We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in d ≥ 1 dimensions, for focusing and defocusing interactions. We present a new proof of existence of solutions that does not require the a priori bound on the spacetime norm, which was introduced in the work of Klainerman and Machedon, [19], and used in our earlier work, [6].

where T1, . . . , Tn, S1, . . . , Sn : [a, b] → R. (2) Suppose K 6= 0. Then we may consider each of the systems {T1, . . . , Tn}, {S1, . . . , Sn} linearly independent. Indeed, starting from an expression of kind (1), we consequently reduce the number of items while it is needed. Assume that we need to express the functions (2) in terms of K. In order to do it, we find such points (a proof of e...

In this survey, we review fundamental properties of coalgebras and their representation theory. Following J.A. Green we present the block theory of coalgebras using indecomposable injectives comodules. Using the cohom and cotensor functors we state Takeuchi-Morita equivalence and use it to sketch the proof of existence of “basic” coalgebras, due to the author and S. Montgomery. This leads to a ...

Abstract. The approach, referred to as ”monodromy transform”, provides some general base for solution of all known integrable space time symmetry reductions of Einstein equations for the case of pure vacuum gravitational fields, in the presence of gravitationally interacting massless fields, as well as for some string theory induced gravity models. In this communication we present the key point...

This review demonstrates the unique role of neutrino by discussing in detail physics and with neutrinos. We deal sources, oscillations, absolute masses, interactions, possible existence sterile neutrinos, theoretical implications. In addition, synergies other research fields are found, requirements to continue successful future, terms technological developments adequate infrastructures, stressed.

Generic absoluteness. Paul Cohen’s work in the 1960s struck a blow against the idea of mathematics as absolute truth. Cohen showed that there is a natural mathematical statement, namely the Continuum Hypothesis, whose truth cannot be proved or disproved in ZFC. (Gödel sentences were discovered earlier but are arguably not natural mathematical statements.) Cohen’s method, called forcing, enlarge...

The current paper proves the results announced in [6]. We isolate a new large cardinal concept, “remarkability.” Consistencywise, remarkable cardinals are between ineffable and ω-Erdös cardinals. They are characterized by the existence of “0-like” embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(R) absoluteness for...