ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES

Let X and Y be two normed spaces.Let U be a non-principal ultrafilter on N.Let g:X→Y be a standard ε-phase isometry for someε≥ 0,i.e.,g(0)=0,and for all u.v ∈ X,||‖g(u)+g(v)‖±‖g(u)-g(v)‖|-|‖u+v‖±‖u-v‖| |≤ε.The mapping g is said to be a phase isometry provided that ε=0.In this paper,we show the following universal inequality of g:for each u^(*) ∈ w^(*)-exp ‖u^(*)‖B_(x^(*)),there exist a phase function σ_(u^(*)):X→{-1,1} and φ ∈ Y^(*) with ‖φ‖=‖u^(*)‖≡α satisfying that|(u^(*),u)-σ_(u^(*))(u)<φ,g(u)>)|≤5/2εα,for all u ∈ X.In particular,let X be a smooth Banach space.Then we show the following:(1) the universal inequality holds for all u^(*) ∈ X^(*);(2) the constant 5/2 can be reduced to 3/2 provided that Y~*is strictly convex;(3) the existence of such a g implies the existence of a phase isometryΘ:X→Y such that■ provided that Y^(**) has the w^(*)-Kadec-Klee property(for example,Y is both reflexive and locally uniformly convex).

ON ALMOST ISOMETRIES FROM L^1 (μ) INTO L~∞(v) OR C_b (△)

Let μ,ν be measures with diem L^1(μ) =dim L~∞(ν)=∞ and ⊿ be a 'subinterval' of the real line. Let E=L~∞(ν) or C_b(⊿), in this paper it turns out that the IAP for the space B(L^1(μ)→E) has a negative answer.

Boolean Automorphisms of a Hypercube Coincide with the Linear Isometries

Boolean homomorphisms of a hypercube, which correspond to the morphisms in the category of finite Boolean algebras, coincide with the linear isometries of the category of finite binary metric vector spaces.

Patterns and Technology -- A Creative Approach to Isometries

Today＇s math education basic syllabus gives an ever increasing emphasis to Geometry and Patterns. Curricula also advocate an approach that allows student to understand the concepts involved supported by dynamical computer tools. However, there is not much research work joining these paradigms. Therefore, a case study was developed, with 9th grade students （14-15 years old）, to evaluate the impact of a creative approach to isometries and symmetries-friezes, rosaceas （rosettes） and wallpapers-centered in patterns （reproduction, continuation, completion, description and creation） and using Geometer＇s Sketchpad software to solve, mainly in pairs, and discuss, with the whole class, the challenging tasks proposed, involving the formulation of hypothesis, argumentation and justification of the reasoning. The statistical analysis of the quantifiable data and content analysis of the qualitative data, collecting trough enquiry, direct observation and documental analysis （involving questionnaires, field notes, logbook, pre-tests and post-test, other works of the students including those computer related, and internal documents of the school） enable to conclude positively regarding the main research question underlying the study. In fact, it led to the conclusion that the teaching strategy implemented has contributed to deepen the student＇s knowledge and skills on geometry, mathematical communication and autonomy as well as to develop a closer relation with the field of geometry itself. This article focuses on one of the cases studied. The pair was selected due to be representative of most students and due to their communication skills.

ON THE ASYMPTOTIC BEHAVIOR OF HOLOMORPHIC ISOMETRIES OF THE POINCAR DISK INTO BOUNDED SYMMETRIC DOMAINS

In this article we bounded symmetric domains study holomorphic isometries of the Poincare disk into Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometrics up to normalizing constants with respect to the Bergman metric, showing in particular that the graph 170 of any germ of holomorphic isometry of the Poincar6 disk A into an irreducible bounded symmetric domain Ω belong to C^N in its Harish-Chandra realization must extend to an affinealgebraic subvariety V belong to C × C^N = C^N＋1, and that the irreducible component of V ∩ （△ × Ω） containing V0 is the graph of a proper holomorphic isometric embedding F ： A→ Ω. In this article we study holomorphie isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δ△. Starting with the structural equation for holomorphic isometrics arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form σ of the holomorphie isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ‖σ‖ must vanish at a general boundary point either to the order 1 or to the order 1/2, called a holomorphie isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes.

COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES

Assume that X and Y are real Banach spaces with the same finite dimension.In this paper we show that if a standard coarse isometry f:X→Y satisfies an integral convergence condition or weak stability on a basis,then there exists a surjective linear isometry U:X→Y such that∥f(x)−Ux∥=o(∥x∥)as∥x∥→∞.This is a generalization about the result of Lindenstrauss and Szankowski on the same finite dimensional Banach spaces without the assumption of surjectivity.As a consequence,we also obtain a stability result forε-isometries which was established by Dilworth.

Invariant measures for planar piecewise isometries

In this paper,we discuss the invariant measures for planar piecewise isometries.It is shown that the Hausdorff measure restricted to an almost invariant set with respect to the Hausdorff measure is invariant.

Hyers-Ulam-Rassias stability of approximate isometries on restricted domains

Let X and Y be real Banach spaces. The stability of Hyers Ulam Rassias approximate isometries on restricted domains S (unbounded or bounded) for into mapping f: S→Y satisfying ‖ f(x)-f(y)‖-‖x-y‖≤ε(x,y) for all x,y∈S is studied in case that the target space Y is uniformly convex Banach space of the modulus of convexity of power type q ≥2 or Y is the L q(Ω,,μ) (1<q <+∞) space or Y is a Hilbert space. Furthermore, the stability of approximate isometries for the case that (x,y)=‖x‖ p+‖y‖ p or (x,y)=‖x-y‖ p for p ≠1 is investigated.

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.

Boundary Regularity of Isometries Between Infinitely Flat Complex Domains

In this paper we prove that isometries with respect to the Kobayashi metric between certain domains having boundary points at which the boundary is infinitely flat extend continuously to the boundary.The strategy is to reestablish the Gehring-Hayman-type Theorem for these complex domains.Furthermore,the regularity of boundary extension map is given.

A solution to Tingley's problem for isometries between the unit spheres of compact C~*-algebras and JB~*-triples

Let f : S(E) → S(B) be a surjective isometry between the unit spheres of two weakly compact JB*-triples not containing direct summands of rank less than or equal to 3. Suppose E has rank greater than or equal to 5. Applying techniques developed in JB*-triple theory, we prove that f admits an extension to a surjective real linear isometry T : E → B. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C*-algebras A and B, without assuming any restriction on the rank of their direct summands(and in particular when A = K(H) and B = K(H′)), extends to a surjective real linear isometry from A into B. These results provide new examples of infinite-dimensional Banach spaces where Tingley's problem admits a positive answer.

The Dantzig Selector:Sparse Signals Recovery via l_(1-q)Minimization Model

We propose the Dantzig selector based on the l_(1-q)(1<q≤2)minimization model for the sparse signal recovery.First,we discuss some properties of l_(1-q)minimization model and give some useful inequalities.Then,we give a sufficient condition based on the restricted isometry property for the stable recovery of signals.The l_(1-2)minimization model of Yin-Lou-He is extended to the l_(1-q)minimization model.

On the Extension of Isometries between Unit Spheres of E and C（Ω）

In this paper, we study the extension of isometries between the unit spheresof some Banach spaces E and the spaces C(Ω). We obtain that if the set sm.S_1(E) of all smoothpoints of the unit sphere S_1(E) is dense in S_1(E), then under some condition, every surjectiveisometry V_0 from S_1(E) onto S_1(C(Ω)) can be extended to be a real linearly isometric map V of Eonto C(Ω). From this result we also obtain some corollaries. This is the first time we study thisproblem on different typical spaces, and the method of proof is also very different too.

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.

On Extension of Isometries in (F)-Spaces

An(F)-space E is said to be locally midpoint constricted (in short, Imp-constricted) if there exists some δ】0 such that D(A/2)【D(A) for every subset A of E with 0【D(A)≤δ, where D(A) denotes the diameter of A. Our main result goes as follow: Let E be an Imp-constricted (F)-space and U an open connected subset of E. Assume that T:U→F is an isometry (i.e., a distance-preserving map) which maps U onto an open subset of the (F)-space F. Then T can be extended to an affine homeomorphism from E to F. Also, some other results about the question whether each isometry between two (F)-spaces is affine are obtained.

Extension of Isometries on the Unit Sphere of L^p Spaces

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L^p（μ） （1 〈 p 〈∞, p ≠ 2） and a Banach space E can be extended to a linear isometry from L^p（μ） onto E, which means that if the unit sphere of E is （metrically） isometric to the unit sphere of L^P（μ）, then E is linearly isometric to L^p（μ）. We also prove that every surjective 1-Lipschitz or anti-l-Lipschitz map between the unit spheres of L^p（μ1, H1） and L^p（μ2, H2） must be an isometry and can be extended to a linear isometry from L^p（μ2, H2） to L^p（μ2, H2）, where H1 and H2 are Hilbert spaces.

Isometries in hyperbolic spaces

Suppose that f:Hn → Hn (n≥2) maps any r-dimensional hyperplane (1≤r【n) into an r-dimensional hyperplane. In this paper, we prove that f is an isometry if and only if f is a surjective map. This result gives an affirmative answer to a recent conjecture due to Li and Yao.

Second fundamental forms of holomorphic isometries of the Poincar disk into bounded symmetric domains and their boundary behavior along the unit circle

Motivated by problems arising from Arithmetic Geometry,in an earlier article one of the authors studied germs of holomorphic isometries between bounded domains with respect to the Bergman metric.In the case of a germ of holomorphic isometry f:(Δ,λ ds 2Δ ;0) → (Ω,ds 2Ω ;0) of the Poincar disk Δ into a bounded symmetric domain Ω C N in its Harish-Chandra realization and equipped with the Bergman metric,f extends to a proper holomorphic isometric embedding F:(Δ,λ ds 2Δ) → (Ω,ds 2Ω) and Graph(f) extends to an affine-algebraic variety V C × C N.Examples of F which are not totally geodesic have been constructed.They arise primarily from the p-th root map ρ p:H → H p and a non-standard holomorphic embedding G from the upper half-plane to the Siegel upper half-plane H 3 of genus 3.In the current article on the one hand we examine second fundamental forms σ of these known examples,by computing explicitly σ 2.On the other hand we study on the theoretical side asymptotic properties of σ for arbitrary holomorphic isometries of the Poincar disk into polydisks.For such mappings expressing via the inverse Cayley transform in terms of the Euclidean coordinate τ=s + it on the upper half-plane H,we have φ(τ)=t 2 u(τ),where u t=0 ≡ 0.We show that u must satisfy the first order differential equation u t | t=0 ≡ 0 on the real axis outside a finite number of points at which u is singular.As a by-product of our method of proof we show that any non-standard holomorphic isometric embedding of the Poincar disk into the polydisk must develop singularities along the boundary circle.The equation φuφt | t=0 ≡ 0 along the real axis for holomorphic isometries into polydisks distinguishes the latter maps from holomorphic isometries into Siegel upper half-planes arising from G.Towards the end of the article we formulate characterization problems for holomorphic isometries suggested both by the theoretical and the computational results of the article.

Isometries for the modulus metric in higher dimensions are conformal mappings

for a proper subdomain D of R^(n) and for all x,y∈D defineμD(x,y)=infC_(xy)Cap(D,C_(xy)),where the infimum is taken over all curves Cxy=γ[0,1]in D withγ(0)=x andγ(1)=y,and Cap denotes the conformal capacity of condensers.The quantityμD is a metric if and only if the domain D has a boundary of positive conformal capacity.If Cap(∂D)>0,we callμD the modulus metric of D.Ferrand et al.(1991)have conjectured that isometries for the modulus metric are conformal mappings.Very recently,this conjecture has been proved for n=2 by Betsakos and Pouliasis(2019).In this paper,we prove that the conjecture is also true in higher dimensions n⩾3.

Partial Isometries and an Invariant of C＊-algebras

In this paper, we will discuss some properties of biprojection-commutative elements which are relevant to the classification of certain infinite C＊-algebras,, and define an important invariant s（A） of C＊-algebra A as well as give some basic properties with regard to s（A）. Moreover we prove that the invariant s（A） has continuity.