Rado Numbers and SAT Computations

Given a linear equation $\mathcal{E}$, the $k$-color Rado number $R_k(\mathcal{E})$ is the smallest integer $n$ such that every $k$-coloring of $\{1,2,3,\dots,n\}$ contains a monochromatic solution to $\mathcal E$. The degree of regularity of $\mathcal E$, denoted $dor(\mathcal E)$, is the largest value $k$ such that $R_k(\mathcal E)$ is finite. In this article we present new theoretical and computational results about the Rado numbers $R_3(\mathcal{E})$ and the degree of regularity of three-variable equations $\mathcal{E}$. % We use SAT solvers to compute many new values of the three-color Rado numbers $R_3(ax+by+cz = 0)$ for fixed integers $a,b,$ and $c$. We also give a SAT-based method to compute infinite families of these numbers. In particular, we show that the value of $R_3(x-y = (m-2) z)$ is equal to $m^3-m^2-m-1$ for $m\ge 3$. This resolves a conjecture of Myers and implies the conjecture that the generalized Schur numbers $S(m,3) = R_3(x_1+x_2 + \dots x_{m-1} = x_m)$ equal $m^3-m^2-m-1$ for $m\ge 3$. Our SAT solver computations, combined with our new combinatorial results, give improved bounds on $dor(ax+by = cz)$ and exact values for $1\le a,b,c\le 5$. We also give counterexamples to a conjecture of Golowich

Critical Percolation Exploration Path and SLE(6): a Proof of Convergence

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE(6). We provide here a detailed proof, which relies on Smirnov's theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy's formula). The version of convergence to SLE(6) that we prove suffices for the Smirnov-Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.Comment: 45 pages, 14 figures; revised version following the comments of a refere

Equation-regular sets and the Fox–Kleitman conjecture

Given k ≥ 1, the Fox–Kleitman conjecture from 2006 states that there exists a nonzero integer b such that the 2k-variable linear Diophantine equation ∑k i=1 (xi − yi) = b is (2k − 1)-regular. This is best possible, since Fox and Kleitman showed that for all b ≥ 1, this equation is not 2k-regular. While the conjecture has recently been settled for all k ≥ 2, here we focus on the case k = 3 and determine the degree of regularity of the corresponding equation for all b ≥ 1. In particular, this independently confirms the conjecture for k = 3. We also briefly discuss the case k = 4

Measurement of the complete nuclide production and kinetic energies of the system 136Xe + hydrogen at 1 GeV per nucleon

We present an extensive overview of production cross sections and kinetic energies for the complete set of nuclides formed in the spallation of 136Xe by protons at the incident energy of 1 GeV per nucleon. The measurement was performed in inverse kinematics at the FRagment Separator (GSI, Darmstadt). Slightly below the Businaro-Gallone point, 136Xe is the stable nuclide with the largest neutron excess. The kinematic data and cross sections collected in this work for the full nuclide production are a general benchmark for modelling the spallation process in a neutron-rich nuclear system, where fission is characterised by predominantly mass-asymmetric splits.Comment: 18 pages, 14 figure

A regional land use survey based on remote sensing and other data: A report on a LANDSAT and computer mapping project, volume 1

The author has identified the following significant results. New LANDSAT analysis software and linkages with other computer mapping software were developed. Significant results were also achieved in training, communication, and identification of needs for developing the LANDSAT/computer mapping technologies into operational tools for use by decision makers

The New Mexico Daily Lobo, Volume 053, No 20, 10/18/1950

The New Mexico Daily Lobo, Volume 053, No 20, 10/18/1950https://digitalrepository.unm.edu/daily_lobo_1950/1082/thumbnail.jp

An interdisciplinary analysis of Colorado Rocky Mountain environments using ADP techniques

The author has identified the following significant results. Good ecological, classification accuracy (90-95%) can be achieved in areas of rugged relief on a regional basis for Level 1 cover types (coniferous forest, deciduous forest, grassland, cropland, bare rock and soil, and water) using computer-aided analysis techniques on ERTS/MSS data. Cost comparisons showed that a Level 1 cover type map and a table of areal estimates could be obtained for the 443,000 hectare San Juan Mt. test site for less than 0.1 cent per acre, whereas photointerpretation techniques would cost more than 0.4 cent per acre. Results of snow cover mapping have conclusively proven that the areal extent of snow in mountainous terrain can be rapidly and economically mapped by using ERTS/MSS data and computer-aided analysis techniques. A distinct relationship between elevation and time of freeze or thaw was observed, during mountain lake mapping. Basic lithologic units such as igneous, sedimentary, and unconsolidated rock materials were successfully identified. Geomorphic form, which is exhibited through spatial and textual data, can only be inferred from ERTS data. Data collection platform systems can be utilized to produce satisfactory data from extremely inaccessible locations that encounter very adverse weather conditions, as indicated by results obtained from a DCP located at 3,536 meters elevation that encountered minimum temperatures of -25.5 C and wind speeds of up to 40.9m/sec (91 mph), but which still performed very reliably