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.

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.

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.

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.

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).

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.

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.

GLOBAL RIGIDITY THEOREMS FOR SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE

In this paper,we mainly study the global rigidity theorem of Riemannian submanifolds in space forms.Let Mn(n≥3)be a complete minimal submanifold in the unit sphere Sn+p(1).Forλ∈[0,n2−1/p),there is an explicit positive constant C(n,p,λ),depending only on n,p,λ,such that,if∫MSn/2dM<∞,∫M(S−λ)n/2+dM<C(n,p,λ),then Mn is a totally geodetic sphere,where S denotes the square of the second fundamental form of the submanifold and∫+=max{0,f}.Similar conclusions can be obtained for a complete submanifold with parallel mean curvature in the Euclidean space Rn+p.

Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators

Let a:=(a_(1),...,a_(n))2[1,∞)^(n),p∈(0,1),andα:=1/p-1.For any x∈R^(n)and t∈[0,∞),letΦ_(p)(x,t):={t/1+(t[x]_(a)^(ν))^(1-p)if να■N,t/1+(t[x]_(a)^(ν))^(1-p)[log(e+|x|a)]^(p)if να∈N,let where [·]a:=1+|·|a,|·|a denotes the anisotropic quasi-homogeneous norm with respect to a,and ν:=a_(1)+…+a_(n).Let H_(a)^(p)(R^(n)),L_(a)^(a)(R^(n)),and H_(a)^(Φ_(p))(R^(n))be,respectively,the anisotropic Hardy space,the anisotropic Campanato space,and the anisotropic Musielak-Orlicz Hardy space associated with Φ_(p) on R^(n).In this article,via first establishing the wavelet characterization of anisotropic Campanato spaces,we prove that for any f∈H_(a)^(p)(R^(n))and g∈L_(a)^(a)(R^(n)),the product of f and g can be decomposed into S(f,g)+T(f,g) in the sense of tempered distributions,where S is a bilinear operator bounded from H_(a)^(p)(R^(n))*L_(a)^(a)(R^(Φ_(p))) to L^(1)(R^(n)) and T is a bilinear operator bounded from H_(a)^(p)(R^(n))*L_(a)^(a)(R^(n)) to H_(a)^(Φ_(p))(R^(n)) .Moreover,this bilinear decomposition is sharp in the dual sense that any y■H_(a)^(Φ_(p))(R^(n)) that fits into the above bilinear decomposition should satisfy(L^(1)(R^(n))+y)*=(L^(1)(R^(n)+H_(a)^(Φ_(p))(R^(n))*.As applications,for any non-constant b∈L_(a)^(a)(R^(n)) and any sublinear operator T satisfying some mild bounded assumptions,we find the largest subspace of H_(a)^(p)(R^(n)),denoted by H_(a,b)^(p)(R^(n)),such that the commutator [b,T] is bounded from H_(a,b)^(p)(R^(n))to L^(1)(R^(n)).In addition,when T is an anisotropic CalderónZygmund operator,the boundedness of [b,T] from H_(a,b)^(p)(R^(n))to L^(1)(R^(n))(or to H_(a)^(1)(R^(n)) is also presented.The key of their proofs is the wavelet characterization of function spaces under consideration.

m-degree Center-connecting Line of Finite Points Set and Its Properties

In this paper, we first give the concept of m-degree center-connecting line in n-dimensional Euclidean space Enand investigate its several properties, then we obtain the length of m-degree center-connecting line formula in finite points set. As its application,we extend the Leibniz formula and length of medians formula in n-dimensional simplex to polytope.

Inequalities for pedal simplices

In this paper,we present inequalities for volumes of subsimplices of a simplex and its pedal simplex and generalize them to m ＋ 1 simplices and their pedal simplices.

Some Geometric Inequalities on the Radii of Inscrib ed Sphere of a Simplex and Its Subsimplex

In this paper, we obtain some geometric inequalities on the radii of inscribed sphere of a simplex and its subsimplex, as particular case of this paper, we obtain some main results of [1].

Preliminary Exploration on Properties of the Equidistant Conjugate Points of Higher Dimensional Simplex

In this paper, the concept of the equidistant conjugate points of a triangle to the n-dimensional Euclidean space is extended. The concept of equidistant conjugate point in high dimensional simplex is defined, and the property of the equidistant conjugate points of a triangle is generalized to high dimensional simplex.

A Note on Minimal Surfaces in Euclidean 3-Space

In this note, a construction of minimal surfaces in Euclidean 3-space is given. By using the product of Weierstrass data of two known minimal surfaces, one gets a new Weierstrass data and a corresponding minimal surface from the Weierstrass representation.

ON THE RECOVERY OF A CURVE ISOMETRICALLY IMMERSED IN E^n

It is known from classical differential geometry that one can reconstruct a curve with (n - 1) prescribed curvature functions, if these functions can be differentiated a certain number of times in the usual sense and if the first (n - 2) functions are strictly positive. It is established here that this result still holds under the assumption that the curvature functions belong to some Sobolev spaces, by using the notion of derivative in the distributional sense. It is also shown that the mapping which associates with such prescribed curvature functions the reconstructed curve is of class C∞.