry method, which derives from the fact of the Principle of Homology-Continuity (PHC). Begin with the original blurred image, we get two further blurred images. Through the regressive deducing curve fitted by these three images, the first iterative deblured image could be obtained. This iterative "blurring-debluring-blurring" process is performed till reach the deblured image. Experiments have proved the availability of the proposed approach and achieved not only common image restoration but also blind image restoration which represents the majority of real problems.

A face detection algorithm based on high dimensional space geometry

In this paper, a face detection algorithm which is based on high dimensional space geometry has been proposed. Then after the simulation experiment of Euclidean Distance and the introduced algorithm, it was theoretically analyzed and discussed that the proposed algorithm has apparently advantage over the Euclidean Distance. Furthermore, in our experiments in color images, the proposed algorithm even gives more surprises.

High-Dimensional Space Geometrical Informatics and its applications to image restoration

With a view to solve the problems in modern information science, we put forward a new subject named High-Dimensional Space Geometrical Informatics (HDSGI). It builds a bridge between information science and point distribution analysis in high-dimensional space. A good many experimental results certified the correctness and availability of the theory of HDSGI. The proposed method for image restoration is an instance of its application in signal processing. Using an iterative "further blurring-debluring-further blurring" algorithm, the deblured image could be obtained.

A novel image restoration algorithm based on high-dimensional space geometry

A novel image restoration approach based on high-dimensional space geometry is proposed, which is quite different from the existing traditional image restoration techniques. It is based on the homeomorphisms and "Principle of Homology Continuity" (PHC), an image is mapped to a point in high-dimensional space. Begin with the original blurred image, we get two further blurred images, then the restored image can be obtained through the regressive curve derived from the three points which are mapped form the images. Experiments have proved the availability of this "blurred-blurred-restored" algorithm, and the comparison with the classical Wiener Filter approach is presented in final.

A Novel Geometric Algorithm for Blind Image Restoration Based on High-Dimensional Space

A novel geometric algorithm for blind image restoration is proposed in this paper, based on High-Dimensional Space Geometrical Informatics (HDSGI) theory. In this algorithm every image is considered as a point, and the location relationship of the points in high-dimensional space, i.e. the intrinsic relationship of images is analyzed. Then geometric technique of "blurring-blurring-deblurring" is adopted to get the deblurring images. Comparing with other existing algorithms like Wiener filter, super resolution image restoration etc., the experimental results show that the proposed algorithm could not only obtain better details of images but also reduces the computational complexity with less computing time. The novel algorithm probably shows a new direction for blind image restoration with promising perspective of applications.