ON LINEAR EXTENSION OF 1-LIPSCHITZ MAPPING FROM HILBERT SPACE INTO A NORMED SPACE

In this article, we prove that an into 1-Lipschitz mapping from the unit sphere of a Hilbert space to the unit sphere of an arbitrary normed space, which under some conditions, can be extended to be a linear isometry on the whole space.

One Roper-Suffridge Extension Operator in Complex Banach Spaces

In this paper, by the definition of almost spirallike mappings of type β and order α and the geometric property of the spirallike mapping of type β, we prove that the generalized Roper-Suffridge extension operator preserves almost spirallikeness of type β and order α in complex Banach spaces. Key words：

Loewner Chains Associated with the Generalized Roper-Suffridge Extension Operator on the Unit Ball in Hilbert Spaces

In this paper, we prove that the generalized Roper-Suffridge extension operator can be embeded in Loewner chains on the unit ball in Hibert spaces, and obtain the fact that the operator keeps the properties of almost spirallike mapping of typeβ and order α, almost starlikeness of order α, spirallikeness of type ofβ and starlikeness.

Expanders, Group Extensions, Hadamard Manifolds and Certain Banach Spaces

In this note,we prove that expanders cannot be coarsely embedded into group extensions of sequences of groups which are coarsely embeddable into Hardamad manifolds and certain Banach spaces due to the similar concentration theorems.

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.

Analysis on Micro Creative Space Development from the Perspective of New Urbanization View Field——Taking Micro Creative Garden of Gongshu District as an Example

With the rapid developmem of new urbanization, a number of micro creative space has emerged in China. The micro creative space of Gongshu District in Hangzhou has experienced a rapid and mixed development. Considering the existing problems of those space, the following measures can help those space realize an organic urban growth： arranging the layout comprehensively, optimizing industrial agglomeration, improving the planning management quality and optimizing public platform serx, ice support. The good development of micro creative garden can promote the full flow of creative element, so as to accelerate city economic structure transformation, increase city space efficiency and promote the city innovation development.

SIR phasing by combination of SOLVE/RESOLVE and dual-space fragment extension involving OASIS

A new phasing procedure has been proposed for dealing with single isomorphous replacement （SIR） x-ray diffraction data. The procedure combines SOLVE/RESOLVE with the dual-space fragment extension involving OASIS. Two sets of SIR data at 0.28 nm resolution taken from the protein （R）-phycoerythrin （PDB code： 1LIA） were used in the test. For one of the two SIR data sets, a default run of SOLVE/RESOLVE based on the heavy-atom substructure found by SHLEXD led automatically to an interpretable electron density map. OASIS could not effectively improve the result. For the other set of SIR data, SOLVE/RESOLVE resulted in a fragmented model consisting of 454 of the total 668 residues, in which only 29 residues were docked into the sequence. Based on this model, 7 iteration cycles of OASIS-DM- RESOLVE （build only） yielded automatically a model of 547 residues with 133 residues docked into the sequence. The overall-averaged phase error decreased considerably and the quality of electron density map was improved significantly. Two more cycles of iterative OASIS-DM-RESOLVE were carried out, in which the output phases and figures of merit from DM were merged with that from the original run of SOLVE/RESOLVE before they were passed onto RESOLVE （build only）. This led automatically to a model containing 452 residues with 173 docked into the sequence. The resultant electron density map is manually traceable. It is concluded that when results of SOLVE/RESOLVE are not sufficiently satisfactory, the combination of SOLVE/RESOLVE and OASIS-DM-RESOLVE （build only） may significantly improve them.

HARNACK EXTENSION FOR THE HENSTOCK INTEGRAL IN THE EUCLIDEAN SPACE

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ON CONDITIONS FOR THE NON LOCALLY CONVEX LINEAR TOPOLOGICAL SPACE TO HAVE THE H．－B． EXTENSION PROPERTY

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Harmonic extension of Q-type space related to logarithmic functions

This paper studies a class of Q-type spaces Q^(m)_(log,λ)(R^(n))related to logarithmic functions.We first investigate some basic properties ofQ^(m)_(log,λ)(R^(n)).Further,by the aid of Poisson integral and harmonic function spaces H^(m)_(log,λ)(R_(+)^(n+1)),the harmonic extension of Q^(m)_(log,λ)(R^(n))and the boundary value problem of H^(m)_(log,λ)(R_(+)^(n+1))are obtained.

A Theorem for Generalized Extension of Operators and Some Applications

In this note it is shown that every bounded linear operator T∈B(Y, Z) defined on a closed subspace Y of a Banach space X admits a generalized extension T∈B(X, V). Some examples of the applications are given; especially, a character-ization of H.I. spaces is obtained.

Extension of isometries on the unit sphere of l^p (Γ) space

In this paper,we obtain that every isometry from the unit sphere S(l p (Γ)) of l p (Γ) (1 【 p 【 ∞,p≠2) onto the unit sphere S(E) of a Banach space E can be extended to be a (real) linear isometry of l p (Γ) onto E,so,we give an affirmative answer to the corresponding Tingley’s problem.

Polynomials,Higher Order Sobolev Extension Theorems and Interpolation Inequalities on Weighted Folland-Stein Spaces on Stratified Groups

This paper consists of three main parts.One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups.Despite the extensive research after Jerison’s work[3]on Poincaré-type inequalities for Hrmander’s vector fields over the years,our results given here even in the nonweighted case appear to be new.Such interpolation inequalities have crucial applications to subelliptic or parabolic PDE’s involving vector fields.The main tools to prove such inequalities are approximating the Sobolev func- tions by polynomials associated with the left invariant vector fields on G.Some very useful properties for polynomials associated with the functions are given here and they appear to have independent interests in their own rights.Finding the existence of such polynomials is the second main part of this paper.Main results of these two parts have been announced in the author’s paper in Mathematical Research Letters[38]. The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on(εδ)domains. Some results of weighted Sobolev spaces are also given here.We construct a linear extension operator which is bounded on different Sobolev spaces simultaneously.In particular,we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev functions.Theorems are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.

Leakage Is Prohibited:Memory Protection Extensions Protected Address Space Randomization

Code reuse attacks pose a severe threat to modern applications. These attacks reuse existing code segments of vulnerable applications as attack payloads and hijack the control flow of a victim application. With high code entropy and a relatively low performance overhead, Address Space Layout Randomization(ASLR) has become the most widely explored defense against code reuse attacks. However, a single memory disclosure vulnerability is able to compromise this defense. In this paper, we present Memory Protection Extensions(MPX)-assisted Address Space Layout Randomization(M-ASLR), a novel code-space randomization scheme. M-ASLR uses several characteristics of Intel MPX to restrict code pointers in memory. We have developed a fully functioning prototype of M-ALSR, and our evaluation results show that M-ASLR:(1) offers no interference with normal operation;(2) protects against buffer overflow attacks, code reuse attacks, and other sophisticated modern attacks;and(3) adds a very low performance overhead(3.3%) to C/C++ applications.

The isometrical extensions of 1-Lipschitz mappings on Gteaux differentiability spaces

Let X and Y be real Banach spaces.Suppose that the subset sm[S1(X)] of the smooth points of the unit sphere [S1(X)] is dense in S1(X).If T0 is a surjective 1-Lipschitz mapping between two unit spheres,then,under some condition,T0 can be extended to a linear isometry on the whole space.

The isometric extension of “into” mappings on unit spheres of AL-spaces

In this paper, we show that if V0 is an isometric mapping from the unit sphere of an AL-space onto the unit sphere of a Banach space E, then V0 can be extended to a linear isometry defined on the whole space.

On Linearly Isometric Extensions for 1-Lipschitz Mappings Between Unit Spheres of ALP-spaces (p>2)

In this paper, we show that if Vo is a 1-Lipschitz mapping between unit spheres of two ALP-spaces with p 〉 2 and -Vo（S1（LP）） C Vo（S1（LP））, then V0 can be extended to a linear isometry defined on the whole space. If 1 〈 p 〈 2 and Vo is an ＂anti-l-Lipschitz＂ mapping, then Vo can also be linearly and isometrically extended.

THE REALIZATION OF MULTIPLIER HILBERT BIMODULE ON BIDUAL SPACE AND TIETZE EXTENSION THEOREM

The multiplier bimodule of Hilbert bimodule is introduced in a way similar to [1], and its realization on a quotient of bidual space and Tietze extension theorem are obtained similar to that in C-algebra case. As a result, the multiplier bimodule here is also a Hilbert bimodule.

On Extension of 1-Lipschitz Mappings between Two Unit Spheres of ~p(Γ) Type Spaces(1

Let T be a mapping from the unit sphere S[l^p（Г）] into S[l^p（△）] of two atomic AL^p- spaces. We prove that if T is a 1-Lipschitz mapping such that -T[S[l^p（Г）]] belong to T[S[l^p（Г）]], then T can be linearly isometrically extended to the whole space for p 〉 2; if T is injective and the inverse mapping T^-1 is a 1-Lipschitz mapping, then T can be extended to be a linear isometry from l^p（Г） into l^p（△） for 1 〈 p ≤ 2.

A Note on Linear Extension of Into-Isometries Between Two Unit Spheres of Atomic AL^p-Space （0 〈 p 〈∞）

In this paper, we shall present a short and simple proof on the isometric linear extension problem of into-isometries between two unit spheres of atomic abstract L^p-spaces （0 〈 p 〈 ∞）.