237 research outputs found
Introduction to Riemannian Geometry and Geometric Statistics: from basic theory to implementation with Geomstats
International audienceAs data is a predominant resource in applications, Riemannian geometry is a natural framework to model and unify complex nonlinear sources of data.However, the development of computational tools from the basic theory of Riemannian geometry is laborious.The work presented here forms one of the main contributions to the open-source project geomstats, that consists in a Python package providing efficient implementations of the concepts of Riemannian geometry and geometric statistics, both for mathematicians and for applied scientists for whom most of the difficulties are hidden under high-level functions. The goal of this monograph is two-fold. First, we aim at giving a self-contained exposition of the basic concepts of Riemannian geometry, providing illustrations and examples at each step and adopting a computational point of view. The second goal is to demonstrate how these concepts are implemented in Geomstats, explaining the choices that were made and the conventions chosen. The general concepts are exposed and specific examples are detailed along the text.The culmination of this implementation is to be able to perform statistics and machine learning on manifolds, with as few lines of codes as in the wide-spread machine learning tool scikit-learn. We exemplify this with an introduction to geometric statistics
Algebraic K-theory of reductive p-adic groups
Motivated by the Farrell-Jones Conjecture for group rings, we formulate the
op-Farrell-Jones Conjecture for the K-theory of Hecke algebras of
td-groups. We prove this conjecture for (closed subgroups of) reductive p-adic
groups G. In particular, the projective class group for a
(closed subgroup) of a reductive p-adic group G can be computed as a colimit of
projective class groups where U varies over the compact
open subgroups of G. This implies that all finitely generated smooth complex
representations of a reductive p-adic G admit finite projective resolutions by
compactly induced representations. For SL we translate the colimit
formula for to a more concrete cokernel description in
terms of stabilizers for the action on the Bruhat-Tits building.
For negative K-theory we obtain vanishing results, while we identify the
higher K-groups with the value of G-homology theory on
the extended Bruhat-Tits building. Our considerations apply to general Hecke
algebras of the form , where we allow a central
character and a twist by an action of G on R. For the
op-Farrell-Jones Conjecture we need to assume and a regularity assumption. As a key intermediate step we
introduce the $\mathcal{C}vcy-Farrell-Jones conjecture. For the latter no
regularity assumptions on R are needed.Comment: 91 page
Collapsibility and Z-Compactifications of CAT(0) Cube Complexes
We extend the notion of collapsibility to non-compact complexes and prove collapsibility of locally-finite CAT(0) cube complexes. Namely, we construct such a cube complex out of nested convex compact subcomplexes with the properties that and collapses to for all .
We then define bonding maps between the compacta and construct an inverse sequence yielding the inverse limit space . This will provide a new way of Z-compactifying . In particular, the process will yield a new Z-boundary, called the cubical boundary
Breaking together: a freedom-loving response to collapse
The collapse of modern societies has already begun. That is the conclusion of two years of research by the interdisciplinary team behind the book 'Breaking Together'. How did it come to this? Because monetary systems caused us to harm each other and nature to such an extent it broke the foundations of our societies. So what can we do? This book describes people allowing the full pain of our predicament to liberate them into living more courageously and creatively. They demonstrate we can be breaking together, not apart, in this era of collapse. Professor Jem Bendell argues that reclaiming our freedoms is essential to soften the fall and regenerate the natural world. Escaping the efforts of panicking elites, we can advance an ecolibertarian agenda for both politics and practical action in a broken world. Endorsing the text, the founder of Schumacher College, Satish Kumar, remarked: “this is a prophetic book.
Natural or anthropogenic variability? A long-term pattern of the zooplankton communities in an ever-changing transitional ecosystem
The Venice Lagoon is an important site belonging to the Italian Long-Term Ecological Research Network (LTER). Alongside with the increasing trend of water temperature and the relevant morphological changes, in recent years, the resident zooplankton populations have also continued to cope with the colonization by alien species, particularly the strong competitor Mnemiopsis leidyi. In this work, we compared the dynamics of the lagoon zooplankton over a period of 20 years. The physical and biological signals are analyzed and compared to evaluate the hypothesis that a slow shift in the environmental balance of the site, such as temperature increase, sea level rise (hereafter called “marinization”), and competition between species, is contributing to trigger a drift in the internal equilibrium of the resident core zooplankton. Though the copepod community does not seem to have changed its state, some important modifications of structure and assembly mechanisms have already been observed. The extension of the marine influence within the lagoon has compressed the spatial gradients of the habitat and created a greater segregation of the niches available to some typically estuarine taxa and broadened and strengthened the interactions between marine species
Oka-1 manifolds
We introduce a new class of complex manifolds: Oka-1 manifolds. They are
characterized by the property that holomorphic maps from any open Riemann
surface satisfy the Runge approximation and the Weierstrass interpolation
condition. We prove that every complex manifold which is dominable by tubes of
complex lines is an Oka-1 manifold. In particular, a manifold dominable by
at most points is an Oka-1 manifold. This provides many examples
of Oka-1 manifolds among compact algebraic surfaces, including all Kummer and
all elliptic K3 surfaces. We also show that every compact rationally connected
manifold is an Oka-1 manifold. The class of Oka-1 manifolds is invariant under
Oka maps inducing a surjective homomorphism of fundamental groups; this
includes holomorphic fibre bundles with connected Oka fibres. In another
direction, we prove that every bordered Riemann surface admits a holomorphic
map with dense image in any connected complex manifold
Relationship between synoptic circulations and the spatial distributions of rainfall in Zimbabwe
This study examines how the atmospheric circulation patterns in Africa south of the equator govern the spatial distribution of precipitation in Zimbabwe. The moisture circulation patterns are designated by an ample set of eight classified circulation types (CTs). Here it is shown that all wet CTs over Zimbabwe features enhanced cyclonic/convective activity in the southwest Indian Ocean. Therefore, enhanced moisture availability in the southwest Indian Ocean is necessary for rainfall formation in parts of Zimbabwe. The wettest CT in Zimbabwe is characterized by a ridging South Atlantic Ocean high-pressure, south of South Africa, driving an abundance of southeast moisture fluxes, from the southwest Indian Ocean into Zimbabwe. Due to the proximity of Zimbabwe to the Agulhas and Mozambique warm current, the activity of the ridging South Atlantic Ocean anticyclone is a dominant synoptic feature that favors above-average rainfall in Zimbabwe. Also, coupled with a weaker state of the Mascarene high, it is shown that a ridging South Atlantic Ocean high-pressure, south of South Africa, can be favorable for the southwest movement of tropical cyclones into the eastern coastal landmasses resulting in above-average rainfall in Zimbabwe. The driest CT is characterized by the northward track of the Southern Hemisphere mid-latitude cyclones leading to enhanced westerly fluxes in the southwest Indian Ocean, limiting moist southeast winds into Zimbabwe
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Computational Methods in Multi-Messenger Astrophysics using Gravitational Waves and High Energy Neutrinos
This dissertation seeks to describe advancements made in computational methods for multi-messenger astrophysics (MMA) using gravitational waves GW and neutrinos during Advanced LIGO (aLIGO)’s first through third observing runs (O1-O3) and, looking forward, to describe novel computational techniques suited to the challenges of both the burgeoning MMA field and high-performance computing as a whole.
The first two chapters provide an overview of MMA as it pertains to gravitational wave/high energy neutrino (GWHEN) searches, including a summary of expected astrophysical sources as well as GW, neutrino, and gamma-ray detectors used in their detection. These are followed in the third chapter by an in-depth discussion of LIGO’s timing system, particularly the diagnostic subsystem, describing both its role in MMA searches and the author’s contributions to the system itself.
The fourth chapter provides a detailed description of the Low-Latency Algorithm for Multi-messenger Astrophysics (LLAMA), the GWHEN pipeline developed by the author and used in O2 and O3. Relevant past multi-messenger searches are described first, followed by the O2 and O3 analysis methods, the pipeline’s performance, scientific results, and finally, an in-depth account of the library’s structure and functionality. In particular, the author’s high-performance multi-order coordinates (MOC) HEALPix image analysis library, HPMOC, is described. HPMOC increases performance of HEALPix image manipulations by several orders of magnitude vs. naive single-resolution approaches while presenting a simple high-level interface and should prove useful for diverse future MMA searches. The performance improvements it provides for LLAMA are also covered.
The final chapter of this dissertation builds on the approaches taken in developing HPMOC, presenting several novel methods for efficiently storing and analyzing large data sets, with applications to MMA and other data-intensive fields. A family of depth-first multi-resolution ordering of HEALPix images — DEPTH9, DEPTH19, and DEPTH40 — is defined, along with algorithms and use cases where it can improve on current approaches, including high-speed streaming calculations suitable for serverless compute or FPGAs.
For performance-constrained analyses on HEALPix data (e.g. image analysis in multi-messenger search pipelines) using SIMD processors, breadth-first data structures can provide short-circuiting calculations in a data-parallel way on compressed data; a simple compression method is described with application to further improving LLAMA performance.
A new storage scheme and associated algorithms for efficiently compressing and contracting tensors of varying sparsity is presented; these demuxed tensors (D-Tensors) have equivalent asymptotic time and space complexity to optimal representations of both dense and sparse matrices, and could be used as a universal drop-in replacement to reduce code complexity and developer effort while improving performance of existing non-optimized numerical code. Finally, the big bucket hash table (B-Table), a novel type of hash table making guarantees on data layout (vs. load factor), is described, along with optimizations it allows for (like hardware acceleration, online rebuilds, and hard realtime applications) that are not possible with existing hash table approaches. These innovations are presented in the hope that some will prove useful for improving future MMA searches and other data-intensive applications
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