Primitive polynomials with prescribed second coefficient

The Hansen-Mullen Primitivity Conjecture (HMPC) (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree n over any finite fieldwith any coefficient arbitrarily prescribed. This has recently been provedwhenever n ≥ 9. It is also known to be truewhen n ≤ 3.We showthat there exists a primitive polynomial of any degree n ≥ 4 over any finite field with its second coefficient (i.e., that of xn−2) arbitrarily prescribed. In particular, this establishes the HMPC when n = 4. The lone exception is the absence of a primitive polynomial of the form x4 + a1x3 + x2 + a3x + 1 over the binary field. For n ≥ 6 we prove a stronger result, namely that the primitive polynomialmay also have its constant termprescribed. This implies further cases of the HMPC. When the field has even cardinality 2-adic analysis is required for the proofs

Primitive free cubics with specified norm and trace

The existence of a primitive free (normal) cubic x3 - ax2 + cx - b over a finite field F with arbitrary specified values of a (≠0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed

Beyond Consistency and Substitutability

International audienceElimination of inconsistent values in instances of the constraint satisfaction problem (CSP) conserves all solutions. Elimination of substitutable values conserves at least one solution. We show that certain values which are neither inconsistent nor substitutable can also be deleted while conserving at least one solution. This allows us to state novel rules for the elimination of values in binary CSP. From a practical point of view, we show that one such rule can be applied in the same asymptotic time complexity as neighbourhood substitution but is strictly stronger. An alternative to the elimination of values from domains is the elimination of variables. We give novel satisfiability-preserving variable elimination operations. In each case we show that if the instance is satisfiable, then a solution to the original instance can always be recovered in low-order polynomial time from a solution to the reduced instance

Extraction of Step-Repulsion Strengths from Terrace Width Distributions: Statistical and Analytic Considerations

Recently it has been recognized that the so-called generalized Wigner distribution may provide at least as good a description of terrace width distributions (TWDs) on vicinal surfaces as the standard Gaussian fit and is particularly applicable for weak repulsions between steps, where the latter fails. Subsequent applications to vicinal copper surfaces at various temperatures confirmed the serviceability of the new analysis procedure but raised some theoretical questions. Here we address these issues using analytical, numerical, and statistical methods. We propose an extension of the generalized Wigner distribution to a two-parameter fit that allows the terrace widths to be scaled by an optimal effective mean width. We discuss quantitatively the approach of a Wigner distribution to a Gaussian form for strong repulsions, how errors in normalization or mean affect the deduced interaction, and how optimally to extract the interaction from the variance and mean of the TWD. We show that correlations reduce by two orders of magnitude the number of {\em independent} measurements in a typical STM image. We also discuss the effect of the discreteness ("quantization") of terrace widths, finding that for high misorientation (small mean width) the standard continuum analysis gives faulty estimates of step interactions.Comment: 13 pages, 7 figures; info added on # ind. measurements/STM imag

Hardware and software status of QCDOC

QCDOC is a massively parallel supercomputer whose processing nodes are based on an application-specific integrated circuit (ASIC). This ASIC was custom-designed so that crucial lattice QCD kernels achieve an overall sustained performance of 50% on machines with several 10,000 nodes. This strong scalability, together with low power consumption and a price/performance ratio of $1 per sustained MFlops, enable QCDOC to attack the most demanding lattice QCD problems. The first ASICs became available in June of 2003, and the testing performed so far has shown all systems functioning according to specification. We review the hardware and software status of QCDOC and present performance figures obtained in real hardware as well as in simulation.Comment: Lattice2003(machine), 6 pages, 5 figure

Towards lattice simulation of the gauge theory duals to black holes and hot strings

A generalization of the AdS/CFT conjecture postulates a duality between IIA string theory and 16 supercharge Yang-Mills quantum mechanics in the large N 't Hooft limit. At low temperatures string theory describes black holes, whose thermodynamics may hence be studied using the dual quantum mechanics. This quantum mechanics is strongly coupled which motivates the use of lattice techniques. We argue that, contrary to expectation, the theory when discretized naively will nevertheless recover continuum supersymmetry as the lattice spacing is sent to zero. We test these ideas by studying the 4 supercharge version of this Yang-Mills quantum mechanics in the 't Hooft limit. We use both a naive lattice action and a manifestly supersymmetric action. Using Monte Carlo methods we simulate the Euclidean theories, and study the lattice continuum limit, for both thermal and non-thermal periodic boundary conditions, confirming continuum supersymmetry is recovered for the naive action when appropriate. We obtain results for the thermal system with N up to 12. These favor the existence of a single deconfined phase for all non-zero temperatures. These results are an encouraging indication that the 16 supercharge theory is within reach using similar methods and resources.Comment: 49 pages, 14 figure

A progressive refinement approach for the visualisation of implicit surfaces

Visualising implicit surfaces with the ray casting method is a slow procedure. The design cycle of a new implicit surface is, therefore, fraught with long latency times as a user must wait for the surface to be rendered before being able to decide what changes should be introduced in the next iteration. In this paper, we present an attempt at reducing the design cycle of an implicit surface modeler by introducing a progressive refinement rendering approach to the visualisation of implicit surfaces. This progressive refinement renderer provides a quick previewing facility. It first displays a low quality estimate of what the final rendering is going to be and, as the computation progresses, increases the quality of this estimate at a steady rate. The progressive refinement algorithm is based on the adaptive subdivision of the viewing frustrum into smaller cells. An estimate for the variation of the implicit function inside each cell is obtained with an affine arithmetic range estimation technique. Overall, we show that our progressive refinement approach not only provides the user with visual feedback as the rendering advances but is also capable of completing the image faster than a conventional implicit surface rendering algorithm based on ray casting

Neutrino-Deuteron Scattering in Effective Field Theory at Next-to-Next-to Leading Order

We study the four channels associated with neutrino-deuteron breakup reactions at next-to-next to leading order in effective field theory. We find that the total cross-section is indeed converging for neutrino energies up to 20 MeV, and thus our calculations can provide constraints on theoretical uncertainties for the Sudbury Neutrino Observatory. We stress the importance of a direct experimental measurement to high precision in at least one channel, in order to fix an axial two-body counterterm.Comment: 32 pages, 14 figures (eps

Scalar-Tensor Theory of Gravity and Generalized Second Law of Thermodynamics on the Event Horizon

In blackhole physics, the second law of thermodynamics is generally valid whether the blackhole is a static or a non-static one. Considering the universe as a thermodynamical system the second law of blackhole dynamics extends to the non-negativity of the sum of the entropy of the matter and the horizon, known as generalized second law of thermodynamics(GSLT). Here, we have assumed the universe to be bounded by the event-horizon or filled with perfect fluid and holographic dark energy in two cases. Thus considering entropy to be an arbitrary function of the area of the event-horizon, we have tried to find the conditions and the restrictions over the scalar field and equation of state for the validity of the GSLT and both in quintessence-era and in phantom-era in scalar tensor theory.Comment: 8 page

Generalized Second Law of Thermodynamics on the Event Horizon for Interacting Dark Energy

Here we are trying to find the conditions for the validity of the generalized second law of thermodynamics (GSLT) assuming the first law of thermodynamics on the event horizon in both cases when the FRW universe is filled with interacting two fluid system- one in the form of cold dark matter and the other is either holographic dark energy or new age graphic dark energy. Using the recent observational data we have found that GSLT holds both in quintessence era as well as in phantom era for new age graphic model while for holographic dark energy GSLT is valid only in phantom era.Comment: 8 pages, 2 figure