Let A be an n n complex matrix and r be the maximum size of its principal submatrices with no o¤-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting kAk be the Frobenius norm of A; we show that jhAx;xij (1 1=2r 1=2n) kAk : This inequality is tight within an additive term O n 2 : If the matrix A is Hermitian, then jhAx;xij (1 1=r) kAk : This inequality...

A coherent F -module on X is a coherent sheafM on X, together with a Frobenius action onM, that is, a morphism of sheaves of OX-modules FM : M→ F∗(M). In other words, FM is a morphism of sheaves of Fq-vector spaces OX → OX such that FM(am) = aFM(m) for every a ∈ OX(U) and m ∈ M(U), where U is any open subset of X. As above, since FM is Fq-linear, it follows that it induces Fq-linear maps on coh...

By invoking the concept of twisted Poincaré symmetry of the algebra of functions on a Minkowski space-time, we demonstrate that the noncommutative space-time with the commutation relations [xμ ,xν ] = iθμν , where θμν is a constant real antisymmetric matrix, can be interpreted in a Lorentzinvariant way [1]. The implications of the twisted Poincaré symmetry on QFT on such a spacetime is briefly ...

These notes were taken at the NSF-CBMS conference on topological and geometric Methods in QFT at Montana State University in summer 2017. Most of the lectures were given by Dan Freed. I live-TEXed these notes using vim, and as such there may be typos; please send questions, comments, complaints, and corrections to [email protected]; any mistakes in the notes are my own. Thanks to Nilay K...

Expressions, such as add fuel to the fire, can be interpreted literally or idiomatically depending on the context they occur in. Many Natural Language Processing applications could improve their performance if idiom recognition were improved. Our approach is based on the idea that idioms and their literal counterparts do not appear in the same contexts. We propose two approaches: (1) Compute in...

We construct a holographic map between asymptotically AdS5 × S5 solutions of 10d supergravity and vacuum expectation values of gauge invariant operators of the dual QFT. The ingredients that enter in the construction are (i) gauge invariant variables so that the KK reduction is independent of any choice of gauge fixing; (ii) the non-linear KK reduction map from 10 to 5 dimensions (constructed p...

Philosophers of physics should be more attentive to the role energy conditions play in GR. I review the changing status of energy conditions for quantum fields – presently there are no singularity theorems for semiclassical GR. So we must re-evaluate how we understand the relationship between GR, QFT and singularities. Moreover, on our present understanding of what it is to be a “reasonable” fi...

We define a notion of “Frobenius pair”, which is a mild generalization of the notion of “Frobenius object” in a monoidal category. We then show that Atiyah duality for smooth manifolds can be encapsulated in the statement that a certain collection of structure obtained from a manifold forms a “commutative Frobenius pair” in the stable homotopy category of spectra.

Frobenius algebras play an important role in the representation theory of finite groups. In the present work, we investigate the (quasi) Frobenius property of n-group algebras. Using the (quasi-) Frobenius property of ring, we can obtain some information about constructions of module category over this ring ([2], p. 66–67).