共查询到20条相似文献,搜索用时 484 毫秒
1.
Manuel González 《Archiv der Mathematik》2011,97(4):345-352
The perturbation classes problem for semi-Fredholm operators asks when the equalities SS(X,Y)=PF+(X,Y){\mathcal{SS}(X,Y)=P\Phi_+(X,Y)} and SC(X,Y)=PF-(X,Y){\mathcal{SC}(X,Y)=P\Phi_-(X,Y)} are satisfied, where SS{\mathcal{SS}} and SC{\mathcal{SC}} denote the strictly singular and the strictly cosingular operators, and PΦ+ and PΦ− denote the perturbation classes for upper semi-Fredholm and lower semi-Fredholm operators. We show that, when Y is a reflexive Banach space, SS(Y*,X*)=PF+(Y*,X*){\mathcal{SS}(Y^*,X^*)=P\Phi_+(Y^*,X^*)} if and only if SC(X,Y)=PF-(X,Y),{\mathcal{SC}(X,Y)=P\Phi_-(X,Y),} and SC(Y*,X*)=PF-(Y*,X*){\mathcal{SC}(Y^*,X^*)=P\Phi_-(Y^*,X^*)} if and only if SS(X,Y)=PF+(X,Y){\mathcal{SS}(X,Y)=P\Phi_+(X,Y)}. Moreover we give examples showing that both direct implications fail in general. 相似文献
2.
Let M{\mathcal M} be a σ-finite von Neumann algebra and
\mathfrak A{\mathfrak A} a maximal subdiagonal algebra of M{\mathcal M} with respect to a faithful normal conditional expectation F{\Phi} . Based on Haagerup’s noncommutative L
p
space Lp(M){L^p(\mathcal M)} associated with M{\mathcal M} , we give a noncommutative version of H
p
space relative to
\mathfrak A{\mathfrak A} . If h
0 is the image of a faithful normal state j{\varphi} in L1(M){L^1(\mathcal M)} such that j°F = j{\varphi\circ \Phi=\varphi} , then it is shown that the closure of
{\mathfrak Ah0\frac1p}{\{\mathfrak Ah_0^{\frac1p}\}} in Lp(M){L^p(\mathcal M)} for 1 ≤ p < ∞ is independent of the choice of the state preserving F{\Phi} . Moreover, several characterizations for a subalgebra of the von Neumann algebra M{\mathcal M} to be a maximal subdiagonal algebra are given. 相似文献
3.
Hiroaki Minami 《Archive for Mathematical Logic》2010,49(4):501-518
We investigate splitting number and reaping number for the structure (ω)
ω
of infinite partitions of ω. We prove that
\mathfrakrd £ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and
\mathfraksd 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
\mathfrakrd < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and
\mathfraksd < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants
\mathfrakrpair{\mathfrak{r}_{pair}} and
\mathfrakspair{\mathfrak{s}_{pair}} . We also study the relation between
\mathfrakrpair, \mathfrakspair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that
cov(M),cov(N) £ \mathfrakrpair £ \mathfraksd,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and
\mathfraks £ \mathfrakspair £ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} . 相似文献
4.
In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous 2-cochain
ω on a Lie algebra
\mathfrak h{\mathfrak h} with values in an
\mathfrak h{\mathfrak h}-module V, we associate subalgebras
\mathfrak sp(\mathfrak h,w) ê \mathfrak ham(\mathfrak h,w){\mathfrak {sp}(\mathfrak h,\omega) \supseteq \mathfrak {ham}(\mathfrak h,\omega)} of symplectic, resp., hamiltonian elements. Then
\mathfrak ham(\mathfrak h,w){\mathfrak {ham}(\mathfrak h,\omega)} has a natural central extension which in turn is contained in a larger abelian extension of
\mathfrak sp(\mathfrak h,w){\mathfrak {sp}(\mathfrak h,\omega)}. In this setting, we study linear actions of a Lie group G on V which are compatible with a homomorphism
\mathfrak g ? \mathfrak ham(\mathfrak h,w){\mathfrak g \to \mathfrak {ham}(\mathfrak h,\omega)}, i.e., abstract hamiltonian actions, corresponding central and abelian extensions of G and momentum maps
J : \mathfrak g ? V{J : \mathfrak g \to V}. 相似文献
5.
The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space ${\Upsilon\colon\widehat X\to X}The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space
U\colon[^(X)]? X{\Upsilon\colon\widehat X\to X} in which [^(Y)] o U-1(Y){\widehat Y\equiv\Upsilon^{-1}(Y)} has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a C
∞ exhaustion function j{\varphi} on [^(X)]{\widehat X} which is strongly q-convex on [^(W)]=U-1(W){\widehat\Omega=\Upsilon^{-1}(\Omega)} outside a uniform neighborhood of the q-dimensional compact irreducible components of [^(Y)]{\widehat Y}. 相似文献
6.
Constantin Costara 《Integral Equations and Operator Theory》2012,73(1):7-16
Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let rT(x) = limsupn ? ¥ || Tnx|| 1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r
T
(x) = 0 if and only if rj(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r
T
(By) = 0 if and only if rj( T) (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}. 相似文献
7.
In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map F X between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres ${\mathbb{S}^k}In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map F
X
between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres
\mathbbSk{\mathbb{S}^k}, the forgetful map
F\mathbbSk{F_{\mathbb{S}^k}} is not one-to-one. This result follows from Theorem A, which proves that the quotient map MET sec < 0 (M)?T sec < 0 (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} is not trivial at some homotopy levels, provided the hyperbolic manifold M satisfies certain conditions. Here MET sec < 0 (M){\mathcal{MET}^{\,\,sec <0 }(M)} is the space of negatively curved metrics on M and T sec < 0 (M) = MET sec < 0 (M)/ DIFF0(M){\mathcal{T}^{\,\,sec <0 }(M) = \mathcal{MET}^{\,\,sec <0 }(M)/ {\rm DIFF}_0(M)} is, as defined in [FO2], the Teichmüller space of negatively curved metrics on M. In particular we conclude that T sec < 0 (M){\mathcal{T}^{\,\,sec <0 }(M)} is, in general, not connected. Two remarks: (1) the nontrivial elements in pkMET sec < 0 (M){\pi_{k}\mathcal{MET}^{\,\,sec <0 }(M)} constructed in [FO3] have trivial image by the map induced by MET sec < 0 (M)?T sec < 0 (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} ; (2) the nonzero classes in pkT sec < 0 (M){\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)} constructed in [FO2] are not in the image of the map induced by MET sec < 0 (M)?T sec < 0 (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} ; the nontrivial classes in pkT sec < 0 (M){\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)} given here, besides coming from MET sec < 0 (M){\mathcal{MET}^{\,\,sec <0 }(M)} and being harder to construct, have a different nature and genesis: the former classes – given in [FO2] – come from the existence
of exotic spheres, while the latter classes – given here – arise from the non-triviality and structure of certain homotopy
groups of the space of pseudo-isotopies of the circle
\mathbbS1{\mathbb{S}^1}. The strength of the new techniques used here allowed us to prove also a homology version of Theorem A, which is given in
Theorem B. 相似文献
8.
In this paper we present new structural information about the multiplier algebra M (A ){\mathcal M (\mathcal A )} of a σ-unital purely infinite simple C*-algebra A{\mathcal {A}}, by characterizing the positive elements A ? M (A ){A\in \mathcal M (\mathcal A )} that are strict sums of projections belonging to A{\mathcal A } . If A ? A{A\not\in \mathcal {A}} and A itself is not a projection, then the necessary and sufficient condition for A to be a strict sum of projections belonging to A{\mathcal {A} } is that ${\|A\| >1 }${\|A\| >1 } and that the essential norm ||A||ess 3 1{\|A\|_{ess} \geq 1}. Based on a generalization of the Perera–Rordam weak divisibility of separable simple C*-algebras of real rank zero to all σ-unital simple C*-algebras of real rank zero, we show that every positive element of A{\mathcal {A}} with norm >1 can be approximated by finite sums of projections. Based on block tri-diagonal approximations, we decompose
any positive element A ? M (A ){A\in \mathcal M (\mathcal {A} )} with ${\| A\| >1 }${\| A\| >1 } and || A||ess 3 1{\| A\|_{ess} \geq 1} into a strictly converging sum of positive elements in A{\mathcal A} with norm >1. 相似文献
9.
We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck’s notion of differential operators on a commutative algebra in such a way that derivations of the
commutative algebra are replaced by
\mathbbDer(A){\mathbb{D}{\rm er}(A)}, the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A){\mathcal{F}(A)}, a certain ‘Fock space’ associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)){\mathcal{D}(\mathcal{F}(A))} of differential operators is filtered and gr D(F(A)){\mathcal{D}(\mathcal{F}(A))}, the associated graded algebra, is commutative in some ‘wheeled’ sense. The resulting ‘wheeled’ Poisson structure on gr D(F(A)){\mathcal{D}(\mathcal{F}(A))} is closely related to the double Poisson structure on
TA \mathbbDer(A){T_{A} \mathbb{D}{\rm er}(A)} introduced by Van den Bergh. Specifically, we prove that gr
D(F(A)) @ F(TA(\mathbbDer(A)),{\mathcal{D}(\mathcal{F}(A))\cong\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)),} provided the algebra A is smooth. Our construction is based on replacing vector spaces by the new symmetric monoidal category of wheelspaces. The Fock space F(A){\mathcal{F}(A)} is a commutative algebra in this category (a “commutative wheelgebra”) which is a structure closely related to the notion of wheeled PROP. Similarly, we have Lie, Poisson, etc., wheelgebras.
In this language, D(F(A)){\mathcal{D}(\mathcal{F}(A))} becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper,
we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule
connection on
\mathbbDer(A){\mathbb{D}{\rm er}(A)} gives rise to a second-order (wheeled) differential operator, a noncommutative analogue of the Batalin-Vilkovisky (BV) operator,
that makes
F(TA(\mathbbDer(A))){\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)))} a BV wheelgebra. In the final section, we explain how the wheeled differential operators D(F(A)){\mathcal{D}(\mathcal{F}(A))} produce ordinary differential operators on the varieties of n-dimensional representations of A for all n ≥ 1. 相似文献
10.
Soit _boxclose{\mathcal V} un anneau de valuation discrète complet d’inégales caractéristiques (0, p), de corps résiduel parfait k, de corps des fractions K. Soient X une variété sur k, Y un ouvert de X. Nous prolongeons le théorème de pleine fidélité de Kedlaya de la manière suivante (en effet, nous ne supposons pas Y lisse): le foncteur canonique
F\text-Isoc f (Y,X/K) ? F\text-Isoc f (Y,Y/K) {F\text{-}\mathrm{Isoc} ^{\dag} (Y,X/K) \to F\text{-}\mathrm{Isoc} ^{\dag} (Y,Y/K) } est pleinement fidèle. Supposons à présent Y lisse. Nous construisons la catégorie Isoc ff (Y,X/K){\mathrm{Isoc} ^{\dag\dag} (Y,X/K) } des isocristaux partiellement surcohérents sur (Y, X) dont les objets sont certains D{\mathcal D} -modules arithmétiques. De plus, nous vérifions l’équivalence de catégories sp (Y,X),+: Isoc f (Y,X/K) @ Isoc ff (Y,X/K){{\rm sp} _{(Y,X),+}: \mathrm{Isoc} ^{\dag} (Y,X/K) \cong \mathrm{Isoc} ^{\dag\dag} (Y,X/K)} . 相似文献
11.
Martin Reiris 《Annales Henri Poincare》2010,10(8):1559-1604
Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume
V(k)=(\frac-k3)3Volg(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by
Vinf=(\frac-16Y(S))\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g
i
, K
i
)} satisfying: (i) k
i
= −3, (ii) Viˉ Vinf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q
0((g
i
, K
i
)) ≤ Λ, where Q
0 is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state
is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples
for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and
cosmologically normalized flow
([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)2g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,¥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E1)\tilde]=E1(([(g)\tilde],[(K)\tilde])) £ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E1=Q0+Q1{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ Vinf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}. 相似文献
12.
Marcos M. Alexandrino 《Geometriae Dedicata》2010,149(1):397-416
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})\}}. 相似文献
13.
Jordi Juan-Huguet 《Integral Equations and Operator Theory》2010,68(2):263-286
Let P be a linear partial differential operator with constant coefficients. For a weight function ω and an open subset Ω of
\mathbbRN{\mathbb{R}^N} , the class EP,{w}(W){\mathcal{E}_{P,\{\omega\}}(\Omega)} of Roumieu type involving the successive iterates of the operator P is considered. The completeness of this space is characterized in terms of the hypoellipticity of P. Results of Komatsu and Newberger-Zielezny are extended. Moreover, for weights ω satisfying a certain growth condition, this class coincides with a class of ultradifferentiable functions if and only if
P is elliptic. These results remain true in the Beurling case EP,(w)(W){\mathcal{E}_{P,(\omega)}(\Omega)}. 相似文献
14.
Let ${\mathcal {H}_{1}}Let H1{\mathcal {H}_{1}} and H2{\mathcal {H}_{2}} be separable Hilbert spaces, and let A ? B(H1), B ? B(H2){A \in \mathcal {B}(\mathcal {H}_{1}),\, B \in \mathcal {B}(\mathcal {H}_{2})} and C ? B(H2, H1){C \in \mathcal {B}(\mathcal {H}_{2},\, \mathcal {H}_{1})} be given operators. A necessary and sufficient condition is given for ${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)}${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)} to be a right (left) invertible operator for some X ? B(H1, H2){X \in \mathcal {B}(\mathcal {H}_{1},\, \mathcal {H}_{2})}. Furthermore, some related results are obtained. 相似文献
15.
For a locally compact group G, we present some characterizations for f{\phi}-contractibility of the Lebesgue–Fourier algebra LA(G){\mathcal{L}A(G)} endowed with convolution or pointwise product. 相似文献
16.
Let ${\mathbb {F}}Let
\mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U
k
(X) for an ergodic action
(Tg)g ? \mathbb Fw{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group
\mathbb Fw{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S1{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where
k £ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for
\mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle
to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case
k £ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic. 相似文献
17.
We propose an algorithm to sample and mesh a k-submanifold M{\mathcal{M}} of positive reach embedded in
\mathbbRd{\mathbb{R}^{d}} . The algorithm first constructs a crude sample of M{\mathcal{M}} . It then refines the sample according to a prescribed parameter e{\varepsilon} , and builds a mesh that approximates M{\mathcal{M}} . Differently from most algorithms that have been developed for meshing surfaces of
\mathbbR 3{\mathbb{R} ^3} , the refinement phase does not rely on a subdivision of
\mathbbR d{\mathbb{R} ^d} (such as a grid or a triangulation of the sample points) since the size of such scaffoldings depends exponentially on the
ambient dimension d. Instead, we only compute local stars consisting of k-dimensional simplices around each sample point. By refining the sample, we can ensure that all stars become coherent leading
to a k-dimensional triangulated manifold [^(M)]{\hat{\mathcal{M}}} . The algorithm uses only simple numerical operations. We show that the size of the sample is O(e-k){O(\varepsilon ^{-k})} and that [^(M)]{\hat{\mathcal{M}}} is a good triangulation of M{\mathcal{M}} . More specifically, we show that M{\mathcal{M}} and [^(M)]{\hat{\mathcal{M}}} are isotopic, that their Hausdorff distance is O(e2){O(\varepsilon ^{2})} and that the maximum angle between their tangent bundles is O(e){O(\varepsilon )} . The asymptotic complexity of the algorithm is T(e) = O(e-k2-k){T(\varepsilon) = O(\varepsilon ^{-k^2-k})} (for fixed M, d{\mathcal{M}, d} and k). 相似文献
18.
Let ${\mathcal{M}_g}Let Mg{\mathcal{M}_g} denote the moduli space of compact Riemann surfaces of genus g and let Ag{\mathcal{A}_g} be the moduli space of principally polarized abelian varieties of dimension g. Let J : Mg ? Ag{J : \mathcal{M}_g \rightarrow \mathcal{A}_g} be the map which associates to a Riemann surface its Jacobian. The map J is injective, and the image Jg : = J(Mg){\mathcal{J}_g := J(\mathcal{M}_g)} is contained in a proper subvariety of Ag{\mathcal{A}_g} when g ≥ 4. The classical and long-studied Schottky problem is to characterize the Jacobian locus Jg{\mathcal{J}_g} in Ag{\mathcal{A}_g}. In this paper we address a large scale version of this problem posed by Farb and called the coarse Schottky problem: What does Jg{\mathcal{J}_g} look like “from far away”, or how “dense” is Jg{\mathcal{J}_g} in the sense of coarse geometry? The large scale geometry of Ag{\mathcal{A}_g} is encoded in its asymptotic cone, Cone¥(Ag){{\rm Cone}_\infty(\mathcal{A}_g)}, which is a Euclidean simplicial cone of real dimension g. Our main result asserts that the Jacobian locus Jg{\mathcal{J}_g} is “coarsely dense” in Ag{\mathcal{A}_g}, which implies that the subset of Cone¥(Ag){{\rm Cone}_\infty(\mathcal{A}_g)} determined by Jg{\mathcal{J}_g} actually coincides with this cone. The proof shows that the Jacobian locus of hyperelliptic curves is coarsely dense in Ag{\mathcal{A}_g} as well. We also study the boundary points of the Jacobian locus Jg{\mathcal{J}_g} in Ag{\mathcal{A}_g} and in the Baily–Borel and the Borel–Serre compactification. We show that for large genus g the set of boundary points of Jg{\mathcal{J}_g} in these compactifications is “small”. 相似文献
19.
Wolfgang M. Ruppert 《Archiv der Mathematik》1999,72(4):278-281
We give an elementary argument for the well known fact that the endomorphism algebra
End(A)?\Bbb Q {\rm {End}}(A)\otimes {\Bbb Q } of a simple complex abelian surface A can neither be an imaginary quadratic field nor a definite quaternion algebra. Another consequence of our argument is that a two-dimensional complex torus T with
\Bbb Q (?d)\hookrightarrow End\Bbb Q (T){\Bbb Q }(\sqrt {d})\hookrightarrow {\rm{End_{{\Bbb Q }}}}(T) where
\Bbb Q (?d){\Bbb Q }(\sqrt {d}) is real quadratic, is algebraic. 相似文献
20.
Let ${\Gamma < {\rm SL}(2, {\mathbb Z})}Let
G < SL(2, \mathbb Z){\Gamma < {\rm SL}(2, {\mathbb Z})} be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors
v0, w0 ? \mathbb Z2 \ {0}{v_{0}, w_{0} \in \mathbb {Z}^{2} \, {\backslash} \, \{0\}}. We consider the set S{\mathcal {S}} of all integers occurring in áv0g, w0?{\langle v_{0}\gamma, w_{0}\rangle}, for g ? G{\gamma \in \Gamma} and the usual inner product on
\mathbb R2{\mathbb {R}^2}. Assume that the critical exponent δ of Γ exceeds 0.99995, so that Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates
and Gamburd’s 5/6-th spectral gap in infinite-volume, we show that S{\mathcal {S}} contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we
show that the exceptional set
\mathfrak E(N){\mathfrak {E}(N)} of integers |n| < N which are locally admissible (n ? S (mod q) for all q 3 1){(n \in \mathcal {S} \, \, ({\rm mod} \, q) \, \, {\rm for\,all} \,\, q \geq 1)} but fail to be globally represented, n ? S{n \notin \mathcal {S}}, has a power savings,
|\mathfrak E(N)| << N1-e0{|\mathfrak {E}(N)| \ll N^{1-\varepsilon_{0}}} for some ${\varepsilon_{0} > 0}${\varepsilon_{0} > 0}, as N → ∞. 相似文献