Homotopy Categories for Simply Connected Torsion Spaces

For each n > 1 and each multiplicative closed set of integers S, we study closed model category structures on the pointed category of topological spaces, where the classes of weak equivalences are classes of maps inducing isomorphism on homotopy groups with coefficients in determined torsion abelian groups, in degrees higher than or equal to n. We take coefficients either on all the cyclic groups with s ∈ S, or in the abelian group where is the group of fractions of the form with s ∈ S. In the first case, for n > 1 the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion. In the second case, for n > 1 we obtain that the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion and the nth homotopy group is divisible. These equivalences of categories are given by colocalizations , obtained by cofibrant approximations on the model structures. These colocalization maps have nice universal properties. For instance, the map is final (in the homotopy category) among all the maps of the form Y → X with Y an (n − 1)-connected CW-complex whose homotopy groups are S-torsion and its nth homotopy group is divisible. The spaces , are constructed using the cones of Moore spaces of the form M(T, k), where T is a coefficient group of the corresponding structure of models, and homotopy colimits indexed by a suitable ordinal. If S is generated by a set P of primes and S ^p is generated by a prime p ∈ P one has that for n > 1 the category is equivalent to the product category . If the multiplicative system S is generated by a finite set of primes, then localized category is equivalent to the homotopy category of n-connected Ext-S-complete CW-complexes and a similar result is obtained for .     

On Linear Fractional Transformations Associated with Generalized J-Inner Matrix Functions

A class U_k1 (J){\mathcal{U}}_{\kappa 1} (J) of generalized J-inner mvf’s (matrix valued functions) W(λ) which appear as resolvent matrices for bitangential interpolation problems in the generalized Schur class of p ×q  mvf￠s S_k^{p ×q}p \times q \, {\rm mvf's}\, {\mathcal{S}}_{\kappa}^{p \times q} and some associated reproducing kernel Pontryagin spaces are studied. These spaces are used to describe the range of the linear fractional transformation T_W based on W and applied to S_k2^{p ×q}{\mathcal{S}}_{\kappa 2}^{p \times q}. Factorization formulas for mvf’s W in a subclass U^°_k1 (J) of U_k1(J){\mathcal{U}^{\circ}_{\kappa 1}} (J)\, {\rm of}\, {\mathcal{U}}_{\kappa 1}(J) found and then used to parametrize the set S_k1+k2^{p ×q} ?T_W [ S_k2^{p ×q} ]{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} \cap T_{W} \left[ {\mathcal{S}}_{\kappa 2}^{p \times q} \right]. Applications to bitangential interpolation problems in the class S_k1+k2^{p ×q}{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} will be presented elsewhere.     

Applications of differential calculus to quasilinear elliptic boundary value problems with non-smooth data

This paper concerns boundary value problems for quasilinear second order elliptic systems which are, for example, of the type

Rigidity Theorem of Hypersurfaces with Constant Scalar Curvature in a Unit Sphere

In this paper, we give a characterization of tori S^1 （ √ nr＋2-n/nr）×S^n-1（√ n-2/nr） and S^m （ √n/m ） ×S^n-m （√n-m/n）. Our result extends the result due to Li （1996）on the condition that M is an n-dimensional complete hypersurface in Sn＋1 with two distinct principal curvatures. Keywords principal curvature, Clifford torus, Gauss equations     

The Average Widths of Sobolev-Wiener Classes and Besov-Wiener Classes

This paper concerns the problem of average σ-width of Sobolev-Wiener classes W^rpq(R^d),W^rpq(M,R^d),and Besov-Wiener classes S^rpqθb(R^d).S^rpqθB(R^d),S^rpqθb(M,R^d),S^rpqθB(R^d)in the metric Lq(R^d) for 1≤q≤p≤∞.The weak asymptotic results concerning the average linear widths,the average Bernstein widths and the infinite-dimensional Gel‘fand widths are obtained,respectively.     

Micro-Local Analysis with Fourier Lebesgue Spaces. Part?I

Let ω,ω ₀ be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, WF_FL^q(w)(f)\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f) of f ? S￠f\in \mathcal{S}' with respect to weighted Fourier Lebesgue space FL^q_(w)\mathcal{F}L^{q}_{(\omega )}. We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in S^(w₀)_r,0S^{(\omega _{0})}_{\rho ,0} hold for such wave-front sets. Especially we prove that

Su un problema di tipo nuovo relativo alle equazioni paraboliche d'ordine superiore al secondo

Sunto Si studia il problema della determinazione di una soluzione dell'equazione a_k(x)∂^ku/∂x^k=f(x, y) entro la semistriscia a≤x≤b, y≥0, che assuma assegnati valori per y=0 e per x=a, x₁, x₂, b (a<x₁<x₂<b). Analogamente si studia il problema della determinazione di una soluzione dell' equazione a_k(x)∂^ku/∂x^k+b(x)∂u/∂y=f(x,y), entro la medesima semistriscia, cha assuma assegnati valori per y=0 e per x=a, x₁, x₂, b e la cui ∂/∂y assuma assegnati valori per y=0. A Giovanni Sansone nel suo 70^mo compleanno.     

Compact embedded rotation hypersurfaces of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

In this paper, we prove that and round geodesic spheres are the only n-dimensional compact embedded rotation hypersurfaces with H_m = 0 (1 ≤ m ≤ n − 1) in a unit sphere Sⁿ⁺¹(1). When m = 1, our result reduces to the result of T. Otsuki [O1], [O2], Brito and Leite [BL]. The project is supported by the grant No. 10531090 of NSFC.     

Around splitting and reaping for partitions of ω

We investigate splitting number and reaping number for the structure (ω) ^ω of infinite partitions of ω. We prove that \mathfrakr_d ￡ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and \mathfraks_d 3 \mathfrakb{\mathfrak{s}_{d}\geq\mathfrak{b}} . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})} . To prove the consistency \mathfrakr_d < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and \mathfraks_d < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants \mathfrakr_pair{\mathfrak{r}_{pair}} and \mathfraks_pair{\mathfrak{s}_{pair}} . We also study the relation between \mathfrakr_pair, \mathfraks_pair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that cov(M),cov(N) ￡ \mathfrakr_pair ￡ \mathfraks_d,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and \mathfraks ￡ \mathfraks_pair ￡ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} .     

Invariant subspaces for operators in a general II<Subscript>1</Subscript>-factor

Let ℳ be a von Neumann factor of type II₁ with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ _T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace affiliated with ℳ, such that the Brown measure of is concentrated on B and the Brown measure of is concentrated on ℂ∖B. Moreover, is T-hyperinvariant and the trace of is equal to μ _T(B). In particular, if T∈ℳ has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T∈ℳ the limit exists in the strong operator topology, and the projection onto is equal to 1_[0,r](A), for every r>0. Supported by The Danish National Research Foundation.     

Analytic Extension of Functions from Analytic Hilbert Spaces

Let M be an invariant subspace of Hv2. It is shown that for each f∈M⊥, f can be analytically extended across (?)Bd\σ(Sz1,…, Szd).     

A Representation of the Lorentz Spin Group and Its Application

For an integer m ≥ 4, we define a set of 2[m/2] × 2[m/2] matrices γj （m）, （j = 0, 1,..., m - 1） which satisfy γj （m）γk （m） ＋γk （m）γj （m） = 2ηjk （m）I[m/2], where （ηjk （m）） 0≤j,k≤m-1 is a diagonal matrix, the first diagonal element of which is 1 and the others are -1, I[m/2] is a 2[m/1] × 2[m/2] identity matrix with [m/2] being the integer part of m/2. For m = 4 and 5, the representation （m） of the Lorentz Spin group is known. For m≥ 6, we prove that （i） when m = 2n, （n ≥ 3）, （m） is the group generated by the set of matrices {T｜T=1/√ξ（（I＋k） 0 ＋ 0 I-K） （ U 0 0 U）, （ii） when m = 2n ＋ 1 （n≥ 3）, （m） is generated by the set of matrices {T｜T=1/√ξ（I -k^- k I）U,U∈ （m-1）,ξ=1-m-2 ∑k,j=0 ηkja^k a^j〉0, K=i[m-3 ∑j=0 a^j γj（m-2）＋a^（m-2） In],K^-=i[m-3∑j=0 a^j γj（m-2）-a^（m-2） In]}     

Desingularization of singular Riemannian foliation

Let F{\mathcal{F}} be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F{\mathcal{F}} we construct a regular Riemannian foliation [^(F)]{\hat{\mathcal{F}}} on a compact Riemannian manifold [^(M)]{\hat{M}} and a desingularization map [^(r)]:[^(M)]? M{\hat{\rho}:\hat{M}\rightarrow M} that projects leaves of [^(F)]{\hat{\mathcal{F}}} into leaves of F{\mathcal{F}}. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F{\mathcal{F}} are compact, then, for each small ${\epsilon >0 }${\epsilon >0 }, we can find [^(M)]{\hat{M}} and [^(F)]{\hat{\mathcal{F}}} so that the desingularization map induces an e{\epsilon}-isometry between M/F{M/\mathcal{F}} and [^(M)]/[^(F)]{\hat{M}/\hat{\mathcal{F}}}. This implies in particular that the space of leaves M/F{M/\mathcal{F}} is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {([^(M)]_n/[^(F)]_n)}{\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}.     

Some Inequalities for Tree Martingales

In this paper we study tree martingales and proved that if 1≤α,β〈∞,1≤p〈∞ then for every predictable tree martingale f=（ft,t∞T）and E[σ^（P）（f）]〈∞,E[S^（P）（f）]〈∞,it holds that ‖（St^（p）（f）,t∈T）‖M^α∞≤Cαβ‖f‖p^αβ,‖（σt^（p）（f）,t∈T）‖M^α,β‖f‖P^αβ,where Cαβ depends only on α and β.     

Ultraproducts of real interpolation spaces between L p -spaces

Let , be a family of compatible couples of L^p-spaces. We show that, given a countably incomplete ultrafilter in , the ultraproduct of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type , an intermediate K?the space between and being a purely atomic measure space, and a K?the function space K(Ω₃) defined on some purely non atomic measure space (Ω₃, ν₃) in such a way that Ω₂ ∪ Ω₃ ≠∅. The research of first and third authors is partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      18. A Note on Certain Block Spaces on the Unit Sphere   总被引：1，自引：0，他引：1   Xiao Feng YE Xiang Rong ZHU 《数学学报(英文版)》2006,22(6):1843-1846 In this note, we clarify a relation between block spaces and the Hardy space. We obtain Bq^0.v belong to H^1（S^n-1）＋L（ln＋L）^1＋v（s^n-1）,v〉-1,q〉1,Furthermore,if v≥ 0, q 〉 1. we verify that block spaces Rq^0.v（S^n-1）are proper subspaces of H1 （S^n- 1）,      19. Weakly Singular and Microscopically Hypersingular Load Perturbation for a Nonlinear Traction Boundary Value Problem: a Functional Analytic Approach      Matteo Dalla Riva  Massimo Lanza de Cristoforis 《Complex Analysis and Operator Theory》2011,5(3):811-833 Let Ω i and Ω o be two bounded open subsets of \mathbbRn{{\mathbb{R}}^{n}} containing 0. Let G i be a (nonlinear) map from ?Wi×\mathbbRn{\partial\Omega^{i}\times {\mathbb{R}}^{n}} to \mathbbRn{{\mathbb{R}}^{n}} . Let a o be a map from ∂Ω o to the set Mn(\mathbbR){M_{n}({\mathbb{R}})} of n × n matrices with real entries. Let g be a function from ∂Ω o to \mathbbRn{{\mathbb{R}}^{n}} . Let γ be a positive valued function defined on a right neighborhood of 0 in the real line. Let T be a map from ]1-(2/n),+￥[×Mn(\mathbbR){]1-(2/n),+\infty[\times M_{n}({\mathbb{R}})} to Mn(\mathbbR){M_{n}({\mathbb{R}})} . Then we consider the problem $\left\{ {ll} {{\rm div}}\, (T(\omega,Du))=0 &\quad {{\rm in}} \;\Omega^{o} \setminus\epsilon{{\rm cl}} \Omega^{i},\\ -T(\omega,Du(x))\nu_{\epsilon\Omega^{i}}(x)=\frac{1}{\gamma(\epsilon)}G^{i}({x}/{\epsilon}, \gamma(\epsilon)\epsilon^{-1} ({\rm log} \, \epsilon)^{-\delta_{2,n}} u(x)) & \quad \forall x \in \epsilon\partial\Omega^{i},\\ T(\omega, Du(x)) \nu^{o}(x)=a^{o}(x)u(x)+g(x) & \quad \forall x \in \partial \Omega^{o}, \right.$\left\{ \begin{array}{ll} {{\rm div}}\, (T(\omega,Du))=0 &\quad {{\rm in}} \;\Omega^{o} \setminus\epsilon{{\rm cl}} \Omega^{i},\\ -T(\omega,Du(x))\nu_{\epsilon\Omega^{i}}(x)=\frac{1}{\gamma(\epsilon)}G^{i}({x}/{\epsilon}, \gamma(\epsilon)\epsilon^{-1} ({\rm log} \, \epsilon)^{-\delta_{2,n}} u(x)) & \quad \forall x \in \epsilon\partial\Omega^{i},\\ T(\omega, Du(x)) \nu^{o}(x)=a^{o}(x)u(x)+g(x) & \quad \forall x \in \partial \Omega^{o}, \end{array} \right.      20. On the ring of local polynomial invariants for a pair of entangled qubits      V. Gerdt  A. Khvedelidze  Yu. Palii 《Journal of Mathematical Sciences》2010,168(3):368-378 The entanglement characteristics of two qubits are encoded in the invariants of the adjoint action of the group SU(2) ⊗ SU(2) on the space of density matrices \mathfrakP+ {\mathfrak{P}_{+} } , defined as the space of 4 × 4 positive semidefinite Hermitian matrices. The corresponding ring \textC[ \mathfrakP+ ]\textSU( 2 ) ?\textSU ?( 2 ) {\text{C}}{\left[ {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} of polynomial invariants is studied. A special integrity basis for \textC[ \mathfrakP+ ]\textSU( 2 ) ?\textSU ?( 2 ) {\text{C}}{\left[ {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} is described, and the constraints on its elements imposed by the positive semidefiniteness of density matrices are given explicitly in the form of polynomial inequalities. The suggested basis is characterized by the property that the minimum number of invariants, namely, two primary invariants of degree 2, 3 and one secondary invariant of degree 4 appearing in the Hironaka decomposition of \textC[ \mathfrakP+ ]\textSU( 2 ) ?\textSU ?( 2 ) {\text{C}}{\left[ {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} , are subject to the polynomial inequalities. Bibliography: 32 titles.

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号

A Note on Certain Block Spaces on the Unit Sphere

In this note, we clarify a relation between block spaces and the Hardy space. We obtain Bq^0.v belong to H^1（S^n-1）＋L（ln＋L）^1＋v（s^n-1）,v〉-1,q〉1,Furthermore,if v≥ 0, q 〉 1. we verify that block spaces Rq^0.v（S^n-1）are proper subspaces of H1 （S^n- 1）,     

Weakly Singular and Microscopically Hypersingular Load Perturbation for a Nonlinear Traction Boundary Value Problem: a Functional Analytic Approach

Let Ω ⁱ and Ω ^o be two bounded open subsets of \mathbbRⁿ{{\mathbb{R}}^{n}} containing 0. Let G ⁱ be a (nonlinear) map from ?Wⁱ×\mathbbRⁿ{\partial\Omega^{i}\times {\mathbb{R}}^{n}} to \mathbbRⁿ{{\mathbb{R}}^{n}} . Let a ^o be a map from ∂Ω ^o to the set M_n(\mathbbR){M_{n}({\mathbb{R}})} of n × n matrices with real entries. Let g be a function from ∂Ω ^o to \mathbbRⁿ{{\mathbb{R}}^{n}} . Let γ be a positive valued function defined on a right neighborhood of 0 in the real line. Let T be a map from ]1-(2/n),+￥[×M_n(\mathbbR){]1-(2/n),+\infty[\times M_{n}({\mathbb{R}})} to M_n(\mathbbR){M_{n}({\mathbb{R}})} . Then we consider the problem

On the ring of local polynomial invariants for a pair of entangled qubits

The entanglement characteristics of two qubits are encoded in the invariants of the adjoint action of the group SU(2) ⊗ SU(2) on the space of density matrices \mathfrakP₊ {\mathfrak{P}_{+} } , defined as the space of 4 × 4 positive semidefinite Hermitian matrices. The corresponding ring \textC[ \mathfrakP₊ ]^{\textSU( 2 ) ?\textSU ?( 2 )} {\text{C}}{\left[ {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} of polynomial invariants is studied. A special integrity basis for \textC[ \mathfrakP₊ ]^{\textSU( 2 ) ?\textSU ?( 2 )} {\text{C}}{\left[ {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} is described, and the constraints on its elements imposed by the positive semidefiniteness of density matrices are given explicitly in the form of polynomial inequalities. The suggested basis is characterized by the property that the minimum number of invariants, namely, two primary invariants of degree 2, 3 and one secondary invariant of degree 4 appearing in the Hironaka decomposition of \textC[ \mathfrakP₊ ]^{\textSU( 2 ) ?\textSU ?( 2 )} {\text{C}}{\left[ {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} , are subject to the polynomial inequalities. Bibliography: 32 titles.