Abstract:
The Aharonov-Bohm effect is a fundamental and controversial issue in physics. At stake are what are the fundamental electromagnetic quantities in quantum physics, if magnetic fields can act at a distance on charged particles and if the magnetic potentials have a real physical significance. From the experimental side the issues were settled by the remarkable experiments of Tonomura et al. in 1982 and 1986 with toroidal magnets that gave a strong experimental evidence of the physical existence of the Aharonov-Bohm effect, and by the recent experiment of Caprez et al. in 2007 that shows that the results of these experiments can not be explained by a force. The Aharonov-Bohm Ansatz of 1959 predicts the results of the experiments of Tonomura et al. and of Caprez et al. In 2009 we gave the first rigorous proof that the Aharonov-Bohm Ansatz is a good approximation to the exact solution for toroidal magnets under the conditions of the experiments of Tonomura et al.. In this paper we prove that our results do not depend on the particular geometry of the magnets and on the velocities of the incoming electrons used on the experiments, and on the gaussian shape of the wave packets used to obtain our quantitative error bound. We consider a general class of magnets that are a finite union of handle bodies. We formulate the Aharonov-bohm Ansatz that is appropriate to this general case and we prove that the exact solution to the Schroedinger equation is given by the Aharonov-Bohm Ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by $v^\rho, 0 < \rho <1$, with $v$ the velocity. The results of Tonomura et al., of Caprez et al., our previous results and the results of this paper give a firm experimental and theoretical basis to the existence of the Aharonov-Bohm effect and to its quantum nature.

Abstract:
We obtain high-velocity estimates with error bounds for the scattering operator of the Schr\"odinger equation in three dimensions with electromagnetic potentials in the exterior of bounded obstacles that are handlebodies. A particular case is a finite number of tori. We prove our results with time-dependent methods. We consider high-velocity estimates where the direction of the velocity of the incoming electrons is kept fixed as its absolute value goes to infinity. In the case of one torus our results give a rigorous proof that quantum mechanics predicts the interference patterns observed in the fundamental experiments of Tonomura et al. that gave a conclusive evidence of the existence of the Aharonov-Bohm effect using a toroidal magnet. We give a method for the reconstruction of the flux of the magnetic field over a cross-section of the torus modulo $2\pi$. Equivalently, we determine modulo $2\pi$ the difference in phase for two electrons that travel to infinity, when one goes inside the hole and the other outside it. For this purpose we only need the high-velocity limit of the scattering operator for one direction of the velocity of the incoming electrons. When there are several tori -or more generally handlebodies- the information that we obtain in the fluxes, and on the difference of phases, depends on the relative position of the tori and on the direction of the velocities when we take the high-velocity limit of the incoming electrons. For some locations of the tori we can determine all the fluxes modulo 2$\pi$ by taking the high-velocity limit in only one direction. We also give a method for the unique reconstruction of the electric potential and the magnetic field outside the handlebodies from the high-velocity limit of the scattering operator.

Abstract:
We study the Aharonov-Bohm effect under the conditions of the Tonomura et al. experiments, that gave a strong evidence of the physical existence of the Aharonov-Bohm effect, and we give the first rigorous proof that the classical Ansatz of Aharonov and Bohm is a good approximation to the exact solution of the Schroedinger equation. We provide a rigorous, quantitative, error bound for the difference in norm between the exact solution and the approximate solution given by the Aharonov-Bohm Ansatz. Our error bound is uniform in time. Using the experimental data, we rigorously prove that the results of the Tonomura et al. experiments, that were predicted by Aharonov and Bohm, actually follow from quantum mechanics. Furthermore, our results show that it would be quite interesting to perform experiments for intermediate size electron wave packets (smaller than the ones used in the Tonomura et al. experiments, that were much larger than the magnet) whose variance satisfies appropriate lower and upper bounds that we provide. One could as well take a larger magnet. In this case, the interaction of the electron wave packet with the magnet is negligible -the probability that the electron wave packet interacts with the magnet is smaller than $10^{-199}$- and, moreover, quantum mechanics predicts the results observed by Tonomura et al. with an error bound smaller than $10^{-99}$, in norm. Our error bound has a physical interpretation. For small variances it is due to Heisenberg's uncertainty principle and for large variances to the interaction with the magnet.

Abstract:
We study obstacle scattering for the Klein-Gordon equation in the presence of long-range magnetic potentials. We extend our previous results for the Klein-Gordon equation to the long-range case and bring to the relativistic scenario the results we previously proved for high-momenta long-range scattering for the Schr\"odinger equation. Interestingly, we show that there are important differences between relativistic and non-relativistic scattering concerning long-range. In particular, we prove that the electric potential can be recovered without assuming that we know the long-range part of the magnetic potential, which has to be supposed in the non-relativistic case. We prove that the electric potential and the magnetic field can be recovered from the high momenta limit of the scattering operator, as well as fluxes modulo $2 \pi $ around handles of the obstacle. Moreover, we prove that, for every $\hat{\mathbf v} \in \mathbb{S}^2$, $ A_\infty(\hat{\mathbf v}) + A_\infty(-\hat{\mathbf v})$ can be reconstructed, where $A_\infty$ is the long-range part of the magnetic potential. We additionally give a simple formula of the high momenta limit of the scattering operator in terms of magnetic fluxes over handles of the obstacle and long-range magnetic fluxes at infinity, that we introduce in this paper. The appearance of these long-range magnetic fluxes is a new effect in scattering theory.

Abstract:
We analyze spin-0 relativistic scattering of charged particles propagating in the exterior, $\Lambda \subset \mathbb{R}^3$, of a compact obstacle $K \subset \mathbb{R}^3$. The connected components of the obstacle are handlebodies. The particles interact with an electro-magnetic field in $\Lambda$ and an inaccessible magnetic field localized in the interior of the obstacle (through the Aharonov-Bohm effect). We obtain high-momenta estimates, with error bounds, for the scattering operator that we use to recover physical information: We give a reconstruction method for the electric potential and the exterior magnetic field and prove that, if the electric potential vanishes, circulations of the magnetic potential around handles (or equivalently, by Stokes' theorem, magnetic fluxes over transverse sections of handles) of the obstacle can be recovered, modulo $2 \pi$. We additionally give a simple formula for the high-momenta limit of the scattering operator in terms of certain magnetic fluxes, in the absence of electric potential. If the electric potential does not vanish, the magnetic fluxes on the handles above referred can be only recovered modulo $\pi$ and the simple expression of the high-momenta limit of the scattering operator does not hold true.

Abstract:
We introduce a general class of long-range magnetic potentials and derive high velocity limits for the scattering operators in quantum mechanics, in the case of two dimensions. We analyze the high velocity limits in the presence of an obstacle and we uniquely reconstruct from them the electric potential and the magnetic field outside the obstacle. We also reconstruct the inaccessible magnetic fluxes produced by fields inside the obstacle modulo $2 \pi$. For every magnetic potential $A$ in our class we prove that its behavior at infinity ($A_\infty(\hat{\mathbf v}), \hat{\mathbf v} \in \mathbb{S}^1$) can be characterized in a natural way. Under very general assumptions we prove that $A_\infty(\hat{\mathbf v}) + A_\infty(- \hat{\mathbf v})$ can be uniquely reconstructed for every $\hat{\mathbf v} \in \mathbb{S}^1$. We characterize properties of the support of the magnetic field outside the obstacle that permit us to uniquely reconstruct $A_\infty(\hat{\mathbf v})$ either for all $ \hat{\mathbf v} \in \mathbb{S}^1$ or for $\hat{\mathbf v}$ in a subset of $\mathbb{S}^1$. We also give a wide class of magnetic fields outside the obstacle allowing us to uniquely reconstruct the total magnetic flux (and $A_\infty(\hat{\mathbf v})$ for all $ \hat{\mathbf v} \in \mathbb{S}^1$). This is relevant because, as it is well-known, in general the scattering operator (even if is known for all velocities or energies) does not define uniquely the total magnetic flux (and $A_\infty(\hat{\mathbf v})$ ). We analyze additionally injectivity (i.e., uniqueness without giving a method for reconstruction) of the high velocity limits of the scattering operator with respect to $A_\infty(\hat{\mathbf v})$. Assuming that the magnetic field outside the obstacle is not identically zero, we provide a class of magnetic potentials for which injectivity is valid.

Abstract:
Tabled evaluation is a recognized and powerful technique that overcomes some limitations of traditional Prolog systems in dealing with recursion and redundant sub-computations. We can distinguish two main categories of tabling mechanisms: suspension-based tabling and linear tabling. While suspension-based mechanisms are considered to obtain better results in general, they have more memory space requirements and are more complex and harder to implement than linear tabling mechanisms. Arguably, the SLDT and DRA strategies are the two most successful extensions to standard linear tabled evaluation. In this work, we propose a new strategy, named DRS, and we present a framework, on top of the Yap system, that supports the combination of all these three strategies. Our implementation shares the underlying execution environment and most of the data structures used to implement tabling in Yap. We thus argue that all these common features allows us to make a first and fair comparison between these different linear tabling strategies and, therefore, better understand the advantages and weaknesses of each, when used solely or combined with the others.

Abstract:
Multi-threading is currently supported by several well-known Prolog systems providing a highly portable solution for applications that can benefit from concurrency. When multi-threading is combined with tabling, we can exploit the power of higher procedural control and declarative semantics. However, despite the availability of both threads and tabling in some Prolog systems, the implementation of these two features implies complex ties to each other and to the underlying engine. Until now, XSB was the only Prolog system combining multi-threading with tabling. In XSB, tables may be either private or shared between threads. While thread-private tables are easier to implement, shared tables have all the associated issues of locking, synchronization and potential deadlocks. In this paper, we propose an alternative view to XSB's approach. In our proposal, each thread views its tables as private but, at the engine level, we use a common table space where tables are shared among all threads. We present three designs for our common table space approach: No-Sharing (NS) (similar to XSB's private tables), Subgoal-Sharing (SS) and Full-Sharing (FS). The primary goal of this work was to reduce the memory usage for the table space but, our experimental results, using the YapTab tabling system with a local evaluation strategy, show that we can also achieve significant reductions on running time.

Abstract:
A critical component in the implementation of a concurrent tabling system is the design of the table space. One of the most successful proposals for representing tables is based on a two-level trie data structure, where one trie level stores the tabled subgoal calls and the other stores the computed answers. In this work, we present a simple and efficient lock-free design where both levels of the tries can be shared among threads in a concurrent environment. To implement lock-freedom we took advantage of the CAS atomic instruction that nowadays can be widely found on many common architectures. CAS reduces the granularity of the synchronization when threads access concurrent areas, but still suffers from low-level problems such as false sharing or cache memory side-effects. In order to be as effective as possible in the concurrent search and insert operations over the table space data structures, we based our design on a hash trie data structure in such a way that it minimizes potential low-level synchronization problems by dispersing as much as possible the concurrent areas. Experimental results in the Yap Prolog system show that our new lock-free hash trie design can effectively reduce the execution time and scale better than previous designs.