We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant. AMS Subject Classification: 57M27 Frobenius systems. Suppose ι : R −→ A is an inclusion of commutative rings, and ι(1) = 1. The restriction functor Res : A−mod −→ R−mod has left and ...

Abstract Gendo-Frobenius algebras are a common generalisation of Frobenius and gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on In addition, shown be precisely those that have counit compatible with this comultiplication. Moreover, new characterisation given. This key constructing

Let X be a smooth projective variety, defined over an algebraically closed field of positive characteristic, such that the tangent bundle TX is trivial. Let FX : X −→ X be the absolute Frobenius morphism of X. We prove that for any n ≥ 1, the n–fold composition Fn X is a torsor over X for a finite group–scheme that depends on n. For any vector bundle E −→ X, we show that the direct image (Fn X)...

I f G induces a primitive permutation group on the set X of points (l-spaces) of the underlying vector space V, then well-known results of Burnside [I, p. 3411 and Schur [6] imply that G acts on X 2-transitively or as a regular or Frobenius group of prime degree; the theorem then follows from [2; 51. We must thus take a nontrivial block of imprimitivity d for G on X, and analyze the action of G...

We give dimension-free and data-dependent bounds for linear multi-task learning where a common linear operator is chosen to preprocess data for a vector of task specific linear-thresholding classifiers. The complexity penalty of multi-task learning is bounded by a simple expression involving the margins of the task-specific classifiers, the Hilbert-Schmidt norm of the selected preprocessor and ...

Conjecture 1. Let F be a number field, let p be some prime number, and fix an isomorphism ι : C ∼= Qp. Then there is a bijection between the set of algebraic cuspidal automorphic representations of GLn(AF ), and the set of isomorphism classes of irreducible continuous representations of the absolute Galois group of F on n-dimensional Qp-vector spaces which are almost everywhere unramified, and ...

We use the theory of p-curvature of connections to analyze stable vector bundles of rank 2 on curves of genus 2 which pull back to unstable bundles under the Frobenius morphism. We take two approaches, first using explicit formulas for p-curvature to analyze low-characteristic cases, and then using degeneration techniques to obtain an answer for a general curve by degenerating to an irreducible...

Let R be an integral domain of finite type over Z and let f : X → SpecR be a smooth projective morphism of relative dimension d ≥ 1. We investigate, for a vector bundle E on the total space X , under what arithmetical properties of a sequence (pn, en)n∈N, consisting of closed points pn in SpecR and Frobenius descent data Epn ∼= F n(F) on the closed fibers Xpn , the bundle E0 on the generic fibe...

Let V be a vector space over a fully ordered field F. In Sec. 2 we characterize cones K with ascending chain condition (ACC) on faces of Ii. In Sec. 3 we show that if K has ACC on faces, then an operator A is strongly irreducible if and only if A is irreducible. In Sec. 4 we prove theorems of Perron-Frobenius type for a strongly irreducible operator A in the case that FR, the real field, and K ...