Related papers: Dependence spaces
The replacement property (or Steinitz Exchange Lemma) for vector spaces has a natural analog for finite groups and their generating sets. For the special case of the groups PSL(2, p), where p is a prime larger than 5, first partial results…
In this Note we show that the notion of a basis of a finite-dimensional vector space could be introduced by an argument much weaker than Gauss' reduction method. Our aim is to give a short proof of a simply formulated lemma, which in fact…
A fundamental theorem of linear algebra asserts that every basis for the vector space $\mathbb{R}^n$ has $n$ elements. In this expository note we present a theorem of W. G. Leavitt describing one way in which this invariant basis number…
The Steinitz lemma, a classic from 1913, states that $a_1,\ldots,a_n$, a sequence of vectors in $\R^d$ with $\sum_1^n a_i=0$, can be rearranged so that every partial sum of the rearranged sequence has norm at most $2d\max \|a_i\|$. In the…
We give a new proof of the base change fundamental lemma for GL_n for certain Hecke functions by comparing the cohomology of two different moduli space of $D$-shtukas. The original proof, due to Clozel and Labesse, make use of the trace…
The extended de Finetti theorem characterizes exchangeable infinite random sequences as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is…
Lattice theoretical generalizations of some classical linear algebra results are formulated. A vector space is replaced by its subspace lattice and a linear map is replaced by the induced lattice map. This map is a complete join…
The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence $\alpha,2\alpha,\ldots,N\alpha$, for any integer $N$ and real number $\alpha$. This statement…
Special relativity is reformulated as a symmetry property of space-time: Space-Time Exchange Invariance. The additional hypothesis of spatial homogeneity is then sufficient to derive the Lorentz transformation without reference to the…
De Finetti's theorem, also called the de Finetti-Hewitt-Savage theorem, is a foundational result in probability and statistics. Roughly, it says that an infinite sequence of exchangeable random variables can always be written as a mixture…
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…
Given a quantum system consisting of many parts, we show that symmetry of the system's state, i.e., invariance under swappings of the subsystems, implies that almost all of its parts are virtually identical and independent of each other.…
Zorn's Lemma is a well-known equivalent of the Axiom of Choice. It is usually regarded as a topic in axiomatic set theory, and its historically standard proof (from the Axiom of Choice) relies on transfinite recursion, a non-elementary…
An alternative (simplified) derivation of the dispersion relation and the expressions for the momentum-energy 4-vector $p_i,p_0$ given initially in [1] is provided. It has turned out that in a rather "pedestrian" manner one can obtain in…
In 2012 Raghavan, Samuel, and Subrahmanyam showed that the Kazhdan--Lusztig basis for the Iwahori--Hecke algebra in type A provides a ``canonical'' basis for the centraliser algebra of the Schur algebra acting on tensor space. In 2022 the…
We revisit Eisenstein's geometric proof of quadratic reciprocity and make explicit the involutive symmetry underlying Eisenstein's lattice-point argument. Building on Gauss's lemma, we interpret the Legendre symbols as counts of lattice…
We prove a de Finetti theorem for exchangeable sequences of states on test spaces, where a test space is a generalization of the sample space of classical probability theory and the Hilbert space of quantum theory. The standard classical…
Given a base point free linear system on an algebraic variety, many classes of singularities are stable under taking suitable members after enlarging the base field. We establish analogous results when the base ring is an excellent ring.
We present a framework for studying the concept of independence in a general context covering database theory, algebra and model theory as special cases. We show that well-known axioms and rules of independence for making inferences…
The raise of the symmetry breaking mechanism by Landau[1] is a landmark in the studies of phase transitions. The Kosterlitz-Thouless phase transition[2-3] and the fractional quantum Hall effect[4], however, are believed to be induced by…