Related papers: Murphy's {\em Positive definite kernels and Hilber…
Let $R$ be a commutative unital ring, $\mathfrak{ a}$ an ideal of $R$ and $M$ a fixed $R$-module. We introduce and study generalisations of $\mathfrak{a}$-reduced modules, $\mathfrak{R}_{\mathfrak{ a}}$ and $\mathfrak{a}$-coreduced modules,…
In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $\mathbb{Q}$, by…
We introduce mollified Christoffel-Darboux (CD) kernels on varieties, a systematic regularization of the classical CD kernel associated with a probability measure on a compact domain. The main motivations are twofold: first, to sharpen the…
As a partial generalisation of the Uhlhorn theorem to Hilbert $C^*$-modules, we show in this article that the module structure and the orthogonality structure of a Hilbert $C^*$-module determine its Hilbert $C^*$-module structure. In fact,…
In this paper, we use the parametrised strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H_3-twisted noncommutative…
Our main goal in this paper, is to generalize to Hilbert C*-modules the concept of fusion frames. Indeed we introduce the notion of *\~nfusion frames associated to weighted sequences of orthogonally complemented submodules of a Hilbert…
It is shown that the metric on the union of the sets $X$ and $Y$ defines a Hilbert $C^*$-module over the uniform Roe algebra of the space $X$ with a fixed metric $d_X$. A number of examples of such Hilbert $C^*$-modules are described.
This is a short review of the algebraic properties of Clifford algebras and spinors. Their use in the description of fundamental physics (elementary particles) is also summarized. Lecture given at the ICCA7 conference, Toulouse (23/05/2005)
I provide a unifying perspective on forecast evaluation, characterizing accurate forecasts of all types, from simple point to complete probabilistic forecasts, in terms of two fundamental underlying properties, autocalibration and…
The goal of this paper is to develop the theory of Deligne-Beilinson cohomology of affine groups with a mixed Hodge structure. The motivation comes from Hodge theory and the study of motives, where such groups appear. Several of Francis…
The discovery of the proton-radius puzzle and the subsequent deuteron-radius puzzle is fueling an on-going debate on possible explanations for the difference in the observed radii obtained from muonic atoms and from electron-nucleus…
They run our lives, if you believe the hype in the news, but there is no precise definition of "algorithms" which is generally accepted by the mathematicians, logicians and computer scientists who create and study them. My main aims here…
We reflect on the notions of positivity and square roots. We review many examples which underline our thesis that square roots of positive maps related to *-algebras are Hilbert modules. As a result of our considerations we discuss…
This is a rough write-up of my lecture at Kinosaki and two lectures at RIMS workshops in Dec 1996, on work in progress that has not yet reached any really worthwhile conclusion, but contains lots of fun calculations. History of Vafa's…
Quantum theory's Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The reconstruction process is not complete, since it…
In kernel methods, temporal information on the data is commonly included by using time-delayed embeddings as inputs. Recently, an alternative formulation was proposed by defining a gamma-filter explicitly in a reproducing kernel Hilbert…
It is well known that the ring radical theory can be approached via language of modules. In this work, we present some generalizations of classical results from module theory, in the two-sided and graded sense. Let $\mathsf{G}$ be a group,…
Almost from the inception of Hilbert's program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various…
This is a survey article on Mercer's Theorem in its most general form and its relations with the theory of reproducing kernel Hilbert spaces and the spectral theory of compact operators. We provide a modern introduction to the basics of the…
This paper addresses the question why quantum mechanics is formulated in a unitary Hilbert space, i.e. in a manifestly complex setting. Investigating the linear dynamics of real quantum theory in a finite-dimensional Euclidean Hilbert space…