Spatial and temporal variability of soil fertility in relation to crop yield zones on hummocky terrain

Non-Peer ReviewedA field study was conducted on hummocky terrain at the Manitoba Zero Tillage Association Research Farm to determine the variability of crop yield as related to landscape position, soil properties, weed populations and plant disease. This information was also used to evaluate technology required for delineation of management units related to precision farming. Variable rate fertiliser management systems can improve efficiency of fertilizer use and environmental sustainability. Adoption of this technology has been hampered due to the difficulty of classifying fields into management units, the high cost of sampling soils on a grid basis, and the variability of soil and plant properties in the landscape. Technology for variable rate fertilizer systems is available, but there is little information available related to yield response in clay soils on hummocky terrain, and the relationship of plant tissue test levels in relation to soil fertility as measured by soil test nitrate nitrogen. Current soil test recommendations for nitrogen are based on soil test nitrate nitrogen from samples bulked from samples in several locations in the field preferably grouped according to topography. Yield data for 1997-2001 were classified into groups with the fuzzy k means, normal mixtures and self-organizing map variants of cluster analysis. Although fuzzy k means commonly used for classification of crop yield and soil properties, a method based on self-organizing maps provided consistent classes when compared across years. Soil nitrate nitrogen varied considerably across the landscape at the site, but was not significantly different (P<0.05) between classes based on crop yield. Yield data can be used to delineate zones for variable management, although fertilizer inputs may be a function of spring soil moisture, runoff and growing season precipitation as they affect seeding, crop emergence and establishment

New satellite climate data records indicate strong coupling between recent frozen season changes and snow cover over high northern latitudes

We examined new satellite climate data records documenting frozen (FR) season and snow cover extent (SCE) changes from 1979 to 2011 over all northern vegetated land areas (≥45 °N). New insight on the spatial and temporal characteristics of seasonal FR ground and snowpack melt changes were revealed by integrating the independent FR and SCE data records. Similar decreasing trends in annual FR and SCE durations coincided with widespread warming (0.4 °C decade−1). Relatively strong declines in FR and SCE durations in spring and summer are partially offset by increasing trends in fall and winter. These contrasting seasonal trends result in relatively weak decreasing trends in annual FR and SCE durations. A dominant SCE retreat response to FR duration decreases was observed, while the sign and strength of this relationship was spatially complex, varying by latitude and regional snow cover, and climate characteristics. The spatial extent of FR conditions exceeds SCE in early spring and is smaller during snowmelt in late spring and early summer, while FR ground in the absence of snow cover is widespread in the fall. The integrated satellite record, for the first time, reveals a general increasing trend in annual snowmelt duration from 1.3 to 3.3 days decade−1 (p \u3c 0.01), occurring largely in the fall. Annual FR ground durations are declining from 0.8 to 1.3 days decade−1. These changes imply extensive biophysical impacts to regional snow cover, soil and permafrost regimes, surface water and energy budgets, and climate feedbacks, while ongoing satellite microwave missions provide an effective means for regional monitoring

Cluster algebras in algebraic Lie theory

We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.Comment: Invited survey; to appear in Transformation Group

Linear resolutions of powers and products

The goal of this paper is to present examples of families of homogeneous ideals in the polynomial ring over a field that satisfy the following condition: every product of ideals of the family has a linear free resolution. As we will see, this condition is strongly correlated to good primary decompositions of the products and good homological and arithmetical properties of the associated multi-Rees algebras. The following families will be discussed in detail: polymatroidal ideals, ideals generated by linear forms and Borel fixed ideals of maximal minors. The main tools are Gr\"obner bases and Sagbi deformation

The Theory of the Interleaving Distance on Multidimensional Persistence Modules

In 2009, Chazal et al. introduced $\epsilon$-interleavings of persistence modules. $\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $\epsilon$-interleavings and $d_I$ generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $d_I$ is equal to the bottleneck distance $d_B$. This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the $\epsilon$-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two $\epsilon$-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, $d_I$ satisfies a universality property. This universality result is the central result of the paper. It says that $d_I$ satisfies a stability property generalizing one which $d_B$ is known to satisfy, and that in addition, if $d$ is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then $d\leq d_I$. We also show that a variant of this universality result holds for $d_B$, over arbitrary fields. Finally, we show that $d_I$ restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in Foundations of Computational Mathematics. 36 page

Persistent Homology Over Directed Acyclic Graphs

We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of more general families of topological spaces or point-cloud data. We give an algorithm to compute the persistent homology groups simultaneously for all subgraphs which contain a single source and a single sink in $O(n^4)$ arithmetic operations, where $n$ is the number of vertices in the graph. We then demonstrate as an application of these tools a method to overlay two distinct filtrations of the same underlying space, which allows us to detect the most significant barcodes using considerably fewer points than standard persistence.Comment: Revised versio

A functorial construction of moduli of sheaves

We show how natural functors from the category of coherent sheaves on a projective scheme to categories of Kronecker modules can be used to construct moduli spaces of semistable sheaves. This construction simplifies or clarifies technical aspects of existing constructions and yields new simpler definitions of theta functions, about which more complete results can be proved.Comment: 52 pp. Dedicated to the memory of Joseph Le Potier. To appear in Inventiones Mathematicae. Slight change in the definition of the Kronecker algebra in Secs 1 (p3) and 2.2 (p6), with corresponding small alterations elsewhere, to make the constructions work for non-reduced schemes. Section 6.5 rewritten. Remark 2.6 and new references adde

On Syzygies for Rings of Invariants of Abelian Groups

It is well known that results on zero-sum sequences over a finitely generated abelian group can be translated to statements on generators of rings of invariants of the dual group. Here the direction of the transfer of information between zero-sum theory and invariant theory is reversed. First it is shown how a presentation by generators and relations of the ring of invariants of an abelian group acting linearly on a finite-dimensional vector space can be obtained from a presentation of the ring of invariants for the corresponding multiplicity free representation. This combined with a known degree bound for syzygies of rings of invariants yields bounds on the presentation of a block monoid associated to a finite sequence of elements in an abelian group. The results have an equivalent formulation in terms of binomial ideals, but here the language of monoid congruences and the notion of catenary degree is used

Network and Seiberg Duality

We define and study a new class of 4d N=1 superconformal quiver gauge theories associated with a planar bipartite network. While UV description is not unique due to Seiberg duality, we can classify the IR fixed points of the theory by a permutation, or equivalently a cell of the totally non-negative Grassmannian. The story is similar to a bipartite network on the torus classified by a Newton polygon. We then generalize the network to a general bordered Riemann surface and define IR SCFT from the geometric data of a Riemann surface. We also comment on IR R-charges and superconformal indices of our theories.Comment: 28 pages, 28 figures; v2: minor correction

Categorification of skew-symmetrizable cluster algebras

We propose a new framework for categorifying skew-symmetrizable cluster algebras. Starting from an exact stably 2-Calabi-Yau category C endowed with the action of a finite group G, we construct a G-equivariant mutation on the set of maximal rigid G-invariant objects of C. Using an appropriate cluster character, we can then attach to these data an explicit skew-symmetrizable cluster algebra. As an application we prove the linear independence of the cluster monomials in this setting. Finally, we illustrate our construction with examples associated with partial flag varieties and unipotent subgroups of Kac-Moody groups, generalizing to the non simply-laced case several results of Gei\ss-Leclerc-Schr\"oer.Comment: 64 page