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This work seeks to make explicit the operational connection between the preparation of two-level quantum systems with their corresponding description (as states) in a Hilbert space. This may sound outdated, but we show there is more to this…
The Goodwillie tower is based on the idea of approximating a functor F by a series of functors satisfying the strong property of "n-excision". In this dissertation, we study a weaker property of "n-additivity" and compare the two. Theorem…
The space of E-infinity structures on an simplicial operad C is the limit of a tower of fibrations, so its homotopy is the abutment of a Bousfield-Kan fringed spectral sequence. The spectral sequence begins (under mild restrictions) with…
Assume that there exists a smooth map between two closed manifolds $M^m\to N^k$ with only finitely many cone-like singular points, where $2\leq k\leq m\leq 2k-1$. If $(m,k)\not\in\{(2,2), (4,3), (5,3), (8,5), (16,9)\}$, then $M^m$ admits a…
We prove fibration theorems \`a la Milnor for differentiable real maps with non isolated critical values. We study the situation for maps with linear discriminant, and prove that the concept of d-regularity is the key point for the…
Finitely many hypersurfaces are removed from unordered configuration spaces of $n$ points in $\mathbb{C}$ to obtain a fibration over unordered configuration spaces of $n-1$ complex points. Fundamental groups of these restricted…
The degree of a map between orientable manifolds is a fundamental concept in topology, providing deep insights into the structure of manifolds and the behavior of maps between them. Recently, this notion has been extensively studied,…
A major ambition of systems science is to uncover the building blocks of any biological network to decipher how cellular function emerges from their interactions. Here, we introduce a graph representation of the information flow in these…
We establish a straightforward estimate for the number of open sets with fundamental group constraints needed to cover the total space of fibrations. This leads to vanishing results for simplicial volume and minimal volume entropy, e.g.,…
Milnor's fibration theorem is about the geometry and topology of real and complex analytic maps near their critical points, a ubiquitous theme in mathematics. As such, after 50 years, this has become a whole area of research on its own,…
In the present paper we consider fibrations $f: S \ra B$ of an algebraic surface onto a curve $B$, with general fibre a curve of genus $g$. Our main results are: 1) A structure theorem for such fibrations in the case $g=2$ 2) A structure…
Categories of lenses/optics and Dialectica categories are both comprised of bidirectional morphisms of basically the same form. In this work we show how they can be considered a special case of an overarching fibrational construction,…
We consider partial liftings of maps at fibrations and compare the primary obstruction to extend the lifting with the obstruction to extend the lifting as a simple map into the total space. A relation between these two obstructions is…
We define a construction on operads which yields a new description of the minimal model. The construction also allows us to define algebraic structures on the homology of chain complexes with homologously trivial operad algebra structures,…
An analysis of a tower of hidden sectors coupled to each other, with one of these hidden sectors coupled to the visible sector, is given and the implications of such couplings on physics in the visible sector are investigated. Thus the…
Microwave, submillimetre-wave, and far-infrared phased arrays are of considerable importance for astronomy. We consider the behaviour imaging phased arrays and interferometric phased arrays from a functional perspective. It is shown that…
Let F_0=B,...,F_n be a sequence of differentiable manifolds, G_i a Lie subgroup of diffeomorphisms of F_i, and H_i a subgroup of G_i central in G_i. We suppose also given a locally trivial bundle p_{K_i} over F_{i-1} which typical fiber is…
For differential calculi on noncommutative algebras, we construct a twisted de Rham cohomology using flat connections on modules. This has properties similar, in some respects, to sheaf cohomology on topological spaces. We also discuss…
We propose a new framework for the study of continuous time dynamical systems on networks. We view such dynamical systems as collections of interacting control systems. We show that a class of maps between graphs called graph fibrations…
We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation,…