25 research outputs found
On exact categories and applications to triangulated adjoints and model structures
We show that Quillen's small object argument works for exact categories under
very mild conditions. This has immediate applications to cotorsion pairs and
their relation to the existence of certain triangulated adjoint functors and
model structures. In particular, the interplay of different exact structures on
the category of complexes of quasi-coherent sheaves leads to a streamlined and
generalized version of recent results obtained by Estrada, Gillespie, Guil
Asensio, Hovey, J{\o}rgensen, Neeman, Murfet, Prest, Trlifaj and possibly
others.Comment: 38 pages; version 2: major revision, more explanation added at
several places, reference list updated and extended, misprints correcte
Homotopy theory of higher categories
This is the first draft of a book about higher categories approached by
iterating Segal's method, as in Tamsamani's definition of -nerve and
Pelissier's thesis. If is a tractable left proper cartesian model category,
we construct a tractable left proper cartesian model structure on the category
of -precategories. The procedure can then be iterated, leading to model
categories of -categories
Pick Interpolation and the Distance Formula
The classical interpolation theorem for the open complex unit disk, due to Nevanlinna and Pick in the early 20th century, gives an elegant criterion for the solvability of the problem as an eigenvalue problem. In the 1960s, Sarason reformulated problems of this type firmly in the language of operator theoretic function theory. This thesis will explore connections between interpolation problems on various domains (both single and several complex variables) with the viewpoint that Sarason’s work suggests.
In Chapter 1, some essential preliminaries on bounded operators on Hilbert space and the functionals that act on them will be presented, with an eye on the various ways distances can be computed between operators and a certain type of ideal. The various topologies one may define on B(H) will play a prominent role in this development. Chapter 2 will introduce the concept of a reproducing kernel Hilbert space, and a distinguished operator algebra that we associate to such spaces know as the multiplier algebra. The various operator theoretic properties that multiplier algebras enjoy will be presented, with a particular emphasis on their invariant subspace lattices and the connection to distance formulae.
In Chapter 3, the Nevanlinna-Pick problem will be invested in general for any repro- ducing kernel Hilbert space, with the basic heuristic for distance formulae being presented. Chapter 4 will treat a large class of reproducing kernel Hilbert spaces associated to measure spaces, where a Pick-like theorem will be established for many members of this class. This approach will closely follow similar results in the literature, including recent treatments by McCullough and Cole-Lewis-Wermer.
Reproducing kernel Hilbert spaces where the analogue of the Nevanlinna-Pick theorem holds are particularly nice. In Chapter 5, the operator theory of these so-called complete Pick spaces will be developed, and used to tackle certain interpolation problems where additional constraints are imposed on the solution. A non-commutative view of interpola- tion will be presented, with the non-commutative analytic Toeplitz algebra of Popescu and Davidson-Pitts playing a prominent role.
It is often useful to consider reproducing kernel Hilbert spaces which arise as natural products of other spaces. The Hardy space of the polydisk is the prime example of this. A general commutative and non-commutative view of such spaces will be presented in Chapter 6, using the left regular representation of higher-rank graphs, first introduced by Kribs-Power. A recent factorization theorem of Bercovici will be applied to these algebras, from which a Pick-type theorem may be deduced. The operator-valued Pick problem for these spaces will also be discussed.
In Chapter 7, the various tools developed in this thesis will be applied to two related problems, known as the Douglas problem and the Toeplitz corona problem
Three models of ordinal computability
In this thesis we expand the scope of ordinal computability, i.e., the study of models of computation that are generalized to infinite domains. The discipline sets itself apart from classical work on generalized recursion theory by focusing strongly on the computational paradigm and an analysis in elementary computational steps. In the present work, two models of classical computability of which no previous generalizations to ordinals are known to the author are lifted to the ordinal domain, namely λ-calculus and Blum-Shub-Smale machines. One of the multiple generalizations of a third model relevant to this thesis, the Turing machine, is employed to further study classical descriptive set theory. The main results are: An ordinal λ-calculus is defined and confluency properties in the form of a weak Church-Rosser theorem are established. The calculus is proved to be strongly related to the constructible hierarchy of sets, a feature typical for an entire subfamily of models of ordinal computation. Ordinal Turing machines with input restricted to subsets of ω are shown to compute the Δ12 sets of reals. Conversely, the machines can be employed to reprove the absoluteness of Σ12 sets (Shoenfield absoluteness) and basic properties of Σ12 sets. New tree representations and new pointclasses defined by the means of ordinal Turing computations are introduced and studied. The Blum-Shub-Smale model for computation on the real numbers is lifted to transfinite running times. The supremum of possible runtimes is determined and an upper bound on the computational strength is given. Summarizing, this thesis both expands the field of ordinal computability by enlarging its palette of computational models and also connects the field further by tying in the new models into the existing framework. Questions that have been raised in the community, e.g. on the possibility of generalizations of λ-calculus and Blum-Shub-Smale machines, are addressed and answered
Formal approaches to number in Slavic and beyond (Volume 5)
The goal of this collective monograph is to explore the relationship between the cognitive notion of number and various grammatical devices expressing this concept in natural language with a special focus on Slavic. The book aims at investigating different morphosyntactic and semantic categories including plurality and number-marking, individuation and countability, cumulativity, distributivity and collectivity, numerals, numeral modifiers and classifiers, as well as other quantifiers. It gathers 19 contributions tackling the main themes from different theoretical and methodological perspectives in order to contribute to our understanding of cross-linguistic patterns both in Slavic and non-Slavic languages
Advances in Trans-dimensional Geophysical Inference
This research presents a series of novel Bayesian
trans-dimensional
methods for geophysical inversion. A first example illustrates
how
Bayesian prior information obtained from theory and numerical
experiments can be used to better inform a difficult
multi-modal inversion of dispersion information from empirical
Greens
functions obtained from ambient noise cross-correlation. This
approach
is an extension of existing partition modeling schemes.
An entirely new class of trans-dimensional algorithm, called the
trans-dimensional tree method is introduced. This new method is
shown
to be more efficient at coupling to a forward model, more
efficient at
convergence, and more adaptable to different dimensions and
geometries
than existing approaches. The efficiency and flexibility of the
trans-dimensional tree method is demonstrated in two different
examples: (1) airborne electromagnetic tomography (AEM) in a 2D
transect inversion, and (2) a fully non-linear inversion of
ambient
noise tomography. In this latter example the resolution at depth
has
been significantly improved by inverting a contiguous band of
frequencies jointly rather than as independent phase velocity
maps,
allowing new insights into crustal architecture beneath Iceland.
In a first test case for even larger scale problems, an
application of
the trans-dimensional tree approach to large global data set is
presented. A global database of nearly 5 million multi-model
path
average Rayleigh wave phase velocity observations has been used
to
construct global phase velocity maps. Results are comparable to
existing published phase velocity maps, however, as the
trans-dimensional approach adapts the resolution appropriate to
the
data, rather than imposing damping or smoothing constraints to
stabilize the inversion, the recovered anomaly magnitudes are
generally higher with low uncertainties. While further
investigation is
needed, this early test case shows that trans-dimensional
sampling can
be applied to global scale seismology problems and that previous
analyses may, in some locales, under estimate the heterogeneity
of the
Earth.
Finally, in a further advancement of partition modelling with
variable
order polynomials, a new method has been developed called
trans-dimensional spectral elements. Previous applications
involving
variable order polynomials have used polynomials that are both
difficult to work with in a Bayesian framework and unstable at
higher orders. By using the orthogonal polynomials typically used
in
modern full-waveform solvers, the useful properties of this type
of
polynomial and its application in trans-dimensional inversion
are
demonstrated. Additionally, these polynomials can be directly
used in
complex differential solvers and an example of this for 1D
inversion
of surface wave dispersion curves is given
Number Theory, Analysis and Geometry: In Memory of Serge Lang
Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future.
In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas, namely number theory, analysis and geometry, representing Lang’s own breadth of interests. A special introduction by John Tate includes a brief and engaging account of Serge Lang’s life
Techniques in Active and Generic Software Libraries
Reusing code from software libraries can reduce the time and effort to construct software
systems and also enable the development of larger systems. However, the benefits
that come from the use of software libraries may not be realized due to limitations in
the way that traditional software libraries are constructed. Libraries come equipped
with application programming interfaces (API) that help enforce the correct use of
the abstractions in those libraries. Writing new components and adapting existing
ones to conform to library APIs may require substantial amounts of "glue" code that
potentially affects software's efficiency, robustness, and ease-of-maintenance. If, as a
result, the idea of reusing functionality from a software library is rejected, no benefits
of reuse will be realized.
This dissertation explores and develops techniques that support the construction
of software libraries with abstraction layers that do not impede efficiency. In many
situations, glue code can be expected to have very low (or zero) performance overhead.
In particular, we describe advances in the design and development of active libraries
- software libraries that take an active role in the compilation of the user's code.
Common to the presented techniques is that they may "break" a library API (in a
controlled manner) to adapt the functionality of the library for a particular use case.
The concrete contributions of this dissertation are: a library API that supports
iterator selection in the Standard Template Library, allowing generic algorithms to
find the most suitable traversal through a container, allowing (in one case) a 30-fold improvement in performance; the development of techniques, idioms, and best practices
for concepts and concept maps in C++, allowing the construction of algorithms
for one domain entirely in terms of formalisms from a second domain; the construction
of generic algorithms for algorithmic differentiation, implemented as an active
library in Spad, language of the Open Axiom computer algebra system, allowing algorithmic
differentiation to be applied to the appropriate mathematical object and
not just concrete data-types; and the description of a static analysis framework to
describe the generic programming notion of local specialization within Spad, allowing
more sophisticated (value-based) control over algorithm selection and specialization
in categories and domains.
We will find that active libraries simultaneously increase the expressivity of the
underlying language and the performance of software using those libraries