Binomial Ideals and Congruences on Nn

Producción CientíficaA congruence on Nn is an equivalence relation on Nn that is compatible with the additive structure. If k is a field, and I is a binomial ideal in k[X1,…,Xn] (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on Nn by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of Xu and Xv that belongs to I. While every congruence on Nn arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on Nn are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly (Kahle and Miller Algebra Number Theory 8(6):1297–1364, 2014) with an eye on Eisenbud and Sturmfels (Duke Math J 84(1):1–45, 1996) and Ojeda and Piedra Sánchez (J Symbolic Comput 30(4):383–400, 2000).National Science Foundation (grant DMS-1500832)Ministerio de Economía, Industria y Competitividad (project MTM2015-65764-C3-1)Junta de Extremadura (grupo de investigación FQM-024

The application of remotely sensed data to pedologic and geomorphic mapping on alluvial fan and playa surfaces in Saline Valley, California

Arid and semiarid regions yield excellent opportunities for the study of pedologic and geomorphic processes. The dominance of rock and soil exposure over vegetation not only provides the ground observer with observational possibilities but also affords good opportunities for measurement by aircraft and satellite remote sensor devices. Previous studies conducted in the area of pedologic and geomorphic mapping in arid regions with remotely sensed data have utilized information obtained in the visible to near-infrared portion of the spectrum. Thermal Infrared Multispectral Scanner (TIMS) and Thematic Mapping (TM) data collected in 1984 are being used in comjunction with maps compiled during a Bureau of Land Management (BLM) soil survey to aid in a detailed mapping of alluvial fan and playa surfaces within the valley. The results from this study may yield valuable information concerning the application of thermal data and thermal/visible data combinations to the problem of dating pedologic and geomorphic features in arid regions

Processing of multispectral thermal IR data for geologic applications

Multispectral thermal IR data were acquired with a 24-channel scanner flown in an aircraft over the E. Tintic Utah mining district. These digital image data required extensive computer processing in order to put the information into a format useful for a geologic photointerpreter. Simple enhancement procedures were not sufficient to reveal the total information content because the data were highly correlated in all channels. The data were shown to be dominated by temperature variations across the scene, while the much more subtle spectral variations between the different rock types were of interest. The image processing techniques employed to analyze these data are described

Evaluation of LANDSAT MSS vs TM simulated data for distinguishing hydrothermal alteration

The LANDSAT Follow-On (LFO) data was simulated to demonstrate the mineral exploration capability of this system for segregating different types of hydrothermal alteration and to compare this capability with that of the existing LANDSAT system. Multispectral data were acquired for several test sites with the Bendix 24-channel MSDS scanner. Contrast enhancements, band ratioing, and principal component transformations were used to process the simulated LFO data for analysis. For Red Mountain, Arizona, the LFO data allowed identification of silicified areas, not identifiable with LANDSAT 1 and 2 data. The improved LFO resolution allowed detection of small silicic outcrops and of a narrow silicified dike. For Cuprite - Ralston, Nevada, the LFO spectral bands allowed discrimination of argillic and opalized altered areas; these could not be spectrally discriminated using LANDSAT 1 and 2 data. Addition of data from the 1.3- and 2.2- micrometer regions allowed better discriminations of hydrothermal alteration types

Uniform random colored complexes

We present here random distributions on $(D+1)$-edge-colored, bipartite graphs with a fixed number of vertices $2p$. These graphs are dual to $D$-dimensional orientable colored complexes. We investigate the behavior of quantities related to those random graphs, such as their number of connected components or the number of vertices of their dual complexes, as $p \to \infty$. The techniques involved in the study of these quantities also yield a Central Limit Theorem for the genus of a uniform map of order $p$, as $p \to \infty$.Comment: 36 pages, 9 figures, minor additions and correction

Random geometric complexes

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometr

Evaluation of thermal data for geologic applications

Sensitivity studies using thermal models indicated sources of errors in the determination of thermal inertia from HCMM data. Apparent thermal inertia, with only simple atmospheric radiance corrections to the measured surface temperature, would be sufficient for most operational requirements for surface thermal inertia. Thermal data does have additional information about the nature of surface material that is not available in visible and near infrared reflectance data. Color composites of daytime temperature, nighttime temperature, and albedo were often more useful than thermal inertia images alone for discrimination of lithologic boundaries. A modeling study, using the annual heating cycle, indicated the feasibility of looking for geologic features buried under as much as a meter of alluvial material. The spatial resolution of HCMM data is a major limiting factor in the usefulness of the data for geologic applications. Future thermal infrared satellite sensors should provide spatial resolution comparable to that of the LANDSAT data

Complexity Measures from Interaction Structures

We evaluate new complexity measures on the symbolic dynamics of coupled tent maps and cellular automata. These measures quantify complexity in terms of $k$-th order statistical dependencies that cannot be reduced to interactions between $k-1$ units. We demonstrate that these measures are able to identify complex dynamical regimes.Comment: 11 pages, figures improved, minor changes to the tex

T-duality and Differential K-Theory

We give a precise formulation of T-duality for Ramond-Ramond fields. This gives a canonical isomorphism between the "geometrically invariant" subgroups of the twisted differential K-theory of certain principal torus bundles. Our result combines topological T-duality with the Buscher rules found in physics.Comment: 23 pages, typos corrected, submitted to Comm.Math.Phy

Large random simplicial complexes, I

In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special cases. Topological and geometric properties of a multi-parameter random simplicial complex depend on the whole combination of the probability parameters and the thresholds for topological properties are convex sets rather than numbers (as in all previously known models). We discuss the containment properties, density domains and dimension of the random simplicial complexes.Comment: 21 pages, 6 figure