Embedding of Besov Spaces and the Volterra Integral Operator

The boundedness and compactness of the inclusion mapping from Besov spaces to tent spaces are studied in this paper. Meanwhile, the boundedness, compactness, and essential norm of the Volterra integral operator T g from Besov spaces to a class of general function spaces are also investigated.

Embedding of Dirichlet type spaces into tent spaces and Volterra operators

In this paper, we study the boundedness and compactness of the inclusion mapping from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to tent spaces. Meanwhile, the boundedness, compactness, and essential norm of Volterra integral operators from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to general function spaces are also investigated.

Supplemental Material: Sustainable densification of the deep crust

Materials and methods, Figures S1–S9 (fluid-inclusion mapping and point analyses of garnet and clinopyroxene with FTIR, sensitivity analyses for the two numerical models, results of the diffusion-reaction model in open system conditions, X-ray mapping of amphibole inclusions in clinopyroxene, evolution in space of the plagioclase composition), Tables S1 and S2 (chemical composition of the main minerals, and local bulk composition used for numerical modelling), and Movies S1 and S2 (results in 2-D of the model coupling reaction, fluid flow, and deformation).<br>

Supplemental Material: Sustainable densification of the deep crust

Materials and methods, Figures S1–S9 (fluid-inclusion mapping and point analyses of garnet and clinopyroxene with FTIR, sensitivity analyses for the two numerical models, results of the diffusion-reaction model in open system conditions, X-ray mapping of amphibole inclusions in clinopyroxene, evolution in space of the plagioclase composition), Tables S1 and S2 (chemical composition of the main minerals, and local bulk composition used for numerical modelling), and Movies S1 and S2 (results in 2-D of the model coupling reaction, fluid flow, and deformation).<br>

On the fundamental group of a Lie semigroup

The simplest type of Lie semigroups are closed convex cones in finite dimensional vector spaces. In general one defines a Lie semigroup to be a closed subsemigroup of a Lie group which is generated by one-parameter semigroups. If W is a closed convex cone in a vector space V, then W is convex and therefore simply connected. A similar statement for Lie semigroups is false in general. There exist generating Lie semigroups in simply connected Lie groups which are not simply connected (Example 1.15). To find criteria for cases when this is true, one has to consider the homomorphisminduced by the inclusion mapping i:S→G, where S is a generating Lie semigroup in the Lie group G. Our main results concern the description of the image and the kernel of this mapping. We show that the image is the fundamental group of the largest covering group of G, into which S lifts, and that the kernel is the fundamental group of the inverse image of 5 in the universal covering group G. To get these results we construct a universal covering semigroup S of S. If j: H(S): = S ∩ S-1 →S is the inclusion mapping of the unit group of S into S, then it turns out that the kernel of the induced mappingmay be identfied with the fundamental group of the unit group H(S)of S and that its image corresponds to the intersection H(S)0 ⋂π1(S), where π1(s) is identified with a central subgroup of S.

SOLITON EQUATIONS AND FREE FERMIONS ON RIEMANN SURFACES

Operator formalism for free fermions on general Riemann surfaces is developed based on the Sato theory of the Kadomtsev-Petviashvilli hierarchy. Moduli spaces of various genuses are included in Sato’s universal Grassmannian manifold (UGM), and we have investigated explicitly behaviour of this inclusion mapping near boundaries of moduli spaces.

Another proof that the inclusion I: ι1→ι2 is 0-radonifying

Kwapień(2) and Schwartz (3) have shown that the inclusion mapping I: ι1→ι2 is 0-radonifying. We shall give another proof of this, using an inequality due to Khint-chine(1).