We develop a Harder-Narasimhan theory for Kisin modules generalizing a similar theory for finite flat group schemes due to Fargues [Far10]. We prove the tensor product theorem, i.e., that the tensor product of semistable objects is again semi-stable. We then apply the tensor product theorem to the study of Kisin varieties for arbitrary connected reductive groups.

A square (0, 1)-matrix X of order n ≥ 1 is called fully indecomposable if there exists no integer k with 1 ≤ k ≤ n− 1, such that X has a k by n− k zero submatrix. The reduced adjacency matrix of a bipartite graph G = (A,B,E) (having A ∪ B = {a1, ..., am} ∪ {b1, ..., bn} as vertex set, and E as edge set), is X = [xij ], 1 ≤ i ≤ m, 1 ≤ j ≤ n, where xij = 1 if aibj ∈ E and xij = 0 otherwise. A sta...

Hydrogen peroxide reacts with ferrous Ascaris hemoglobin and converts it to oxyhemoglobin in a sequence of two reactions. The first is the reaction of ferrous Ascaris hemoglobin with hydrogen peroxide to form a stable product; the second is the reaction of this product with hydrogen peroxide to form oxyhemoglobin. Both reactions follow second order kinetics, since they are first order with resp...

Introduction 150 Organization 152 Acknowledgments 152 1. Symmetric spectra 153 1.1. Simplicial sets 153 1.2. Symmetric spectra 154 1.3. Simplicial structure on Sp 156 1.4. Symmetric Ω-spectra 158 2. The smash product of symmetric spectra 159 2.1. Symmetric sequences 159 2.2. Symmetric spectra 162 2.3. The ordinary category of spectra 165 3. Stable homotopy theory of symmetric spectra 165 3.1. S...

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We show how some attractive information–theoretic properties of Gaussians pass over to more general families of stable densities. We define a new score function for symmetric stable laws, and use it to give a stable version of the heat equation. Using this, we derive a version of the de Bruijn identity, allowing us to write the derivative of relative entropy as an inner product of score functio...