The Power of Non-Determinism in Higher-Order Implicit Complexity

We investigate the power of non-determinism in purely functional programming languages with higher-order types. Specifically, we consider cons-free programs of varying data orders, equipped with explicit non-deterministic choice. Cons-freeness roughly means that data constructors cannot occur in function bodies and all manipulation of storage space thus has to happen indirectly using the call stack. While cons-free programs have previously been used by several authors to characterise complexity classes, the work on non-deterministic programs has almost exclusively considered programs of data order 0. Previous work has shown that adding explicit non-determinism to cons-free programs taking data of order 0 does not increase expressivity; we prove that this - dramatically - is not the case for higher data orders: adding non-determinism to programs with data order at least 1 allows for a characterisation of the entire class of elementary-time decidable sets. Finally we show how, even with non-deterministic choice, the original hierarchy of characterisations is restored by imposing different restrictions.Comment: pre-edition version of a paper accepted for publication at ESOP'1

An analysis of students\u27 perceptions of online multimedia tutorials to assist classroom instruction

Multimedia has become the new learning environment. As technology advances and becomes more prominent, traditional education needs to adapt and utilize all the benefits that can be achieved with computer-assisted instruction. Awareness of students\u27 attitudes toward multimedia-based education is critical in the evaluation of multimedia courses and development of multimedia-based curriculum. The purpose of this study was to examine students\u27 perceptions of multimedia-based instruction. Educational institutions are faced with the challenges of delivering valuable training options to meet the varying needs of students and staff; computer technology has become and will continually be integrated into the classroom to make learning easier, more successful and efficient. For this reason, the promotion and monitoring of positive attitudes towards multimedia-based instruction is critical. Four courses were selected by the researcher for analysis because of their use of a multimedia learning environment. A29-item questionnaire was administered to 123 students to measure their attitudes towards online learning with multimedia tutorials. Students were segregated into groups based on their prior experience and analyzed using descriptive statistics (mean, median, mode, standard deviation and frequency) and an independent t test. The data collected from the survey answered questions pertaining to the benefit of additional online tutorials, learner independence, classroom autonomy, effect on their learning, and anxiety. Based on the results of this study, the statistical analysis indicated there were significant differences at the \u3c.05 level between the two groups of students for each of the five hypotheses. Additionally, students\u27 attitudes on average were nearly half a point higher (M = 0.46) for those who had prior experience with multimedia in the classroom than students with no experience based on a Likert five-point scale. However, students\u27 attitudes were lower for students with prior experience (M = 2. 76, SD = 0. 77) over students without experience (M = 2.81, SD = 1.00) for the research question about whether students prefer online multimedia learning tutorials over traditional class handouts

Gravity-driven Lyα blobs from cold streams into galaxies

We use high-resolution cosmological hydrodynamical adaptive mesh refinement (AMR) simulations to predict the characteristics of Lyα emission from the cold gas streams that fed galaxies in massive haloes at high redshift. The Lyα luminosity in our simulations is powered by the release of gravitational energy as gas flows from the intergalactic medium into the halo potential wells. The ultraviolet UV background contributes only <20 per cent to the gas heating. The Lyα emissivity is due primarily to electron-impact excitation cooling radiation in gas at ∼2 × 104 K. We calculate the Lyα emissivities assuming collisional ionization equilibrium at all gas temperatures. The simulated streams are self-shielded against the UV background, so photoionization and recombination contribute negligibly to the Lyα line formation. We produce theoretical maps of the Lyα surface brightnesses, assuming that ∼85 per cent of the Lyα photons are directly observable. We do not consider transfer of the Lyα radiation, nor do we include the possible effects of internal sources of photoionization such as star-forming regions. Dust absorption is expected to obscure a small fraction of the luminosity in the streams. We find that typical haloes of mass Mv∼ 1012-1013 M☉ at z∼ 3 emit as Lyα blobs (LABs) with luminosities 1043-1044 erg s−1. Most of the Lyα comes from the extended (50-100 kpc) narrow, partly clumpy, inflowing, cold streams of (1-5) × 104 K that feed the growing galaxies. The predicted LAB morphology is therefore irregular, with dense clumps and elongated extensions. The integrated area contained within surface brightness isophotes of 2 × 10−18 erg s−1 cm−2 arcsec−2 is ∼2-100 arcsec2, consistent with observations. The linewidth is expected to range from 102 to more than 103 km s−1 with a large variance. The typical Lyα surface brightness profile is ∝r−1.2 where r is the distance from the halo centre. Our simulated LABs are similar in luminosity, morphology and extent to the observed LABs, with distinct kinematic features. The predicted Lyα luminosity function is consistent with observations, and the predicted areas and linewidths roughly recover the observed scaling relations. This mechanism for producing LABs appears inevitable in many high-z galaxies, though it may work in parallel with other mechanisms. Some of the LABs may thus be regarded as direct detections of the cold streams that drove galaxy evolution at high

Tricritical Points in Random Combinatorics: the (2+p)-SAT case

The (2+p)-Satisfiability (SAT) problem interpolates between different classes of complexity theory and is believed to be of basic interest in understanding the onset of typical case complexity in random combinatorics. In this paper, a tricritical point in the phase diagram of the random $2+p$-SAT problem is analytically computed using the replica approach and found to lie in the range $2/5 \le p_0 \le 0.416$. These bounds on $p_0$ are in agreement with previous numerical simulations and rigorous results.Comment: 7 pages, 1 figure, RevTeX, to appear in J.Phys.

Dynamical friction in constant density cores: a failure of the Chandrasekhar formula

Using analytic calculations and N-body simulations we show that in constant density (harmonic) cores, sinking satellites undergo an initial phase of very rapid (super-Chandrasekhar) dynamical friction, after which they experience no dynamical friction at all. For density profiles with a central power law profile, ρ∝r−α, the infalling satellite heats the background and causes α to decrease. For α < 0.5 initially, the satellite generates a small central constant density core and stalls as in the α= 0 case. We discuss some astrophysical applications of our results to decaying satellite orbits, galactic bars and mergers of supermassive black hole binaries. In a companion paper we show that a central constant density core can provide a natural solution to the timing problem for Fornax's globular cluster

Fast Scalable Construction of (Minimal Perfect Hash) Functions

Recent advances in random linear systems on finite fields have paved the way for the construction of constant-time data structures representing static functions and minimal perfect hash functions using less space with respect to existing techniques. The main obstruction for any practical application of these results is the cubic-time Gaussian elimination required to solve these linear systems: despite they can be made very small, the computation is still too slow to be feasible. In this paper we describe in detail a number of heuristics and programming techniques to speed up the resolution of these systems by several orders of magnitude, making the overall construction competitive with the standard and widely used MWHC technique, which is based on hypergraph peeling. In particular, we introduce broadword programming techniques for fast equation manipulation and a lazy Gaussian elimination algorithm. We also describe a number of technical improvements to the data structure which further reduce space usage and improve lookup speed. Our implementation of these techniques yields a minimal perfect hash function data structure occupying 2.24 bits per element, compared to 2.68 for MWHC-based ones, and a static function data structure which reduces the multiplicative overhead from 1.23 to 1.03

Resolving the timing problem of the globular clusters orbiting the Fornax dwarf galaxy

We re-investigate the old problem of the survival of the five globular clusters orbiting the Fornax dwarf galaxy in both standard and modified Newtonian dynamics. For the first time in the history of the topic, we use accurate mass models for the Fornax dwarf, obtained through Jeans modelling of the recently published line of sight velocity dispersion data, and we are also not resigned to circular orbits for the globular clusters. Previously conceived problems stem from fixing the starting distances of the globulars to be less than half the tidal radius. We relax this constraint since there is absolutely no evidence for it and show that the dark matter paradigm, with either cusped or cored dark matter profiles, has no trouble sustaining the orbits of the two least massive globular clusters for a Hubble time almost regardless of their initial distance from Fornax. The three most massive globulars can remain in orbit as long as their starting distances are marginally outside the tidal radius. The outlook for modified Newtonian dynamics is also not nearly as bleak as previously reported. Although dynamical friction inside the tidal radius is far stronger in MOND, outside dynamical friction is negligible due to the absence of stars. This allows highly radial orbits to survive, but more importantly circular orbits at distances more than 85% of Fornax's tidal radius to survive indefinitely. The probability of the globular clusters being on circular orbits at this distance compared with their current projected distances is discussed and shown to be plausible. Finally, if we ignore the presence of the most massive globular (giving it a large line of sight distance) we demonstrate that the remaining four globulars can survive within the tidal radius for the Hubble time with perfectly sensible orbits.Comment: 8 pages, 10 figures, 1 table, MNRAS in pres

Phase coexistence and finite-size scaling in random combinatorial problems

We study an exactly solvable version of the famous random Boolean satisfiability problem, the so called random XOR-SAT problem. Rare events are shown to affect the combinatorial ``phase diagram'' leading to a coexistence of solvable and unsolvable instances of the combinatorial problem in a certain region of the parameters characterizing the model. Such instances differ by a non-extensive quantity in the ground state energy of the associated diluted spin-glass model. We also show that the critical exponent $\nu$, controlling the size of the critical window where the probability of having solutions vanishes, depends on the model parameters, shedding light on the link between random hyper-graph topology and universality classes. In the case of random satisfiability, a similar behavior was conjectured to be connected to the onset of computational intractability.Comment: 10 pages, 5 figures, to appear in J. Phys. A. v2: link to the XOR-SAT probelm adde

Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses

We develop a systematic cluster expansion for dilute systems in the highly dilute phase. We first apply it to the calculation of the entropy of the K-satisfiability problem in the satisfiable phase. We derive a series expansion in the control parameter, the average connectivity, that is identical to the one obtained by using the replica approach with a replica symmetric ({\sc rs}) {\it Ansatz}, when the order parameter is calculated via a perturbative expansion in the control parameter. As a second application we compute the free-energy of the Viana-Bray model in the paramagnetic phase. The cluster expansion allows one to compute finite-size corrections in a simple manner and these are particularly important in optimization problems. Importantly enough, these calculations prove the exactness of the {\sc rs} {\it Ansatz} below the percolation threshold and might require its revision between this and the easy-to-hard transition.Comment: 21 pages, 7 figs, to appear in Phys. Rev.

Simplest random K-satisfiability problem

We study a simple and exactly solvable model for the generation of random satisfiability problems. These consist of $\gamma N$ random boolean constraints which are to be satisfied simultaneously by $N$ logical variables. In statistical-mechanics language, the considered model can be seen as a diluted p-spin model at zero temperature. While such problems become extraordinarily hard to solve by local search methods in a large region of the parameter space, still at least one solution may be superimposed by construction. The statistical properties of the model can be studied exactly by the replica method and each single instance can be analyzed in polynomial time by a simple global solution method. The geometrical/topological structures responsible for dynamic and static phase transitions as well as for the onset of computational complexity in local search method are thoroughly analyzed. Numerical analysis on very large samples allows for a precise characterization of the critical scaling behaviour.Comment: 14 pages, 5 figures, to appear in Phys. Rev. E (Feb 2001). v2: minor errors and references correcte