Measurement of and the dipion contribution to the muon anomaly with the KLOE detector

We have measured the cross section σ(e⁺e⁻→π⁺π⁻γ(γ)) at DAΦNE, the Frascati -factory, using events with initial state radiation photons emitted at small angle and inclusive of final state radiation. We present the analysis of a new data set corresponding to an integrated luminosity of 240 pb⁻¹. We have achieved a reduced systematic uncertainty with respect to previously published KLOE results. From the cross section we obtain the pion form factor and the contribution to the muon magnetic anomaly from two-pion states in the mass range 0.592<M_ππ<0.975 GeV. For the latter we find Δ^ππa_μ=(387.2±0.5_stat±2.4_exp±2.3_th)×10⁻¹⁰.     

On the Blaschke isoparametric hypersurfaces in the unit sphere

Given an immersed submanifold x ： M^M → S^n in the unit sphere S^n without umbilics, a Blaschke eigenvalue of x is by definition an eigenvalue of the Blaschke tensor of x. x is called Blaschke isoparametric if its Mobius form vanishes identically and all of its Blaschke eigenvalues are constant. Then the classification of Blaschke isoparametric hypersurfaces is natural and interesting in the MSbius geometry of submanifolds. When n = 4, the corresponding classification theorem was given by the authors. In this paper, we are able to complete the corresponding classification for n = 5. In particular, we shall prove that all the Blaschke isoparametric hypersurfaces in S^5 with more than two distinct Blaschke eigenvalues are necessarily Mobius isoparametric.     

Normal forms for real quadratic forms

We investigate the non-diagonal normal forms of a quadratic form on , in particular for n = 3. For this case it is shown that the set of normal forms is the closure of a 5-dimensional submanifold in the 6-dimensional Grassmannian of 2-dimensional subspaces of . Received: 27 June 2008     

Solutions to a family of matrix equations by using the Kronecker matrix polynomials

Closed form solutions to a family of generalized Sylvester matrix equation in form of are given by using the so-called Kronecker matrix polynomials. It is found that the structure of the solutions is independent of the orders ?,ψ and φ. This type of uniform closed form solutions includes our early results as special cases. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in linear systems.     

Differential Posets and Smith Normal Forms

We conjecture a strong property for the up and down maps U and D in an r-differential poset: DU + tI and UD + tI have Smith normal forms over . In particular, this would determine the integral structure of the maps U, D, UD, DU, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice Y F studied by Okada and its r-differential generalizations Z(r), as well as verifying many of its consequences for Young’s lattice Y and the r-differential Cartesian products Y ^r .     

Classification of quasi-minimal slant surfaces in Lorentzian complex space forms

From J-action point of views, slant surfaces are the simplest and the most natural surfaces of a (Lorentzian) Kähler surface (\(\tilde M,\tilde g\), J). Slant surfaces arise naturally and play some important roles in the studies of surfaces of Kähler surfaces (see, for instance, [13]). In this article, we classify quasi-minimal slant surfaces in the Lorentzian complex plane C ₁ ² . More precisely, we prove that there exist five large families of quasi-minimal proper slant surfaces in C ₁ ² . Conversely, quasi-minimal slant surfaces in C ₁ ² are either Lagrangian or locally obtained from one of the five families. Moreover, we prove that quasi-minimal slant surfaces in a non-flat Lorentzian complex space form are Lagrangian.     

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,μ) be an abstract Wiener space and let D_V:=VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:=V^*BV in H and in , as well as the realisations of the operators and in L^p(E,μ) and respectively, where 1<p<∞. Our main result asserts that the following four assertions are equivalent:

(1) with for ;

(2) admits a bounded H^∞-functional calculus on ;

(3) with for ;

(4) admits a bounded H^∞-functional calculus on .

Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V=I, ). A one-sided version of (1)–(4), giving L^p-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C₀-contraction semigroup on a Hilbert space H and let −L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an L^p-domain characterisation for the operator L.

Strong continuity of generalized Feynman-Kac semigroups: Necessary and sufficient conditions

Let (E,D(E)) be a strongly local, quasi-regular symmetric Dirichlet form on L²(E;m) and ((Xt)_t?0,(Px)_x∈E) the diffusion process associated with (E,D(E)). For u∈De(E), u has a quasi-continuous version and has Fukushima's decomposition: , where is the martingale part and is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman-Kac semigroup defined by , t?0. Two necessary and sufficient conditions for to be strongly continuous are obtained by considering the quadratic form (Qu,Db(E)), where Qu(f,f):=E(f,f)+E(u,f²) for f∈Db(E), and the energy measure μ〈u〉 of u, respectively. An example is also given to show that is strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant (cf. Definition 4.5).     

Symmetry Classification Using Noncommutative Invariant Differential Operators

Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group G_f or, equivalently, of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated overdetermined "defining system" of differential equations. The usual computer classification method which applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest due to the excessive size of expressions generated in intermediate computations. We provide an alternative classification method which exploits the knowledge of an equivalence group G preserving the class. A noncommutative differential elimination procedure due to Lemaire, Reid, and Zhang, where each step of the procedure is invariant under G, can be applied and an existence and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F. The method is applied to a class of nonlinear diffusion convection equations v_x = u, v_t = B(u) u_x - K(u) which is invariant under a large but easily determined equivalence group G. In this example the complexity of the calculations is much reduced by the use of G-invariant differential operators.     

Free and nonfree Voronoi polyhedra

The Voronoi polyhedron of some point v of a translation lattice is the closure of the set of points in space that are closer to v than to any other lattice points. Voronoi polyhedra are a special case of parallelohedra, i.e., polyhedra whose parallel translates can fill the entire space without gaps and common interior points. The Minkowski sum of a parallelohedron with a segment is not always a parallelohedron. A parallelohedron P is said to be free along a vector e if the sum of P with a segment of the line spanned by e is a parallelohedron. We prove a theorem stating that if the Voronoi polyhedron P _v (f) of a quadratic form f is free along some vector, then the Voronoi polyhedron P _v (g) of each form g lying in the closure of the L-domain of f is also free along some vector. For the dual root lattice E ₆*, and the infinite series of lattices D _2m ⁺ , m ≥ 4, we prove that their Voronoi polyhedra are nonfree in all directions.