35 research outputs found
Software Citation in Theory and Practice
In most fields, computational models and data analysis have become a
significant part of how research is performed, in addition to the more
traditional theory and experiment. Mathematics is no exception to this trend.
While the system of publication and credit for theory and experiment (journals
and books, often monographs) has developed and has become an expected part of
the culture, how research is shared and how candidates for hiring, promotion
are evaluated, software (and data) do not have the same history. A group
working as part of the FORCE11 community developed a set of principles for
software citation that fit software into the journal citation system, allow
software to be published and then cited, and there are now over 50,000 DOIs
that have been issued for software. However, some challenges remain, including:
promoting the idea of software citation to developers and users; collaborating
with publishers to ensure that systems collect and retain required metadata;
ensuring that the rest of the scholarly infrastructure, particularly indexing
sites, include software; working with communities so that software efforts
"count" and understanding how best to cite software that has not been
published
Pairing Along the Line
Pairing energies for the addition of two neutrons on even-even nuclei with are studied. The dependence is attributed to the number and type of
orbitals that are occupied in the valence shell-model space. Properties in the
region from depend on the location of the orbital.Comment: 3 pages and 2 figure
StocHy: automated verification and synthesis of stochastic processes
StocHy is a software tool for the quantitative analysis of discrete-time
stochastic hybrid systems (SHS). StocHy accepts a high-level description of
stochastic models and constructs an equivalent SHS model. The tool allows to
(i) simulate the SHS evolution over a given time horizon; and to automatically
construct formal abstractions of the SHS. Abstractions are then employed for
(ii) formal verification or (iii) control (policy, strategy) synthesis. StocHy
allows for modular modelling, and has separate simulation, verification and
synthesis engines, which are implemented as independent libraries. This allows
for libraries to be easily used and for extensions to be easily built. The tool
is implemented in C++ and employs manipulations based on vector calculus, the
use of sparse matrices, the symbolic construction of probabilistic kernels, and
multi-threading. Experiments show StocHy's markedly improved performance when
compared to existing abstraction-based approaches: in particular, StocHy beats
state-of-the-art tools in terms of precision (abstraction error) and
computational effort, and finally attains scalability to large-sized models (12
continuous dimensions). StocHy is available at www.gitlab.com/natchi92/StocHy
Homotopy continuation methods for coupled-cluster theory in quantum chemistry
Homotopy methods have proven to be a powerful tool for understanding the
multitude of solutions provided by the coupled-cluster polynomial equations.
This endeavor has been pioneered by quantum chemists that have undertaken both
elaborate numerical as well as mathematical investigations. Recently, from the
perspective of applied mathematics, new interest in these approaches has
emerged using both topological degree theory and algebraically oriented tools.
This article provides an overview of describing the latter development
Recent mathematical advances in coupled cluster theory
This article presents an in-depth educational overview of the latest
mathematical developments in coupled cluster (CC) theory, beginning with
Schneider's seminal work from 2009 that introduced the first local analysis of
CC theory. We offer a tutorial review of second quantization and the CC ansatz,
laying the groundwork for understanding the mathematical basis of the theory.
This is followed by a detailed exploration of the most recent mathematical
advancements in CC theory.Our review starts with an in-depth look at the local
analysis pioneered by Schneider which has since been applied to analyze various
CC methods. We then move on to discuss the graph-based framework for CC methods
developed by Csirik and Laestadius. This framework provides a comprehensive
platform for comparing different CC methods, including multireference
approaches. Next, we delve into the latest numerical analysis results analyzing
the single reference CC method developed by Hassan, Maday, and Wang. This very
general approach is based on the invertibility of the CC function's Fr\'echet
derivative. We conclude the article with a discussion on the recent
incorporation of algebraic geometry into CC theory, highlighting how this novel
and fundamentally different mathematical perspective has furthered our
understanding and provides exciting pathways to new computational approaches
On the Connectivity of the Disguised Toric Locus of a Reaction Network
Complex-balanced mass-action systems are some of the most important types of
mathematical models of reaction networks, due to their widespread use in
applications, as well as their remarkable stability properties. We study the
set of positive parameter values (i.e., reaction rate constants) of a reaction
network that, according to mass-action kinetics, generate dynamical systems
that can be realized as complex-balanced systems, possibly by using a different
graph . This set of parameter values is called the disguised toric locus of
. The -disguised toric locus of is defined analogously,
except that the parameter values are allowed to take on any real values. We
prove that the disguised toric locus of is path-connected, and the
-disguised toric locus of is also path-connected. We also show
that the closure of the disguised toric locus of a reaction network contains
the union of the disguised toric loci of all its subnetworks.Comment: 18 pages, 2 figure
Characteristic polynomials and eigenvalues of tensors
We lay the geometric foundations for the study of the characteristic
polynomial of tensors. For symmetric tensors of order and dimension
and symmetric tensors of order and dimension , we prove that only
finitely many tensors share any given characteristic polynomial, unlike the
case of symmetric matrices and the case of non-symmetric tensors. We propose
precise conjectures for the dimension of the variety of tensors sharing the
same characteristic polynomial, in the symmetric and in the non-symmetric
setting.Comment: 25 pages, comments are welcom