392 research outputs found
Subdirect products of groups and the n-(n+1)-(n+2) Conjecture
We analyse the subgroup structure of direct products of groups. Earlier work
on this topic has revealed that higher finiteness properties play a crucial
role in determining which groups appear as subgroups of direct products of free
groups or limit groups. Here, we seek to relate the finiteness properties of a
subgroup to the way it is embedded in the ambient product. To this end we
formulate a conjecture on finiteness properties of fibre products of groups. We
present different approaches to this conjecture, proving a general result on
finite generation of homology groups of fibre products and, for certain special
cases, results on the stronger finiteness properties F_n and FP_n.Comment: 32 page
Konflikte im Raum - Verständnis von gesellschaftlichen Diskursen durch Argumentation im Geographieunterricht
Nuclei embedded in an electron gas
The properties of nuclei embedded in an electron gas are studied within the
relativistic mean-field approach. These studies are relevant for nuclear
properties in astrophysical environments such as neutron-star crusts and
supernova explosions. The electron gas is treated as a constant background in
the Wigner-Seitz cell approximation. We investigate the stability of nuclei
with respect to alpha and beta decay. Furthermore, the influence of the
electronic background on spontaneous fission of heavy and superheavy nuclei is
analyzed. We find that the presence of the electrons leads to stabilizing
effects for both decay and spontaneous fission for high electron
densities. Furthermore, the screening effect shifts the proton dripline to more
proton-rich nuclei, and the stability line with respect to beta decay is
shifted to more neutron-rich nuclei. Implications for the creation and survival
of very heavy nuclear systems are discussed.Comment: 35 pages, latex+ep
Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory
This book aims to provide an introduction to the topic of deep learning
algorithms. We review essential components of deep learning algorithms in full
mathematical detail including different artificial neural network (ANN)
architectures (such as fully-connected feedforward ANNs, convolutional ANNs,
recurrent ANNs, residual ANNs, and ANNs with batch normalization) and different
optimization algorithms (such as the basic stochastic gradient descent (SGD)
method, accelerated methods, and adaptive methods). We also cover several
theoretical aspects of deep learning algorithms such as approximation
capacities of ANNs (including a calculus for ANNs), optimization theory
(including Kurdyka-{\L}ojasiewicz inequalities), and generalization errors. In
the last part of the book some deep learning approximation methods for PDEs are
reviewed including physics-informed neural networks (PINNs) and deep Galerkin
methods. We hope that this book will be useful for students and scientists who
do not yet have any background in deep learning at all and would like to gain a
solid foundation as well as for practitioners who would like to obtain a firmer
mathematical understanding of the objects and methods considered in deep
learning.Comment: 601 pages, 36 figures, 45 source code
An overview on deep learning-based approximation methods for partial differential equations
It is one of the most challenging problems in applied mathematics to
approximatively solve high-dimensional partial differential equations (PDEs).
Recently, several deep learning-based approximation algorithms for attacking
this problem have been proposed and tested numerically on a number of examples
of high-dimensional PDEs. This has given rise to a lively field of research in
which deep learning-based methods and related Monte Carlo methods are applied
to the approximation of high-dimensional PDEs. In this article we offer an
introduction to this field of research, we review some of the main ideas of
deep learning-based approximation methods for PDEs, we revisit one of the
central mathematical results for deep neural network approximations for PDEs,
and we provide an overview of the recent literature in this area of research.Comment: 23 page
Counterexamples to local Lipschitz and local H\"older continuity with respect to the initial values for additive noise driven SDEs with smooth drift coefficient functions with at most polynomially growing derivatives
In the recent article [A. Jentzen, B. Kuckuck, T. M\"uller-Gronbach, and L.
Yaroslavtseva, arXiv:1904.05963 (2019)] it has been proved that the solutions
to every additive noise driven stochastic differential equation (SDE) which has
a drift coefficient function with at most polynomially growing first order
partial derivatives and which admits a Lyapunov-type condition (ensuring the
the existence of a unique solution to the SDE) depend in a logarithmically
H\"older continuous way on their initial values. One might then wonder whether
this result can be sharpened and whether in fact, SDEs from this class
necessarily have solutions which depend locally Lipschitz continuously on their
initial value. The key contribution of this article is to establish that this
is not the case. More precisely, we supply a family of examples of additive
noise driven SDEs which have smooth drift coefficient functions with at most
polynomially growing derivatives whose solutions do not depend on their initial
value in a locally Lipschitz continuous, nor even in a locally H\"older
continuous way.Comment: 27 page
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