Questions about linear spaces

AbstractWe present three themes of interest for future research that require the cooperation of fairly large teams: 1.linear spaces as building blocks;2.data for an Atlas of linear spaces;3.morphisms of linear spaces

Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics

Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner's measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite dimensional vectors spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions

Article the singular value expansion for arbitrary bounded linear operators

The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. It is well known that the SVD extends naturally to a compact linear operator mapping one Hilbert space to another; the resulting representation is known as the singular value expansion (SVE). It is less well known that a general bounded linear operator defined on Hilbert spaces also has a singular value expansion. This SVE allows a simple analysis of a variety of questions about the operator, such as whether it defines a well-posed linear operator equation and how to regularize the equation when it is not well posed

A general theory of self-similarity

A little-known and highly economical characterization of the real interval [0, 1], essentially due to Freyd, states that the interval is homeomorphic to two copies of itself glued end to end, and, in a precise sense, is universal as such. Other familiar spaces have similar universal properties; for example, the topological simplices Delta^n may be defined as the universal family of spaces admitting barycentric subdivision. We develop a general theory of such universal characterizations. This can also be regarded as a categorification of the theory of simultaneous linear equations. We study systems of equations in which the variables represent spaces and each space is equated to a gluing-together of the others. One seeks the universal family of spaces satisfying the equations. We answer all the basic questions about such systems, giving an explicit condition equivalent to the existence of a universal solution, and an explicit construction of it whenever it does exist.Comment: 81 pages. Supersedes arXiv:math/0411344 and arXiv:math/0411345. To appear in Advances in Mathematics. Version 2: tiny errors correcte

Weak product spaces of Dirichlet series

Let H2H2 denote the space of ordinary Dirichlet series with square summable coefficients, and let H20H02 denote its subspace consisting of series vanishing at +∞+∞ . We investigate the weak product spaces H2⊙H2H2⊙H2 and H20⊙H20H02⊙H02 , finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact that H20⊙H20H02⊙H02 does not contain the infinite-dimensional subspace of H2H2 of series which lift to linear functions on the infinite polydisc. The problems considered stem from questions about the dual spaces of these weak product spaces, and are therefore naturally phrased in terms of multiplicative Hankel forms. We show that there are bounded, even Schatten class, multiplicative Hankel forms on H20×H20H02×H02 whose analytic symbols are not in H2H2 . Based on this result we examine Nehari’s theorem for such Hankel forms. We define also the skew product spaces associated with H2⊙H2H2⊙H2 and H20⊙H20H02⊙H02 , with respect to both half-plane and polydisc differentiation, the latter arising from Bohr’s point of view. In the process we supply square function characterizations of the Hardy spaces HpHp , for 0<p<∞0<p<∞ , from the viewpoints of both types of differentiation. Finally we compare the skew product spaces to the weak product spaces, leading naturally to an interesting Schur multiplier problem

Makings of imagination in alternative cultural spaces in Cairo

To speak of space as a concept, how it is produced, de/reconstructed, and imagined is a process that involves multiplicities of understanding about the makings that take place. My concern is in this research is exploring the dynamic relationships that take place between cultural spaces in Cairo, the subjectivities of their participants and the possibilities that might be offered through these relations for a different social imagination that could be manifested in the details of their everydayness. The main question of my thesis is; In which ways and conditions can some of the contemporary cultural spaces in Cairo situate their presence and serve as a liberating spaces that nurture imaginations capable of transfiguring the status quo whether intellectual, social or political. My research questions are anchored in four focal theoretical concepts: space, subjectivity, imagination and how these concepts are manifested in everydayness. I will not deal with them as separate, linear or static concepts but as dimensions that are constantly in dynamic change in relation to each other. I will not attempt a cause/effect analysis and I am not after a comparative or a descriptive analysis of the two cultural spaces I chose ( Nahda Association- Jesuit\u27s culture centre in Cairo, The Choir Project of Cairo). I believe this different and dynamic configuration of theorizing will enable different moods of thinking and greater capacity of exploration to acquire different kind of knowledge about the contemporary moment in Cairo\u27s cultural scene, which is rapidly changing and how they can possibly provide fertile conditions for a different social imagination to take place

Infinite dimensional moment problem: open questions and applications

Infinite dimensional moment problems have a long history in diverse applied areas dealing with the analysis of complex systems but progress is hindered by the lack of a general understanding of the mathematical structure behind them. Therefore, such problems have recently got great attention in real algebraic geometry also because of their deep connection to the finite dimensional case. In particular, our most recent collaboration with Murray Marshall and Mehdi Ghasemi about the infinite dimensional moment problem on symmetric algebras of locally convex spaces revealed intriguing questions and relations between real algebraic geometry, functional and harmonic analysis. Motivated by this promising interaction, the principal goal of this paper is to identify the main current challenges in the theory of the infinite dimensional moment problem and to highlight their impact in applied areas. The last advances achieved in this emerging field and briefly reviewed throughout this paper led us to several open questions which we outline here.Comment: 14 pages, minor revisions according to referee's comments, updated reference

Words are Malleable: Computing Semantic Shifts in Political and Media Discourse

Recently, researchers started to pay attention to the detection of temporal shifts in the meaning of words. However, most (if not all) of these approaches restricted their efforts to uncovering change over time, thus neglecting other valuable dimensions such as social or political variability. We propose an approach for detecting semantic shifts between different viewpoints--broadly defined as a set of texts that share a specific metadata feature, which can be a time-period, but also a social entity such as a political party. For each viewpoint, we learn a semantic space in which each word is represented as a low dimensional neural embedded vector. The challenge is to compare the meaning of a word in one space to its meaning in another space and measure the size of the semantic shifts. We compare the effectiveness of a measure based on optimal transformations between the two spaces with a measure based on the similarity of the neighbors of the word in the respective spaces. Our experiments demonstrate that the combination of these two performs best. We show that the semantic shifts not only occur over time, but also along different viewpoints in a short period of time. For evaluation, we demonstrate how this approach captures meaningful semantic shifts and can help improve other tasks such as the contrastive viewpoint summarization and ideology detection (measured as classification accuracy) in political texts. We also show that the two laws of semantic change which were empirically shown to hold for temporal shifts also hold for shifts across viewpoints. These laws state that frequent words are less likely to shift meaning while words with many senses are more likely to do so.Comment: In Proceedings of the 26th ACM International on Conference on Information and Knowledge Management (CIKM2017