Related papers: Independence of Sets Without Stability
We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that in finite dimensional Hilbert spaces, Parseval frames with norms bounded away from 1…
In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for $\mu$-forking, a certain independence notion arising from splitting. This had been a longstanding technical…
Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for a certain independence relation called…
Decomposable dependency models possess a number of interesting and useful properties. This paper presents new characterizations of decomposable models in terms of independence relationships, which are obtained by adding a single axiom to…
We are concerned with the detection of associations between random vectors of any dimension. Few tests of independence exist that are consistent against all dependent alternatives. We propose a powerful test that is applicable in all…
The standard definition of the dimension of a vector space or rank of a module states that dimension or rank is equal to the cardinality of any basis, which requires an understanding of the concepts of basis, generating set, and linear…
We provide a differential-algebraic description of forking independence in the stable theory DCF$_{p,m}$ of differentially closed fields of characteristic $p>0$ with $m$-many commuting derivations. As a by-product of this description, we…
Classification and invariants, with respect to basis changes, of finite dimensional algebras are considered. An invariant open, dense (in the Zariscki topology) subset of the space of structural constants is defined. The algebras with…
In the literature there are two different notions of lovely pairs of a theory T, according to whether T is simple or geometric. We introduce a notion of lovely pairs for an independence relation, which generalizes both the simple and the…
We introduce a novel choice dataset, called joint choice, in which options and menus are multidimensional. In this general setting, we define a notion of choice separability, which requires that selections from some dimensions are never…
We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP$_1$-like and non-simple, a…
We study the zero sets of the independence polynomial on recursive sequences of graphs. We prove that for a maximally independent starting graph and a stable and expanding recursion algorithm, the zeros of the independence polynomial are…
We propose a definition for background (in)/dependence in dynamical theories of the evolution of configurations that have a continuous symmetry and test this definition on particle models and on gravity. Our definition draws from Barbour's…
In the area of Tame Geometry, different model-theoretic tameness conditions are established and their relationships are analyzed. We construct a subfield $K$ of the real numbers that lacks several of such tameness properties. As our main…
We present a null witness of the dimension of a quantum system, discriminating real, complex and classical spaces, based on equality due to linear independence. The witness involves only a single measurement with sufficiently many outcomes…
We study the independence structure of finitely exchangeable distributions over random vectors and random networks. In particular, we provide necessary and sufficient conditions for an exchangeable vector so that its elements are completely…
We initiate a systematic study of \emph{generic stability independence} and introduce the class of \emph{treeless theories} in which this notion of independence is particularly well-behaved. We show that the class of treeless theories…
Given a functor $F: \mathcal{C} \to \mathcal{D}$ and a model-theoretic independence relation on $\mathcal{D}$, we can lift that independence relation along $F$ to $\mathcal{C}$ by declaring a commuting square in $\mathcal{C}$ to be…
This paper investigates the stability of distance-based \textit{flexible} undirected formations in the plane. Without rigidity, there exists a set of connected shapes for given distance constraints, which is called the ambit. We show that a…
The notion of relative universality with respect to a {\sigma}-field was introduced to establish the unbiasedness and Fisher consistency of an estimator in nonlinear sufficient dimension reduction. However, there is a gap in the proof of…