Isotropic contact forces in arbitrary representation: heterogeneous few-body problems and low dimensions

The Bethe-Peierls asymptotic approach which models pairwise short-range forces by contact conditions is introduced in arbitrary representation for spatial dimensions less than or equal to 3. The formalism is applied in various situations and emphasis is put on the momentum representation. In the presence of a transverse harmonic confinement, dimensional reduction toward two-dimensional (2D) or one-dimensional (1D) physics is derived within this formalism. The energy theorem relating the mean energy of an interacting system to the asymptotic behavior of the one-particle density matrix illustrates the method in its second quantized form. Integral equations that encapsulate the Bethe-Peierls contact condition for few-body systems are derived. In three dimensions, for three-body systems supporting Efimov states, a nodal condition is introduced in order to obtain universal results from the Skorniakov Ter-Martirosian equation and the Thomas collapse is avoided. Four-body bound state eigenequations are derived and the 2D '3+1' bosonic ground state is computed as a function of the mass ratio

C Language Extensions for Hybrid CPU/GPU Programming with StarPU

Modern platforms used for high-performance computing (HPC) include machines with both general-purpose CPUs, and "accelerators", often in the form of graphical processing units (GPUs). StarPU is a C library to exploit such platforms. It provides users with ways to define "tasks" to be executed on CPUs or GPUs, along with the dependencies among them, and by automatically scheduling them over all the available processing units. In doing so, it also relieves programmers from the need to know the underlying architecture details: it adapts to the available CPUs and GPUs, and automatically transfers data between main memory and GPUs as needed. While StarPU's approach is successful at addressing run-time scheduling issues, being a C library makes for a poor and error-prone programming interface. This paper presents an effort started in 2011 to promote some of the concepts exported by the library as C language constructs, by means of an extension of the GCC compiler suite. Our main contribution is the design and implementation of language extensions that map to StarPU's task programming paradigm. We argue that the proposed extensions make it easier to get started with StarPU,eliminate errors that can occur when using the C library, and help diagnose possible mistakes. We conclude on future work

Ramsey-type graph coloring and diagonal non-computability

A function is diagonally non-computable (d.n.c.) if it diagonalizes against the universal partial computable function. D.n.c. functions play a central role in algorithmic randomness and reverse mathematics. Flood and Towsner asked for which functions h, the principle stating the existence of an h-bounded d.n.c. function (DNR_h) implies the Ramsey-type K\"onig's lemma (RWKL). In this paper, we prove that for every computable order h, there exists an~$\omega$-model of DNR_h which is not a not model of the Ramsey-type graph coloring principle for two colors (RCOLOR2) and therefore not a model of RWKL. The proof combines bushy tree forcing and a technique introduced by Lerman, Solomon and Towsner to transform a computable non-reducibility into a separation over omega-models.Comment: 18 page

Controlling iterated jumps of solutions to combinatorial problems

Among the Ramsey-type hierarchies, namely, Ramsey's theorem, the free set, the thin set and the rainbow Ramsey theorem, only Ramsey's theorem is known to collapse in reverse mathematics. A promising approach to show the strictness of the hierarchies would be to prove that every computable instance at level n has a low_n solution. In particular, this requires effective control of iterations of the Turing jump. In this paper, we design some variants of Mathias forcing to construct solutions to cohesiveness, the Erdos-Moser theorem and stable Ramsey's theorem for pairs, while controlling their iterated jumps. For this, we define forcing relations which, unlike Mathias forcing, have the same definitional complexity as the formulas they force. This analysis enables us to answer two questions of Wei Wang, namely, whether cohesiveness and the Erdos-Moser theorem admit preservation of the arithmetic hierarchy, and can be seen as a step towards the resolution of the strictness of the Ramsey-type hierarchies.Comment: 32 page

Modified scalar product for the effective range approach: the molecular contribution

The modified scalar product which permits to restore the self-adjoint character of the effective range approach is derived for one-channel contact models where a more general internal structure is included. In the case of the effective range approach, the modified scalar product is interpreted in the light of a generic two-channel model for a narrow Feshbach resonance as a way to take into account implicitly the molecular contribution of the closed channel