Generalizations of Euler Inequality in the n-dimensional Euclidean Space

The relation between the circum-radius and the in-radius of an n-dimensional simplex in E^n is studied.Two new generalizations of Euler inequality for the n-dimensional simplex are established.Besides,we obtain some stronger generalizations of Euler inequality for the n-dimensional simplex than previously known results.

ON THE REFLECTION OF IMMERSIONS IN EUCLIDEAN SPACES~*

We calculate the first obstruction to regular homotopy of an immersion f: M(n) --> R(n + k) into r-degrees-f for a reflection r of R(n + k). In the case of k = n - 1 we give necessary and sufficient conditions for f to be regularly homotopic to r-degrees-f in terms of obstructions to the existence of a normal vectorfield of f.

CANONICAL REPRESENTATIONS AND DEGREE OF FREEDOM FORMULAE OF ORTHOGONAL TENSORS IN N-DIMENSIONAL EUCLIDEAN SPACE

In this paper, with the help of the eigenvalue properties of orthogonal tensors in n-dimensional Euclidean space and the representations of the orthogonal tensors in 2-dimensional space, the canonical representations of orthogonal tensors in n-dimensional Euclidean space are easily gotten. The paper also gives all the constraint relationships among the principal invariants of arbitrarily given orthogonal tensor by use of Cayley-Hamilton theorem; these results make it possible to solve all the eigenvalues of any orthogonal tensor based on a quite reduced equation of m-th order, where m is the integer part ofn \2. Finally, the formulae of the degree of freedom of orthogonal tensors are given.

ON SPACELIKE AUSTERE SUBMANIFOLDS IN PSEUDO-EUCLIDEAN SPACE

In this article, we construct some spacelike austere submanifolds in pseduoEuclidean spaces. We also get some indefinite special Lagrangian submanifolds by constructing twisted normal bundle of spacelike austere submanifolds in pseduo-Euclidean spaces.

Description of the Nature Using the Models Developed in Euclidean Space

In elucidating the laws of matter motion, it is necessary also to take into account the subjective human possibilities to think and construct models. These possibilities are restricted to the framework of Euclidean space. No problems could arise during the development of the laws of classical science. However, it was established later on that in some areas it was rather difficult to describe the motion of the matter in terms of Euclidean models. In these cases, researchers either introduce a space of higher dimensionality, use complex numbers, or make some deformations of our habitual Euclidean space. Those were exactly the cases for which the pseudo-Euclidean, Hilbertian, reciprocal, micro-Euclidean and other spaces were proposed. Humans are able to think only in terms of Euclidean space. So, to provide a correct description of unusual motion of matter, the necessity arises to transform the information into the understandable Euclidean space. The operators suitable for these purposes are Lorentz transformations, Schrodinger equation, the integral transformations of Fourier and Weierstrass, etc. The features of information transformations between different spaces are illustrated with the examples from the areas of X-ray structural analysis and quantum physics.

An Addressing and Routing Scheme Based on Modified Euclidean Space for Hexagonal Networks

The addressing and routing algorithm on hexagonal networks is still an open problem so far.Although many related works have been done to resolve this problem to some extent,the properties of hexagonal networks are still not explored adequately.In this paper,we first create an oblique coordinate system and redefine the Euclidean space to address the hexagonal nodes.Then an optimal routing algorithm using vectors and angles of the redefined Euclidean space is developed.Compared with the traditional 3-directions scheme and the Cayley graph method,the proposed routing algorithm is more efficient and totally independent of the scale of networks with two-tuples addresses.We also prove that the path(s) obtained by this algorithm is always the shortest one(s).

HARNACK EXTENSION FOR THE HENSTOCK INTEGRAL IN THE EUCLIDEAN SPACE

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Almost Injective Mappings of Totally Bounded Metric Spaces into Finite Dimensional Euclidean Spaces

Let χ= be a metric space and let ε be a positive real number. Then a function f: X→Y is defined to be an ε-map if and only if for all y∈Y, the diameter of f-1(y)?is at most ε. In Theorem 10 we will give a new proof for the following well known fact: if χ is totally bounded, then for all ε there exists a finite number n and a continuous ε-map fε: X→Rn (here Rn is the usual n-dimensional Euclidean space endowed with the Euclidean metric). If ε is “small”, then fε is “almost injective”;and still exists even if χ has infinite covering dimension (in this case, n depends on ε, of course). Contrary to the known proofs, our proof technique is effective in the sense, that it allows establishing estimations for n in terms of ε and structural properties of χ.

On Embeddings of Tori in Euclidean Spaces

Using the relation between the set of embeddings of tori into Euclideanspaces modulo ambient isotopies and the homotopy groups of Stiefel manifolds, we prove new resultson embeddings of tori into Euclidean spaces.

PL Embeddings of PL Manifolds into Some Euclidean Spaces

The main result in this paper is the following:Theorem. Assume that W is a k-connected compact PL n-manifold with boundary, BdW is,（k-Ⅰ）-connect,k≥Ⅰ. （BdW is Ⅰ-connected for k=Ⅰ）, 0≤h≤2k, 2n-h>5 and there exists a normal block（n-h-Ⅰ）-bundle v over W. then（1） There is neat PL embedding W→D2n-hwhich normal block bundle is isomorphic to v（?） ε.（2） There is a PL embedding W→S2n-h-1which normal block bundle is isomorphic to v. Where ε denotes the trivial block l-boundle, D2n-h={x=(x1, x2,…, x2n-h∈R2n-h||xi|≤Ⅰ} and S2n-h-Ⅰ=BdD2n-h.

Euclidean Model of Space and Time

The aim of this work is to show that the currently widely accepted geometrical model of space and time based on the works of Einstein and Minkowski is not unique. The work presents an alternative geometrical model of space and time, a model which, unlike the current one, is based solely on Euclidean geometry. In the new model, the pseudo-Euclidean spacetime is replaced with a specific subset of four-dimensional Euclidean space. The work shows that four-dimensional Euclidean space allows explanation of known relativistic effects that are now explained in pseudo-Euclidean spacetime by Einstein’s Special Theory of Relativity (STR). It also shows simple geometric-kinematical nature of known relativistic phenomena and among others explains why we cannot travel backward in time. The new solution is named the Euclidean Model of Space and Time (EMST).

Reality and Differential Calculus on the q-Euclidean Space

By introducing the left and right derivatives, we establish a real structure for the covariant differential calculus on the N-dimensional quantum Euclidean space. The method is also applicable to the quantum Minkowski space.

The Realization of Elementary Configurations in Euclidean Space

By "an elementary configuration" we mean a set of a finite number of points,oriented byperplanes and oriented hyperspheres.In this paper,a complete solution of the following problems is given.Does there exist in Euclidean space a certain elementary configuration with a prescribed metric for cach pair of its elements? If so.how can one find the coordinate representations of the elements of such a configuration?

Elastic line deformed on a pseudo-hypersurface by an external field in pseudo-Euclidean spaces

We derive intrinsic formulation for elastic line deformed on a pseudo-hypersurface by an external field in the pseudo-Euclidean spaces E_v^n.This formulation determines elastic line deformed on a pseudo-hypersurface.

A Euclidean-Like Discrete Spacetime from the Unification of a Multitude of Hypercubic Lattices

An autonomous discrete space is proposed consisting of a huge number of four dimensional hypercubic lattices, unified along one of the four axes. The unification is such that the properties of the individual lattice are preserved. All the unifying axes are parallel, and the other axes have indeterminate mutual relations. The two kinds of axes are non-interchangeable resembling time and space of reality. The unification constitutes a framework without spatial properties. In case the axes with indeterminate relations are present at regular intervals in the time and the space, a Euclidean-like metric and goniometry can be obtained. In thus defined space-like structure, differences in speed and relativistic relations are only possible within regions of space enclosed by aberrations of the structure.

应用双曲空间特征融合的姓名消歧方法研究

针对传统用户影响力分析等研究遇到姓名重名的挑战,姓名歧义的影响日益严重的问题,本文基于双曲空间结合欧氏空间进行特征融合,提出融合多空间特征的网络对齐方法(geometry interaction network alignment,GINA),有效建模网络结构对用户姓名消歧的主要作用。本文同时使用欧氏空间和双曲空间进行网络表示学习,以获取具有不同空间特点的网络结构信息,使用跨空间网络映射及跨空间特征融合在尽量减少空间映射损失的情况下实现不同空间的信息交互得到最终的网络表示,进行网络对齐,进而实现姓名消歧。本文在中文论文合作网络、英文论文合作网络和中文专利合作网络上,两两对齐构建论文-专利实证数据网络对齐数据集和中文-英文实证数据网络对齐数据集,开展GINA模型在网络对齐数据集上对重名人员身份识别和中外论文身份识别2个实证场景试验验证,双曲空间融合欧氏空间相比单一空间精确率增加了24.9%,验证了GINA方法的有效性。

Top-k空间偏好查询方法研究

随着无线通信技术的发展和智能移动终端的广泛普及,基于位置的服务已经融入了人们生活的方方面面。其中,Top-k空间偏好查询在地理信息系统、城市建设规划、资源调度与分配、旅游规划等领域具有重要的意义。Top-k空间偏好查询是根据空间对象周围的特征对空间对象进行等级评价,并返回具有最高等级评价的k个空间对象。目前,对于Top-k空间偏好查询的研究主要集中在欧式空间和路网环境,本文对欧式空间和路网环境下的Top-k空间偏好查询方法进行分析和比较。

Geometrization of Radial Particles in Non-Empty Space Complies with Tests of General Relativity

Curved space-time 4-interval of any probe particle does not contradict to flat non-empty 3-space which, in turn, assumes the global material overlap of elementary continuous particles or the nonlocal Universe with universal Euclidean geometry. Relativistic particle’s time is the chain function of particles speed and this time differs from the proper time of a motionless local observer. Equal passive and active relativistic energy-charges are employed to match the universal free fall and the Principle of Equivalence in non-empty (material) space, where continuous radial densities of elementary energy-charges obey local superpositions and mutual penetrations. The known planetary perihelion precession, the radar echo delay, and the gravitational light bending can be explained quantitatively by the singularity-free metric without departure from Euclidean spatial geometry. The flatspace precession of non-point orbiting gyroscopes is non- Newtonian one due to the Einstein dilation of local time within the Earth's radial energy-charge rather than due to unphysical warping of Euclidean space.

Geometrization of Radial Particles in Non-Empty Space Complies with Tests of General Relativity

Curved space-time 4-interval of any probe particle does not contradict to flat non-empty 3-space which, in turn, assumes the global material overlap of elementary continuous particles or the nonlocal Universe with universal Euclidean geometry. Relativistic particle’s time is the chain function of particles speed and this time differs from the proper time of a motionless local observer. Equal passive and active relativistic energy-charges are employed to match the universal free fall and the Principle of Equivalence in non-empty (material) space, where continuous radial densities of elementary energy-charges obey local superpositions and mutual penetrations. The known planetary perihelion precession, the radar echo delay, and the gravitational light bending can be explained quantitatively by the singularity-free metric without departure from Euclidean spatial geometry. The flatspace precession of non-point orbiting gyroscopes is non-New- tonian one due to the Einstein dilation of local time within the Earth’s radial energy-charge rather than due to unphysical warping of Euclidean space.

The Relativistic Geometry of the Complex Matter Space

Duality behavior of photons in wave-particle property has posed challenges and opportunities to discover other frontiers of fundamental particles leading to the relativistic and quantum description of matter. The speed of particles faster than the speed of light could not be recognized, and matter was always described as a real number. A new fundamental view on matter as a complex value has been introduced by many authors who present a paradigm that is shifted from real or pure imaginary particles to Complex Matter Space. A new assumption will be imposed that matter has two intrinsic components: i) mass, and ii) charge. The mass will be measured by real number systems and charged by an imaginary unit. The relativistic concept of Complex Matter Space on energy and momentum is investigated and we can conclude that the new Complex Matter Space (CMS) theory will help get one step closer to a better understanding toward: 1) Un-Euclidean description of Minkowski Geometry in the context of the Complex Matter Space, 2) transformation from Euclidean to Minkowski space and its relativistic interpretation. Finally, geometrical foundations are essential to have a real picture of space, matter, and the universe.