In this paper,we study isometries and phase-isometries of non-Archimedean normed spaces.We show that every isometry f:Sr(X)→Sr(X),where X is a finite-dimensional non-Archimedean normed space and Sr(X)is a sphere with...In this paper,we study isometries and phase-isometries of non-Archimedean normed spaces.We show that every isometry f:Sr(X)→Sr(X),where X is a finite-dimensional non-Archimedean normed space and Sr(X)is a sphere with radius r∈||X||,is surjective if and only if is spherically complete and k is finite.Moreover,we prove that if X and Y are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with|2|=1,any phase-isometry f:X→Y is phase equivalent to an isometric operator.展开更多
In this paper the quasi-constant curvature space and the Riemannian manifold contained the totally umbilical hypersurface family are studied,and two theorems are given at the same time.
Let X, Y be two real Banach spaces and ε≥0. A map f ： X → Y is said to be a standard ε-isometry if│││f/（x） - f（y）││ - ]ix - Y││x-y││ ε for all x,y C X and with f（O） = O. We say that a pair of Ban...Let X, Y be two real Banach spaces and ε≥0. A map f ： X → Y is said to be a standard ε-isometry if│││f/（x） - f（y）││ - ]ix - Y││x-y││ ε for all x,y C X and with f（O） = O. We say that a pair of Banach spaces （X, Y） is stable if there exists γ〉 0 such that, for every such ε and every standard v-isometry f ： X → Y, there is a bounded linear operator T ： L（f） → f（X） → X so that ││Tf（x） - x││ ≤γε for all x E X. X（Y） is said to be universally left-stable if （X, Y） is always stable for every Y（X）. In this paper, we show that if a dual Banach space X is universally left-stable, then it is isometric to a complemented w＊-closed subspace of ∞ （1） for some set F, hence, an injective space; and that a Banach space is universally left-stable if and only if it is a cardinality injective space; and universally left-stability spaces are invariant.展开更多
This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 〈 p ≤ 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is ...This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 〈 p ≤ 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant （RIC）. Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to δ4Ar 〈 0.558 and δ3rA 〈 0.4721, respectively. Recently, the upper bounds of RIC can be improved to δ2rA 〈 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to δ2tA 〈 0.4931 and δrA 〈 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with δ2rA 〉1√2 or δrA 〉 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δ2rA and δrA.Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p （0 〈 p 〈 1） quasi norm minimization problem.展开更多
Orthogonal matching pursuit （OMP） algorithm is an efficient method for the recovery of a sparse signal in compressed sensing, due to its ease implementation and low complexity. In this paper, the robustness of the O...Orthogonal matching pursuit （OMP） algorithm is an efficient method for the recovery of a sparse signal in compressed sensing, due to its ease implementation and low complexity. In this paper, the robustness of the OMP algorithm under the restricted isometry property （RIP） is presented. It is shown that 5K＋V/KOK,1 〈 1 is sufficient for the OMP algorithm to recover exactly the support of arbitrary /（-sparse signal if its nonzero components are large enough for both 12 bounded and lz~ bounded noises.展开更多
Orthogonal multi-matching pursuit（OMMP）is a natural extension of orthogonal matching pursuit（OMP）in the sense that N（N≥1）indices are selected per iteration instead of 1.In this paper,the theoretical performance...Orthogonal multi-matching pursuit（OMMP）is a natural extension of orthogonal matching pursuit（OMP）in the sense that N（N≥1）indices are selected per iteration instead of 1.In this paper,the theoretical performance of OMMP under the restricted isometry property（RIP）is presented.We demonstrate that OMMP can exactly recover any K-sparse signal from fewer observations y=φx,provided that the sampling matrixφsatisfiesδKN-N＋1＋√K/NθKN-N＋1,N〈1.Moreover,the performance of OMMP for support recovery from noisy observations is also discussed.It is shown that,for l_2 bounded and l_∞bounded noisy cases,OMMP can recover the true support of any K-sparse signal under conditions on the restricted isometry property of the sampling matrixφand the minimum magnitude of the nonzero components of the signal.展开更多
Let M_(t) be an isoparametric foliation on the unit sphere(S^(n−1)(1),g^(st))with d principal curvatures.Using the spherical coordinatesinduced by M_(t),we construct a Minkowski norm with the representation F=r√2f(t)...Let M_(t) be an isoparametric foliation on the unit sphere(S^(n−1)(1),g^(st))with d principal curvatures.Using the spherical coordinatesinduced by M_(t),we construct a Minkowski norm with the representation F=r√2f(t),which generalizes the notions of(α,β)-norm and(α1,α2)-norm.Using the technique of the spherical local frame,we givean exact and explicit answer to the question when F=r√2 f(t)really defines a Minkowski norm.Using the similar technique,we study the Hessian isometry Φ between two Minkowski norms induced by M_(t),which preservesthe orientation and fixes the spherical ξ-coordinates.There aretwo ways to describe this Φ,either by a system of ODEs,or by its restriction toany normal plane for M_(t),which is then reduced to a Hessian isometry between Minkowski norms on R^(2) satisfying certain symmetry and(d)-properties.When d>2,we prove that this Φ can be obtained by gluing positive scalar multiplications and compositions of the Legendre transformation and positive scalar multiplications,so it must satisfy the(d)-property for any orthogonal decomposition R^(n)=V'+V'',i.e.,for any nonzero x=x'+x'' and Φ(x)=x=x'+x''with x',x'∈V'and x'',x''∈V'',we have g_(x)^(F1)(x'',x)=g_(x)^(F2)x(x'',x).As byproducts,we prove the following results.On the indicatrix(S_(F,g)),where F is a Minkowski norm induced by M_(t) and g is the Hessian metric,the foliation N_(t)=S_(F)∩R>_(0)M_(0) is isoparametric.Laugwitz Conjecture is valid for a Minkowski norm F induced by M_(t),i.e.,if its Hessian metric g is flat on R^(n)\{0}with n>2,then F is Euclidean.展开更多
Dunwoody manifolds are an interesting class of closed connected orientable 3-manifolds, which are defined by means of Heegaard diagrams having a rotational symmetry. They are proved to be cyclic coverings of lens spac...Dunwoody manifolds are an interesting class of closed connected orientable 3-manifolds, which are defined by means of Heegaard diagrams having a rotational symmetry. They are proved to be cyclic coverings of lens spaces （possibly S3） branched over （1, 1）-knots. Here we study the Dunwoody manifolds which are cyclic coverings of the 3-sphere branched over two specified families of Montesinos knots. Then we determine the Dunwoody parameters for such knots and the isometry groups for the considered manifolds in the hyperbolic case. A list of volumes for some hyperbolic Dunwoody manifolds completes the paper.展开更多
This paper solves the isometry problem between two indefinite metrics by using exterior differential systems. We prove that an indefinite metric is either rigid, admits a one parameter group of isometries (rotation...This paper solves the isometry problem between two indefinite metrics by using exterior differential systems. We prove that an indefinite metric is either rigid, admits a one parameter group of isometries (rotation like surfaces), or admits a three parameter group of isometries (K=constant).展开更多
The main result of this paper is to prove Fang and Wang＇s result by another method： Let E be any normed linear space and Vo ： S（E）→ S（l^1） be a surjective isometry. Then V0 can be linearly isometrically extend...The main result of this paper is to prove Fang and Wang＇s result by another method： Let E be any normed linear space and Vo ： S（E）→ S（l^1） be a surjective isometry. Then V0 can be linearly isometrically extended to E.展开更多
A hybrid Compressed Sensing and Primal-Dual Wavelet(CSP-PDW)technique is proposed for the compression and reconstruction of ECG signals.The compression and reconstruction algorithms are implemented using four key conc...A hybrid Compressed Sensing and Primal-Dual Wavelet(CSP-PDW)technique is proposed for the compression and reconstruction of ECG signals.The compression and reconstruction algorithms are implemented using four key concepts:Sparsifying Basis,Restricted Isometry Principle,Gaussian Random Matrix,and Convex Minimization.In addition to the conventional compression sensing reconstruction approach,wavelet-based processing is employed to enhance reconstruction efficiency.A mathematical model of the proposed algorithm is derived analytically to obtain the essential parameters of compression sensing,including the sparsifying basis,measurement matrix size,and number of iterations required for reconstructing the original signal and determining the type and level of wavelet processing.The low time complexity of the proposed algorithm makes it an ideal candidate for ECG monitoring systems in IoT-based e-healthcare applications.A feature extraction algorithm is also developed to show that the important ECG peaks remain unaltered after reconstruction.The clinical relevance of the reconstructed signal and the efficiency of the developed algorithm are evaluated using four validation parameters at three different compression ratios.展开更多
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 ...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.展开更多
If T is an isomorphism of （c0） into C（Ω） （where Ω is a sequentially compact and paracompact space, or a compact metric space in particular）, which satisfies the condition ｜｜T｜｜·｜｜T^-1｜｜ ≤ 1 ＋ε ...If T is an isomorphism of （c0） into C（Ω） （where Ω is a sequentially compact and paracompact space, or a compact metric space in particular）, which satisfies the condition ｜｜T｜｜·｜｜T^-1｜｜ ≤ 1 ＋ε for some ε ∈ （0,1/5）, then T/｜｜T｜｜ is close to an isometry with an error less than 9ε. The proof of this article is simple without using the dual space or adjoint operator.展开更多
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‖±...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).展开更多
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 t...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.展开更多
In this paper, some problems for isometric approximation are discussed. It is shown that an almost isometry from the unit ball of C(X) into the unit ball of C(Y) is near to an isometry. We also give a counterexample t...In this paper, some problems for isometric approximation are discussed. It is shown that an almost isometry from the unit ball of C(X) into the unit ball of C(Y) is near to an isometry. We also give a counterexample to this problem on the unit ball of l1.展开更多
In this paper, one of the Aleksandrov problem was resolved, the proof that a mapping f which preserve unit distance between two real p-normed spaces X and Y is an isometry if Y is a p-strictly convex space and f satis...In this paper, one of the Aleksandrov problem was resolved, the proof that a mapping f which preserve unit distance between two real p-normed spaces X and Y is an isometry if Y is a p-strictly convex space and f satisfies locally Lipschitz condition was shown, and a same result in normed spaces was given. In addition, a proof which there doesn't exist any isometry between some spaces was obtained.展开更多
Anterior cruciate ligament(ACL) rupture is one of the commonest knee sport injuries. The annual incidence of the ACL injury is between 100000-200000 in the United States. Worldwide around 400000 ACL reconstructions ar...Anterior cruciate ligament(ACL) rupture is one of the commonest knee sport injuries. The annual incidence of the ACL injury is between 100000-200000 in the United States. Worldwide around 400000 ACL reconstructions are performed in a year. The goal of ACL reconstruction is to restore the normal knee anatomy and kinesiology. The tibial and femoral tunnel placements are of primordial importance in achieving this outcome. Otherfactors that influence successful reconstruction are types of grafts, surgical techniques and rehabilitation programmes. A comprehensive understanding of ACL anatomy has led to the development of newer techniques supplemented by more robust biological and mechanical concepts. In this review we are mainly focussing on the evolution of tunnel placement in ACL reconstruction, focusing on three main categories, i.e., anatomical, biological and clinical outcomes. The importance of tunnel placement in the success of ACL reconstruction is well researched. Definite clinical and functional data is lacking to establish the superiority of the single or double bundle reconstruction technique. While there is a trend towards the use of anteromedial portals for femoral tunnel placement, their clinical superiority over trans-tibial tunnels is yet to be established.展开更多
This paper establishes a high order condition on the restricted isometry property adapted to a frame D （D-RIF） for the signal recovery. It is shown that if the measurementmatrix A satisfies the D-RIP condition δtk ...This paper establishes a high order condition on the restricted isometry property adapted to a frame D （D-RIF） for the signal recovery. It is shown that if the measurementmatrix A satisfies the D-RIP condition δtk 〈t-1/t for t 〉 1, then all signals f which aresparse in terms of a tight frame D can be recovered stably or exactly via the l1-analysis model based on y= Af ＋ z in 12 and Dantzig selector bounded noise setting.展开更多
基金supported by the Natural Science Foundation of China (12271402)the Natural Science Foundation of Tianjin City (22JCYBJC00420)。
文摘In this paper,we study isometries and phase-isometries of non-Archimedean normed spaces.We show that every isometry f:Sr(X)→Sr(X),where X is a finite-dimensional non-Archimedean normed space and Sr(X)is a sphere with radius r∈||X||,is surjective if and only if is spherically complete and k is finite.Moreover,we prove that if X and Y are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with|2|=1,any phase-isometry f:X→Y is phase equivalent to an isometric operator.
文摘In this paper the quasi-constant curvature space and the Riemannian manifold contained the totally umbilical hypersurface family are studied,and two theorems are given at the same time.
基金Supported by National Natural Science Foundation of China(Grant Nos.11071201 and 11001231)
文摘Let X, Y be two real Banach spaces and ε≥0. A map f ： X → Y is said to be a standard ε-isometry if│││f/（x） - f（y）││ - ]ix - Y││x-y││ ε for all x,y C X and with f（O） = O. We say that a pair of Banach spaces （X, Y） is stable if there exists γ〉 0 such that, for every such ε and every standard v-isometry f ： X → Y, there is a bounded linear operator T ： L（f） → f（X） → X so that ││Tf（x） - x││ ≤γε for all x E X. X（Y） is said to be universally left-stable if （X, Y） is always stable for every Y（X）. In this paper, we show that if a dual Banach space X is universally left-stable, then it is isometric to a complemented w＊-closed subspace of ∞ （1） for some set F, hence, an injective space; and that a Banach space is universally left-stable if and only if it is a cardinality injective space; and universally left-stability spaces are invariant.
基金supported by National Natural Science Foundation of China (Grant Nos.91130009, 11171299 and 11041005)National Natural Science Foundation of Zhejiang Province in China (Grant Nos. Y6090091 and Y6090641)
文摘This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 〈 p ≤ 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant （RIC）. Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to δ4Ar 〈 0.558 and δ3rA 〈 0.4721, respectively. Recently, the upper bounds of RIC can be improved to δ2rA 〈 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to δ2tA 〈 0.4931 and δrA 〈 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with δ2rA 〉1√2 or δrA 〉 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δ2rA and δrA.Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p （0 〈 p 〈 1） quasi norm minimization problem.
基金supported by National Natural Science Foundation of China(Grant Nos.11271060,U0935004,U1135003,11071031,11290143 and 11101096)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry,National Engineering Research Center of Digital Lifethe Guangdong Natural Science Foundation(Grant No.S2012010010376)
文摘Orthogonal matching pursuit （OMP） algorithm is an efficient method for the recovery of a sparse signal in compressed sensing, due to its ease implementation and low complexity. In this paper, the robustness of the OMP algorithm under the restricted isometry property （RIP） is presented. It is shown that 5K＋V/KOK,1 〈 1 is sufficient for the OMP algorithm to recover exactly the support of arbitrary /（-sparse signal if its nonzero components are large enough for both 12 bounded and lz~ bounded noises.
基金supported by the Science Foundation of Guangdong University of Finance & Economics(Grant No.13GJPY11002)National Natural Science Foundation of China(Grant Nos.11071031,11271060,11290143,U0935004 and U1135003)+1 种基金the Guangdong Natural Science Foundation(Grant No.S2012010010376)the Guangdong University and Colleges Technology Innovation Projects(Grant No.2012KJCX0048)
文摘Orthogonal multi-matching pursuit（OMMP）is a natural extension of orthogonal matching pursuit（OMP）in the sense that N（N≥1）indices are selected per iteration instead of 1.In this paper,the theoretical performance of OMMP under the restricted isometry property（RIP）is presented.We demonstrate that OMMP can exactly recover any K-sparse signal from fewer observations y=φx,provided that the sampling matrixφsatisfiesδKN-N＋1＋√K/NθKN-N＋1,N〈1.Moreover,the performance of OMMP for support recovery from noisy observations is also discussed.It is shown that,for l_2 bounded and l_∞bounded noisy cases,OMMP can recover the true support of any K-sparse signal under conditions on the restricted isometry property of the sampling matrixφand the minimum magnitude of the nonzero components of the signal.
基金supported by Beijing Natural Science Foundation(Grant No.Z180004)National Natural Science Foundation of China(Grant Nos.11771331 and 11821101)Capacity Building for SciTech Innovation—Fundamental Scientific Research Funds(Grant No.KM201910028021)。
文摘Let M_(t) be an isoparametric foliation on the unit sphere(S^(n−1)(1),g^(st))with d principal curvatures.Using the spherical coordinatesinduced by M_(t),we construct a Minkowski norm with the representation F=r√2f(t),which generalizes the notions of(α,β)-norm and(α1,α2)-norm.Using the technique of the spherical local frame,we givean exact and explicit answer to the question when F=r√2 f(t)really defines a Minkowski norm.Using the similar technique,we study the Hessian isometry Φ between two Minkowski norms induced by M_(t),which preservesthe orientation and fixes the spherical ξ-coordinates.There aretwo ways to describe this Φ,either by a system of ODEs,or by its restriction toany normal plane for M_(t),which is then reduced to a Hessian isometry between Minkowski norms on R^(2) satisfying certain symmetry and(d)-properties.When d>2,we prove that this Φ can be obtained by gluing positive scalar multiplications and compositions of the Legendre transformation and positive scalar multiplications,so it must satisfy the(d)-property for any orthogonal decomposition R^(n)=V'+V'',i.e.,for any nonzero x=x'+x'' and Φ(x)=x=x'+x''with x',x'∈V'and x'',x''∈V'',we have g_(x)^(F1)(x'',x)=g_(x)^(F2)x(x'',x).As byproducts,we prove the following results.On the indicatrix(S_(F,g)),where F is a Minkowski norm induced by M_(t) and g is the Hessian metric,the foliation N_(t)=S_(F)∩R>_(0)M_(0) is isoparametric.Laugwitz Conjecture is valid for a Minkowski norm F induced by M_(t),i.e.,if its Hessian metric g is flat on R^(n)\{0}with n>2,then F is Euclidean.
文摘Dunwoody manifolds are an interesting class of closed connected orientable 3-manifolds, which are defined by means of Heegaard diagrams having a rotational symmetry. They are proved to be cyclic coverings of lens spaces （possibly S3） branched over （1, 1）-knots. Here we study the Dunwoody manifolds which are cyclic coverings of the 3-sphere branched over two specified families of Montesinos knots. Then we determine the Dunwoody parameters for such knots and the isometry groups for the considered manifolds in the hyperbolic case. A list of volumes for some hyperbolic Dunwoody manifolds completes the paper.
基金the National Natural Science Foundationof China (No.1970 10 17)
文摘This paper solves the isometry problem between two indefinite metrics by using exterior differential systems. We prove that an indefinite metric is either rigid, admits a one parameter group of isometries (rotation like surfaces), or admits a three parameter group of isometries (K=constant).
基金Foundation item： the National Natural Science Foundation of China （No. 10571090） the Research Fund for the Doctoral Program of Higher Education （No. 20060055010） and the Fund of Tianjin Educational Comittee （No. 20060402）.
文摘The main result of this paper is to prove Fang and Wang＇s result by another method： Let E be any normed linear space and Vo ： S（E）→ S（l^1） be a surjective isometry. Then V0 can be linearly isometrically extended to E.
文摘A hybrid Compressed Sensing and Primal-Dual Wavelet(CSP-PDW)technique is proposed for the compression and reconstruction of ECG signals.The compression and reconstruction algorithms are implemented using four key concepts:Sparsifying Basis,Restricted Isometry Principle,Gaussian Random Matrix,and Convex Minimization.In addition to the conventional compression sensing reconstruction approach,wavelet-based processing is employed to enhance reconstruction efficiency.A mathematical model of the proposed algorithm is derived analytically to obtain the essential parameters of compression sensing,including the sparsifying basis,measurement matrix size,and number of iterations required for reconstructing the original signal and determining the type and level of wavelet processing.The low time complexity of the proposed algorithm makes it an ideal candidate for ECG monitoring systems in IoT-based e-healthcare applications.A feature extraction algorithm is also developed to show that the important ECG peaks remain unaltered after reconstruction.The clinical relevance of the reconstructed signal and the efficiency of the developed algorithm are evaluated using four validation parameters at three different compression ratios.
基金supported by the National Natural Science Foundation of China“Variable exponential function spaces on variable anisotropic Euclidean spaces and their applications”(12261083),“Harmonic analysis on affine symmetric spaces”(12161083).
文摘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.
基金National Natural Science Foundation of China(10571090)the Research Fund for the Doctoral Program of Higher Education(20060055010)
文摘If T is an isomorphism of （c0） into C（Ω） （where Ω is a sequentially compact and paracompact space, or a compact metric space in particular）, which satisfies the condition ｜｜T｜｜·｜｜T^-1｜｜ ≤ 1 ＋ε for some ε ∈ （0,1/5）, then T/｜｜T｜｜ is close to an isometry with an error less than 9ε. The proof of this article is simple without using the dual space or adjoint operator.
基金supported by the NSFC(12126329,12171266,12126346)the NSF of Fujian Province of China(2023J01805)+5 种基金the Research Start-Up Fund of Jimei University(ZQ2021017)supported by the NSFC(12101234)the NSF of Hebei Province(A2022502010)the Fundamental Research Funds for the Central Universities(2023MS164)the China Scholarship Councilsupported by the Simons Foundation(585081)。
文摘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).
基金Supported by National Natural Science Foundation of China(11731010 and 12071388)。
文摘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.
文摘In this paper, some problems for isometric approximation are discussed. It is shown that an almost isometry from the unit ball of C(X) into the unit ball of C(Y) is near to an isometry. We also give a counterexample to this problem on the unit ball of l1.
文摘In this paper, one of the Aleksandrov problem was resolved, the proof that a mapping f which preserve unit distance between two real p-normed spaces X and Y is an isometry if Y is a p-strictly convex space and f satisfies locally Lipschitz condition was shown, and a same result in normed spaces was given. In addition, a proof which there doesn't exist any isometry between some spaces was obtained.
文摘Anterior cruciate ligament(ACL) rupture is one of the commonest knee sport injuries. The annual incidence of the ACL injury is between 100000-200000 in the United States. Worldwide around 400000 ACL reconstructions are performed in a year. The goal of ACL reconstruction is to restore the normal knee anatomy and kinesiology. The tibial and femoral tunnel placements are of primordial importance in achieving this outcome. Otherfactors that influence successful reconstruction are types of grafts, surgical techniques and rehabilitation programmes. A comprehensive understanding of ACL anatomy has led to the development of newer techniques supplemented by more robust biological and mechanical concepts. In this review we are mainly focussing on the evolution of tunnel placement in ACL reconstruction, focusing on three main categories, i.e., anatomical, biological and clinical outcomes. The importance of tunnel placement in the success of ACL reconstruction is well researched. Definite clinical and functional data is lacking to establish the superiority of the single or double bundle reconstruction technique. While there is a trend towards the use of anteromedial portals for femoral tunnel placement, their clinical superiority over trans-tibial tunnels is yet to be established.
基金supported by National Natural Science Foundation of China(11271050 and 11371183)
文摘This paper establishes a high order condition on the restricted isometry property adapted to a frame D （D-RIF） for the signal recovery. It is shown that if the measurementmatrix A satisfies the D-RIP condition δtk 〈t-1/t for t 〉 1, then all signals f which aresparse in terms of a tight frame D can be recovered stably or exactly via the l1-analysis model based on y= Af ＋ z in 12 and Dantzig selector bounded noise setting.
