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1.
Let F be a differentiable manifold endowed with an almost K?hler structure (J,ω), α a J-holomorphic action of a compact Lie group on F, and K a closed normal subgroup of which leaves ω invariant.
The purpose of this article is to introduce gauge theoretical invariants for such triples (F,α,K). The invariants are associated with moduli spaces of solutions of a certain vortex type equation on a Riemann surface Σ.
Our main results concern the special case of the triple
where αcan denotes the canonical action of on . We give a complex geometric interpretation of the corresponding moduli spaces of solutions in terms of gauge theoretical
quot spaces, and compute the invariants explicitly in the case r=1.
Proving a comparison theorem for virtual fundamental classes, we show that the full Seiberg–Witten invariants of ruled surfaces,
as defined in [OT2], can be identified with certain gauge theoretical Gromov–Witten invariants of the triple (Hom(ℂ,ℂ<
r
0),αcan, U(1)). We find the following formula for the full Seiberg–Witten invariant of a ruled surface over a Riemann surface of genus
g:
where [F] denotes the class of a fibre. The computation of the invariants in the general case r >1 should lead to a generalized Vafa-Intriligator formula for “twisted”Gromov–Witten invariants associated with sections
in Grassmann bundles.
Received: 22 February 2001 / Accepted: 16 January 2002 相似文献
2.
For operators with a discrete spectrum, {λ
j
2}, the counting function of λ
j
's, N (λ), trivially satisfies N ( λ+δ ) −N ( λ−δ ) =∑
j
δλ
j
((λ−δ,λ+δ]). In scattering situations the natural analogue of the discrete spectrum is given by resonances, λ
j
∈ℂ+, and of N (λ), by the scattering phase, s(λ). The relation between the two is now non-trivial and we prove that
where ωℂ+ is the harmonic measure of the upper of half plane and δ can be taken dependent on λ. This provides a precise high energy
version of the Breit–Wigner approximation, and relates the properties of s (λ) to the distribution of resonances close to the real axis.
Received: 16 October 1998 / Accepted: 28 January 1999 相似文献
3.
We construct approximate solutions to the time–dependent Schr?dinger?equation
for small values of ħ. If V satisfies appropriate analyticity and growth hypotheses and , these solutions agree with exact solutions up to errors whose norms are bounded by
for some C and γ>0. Under more restrictive hypotheses, we prove that for sufficiently small T
′, implies the norms of the errors are bounded by
for some C
′, γ′>0, and σ > 0.
Received: 7 January 1999 / Accepted: 30 April 1999 相似文献
4.
Oded Schramm Scott Sheffield David B. Wilson 《Communications in Mathematical Physics》2009,288(1):43-53
The conformal loop ensembles CLE
κ
, defined for 8/3 ≤ κ ≤ 8, are random collections of loops in a planar domain which are conjectured scaling limits of the O(n) loop models. We calculate the distribution of the conformal radii of the nested loops surrounding a deterministic point.
Our results agree with predictions made by Cardy and Ziff and by Kenyon and Wilson for the O(n) model. We also compute the expectation dimension of the CLE
κ
gasket, which consists of points not surrounded by any loop, to be
, which agrees with the fractal dimension given by Duplantier for the O(n) model gasket.
Partially supported by NSF grant DMS0403182. 相似文献
5.
The characters of the infinite symmetric group are extended to multiplicative positive definite functions on pair partitions by using an explicit representation due to Veršik and Kerov. The von Neumann algebra generated by the fields with f in an infinite dimensional real Hilbert space is infinite and the vacuum vector is not separating. For a family depending on an integer N< - 1 an ``exclusion principle' is found allowing at most ``identical particles' on the same state:
The algebras are type factors. Functors of white noise are constructed and proved to be non-equivalent for different values of N.
Received: 28 September 2001 / Accepted: 10 November 2001 Published online: 31 July 2002 相似文献
6.
Let A
1,…,A
N
be complex self-adjoint matrices and let ρ be a density matrix. The Robertson uncertainty principle
gives a bound for the quantum generalized covariance in terms of the commutators [A
h
,A
j
]. The right side matrix is antisymmetric and therefore the bound is trivial (equal to zero) in the odd case N=2m+1.
Let f be an arbitrary normalized symmetric operator monotone function and let 〈⋅,⋅〉
ρ,f
be the associated quantum Fisher information. Based on previous results of several authors, we propose here as a conjecture
the inequality
whose validity would give a non-trivial bound for any N∈ℕ using the commutators i[ρ,A
h
]. 相似文献
7.
Ricardo Weder 《Communications in Mathematical Physics》2000,215(2):343-356
In this paper we study the following nonlinear Schr?dinger equation on the line,
where f is real-valued, and it satisfies suitable conditions on regularity, on growth as a function of u and on decay as x→±∞. The generic potential, V, is real-valued and it is chosen so that the spectrum of consists of one simple negative eigenvalue and absolutely-continuous spectrum filling [0, ∞). The solutions to this equation
have, in general, a localized and a dispersive component. The nonlinear bound states, that bifurcate from the zero solution
at the energy of the eigenvalue of H, define an invariant center manifold that consists of the orbits of time-periodic localized solutions. We prove that all
small solutions approach a particular periodic orbit in the center manifold as t→±∞. In general, the periodic orbits are different for t→±∞. Our result implies also that the nonlinear bound states are asymptotically stable, in the sense that each solution with
initial data near a nonlinear bound state is asymptotic as t→±∞ to the periodic orbits of nearby nonlinear bound states that are, in general, different for t→±∞.
Received: 20 January 2000 / Accepted: 1 June 2000 相似文献
8.
Let
Let N be a positive integer and define Φ(N) as the number of matrices, C, which are products of A and B, where both A and B must occur, such that the trace, Tr(C)=N. It has been conjectured that
see [10]. In this note we consider the summatory function
and show that
Received: 4 January 2001 / Accepted: 10 April 2001 相似文献
9.
We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝ
n
. They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features.
The new examples include the instanton algebra and the NC-4-spheres S
4
θ. We construct the noncommutative algebras ?=C
∞ (S
4
θ) of functions on NC-spheres as solutions to the vanishing, ch
j
(e) = 0, j < 2, of the Chern character in the cyclic homology of ? of an idempotent e∈M
4 (?), e
2=e, e=e
*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the
corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S
3
θ distinct from quantum group deformations SU
q
(2) of SU (2).
We then construct the noncommutative geometry of S
θ
4 as given by a spectral triple ?, ℋ, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g
μν on S
4 whose volume form is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,
where <␣> is the projection on the commutant of 4 × 4 matrices.
We shall show how to construct the Dirac operator D on the noncommutative 4-spheres S
θ
4 so that the previous equation continues to hold without any change.
Finally, we show that any compact Riemannian spin manifold whose isometry group has rank r≥ 2 admits isospectral deformations to noncommutative geometries.
Received: 5 December 2000 / Accepted: 8 March 2001 相似文献
10.
Qionglei Chen Changxing Miao Zhifei Zhang 《Communications in Mathematical Physics》2007,275(3):861-872
We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition,
we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, namely
where Δ
j
is the frequency localization operator in the Littlewood-Paley decomposition. 相似文献
11.
We study the spectrum of the operator
generating an infinite-dimensional diffusion process Ξ (t), in space . Here ν is a “natural”Ξ (t)-invariant measure on which is a Gibbs distribution corresponding to a (formal) Hamiltonian H of an anharmonic crystal, with a value of the inverse temperature β > 0. For β small enough, we establish the existence of
an L-invariant subspace such that has a distinctive character related to a “quasi-particle” picture. In particular, has a Lebesgue spectrum separated from the rest of the spectrum of L and concentrated near a point κ1>0 giving the smallest non-zero eigenvalue of a limiting problem associated with β= 0.
An immediate corollary of our result is an exponentially fast L
2-convergence to equilibrium for the process Ξ(t) for small values of β.
Received: 6 October 1998 / Accepted: 9 April 1999 相似文献
12.
Let μ
0 be a probability measure on ℝ3 representing an initial velocity distribution for the spatially homogeneous Boltzmann equation for pseudo Maxwellian molecules.
As long as the initial energy is finite, the solution μ
t
will tend to a Maxwellian limit. We show here that if
, then instead, all of the mass “explodes to infinity” at a rate governed by the tail behavior of μ
0. Specifically, for L0, define
Let B
R
denote the centered ball of radius R. Then for every R,
The explicit rate is estimated in terms of the rate of divergence of η
L
. For example, if η
L
≥Const.L
s
, some s>0,
is bounded by a multiple of e
−[κ3s/(10+9s)]t
, where κ is the absolute value of the spectral gap in the linearized collision operator. Note that in this case, letting B
t
denote the ball of radius e
rt
for any r<κ
s/(10+9s), we still have
.
This result shows in particular that the necessary and sufficient condition for lim
t→∞
μ
t
to exist is that the initial data have finite energy. While the “explosion” of the mass towards infinity in the case of infinite
energy may seem to be intuitively clear, there seems not to have been any proof, even without the rate information that our
proof provides, apart from an analogous result, due to the authors, concerning the Kac equation. A class of infinite energy
eternal solutions of the Boltzmann equation have been studied recently by Bobylev and Cercignani. Our rate information is
shown here to provide a limit on the tails of such eternal solutions.
E. Carlen’s work is partially supported by U.S. National Science Foundation grant DMS 06-00037.
E. Gabetta’s and E. Regazzini’s work is partially supported by Cofin 2004 “Probleme matematici delle teorie cinetiche” (MIUR). 相似文献
13.
William Arveson 《Communications in Mathematical Physics》1997,187(1):19-43
An E
0-semigroup acting on is called pure if its tail von Neumann algebra is trivial in the sense that
We determine all pure E
0-semigroups which have a weakly continuous invariant state ω and which are minimal in an appropriate sense. In such cases the dynamics of the state space must stabilize
as follows: for every normal state ρ of there is convergence to equilibrium in the trace norm
A normal state ω with this property is called an absorbing state for α.
Such E
0-semigroups must be cocycle perturbations of CAR/CCR flows, and we develop systematic methods for constructing those perturbations which have absorbing states with prescribed
finite eigenvalue lists.
Received: 28 October 1996 / Accepted: 11
November 1996 相似文献
14.
We study integrable cocycles u(n,x) over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively
curved space Y, e.g. a Cartan–Hadamard space or a uniformly convex Banach space. It is proved that for any y∈Y and almost all x, there exist A≥ 0 and a unique geodesic ray γ (t,x) in Y starting at y such that
In the case where Y is the symmetric space GL
N
(ℝ)/O
N
(ℝ) and the cocycles take values in GL
N
(ℝ), this is equivalent to the multiplicative ergodic theorem of Oseledec.
Two applications are also described. The first concerns the determination of Poisson boundaries and the second concerns Hilbert-Schmidt
operators.
Received: 27 April 1999 / Accepted: 25 May 1999 相似文献
15.
Ricardo E. Gamboa Saraví 《General Relativity and Gravitation》2009,41(7):1459-1473
We present the exact solution of Einstein’s equation corresponding to a static and plane symmetric distribution of matter
with constant positive density located below z = 0. This solution depends essentially on two constants: the density ρ and a parameter κ. We show that these space–times finish down below at an inner singularity at finite depth. We show that for κ ≥ 0.3513 . . . the dominant energy condition is satisfied all over the space–time. We match this solution to the vacuum one
and compute the external gravitational field in terms of slab’s parameters. Depending on the value of κ, these slabs can be attractive, repulsive or neutral. In the first case, the space–time also finishes up above at an empty
repelling singular boundary. In the other cases, they turn out to be semi-infinite and asymptotically flat when z → ∞. We also find solutions consisting of joining an attractive slab and a repulsive one, and two neutral ones. We also discuss
how to assemble a “gravitational capacitor” by inserting a slice of vacuum between two such slabs. 相似文献
16.
Differential equations and maps are the most frequently studied examples of dynamical systems and may be considered as continuous
and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor-like fractal
subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate
framework. A new calculus calledF
α-calculus, is a natural calculus on subsetsF⊂ R of dimension α,0 < α ≤ 1. It involves integral and derivative of order α, calledF
α-integral andF
α-derivative respectively. TheF
α-integral is suitable for integrating functions with fractal support of dimension α, while theF
α-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function
occur naturally as solutions ofF
α-differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems.
We discuss construction and solutions of some fractal differential equations of the form
whereh is a vector field andD
F,t
α
is a fractal differential operator of order α in timet. We also consider some equations of the form
whereL is an ordinary differential operator in the real variablex, and(t,x) ∈F × Rn whereF is a Cantor-like set of dimension α.
Further, we discuss a method of finding solutions toF
α-differential equations: They can be mapped to ordinary differential equations, and the solutions of the latter can be transformed
back to get those of the former. This is illustrated with a couple of examples. 相似文献
17.
J. Swoboda 《Optics and Spectroscopy》2007,103(3):343-348
Given a compact, connected Lie group G with Lie algebra
. We discuss time-optimal control of bilinear systems of the form
where H
d
, H
j
∈
, U ∈ G, and the v
j
act as control variables. The case G = SU(2
n
) has found interesting applications to questions of time-optimal control of spin systems. In this context Eq. (I) describes
the dynamics of an n-particle system with fixed drift Hamiltonian H
d
, which is to be controlled by a number of exterior magnetic fields of variable strength, proportional to the parameters v
j
. The question of interest here is to transfer the system from a given initial state U
0 to a prescribed final state U
1 in least possible time. Denote by
the Lie algebra spanned by H
1, ..., H
m
, and by K the corresponding Lie subgroup of G. After reformulating the optimal control problem for system (I) in terms of an equivalent problem on the homogeneous space
G/K we discuss in detail time-optimal control strategies for system (I) in the case where G/K carries the structure of a Riemannian symmetric space.
The text submitted by the author in English. 相似文献
| ((I)) |
18.
For aL
∞
(ℝ+)∩L
1
(ℝ+) the truncated Bessel operator B
τ
(a) is the integral operator acting on L
2
[0,τ] with the kernel
where J
ν
stands for the Bessel function with ν>−1. In this paper we determine the asymptotics of the determinant det(I+B
τ
(a)) as τ→∞ for sufficiently smooth functions a for which a(x)≠1 for all x[0,∞). The asymptotic formula is of the form det(I+B
τ
(a))∼G
τ
E with certain constants G and E, and thus similar to the well-known Szeg?-Akhiezer-Kac formula for truncated Wiener-Hopf determinants.
Received: 23 April 2002 / Accepted: 25 September 2002 Published online: 24 January 2003
RID="*"
ID="*" Supported in part by NSF Grant DMS-9970879.
Communicated by J.L. Lebowitz 相似文献
19.
P. Contucci M. Degli Esposti C. Giardinà S. Graffi 《Communications in Mathematical Physics》2003,236(1):55-63
Let {E Σ (N)} ΣΣN be a family of |Σ N |=2 N centered unit Gaussian random variables defined by the covariance matrix C N of elements c N (Σ,τ):=Av(E Σ (N)E τ (N)) and the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N=N 1 +N 2 , and all pairs (Σ,τ)Σ N ×Σ N :
where π k (Σ),k=1,2 are the projections of ΣΣ N into Σ Nk . The condition is explicitly verified for the Sherrington-Kirkpatrick, the even p-spin, the Derrida REM and the Derrida-Gardner GREM models. 相似文献
20.
Dong Li 《Journal of statistical physics》2007,129(2):265-287
It has been shown in E and Li (Comm. Pure. Appl. Math., 2007, in press) that the Andersen dynamics is uniformly ergodic. Exponential convergence to the invariant measure is established
with an error bound of the form
where N is the number of particles, ν is the collision frequency and κ(ν)→const as ν→0. In this article we study the dependence on ν of the rate of convergence to equilibrium. In the one dimension and one particle case, we improve the error bound to be
In the d-dimension N-particle free-streaming case, it is proved that the optimal error bound is
It is also shown that as ν→∞, on the diffusive time scale, the Andersen dynamics converges to a Smoluchowski equation. 相似文献