We develop an analogue of the classical Scott analysis for metric structures and infinitary continuous logic. Among our results are the existence of Scott sentences for metric structures and a version of the López-Escobar theorem. We also derive some descriptive set theoretic consequences: most notably, that isomorphism on a class of separable structures is a Borel equivalence relation iff their Scott rank is uniformly bounded below ω 1 . Finally, we apply our methods to study the Gromov-Hausdorff distance between metric spaces and the Kadets distance between Banach spaces, showing that the set of spaces with distance 0 to a fixed space is a Borel set.

In this note we study the structure of Lipschitz-free Banach spaces. We show that every Lipschitz-free Banach space over an infinite metric space contains a complemented copy of ℓ1. This result has many consequences for the structure of Lipschitz-free Banach spaces. Moreover, we give an example of a countable compact metric space K such that F(K) is not isomorphic to a subspace of L1 and we show that whenever M is a subset of R n , then F(M ) is weakly sequentially complete; in particular, c0 does not embed into F(M ).2010 Mathematics Subject Classification. 46B03, 54E35.

No abstract

It is conjectured that the component wave functions of the Rabi Hamiltonian in Bargmann's Hilbert space are terminating series of spheroidal wave functions and generalized spheroidal wave functions of Leitner and Meixner. Numerical calculations strongly support the conjecture.PACS numbers 03.65.Ge, 02.90. + pThe Rabi Hamiltonian has isolated exact solutions for particular values of the interaction constant. 1 In Bargmann's Hilbert space of analytical functions 2 " 4 the component wave functions are, in this case, terminating series of elementary transcendental functions. 1,5 " 7 This property is due to similarities in the pole structure of the differential equations defining the expansion functions and the differential equations of the component wave functions for the particular values of the interaction constant. We asked ourselves whether this observation could be a guide to more complicated but possibly known expansion functions, which allow for terminating expansions of the wave functions in the general case. By applying these principles (and a fair amount of intuition) we have in fact been able to guess the expansion functions and the expansion. We are, however, unable to produce a mathematical proof. However, we have a vast amount of numerical evidence, which, without any exception, supports our conjecture.The Rabi Hamiltonian in BargmamVs method 1,4 is a linear first-order matrix differential operator,whose eigenvalues A, in the excited state / (/ = 0,1,2 . . . ) are determined by the requirement that the up and down components of the wave functions */*»>> -(^/V2)^ + 1 / 2^/ ->(^)| T } + (f/V2)-* + 1 / 2 // (m) (f )l I > (2)(m= ± y) belong to the space of entire functions. We introduce a new independent variable z = y£ 2 , insert (1) and (2) in the Schrodinger equation, and collect the spin-up and -down components. We then obtain the following system of differential equations: d z-/ m) (z) dz (< \{m + \})4>l m Hz) + K ( z -m + l/2. 1 f \m _±U-m + 1/2 )f i (m Hz) + z+ m+ v 2^-rf m Hz) dz" = 0,Here A, = 2€/ + 1 = Vj + y ~ 2* 2and v t is the baseline parameter introduced by Judd. 8 For m = -y Eqs. (3) and (4) are identical with (2.28) and (2.29), for m = + y with (2.28a) and (2.29a), of Ref. 1. Equations (3) and (4) have an irregular singularity at infinity and two regular singularities at Z = K 2 (exponents 0 and v t ) and z = 0 (exponents 0 and -y). Therefore, this singularity is elementary in Ince's classification. 9 The conjecture is that the functions <£/ m) (z), f^m ) (z) can be expanded in / + 4 solutions H> 2 (3) (],V\A\Z) of a second-order differential equation with the same location and character of the singular points. The differential equation is given by dz 2 wPHz) + J±L + ± Z -K* ^2 (3) (Z) + dz K 2 -+• (Z~K 2 ) z\z Wj , <3) (z) = 0,

Abstract. We prove that the Lipschitz-free space over a separable ultrametric space has a monotone Schauder basis and is isomorphic to 1. This extends results of A. Dalet using an alternative approach.Mathematics Subject Classification. 46B03, 46B15, 54E35.

We find general conditions under which Lipschitz-free spaces over metric spaces are isomorphic to their infinite direct 1sum and exhibit several applications. As examples of such applications we have that Lipschitz-free spaces over balls and spheres of the same finite dimensions are isomorphic, that the Lipschitz-free space over Z d is isomorphic to its 1 -sum, or that the Lipschitz-free space over any snowflake of a doubling metric space is isomorphic to 1 . Moreover, following new ideas of Bruè et al. from [10] we provide an elementary self-contained proof that Lipschitz-free spaces over doubling metric spaces are complemented in Lipschitz-free spaces over their superspaces and they have BAP. Everything, including the results about doubling metric spaces, is explored in the more comprehensive setting of p-Banach spaces, which allows us to appreciate the similarities and differences of the theory between the cases p < 1 and p = 1.

This paper initiates the study of the structure of a new class of p-Banach spaces, 0 < p < 1, namely the Lipschitz free p-spaces (alternatively called Arens-Eells p-spaces) F p (M) over pmetric spaces. We systematically develop the theory and show that some results hold as in the case of p = 1, while some new interesting phenomena appear in the case 0 < p < 1 which have no analogue in the classical setting. For the former, we, e.g., show that the Lipschitz free p-space over a separable ultrametric space is isomorphic to p for all 0 < p ≤ 1. On the other hand, solving a problem by the first author and N. Kalton, there are metric spaces N ⊂ M such that the natural embedding from F p (N ) to F p (M) is not an isometry.

We develop tools for proving isomorphisms of normed spaces of Lipschitz functions over various doubling metric spaces and Banach spaces. In particular, we show that LipMore generally, we e.g. show that Lip 0 (Γ) ≃ Lip 0 (G), where Γ is from a large class of finitely generated nilpotent groups and G is its Mal'cev closure; or that Lip 0 (ℓp) ≃ Lip 0 (Lp), for all 1 ≤ p < ∞.We leave a large area for further possible research.2010 Mathematics Subject Classification. 46B03 (primary), and 22E40 (secondary).