We introduce various notions of rank for a high order symmetric tensor, namely: rank, border rank, catalecticant rank, generalized rank, scheme length, border scheme length, extension rank and smoothable rank. We analyze the stratification induced by these ranks. The mutual relations between these stratifications, allow us to describe the hierarchy among all the ranks. We show that strict inequ...

In this paper we prove Vaught's conjecture for superstable theories in which each complete type has nite U? rank. The general idea is to associate with the theory an V ? deenable group G (called the structure group) which controls the isomorphism types of the models. We use module theory to show that when the theory has few countable models and M is a countable model there is a nice decompositi...

the rank-k numerical range has a close connection to the construction of quantum error correction code for a noisy quantum channel. for noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the associated joint rank-k numerical range is non-empty. in this paper the notion of joint rank-k numerical range is generalized and some statements of [2011, generaliz...

February 1, 2017 Version 3.0.0 Date 2016-09-16 Title Rank Product method for identifying differentially expressed genes with application in meta-analysis Author Francesco Del Carratore , Andris Janckevics Fangxin Hong , Ben Wittner , Rainer Breitling <r.breitling...

Classification of harmonic and semiharmonic graphs according to their cyclomatic number became of interest recently. All finite harmonic graphs with up to four independent cycles, as well as all finite semiharmonic graphs with at most one cycle were determined. Here, we determine all finite semiharmonic bicyclic graphs. Besides that, we present several methods to construct semiharmonic graphs f...

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