Nonassociative geometry: Towards discrete structure of spacetime

In the framework of nonassociative geometry (hep-th/0003238) a unified description of continuum and discrete spacetime is proposed. In our approach at the Planck scales the spacetime is described as a so-called "diodular discrete structure" which at large spacetime scales `looks like' a differentiable manifold. After a brief review of foundations of nonassociative geometry,we discuss the nonassociative smooth and discrete de Sitter spacetimes.Comment: RevTex file, 5 pages, typos correcte

Non-associative geometry and discrete structure of spacetime

A new mathematical theory, non-associative geometry, providing a unified algebraic description of continuous and discrete spacetime, is introduced.Comment: LATeX2e file, 11 pages, talk given at "Loop's 99 meeting" (Prague, July 27 - August 1, 1999). To appear in Comment. Math. Univ. Carolin

Making Algebra More Accessible: How Steep Can it be for Teachers?

Teacher educators need to support middle grades teachers in developing mathematical knowledge for teaching algebraic concepts. In particular, teachers should become familiar with possible introductions and sequencing to the concept of slope, and common middle school students’ limited conceptions about measuring the steepness of an incline. Steepness can be expressed directly in terms of an angle or indirectly as a slope. Encouraging middle school students to find a measure of steepness using a ratio may help support students’ transition to multiplicative thinking. This mixed – methods study explores middle school students’ responses in solving a comparison problem involving the steepness of two inclines, in order to gain insight into common student strategies. The quantitative portion of the study involved written surveys distributed to 256 Grade 7 participants in the United States. We examined the frequency and types of solutions offered by these participants. We found that 27% of the participants provided an incorrect solution which was consistent with additive reasoning. The qualitative portion of this study consisted of small group interviews of 19 Grade 7 participants, who were conflicted in the different solutions they produced from using additive reasoning and their geometric knowledge

Intense short-period internal waves in the ocean

Trains of quasi-periodic high-frequency internal waves (IWs) of large amplitude are common in the upper thermocline of the ocean. Sources for these waves may be different ones but it is not always possible to experimentally establish them for certain. We analyzed results of many IW experiments carried out in different representative regions of the World Ocean, including continental margins in the Mid-Atlantic Bight, in the northwestern Pacific at Kamchatka, the Seyshelles-Mascarene bottom rise, and some regions of the open ocean where the intense short-period IWs occur. Comparative analysis of the intense IWs observed in the Mid-Atlantic Bight and at Kamchatka revealed similarity and difference in the IW field in these regions differing by their bottom topography. Most of the observed trains in the Mid-Atlantic Bight propagate shoreward from the shelf break in the form of soliton packets or solibores and do not occur seaward from the shelf. The soliton trains in the northwestern Pacific at Kamchatka are common not only at the shelf edge but also in deep water where they propagate in various directions that seem to be related to the supercritical steepness and complicated form of the continental slope. Observation of generation and evolution of the IW trains at the Seyshelles-Mascarene bottom rise where huge internal solitons have been encountered has shown that the undular bore generated at the lee side of the bottom rise gradually evolves in a train of solitons with the trailing linear waves. Large solitons are generated also in deep water as a result of ray propagation of the internal tide emanated from the rise as happens in the Bay of Biscay. Certain consequences of the IW interaction with the background current leading to intensification of the high-frequency waves were observed in several regions of the open ocean. Revealed dependency of the intense wave propagation direction on the current direction, and closeness of the wave frequency to the frequency at which the waveguide steeply tapers may be regarded as clear evidences for the important role which currents play in the IW intensification

Smooth Loops, Generalized Coherent States and Geometric Phases

A description of generalized coherent states and geometric phases in the light of the general theory of smooth loops is given.Comment: LATeX file, 11 page

On incomplete factorization implicit technique for 2D elliptic FD equations

A new variant of Incomplete Factorization Implicit (IFI) iterative technique for 2D elliptic finite-difference (FD) equations is suggested which is differed by applying the matrix tridiagonal algorithm. Its iteration parameter is shown be linked with the one for Alternating Direction Implicit method. An effective set of values for the parameter is suggested. A procedure for enhancing the set of iteration parameters for IFI is proposed. The technique is applied to a 5-point FD scheme, and to a 9-point FD scheme. It is suggested applying the solver for 5-point scheme to solving boundary-value problems for the 9-point scheme, too. The results of numerical experiment with Dirichlet and Neumann boundary-value problems for Poisson equation in a rectangle, and in a quasi-circle are presented. Mixed boundary-value problems in square are considered, too. The effectiveness of IFI is high, and weakly depends on the type of boundary conditions

The principle of relative locality

We propose a deepening of the relativity principle according to which the invariant arena for non-quantum physics is a phase space rather than spacetime. Descriptions of particles propagating and interacting in spacetimes are constructed by observers, but different observers, separated from each other by translations, construct different spacetime projections from the invariant phase space. Nonetheless, all observers agree that interactions are local in the spacetime coordinates constructed by observers local to them. This framework, in which absolute locality is replaced by relative locality, results from deforming momentum space, just as the passage from absolute to relative simultaneity results from deforming the linear addition of velocities. Different aspects of momentum space geometry, such as its curvature, torsion and non-metricity, are reflected in different kinds of deformations of the energy-momentum conservation laws. These are in principle all measurable by appropriate experiments. We also discuss a natural set of physical hypotheses which singles out the cases of momentum space with a metric compatible connection and constant curvature.Comment: 12 pages, 3 figures; in version 2 one reference added and some minor modifications in sects. II and III mad