Representations of Positive Projections I

In this paper we start the development of a general theory of Maharam-type representation theorems for positive projections on Dedekind complete vector lattices. In the approach to these results the theory off-algebras plays a crucial role. AMS classification: 47B65, 46A40, 06F25 To Dorothy Maharam in admiration     

Representations of Positive Projections II

In this paper we obtain a number of Maharam-type slice integral representations, with respect to scalar measures, for positive projections in Dedekind complete vector lattices and f-algebras. AMS Classification: 47B65, 46A40, 06F25     

Products of Commuting Boolean Algebras of Projections and Banach Space Geometry

New criteria and Banach spaces are presented (for example, GL-spacesand Banach spaces with property () that ensure that the Booleanalgebra generated by a pair of bounded, commuting Boolean algebrasof projections is itself bounded. The notion of R-boundednessplays a fundamental role. It is shown that the strong operatorclosure of any R-bounded Boolean algebra of projections is necessarilyBade complete. Also, for a Dedekind -complete Banach latticeE, the Boolean algebra consisting of all band projections inE is R-bounded if and only if E has finite cotype. In this situation,every bounded Boolean algebra of projections in E is R-boundedand has a Bade complete strong closure. 2000 Mathematics SubjectClassification 46B20, 47L10 (primary), 46B42, 47B40, 47B60 (secondary).     

因子von Neumann代数上导子的等价刻画

设R是含非平凡幂等元P的素环,C∈R,C=PC.本文证明可加映射△:R→R在C可导,即△(AB)=△(A)B+A△(B),A,B∈R,AB=C当且仅当存在导子δ:R→R,使得△(A)=δ(A)+△(I)A,A∈R.没有I_1型中心直和项的von Neumann代数上的可导映射也有类似结论.利用该结论证明了,若非零算子C∈B(X),使得ran(C)或ker(C)在X中可补,则可加映射△:B(X)→B(X)在C可导当且仅当它是导子.特别地,证明了因子von Neumann代数上的可加映射在任意但固定的非零算子可导当且仅当它是导子.     

B(■)的生成元

设H是复数域C上的可分Hilbert空间,B(■)是■上有界线性算子的全体.本文证明B(■)可以由三个投影生成,改进了C.Davis[2]的方法,使得第三个投影E3的取法更为灵活,证明更为简洁,获得了这样{E1,E2,E3}更多的族,并且给出当■是有限维情形时的证明。     

Well-bounded operators on nonreflexive Banach spaces

Every well-bounded operator on a reflexive Banach space is of type (B), and hence has a nice integral representation with respect to a spectral family of projections. A longstanding open question in the theory of well-bounded operators is whether there are any nonreflexive Banach spaces with this property. In this paper we extend the known results to show that on a very large class of nonreflexive spaces, one can always find a well-bounded operator which is not of type (B). We also prove that on any Banach space, compact well-bounded operators have a simple representation as a combination of disjoint projections.

     

Diophantine geometry over groups V2: quantifier elimination II

This paper is the sixth in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the two papers on quantifier elimination we use the iterative procedure that validates the correctness of an AE sentence defined over a free group, presented in the fourth paper, to show that the Boolean algebra of AE sets defined over a free group is invariant under projections, hence, show that every elementary set defined over a free group is in the Boolean algebra of AE sets. The procedures we use for quantifier elimination, presented in this paper, enable us to answer affirmatively some of Tarski’s questions on the elementary theory of a free group in the last paper of this sequence. Received (resubmission): January 2004 Revision: November 2005 Accepted: March 2006 Partially supported by an Israel Academy of Sciences Fellowship.     

Besicovitch-Federer projection theorem and geodesic flows on Riemann surfaces

We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2D measures invariant under the geodesic flow having 2D supports such that their projections to the base manifold are 2D but the supports of the projections are Lebesgue negligible. In particular, the union of complete geodesics has Hausdorff dimension 2 and is Lebesgue negligible.     

Riemannian and Sub-Riemannian Geodesic Flows

We show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This result allows us to describe the sub-Riemannian geodesic flow on totally geodesic Riemannian foliations in terms of the Riemannian geodesic flow. Also, given a submersion \(\pi :M \rightarrow B\), we describe when the projections of a Riemannian and a sub-Riemannian geodesic flow in M coincide.     

On Local Convergence of the Method of Alternating Projections

The method of alternating projections is a classical tool to solve feasibility problems. Here we prove local convergence of alternating projections between subanalytic sets \(A,B\) under a mild regularity hypothesis on one of the sets. We show that the speed of convergence is \({\mathcal {O}}(k^{-\rho })\) for some \(\rho \in (0,\infty )\).     

An equivalent condition for a uniform space to be coverable

We prove that an equivalent condition for a uniform space to be coverable is that the images of the natural projections in the fundamental inverse system are uniformly open in a certain sense. As corollaries we (1) obtain a concrete way to find covering entourage, (2) correct an error in [V. Berestovskii, C. Plaut, Uniform universal covers of uniform spaces, Topology Appl. 154 (2007) 1748-1777], and (3) show that coverability is equivalent to chain connectedness and uniform joinability in the sense of [N. Brodskiy, J. Dydak, B. Labuz, A. Mitra, Rips complexes and universal covers in the uniform category, preprint arXiv:0706.3937.     

The Minimization of the Quadratic Mean of an Integral Dose

In this work, we define the optimal dose as a combination of the projections on orthogonal axes of the absorbed dose and an integer multiple of the integral dose. Here, we show that such optimal dose minimizes the mean square of the total absorbed dose subject to certain conditions of integration. We prove that there is a unique minimizer.     

Projections of orbital measures for classical Lie groups

In this paper we compute the radial parts of the projections of orbital measures for the compact Lie groups of B, C, and D type, extending previous results obtained for the case of the unitary group by Olshanski and Faraut. Applying the method of Faraut, we show that the radial part of the projection of an orbital measure is expressed in terms of a B-spline with knots located symmetrically with respect to zero.     

An infinite sequence of non-realizable weavings

A weaving is a number of lines drawn in the plane so that no three lines intersect at a point, and the intersections are drawn so as to show which of the two lines is above the other. For each integer n4 we construct a weaving of n lines, which is not realizable as a projection of a number of lines in 3-space, all of whose subfigures are realizable as such projections.     

Subspaces,angles and pairs of orthogonal projections

We give a new proof for the Wedin theorem on the simultaneous unitary similarity transformation of two orthogonal projections and show that it is equivalent to Halmos' theorem on the unitary equivalence of projection pairs. As a consequence of these theorems, we derive several results on pairs of orthogonal projections, relative subspace positions and oblique projections as well.     

The effect of gossip on social networks

In this article, we develop a simple model for the effect of gossip spread on social network structure. We define gossip as information passed between two individuals A and B about a third individual C which affects the strengths of all three relationships: it strengthens A‐B and weakens both B‐C and A‐C. We find, in both an analytic derivation and model simulations, that if gossip does not spread beyond simple triads, it destroys them but if gossip propagates through large dense clusters, it strengthens them. Additionally, our simulations show that the effect of gossip on network metrics (clustering coefficient, average‐path‐length, and sum‐of‐strengths) varies with network structure and average‐node‐degree. © 2010 Wiley Periodicals, Inc. Complexity 16: 39‐47, 2011     

The Structure and Enumeration of Link Projections

We define a decomposition of link projections whose pieces we call atoroidal graphs. We describe a surgery operation on these graphs and show that all atoroidal graphs can be generated by performing surgery repeatedly on a family of well-known link projections. This gives a method of enumerating atoroidal graphs and hence link projections by recomposing the pieces of the decomposition.

     

Guided Projections for Analyzing the Structure of High-Dimensional Data

A powerful data transformation method named guided projections is proposed creating new possibilities to reveal the group structure of high-dimensional data in the presence of noise variables. Using projections onto a space spanned by a selection of a small number of observations allows measuring the similarity of other observations to the selection based on orthogonal and score distances. Observations are iteratively exchanged from the selection creating a nonrandom sequence of projections, which we call guided projections. In contrast to conventional projection pursuit methods, which typically identify a low-dimensional projection revealing some interesting features contained in the data, guided projections generate a series of projections that serve as a basis not just for diagnostic plots but to directly investigate the group structure in data. Based on simulated data, we identify the strengths and limitations of guided projections in comparison to commonly employed data transformation methods. We further show the relevance of the transformation by applying it to real-world datasets.     

Algebraic reflexivity of the isometry group of some spaces of Lipschitz functions

We show that the isometry groups of Lip(X,d) and lip(X,dα) with α∈(0,1), for a compact metric space (X,d), are algebraically reflexive. We also prove that the sets of isometric reflections and generalized bi-circular projections on such spaces are algebraically reflexive. In order to achieve this, we characterize generalized bi-circular projections on these spaces.     

Perturbation of Angles Between Linear Subspaces

We consider in this note how the principal angles between column spaces R(A) and R(B) change when the elements in A and B are subject to perturbations. The basic idea in the proof of our results is that the non-zero cosine values of the principal angles between R(A) and R(B) coincide with the non-zero singular values of $P_AP_B$, the product of two orthogonal projections, and consequently we can apply a perturbation theorem of orthogonal projections proved by the author[4].