MARGINAL AGRICULTURAL LAND CLASSIFICATION: A NEW APPROACH

Land Economics/Use,

Self-Similar Graphs

For any graph $G$ on $n$ vertices and for any {\em symmetric} subgraph $J$ of $K_{n,n}$, we construct an infinite sequence of graphs based on the pair $(G,J)$. The First graph in the sequence is $G$, then at each stage replacing every vertex of the previous graph by a copy of $G$ and every edge of the previous graph by a copy of $J$ the new graph is constructed. We call these graphs {\em self-similar} graphs. We are interested in delineating those pairs $(G,J)$ for which the chromatic numbers of the graphs in the sequence are bounded. Here we have some partial results. When $G$ is a complete graph and $J$ is a special matching we show that every graph in the resulting sequence is an {\em expander} graph.Comment: 13 pages, 1 tabl

High temperature measuring device

Ultrasonic pulse technique for measuring average gas temperature in nuclear rocket engine - sound propagation and environmental studie

Flows, Fragmentation, and Star Formation. I. Low-mass Stars in Taurus

The remarkably filamentary spatial distribution of young stars in the Taurus molecular cloud has significant implications for understanding low-mass star formation in relatively quiescent conditions. The large scale and regular spacing of the filaments suggests that small-scale turbulence is of limited importance, which could be consistent with driving on large scales by flows which produced the cloud. The small spatial dispersion of stars from gaseous filaments indicates that the low-mass stars are generally born with small velocity dispersions relative to their natal gas, of order the sound speed or less. The spatial distribution of the stars exhibits a mean separation of about 0.25 pc, comparable to the estimated Jeans length in the densest gaseous filaments, and is consistent with roughly uniform density along the filaments. The efficiency of star formation in filaments is much higher than elsewhere, with an associated higher frequency of protostars and accreting T Tauri stars. The protostellar cores generally are aligned with the filaments, suggesting that they are produced by gravitational fragmentation, resulting in initially quasi-prolate cores. Given the absence of massive stars which could strongly dominate cloud dynamics, Taurus provides important tests of theories of dispersed low-mass star formation and numerical simulations of molecular cloud structure and evolution.Comment: 32 pages, 9 figures: to appear in Ap

Turbulent Cooling Flows in Molecular Clouds

We propose that inward, subsonic flows arise from the local dissipation of turbulent motions in molecular clouds. Such "turbulent cooling flows" may account for recent observations of spatially extended inward motions towards dense cores. These pressure-driven flows may arise from various types of turbulence and dissipation mechanisms. For the example of MHD waves and turbulence damped by ion-neutral friction, sustained cooling flow requires that the outer gas be sufficiently turbulent, that the inner gas have marginal field-neutral coupling, and that this coupling decrease sufficiently rapidly with increasing density. These conditions are most likely met at the transition between outer regions ionized primarily by UV photons and inner regions ionized primarily by cosmic rays. If so, turbulent cooling flows can help form dense cores, with speeds faster than expected for ambipolar diffusion. Such motions could reduce the time needed for dense core formation and could precede and enhance the motions of star-forming gravitational infall.Comment: To appear ApJL, Nov.10, 4 ApJ style pages, Postscrip

Top-down Automated Theorem Proving (Notes for Sir Timothy)

We describe a "top down" approach for automated theorem proving (ATP). Researchers might usefully investigate the forms of the theorems mathematicians use in practice, carefully examine how they differ and are proved in practice, and code all relevant domain concepts. These concepts encode a large portion of the knowledge in any domain. Furthermore, researchers should write programs that produce proofs of the kind that human mathematicians write (and publish); this means proofs that might sometimes have mistakes; and this means making inferences that are sometimes invalid. This approach is meant to contrast with the historically dominant "bottom up" approach: coding fundamental types (typically sets), axioms and rules for (valid) inference, and building up from this foundation to the theorems of mathematical practice and to their outstanding questions. It is an important fact that the actual proofs that mathematicians publish in math journals do not look like the formalized proofs of Russell & Whitehead's Principia Mathematica (or modern computer systems like Lean that automate some of this formalization). We believe some "lack of rigor" (in mathematical practice) is human-like, and can and should be leveraged for ATP.Comment: Cross list with cs.A

Comparison of Shape Derivatives Using CutFEM for Ill-posed Bernoulli Free Boundary Problem

In this paper we study and compare three types of shape derivatives for free boundary identification problems. The problem takes the form of a severely ill-posed Bernoulli problem where only the Dirichlet condition is given on the free (unknown) boundary, whereas both Dirichlet and Neumann conditions are available on the fixed (known) boundary. Our framework resembles the classical shape optimization method in which a shape dependent cost functional is minimized among the set of admissible domains. The position of the domain is defined implicitly by the level set function. The steepest descent method, based on the shape derivative, is applied for the level set evolution. For the numerical computation of the gradient, we apply the Cut Finite Element Method (CutFEM), that circumvents meshing and re-meshing, without loss of accuracy in the approximations of the involving partial differential models. We consider three different shape derivatives. The first one is the classical shape derivative based on the cost functional with pde constraints defined on the continuous level. The second shape derivative is similar but using a discretized cost functional that allows for the embedding of CutFEM formulations directly in the formulation. Different from the first two methods, the third shape derivative is based on a discrete formulation where perturbations of the domain are built into the variational formulation on the unperturbed domain. This is realized by using the so-called boundary value correction method that was originally introduced to allow for high order approximations to be realized using low order approximation of the domain. The theoretical discussion is illustrated with a series of numerical examples showing that all three approaches produce similar result on the proposed Bernoulli problem