The Representation Theorem of onto Isometric Mappings between Two Unit Spheres of l^1(Г) Type Spaces and The Application to the Isometric Extension Problem

In this paper,we first derive the representation theorem of onto isometric mappings in theunit spheres of l~1(F) type spaces,and then conclude that such mappings can be extended to the wholespace as real linear isometrics by using a previous result of the author.

L(Ω,μ) CANNOT ISOMETRICALLY CONTAIN SOME THREE-DIMENSIONAL SUBSPACES OF AM-SPACES

This article presents a novel method to prove that： let E be an AM-space and if dim E ≥ 3, then there does not exist any odd subtractive.isometric mapping from the unit sphere S（E） into S[L（Ω, μ）]. In particular, there does not exist any real linear isometry from E into L（Ω, μ）.

The Isometric Extension of the Into Mapping from the Unit Sphere S_1(E) to S_1(l~∞(Г))

This paper considers the isometric extension problem concerning the mapping from the unitsphere S(E)of the normed space E into the unit sphere S(l~∞(Γ)).We find a condition under whichan isometry from S,(E)into S1(l~∞(Γ))can be linearly and isometrically extended to the whole space.Since l~∞(Γ)is universal with respect to isometry for normed spaces,isometric extension problemson a class of normed spaces are solved.More precisely,if E and F are two normed spaces,and ifV:S(E)→S(F)is a surjective isometry,where c(Γ)■(Γ),then Vcan be extended tobe an isometric operator defined on the whole space.

ISOMETRIES ON THE SPACE s

In this article, the author presents some results of the isometric linear extension from some spheres in the finite dimensional space s（n）. Moreover, the author presents the representation for the onto isometric mappings in the space s. It is obtained that if V is a surjective isometry from the space s onto s with V（0） = 0, then V must be real linear.

EXTENSION OF ISOMETRIES BETWEEN THE UNIT SPHERES OF COMPLEX lp(Γ)(p > 1) SPACES

In this paper, we study the extension of isometries between the unit spheres of complex Banach spaces lp（Γ） and lp（△）（p 〉1）. We first derive the representation of isometries between the unit spheres of complex Banach spaces lp（Γ） and lp（△）. Then we arrive at a conclusion that any surjective isometry between the unit spheres of complex Banach spaces lp（Γ）and lp（△） can be extended to be a linear isometry on the whole space.

The Isometric Extension of an Into Mapping from the Unit Sphere S(e_((2))~∞)to S(L^1(μ))

This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into-isometric mapping from the unit sphere S（t（2）^∞） to S（L^1（μ） can be （real） linearly isometrically extended.

Reaction Coordinates by Nonlinear Dimensionality Reduction

Deriving reaction coordinates for the characterization of chemical reactions has long been a demanding task.In our previous work[ACS Cent.Sci.3,407(2017)],the reaction coordinate of a(retro-)Claisen rearrangement in aqueous solution optimized through a Bayesian measure,a linear combination of bond lengths formation and breakage,was judged to be optimal among all trails.Here,considering the nonlinearity of the transition state,we use isometric mapping and locally linear embedding to obtain one reaction coordinate which is composed of a few collective variables.With these methods,we find a more reasonable and powerful one-dimensional reaction coordinate,which can well describe the reaction progression.To explore the reaction mechanism,we analyze the contribution of intrinsic molecular properties and the solventsolute interactions to the nonlinear reaction coordinate.Furthermore,another coordinate is identified to characterize the heterogeneity of reaction mechanisms.

A Note on Linear Extension of Isometries Between the Unit Spheres in β-normed Spaces

In this paper, we give four general results on linear extension of isometries between the unit spheres in β-normed spaces. These results improve the corresponding theorems in β-normed spaces.

On isometric extension problem between two unit spheres

In this paper we introduce the isometric extension problem of isometric mappings between two unit spheres. Some important results of the related problems are outlined and the recent progress is mentioned.

The isometric extension problem in the unit spheres of l^p(Г)(p >1) type spaces

In this paper, we first derive the representation theorem of onto isometric mappings in the unit spheres of l p (Г) (@#@(p >) 1,p ≠ 2 type spaces, and then we conclude that such mappings can be extended to the whole space as real linear isometries by using the previous result of the author.

A positioning method for maize seed laser-cutting slice using linear discriminant analysis based on isometric distance measurement

It is necessary that vision system should aid laser-cutting manipulator to position the specified part of each maize seed for getting the slice breeding genotype analysis with high throughput.Each of trivial maize seeds should be recognized and positioned in a certain posture.Correlation area ratio(CAR)is defined as the metric of pixel attribute.A large template of round mask is adopted for seed morphological detection to measure the CAR values.We get the feature points extracted from the seed image through the isometric mapping operation.Iterative processes of linear discriminant analysis search the morphological data space to learn non-linear transformations to the space where data are linearly separable.Linear discriminant analysis utilizes the data directional distribution to position the major axis and distinguish different parts of maize seed.The labeling partition operation is applied for picking out the scattered pieces to be finely clustered.Without denoising process,the feature region could be recognized with accuracies by the synthetical methods.Extensive experiments on a large amount of seeds demonstrate the effectiveness of proposed methods.

Extension of Isometries Between the Unit Spheres of Normed Space E and C(Ω)

In this paper, we study the extension of isometries between the unit spheres of normed space E and C（Ω）. We obtain that any surjective isometry between the unit spheres of normed space E and C（Ω） can be extended to be a linear isometry on the whole space E and give an affirmative answer to the corresponding Tingley＇s problem （where Ω be a compact metric space）.

On Extension of Isometries Between the Unit Spheres of Normed Space E and l^p(p>1)

In this paper, we study the extension of isometries between the unit spheres of normed space E and lP（p 〉 1）. We arrive at a conclusion that any surjective isometry between the unit sphere of normed space lP（p 〉 1） and E can be extended to be a linear isometry on the whole space lP（p 〉 1） under some condition.

Isometries on the Quasi-Banach Spaces L^p (0 < p <1)

We study the extension of isometries between the unit spheres of quasi-Banach spaces Lp for 0〈p〈1. We give some sufficient conditions such that an isometric mapping from the the unit sphere of Lp（μ） into that of another LP（ν） can be extended to be a linear isometry defined on the whole space.