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We give a discrete Bonnet Myers type theorem for the effective diameter assuming positive Ollivier curvature. We prove that this diameter bound is attained if and only if the graph is a cocktail party graph, a Johnson graph, a halved cube, a Schl\"afli graph, a Gosset graph, or a cartesian product of the mentioned graphs with same Ollivier curvature. As a key step in the proof, we introduce the notion of reflective graphs as graphs such that for any two neighbors there exists a certain self-inverse automorphism mapping one neighbor to another. We classify these graphs as arbitrary cartesian products of the graphs mentioned before.

In this paper, we prove the second-order Sobolev inequalities on Cayley graphs of groups of polynomial growth. We use the discrete Concentration-Compactness principle to prove the existence of extremal functions for best constants in supercritical cases. As applications, we get the existence of positive ground state solutions to the $p$-biharmonic equations and the Lane-Emden systems., Comment: 25 pages, 1 figure. arXiv admin note: text overlap with arXiv:2106.15982

In this sequence of two papers, we introduce a curvature flow on (mixed) weighted graphs which is based on the Bakry-\'Emery calculus. The flow is described via a time-continuous evolution through the weighting schemes. By adapting this flow to preserve the Markovian property, its limits turn out to be curvature sharp. Our aim is to present the flow in the most general case of not necessarily reversible random walks allowing laziness, including vanishing transition probabilities along some edges ("degenerate" edges). This approach requires to extend all concepts (in particular, the Bakry-\'Emery curvature related notions) to this general case and it leads to a distinction between the underlying topology (a mixed combinatorial graph) and the weighting scheme (given by transition rates). We present various results about curvature sharp vertices and weighted graphs as well as some fundamental properties of this new curvature flow. This paper is accompanied by a second paper discussing the curvature flow implementation in Python for practical use. In this second paper we present examples and exhibit further properties of the flow.

This paper proposes a computationally efficient methodology to predict the damage progression in solder contacts of electronic components using temperature-time curves. For this purpose, two machine learning algorithms, a Multilayer Perceptron and a Long Short-Term Memory network, are trained and compared with respect to their prediction accuracy and the required amount of training data. The training is performed using synthetic, normally distributed data that is realistic for automotive applications. A finite element model of a simple bipolar chip resistor in surface mount technology configuration is used to numerically compute the synthetic data. As a result, both machine learning algorithms show a relevant accuracy for the prediction of accumulated creep strains. With a training data length of 350 hours (12.5% of the available training data), both models show a constantly good fitting performance of $R^2$ of 0.72 for the Multilayer Perceptron and $R^2$ of 0.87 for the Long Short-Term Memory network. The prediction errors of the accumulated creep strains are less than 10% with an amount of 350 hours training data and decreases to less than 5 % when using further data. Therefore, both approaches are promising for the lifetime prediction directly on the electronic device., Comment: 10 pages, 12 figures

In this work we test the numerical behaviour of matrix-valued fields approximated by finite element subspaces of $[\mathit{H}^1]^{3\times 3}$, $[\mathit{H}(\mathrm{curl})]^3$ and $\mathit{H}(\mathrm{sym}\mathrm{Curl})$ for a linear abstract variational problem connected to the relaxed micromorphic model. The formulation of the corresponding finite elements is introduced, followed by numerical benchmarks and our conclusions. The relaxed micromorphic continuum model reduces the continuity assumptions of the classical micromorphic model by replacing the full gradient of the microdistortion in the free energy functional with the Curl. This results in a larger solution space for the microdistortion, namely $[\mathit{H}(\mathrm{curl})]^3$ in place of the classical $[\mathit{H}^1]^{3\times 3}$. The continuity conditions on the microdistortion can be further weakened by taking only the symmetric part of the Curl. As shown in recent works, the new appropriate space for the microdistortion is then $\mathit{H}(\mathrm{sym}\mathrm{Curl})$. The newly introduced space gives rise to a new differential complex for the relaxed micromorphic continuum theory.

The classical Cauchy continuum theory is suitable to model highly homogeneous materials. However, many materials, such as porous media or metamaterials, exhibit a pronounced microstructure. As a result, the classical continuum theory cannot capture their mechanical behaviour without fully resolving the underlying microstructure. In terms of finite element computations, this can be done by modelling the entire body, including every interior cell. The relaxed micromorphic continuum offers an alternative method by instead enriching the kinematics of the mathematical model. The theory introduces a microdistortion field, encompassing nine extra degrees of freedom for each material point. The corresponding elastic energy functional contains the gradient of the displacement field, the microdistortion field and its Curl (the micro-dislocation). Therefore, the natural spaces of the fields are $[\mathit{H}^1]^3$ for the displacement and $[\mathit{H}(\mathrm{curl})]^3$ for the microdistortion, leading to unusual finite element formulations. In this work we describe the construction of appropriate finite elements using N\'ed\'elec and Raviart-Thomas subspaces, encompassing solutions to the orientation problem and the discrete consistent coupling condition. Further, we explore the numerical behaviour of the relaxed micromorphic model for both a primal and a mixed formulation. The focus of our benchmarks lies in the influence of the characteristic length $L_\mathrm{c}$ and the correlation to the classical Cauchy continuum theory.

The electric double layer (EDL) that forms at the interface between a polyelectrolyte gel and a salt bath is studied using asymptotic and numerical methods. Specifically, matched asymptotic expansions, based on the smallness of the Debye length relative to the typical gel dimensions, are used to construct solutions of the governing equations and derive electroneutral models with consistent jump conditions across the gel-bath interface. A general approach for solving the equations of incompressible nonlinear elasticity in a curved boundary layer is developed and used to resolve the gel mechanics in the EDL. A critical feature of the model is that it accounts for phase separation within the gel, which gives rise to diffuse interfaces with a characteristic thickness described by the Kuhn length. We show that the solutions of the electroneutral model can only be asymptotically matched to the solutions in the EDL, in general, when the Kuhn length greatly exceeds the Debye length. Conversely, if the Debye length is similar to or larger than the Kuhn length, then the entire gel can self-organise into periodic, electrically charged domains via phase separation. The breakdown of electroneutrality demonstrates that the commonly invoked electroneutral assumption must be used with caution, as it generally only applies when the Debye length is much smaller than the Kuhn length.

Matrix approximations are a key element in large-scale algebraic machine learning approaches. The recently proposed method MEKA (Si et al., 2014) effectively employs two common assumptions in Hilbert spaces: the low-rank property of an inner product matrix obtained from a shift-invariant kernel function and a data compactness hypothesis by means of an inherent block-cluster structure. In this work, we extend MEKA to be applicable not only for shift-invariant kernels but also for non-stationary kernels like polynomial kernels and an extreme learning kernel. We also address in detail how to handle non-positive semi-definite kernel functions within MEKA, either caused by the approximation itself or by the intentional use of general kernel functions. We present a Lanczos-based estimation of a spectrum shift to develop a stable positive semi-definite MEKA approximation, also usable in classical convex optimization frameworks. Furthermore, we support our findings with theoretical considerations and a variety of experiments on synthetic and real-world data.

Continued observations of the Double Pulsar, PSR J0737-3039A/B, consisting of two radio pulsars (A and B) that orbit each other with a period of 2.45hr in a mildly eccentric (e=0.088) binary system, have led to large improvements in the measurement of relativistic effects in this system. With a 16-yr data span, the results enable precision tests of theories of gravity for strongly self-gravitating bodies and also reveal new relativistic effects that have been expected but are now observed for the first time. These include effects of light propagation in strong gravitational fields which are currently not testable by any other method. We observe retardation and aberrational light-bending that allow determination of the pulsar's spin direction. In total, we have detected seven post-Keplerian (PK) parameters, more than for any other binary pulsar. For some of these effects, the measurement precision is so high that for the first time we have to take higher-order contributions into account. These include contributions of A's effective mass loss (due to spin-down) to the observed orbital period decay, a relativistic deformation of the orbit, and effects of the equation of state of super-dense matter on the observed PK parameters via relativistic spin-orbit coupling. We discuss the implications of our findings, including those for the moment of inertia of neutron stars. We present the currently most precise test of general relativity's (GR's) quadrupolar description of gravitational waves, validating GR's prediction at a level of $1.3 \times 10^{-4}$ (95% conf.). We demonstrate the utility of the Double Pulsar for tests of alternative theories by focusing on two specific examples and discuss some implications for studies of the interstellar medium and models for the formation of the Double Pulsar. Finally, we provide context to other types of related experiments and prospects for the future., Comment: 56 pages, 23 Figures. Published by Physical Review X. Uploaded ArXiv version is authors' (pre-proof) version with abbreviated abstract. Resubmission for updated acknowledgement information. For final published version see https://link.aps.org/doi/10.1103/PhysRevX.11.041050

Assessing the effects of the energy transition and liberalization of energy markets on resource adequacy is an increasingly important and demanding task. The rising complexity in energy systems requires adequate methods for energy system modeling leading to increased computational requirements. Furthermore, with complexity, uncertainty increases likewise calling for probabilistic assessments and scenario analyses. To adequately and efficiently address these various requirements, new methods from the field of data science are needed to accelerate current methods. With our systematic literature review, we want to close the gap between the three disciplines (1) assessment of security of electricity supply, (2) artificial intelligence, and (3) design of experiments. For this, we conduct a large-scale quantitative review on selected fields of application and methods and make a synthesis that relates the different disciplines to each other. Among other findings, we identify metamodeling of complex security of electricity supply models using AI methods and applications of AI-based methods for forecasts of storage dispatch and (non-)availabilities as promising fields of application that have not sufficiently been covered, yet. We end with deriving a new methodological pipeline for adequately and efficiently addressing the present and upcoming challenges in the assessment of security of electricity supply.

We characterize Forman curvature lower bounds via contractivity of the Hodge Laplacian semigroup. We prove that Ollivier and Forman curvature coincide on edges when maximizing the Forman curvature over the choice of 2-cells. To this end, we translate between 2-cells and transport plans. Moreover, we give improved diameter bounds. We explicitly warn the reader that our Forman curvature notion does not coincide with Forman's original definition, but can be seen as generalization of the latter one.

Together with many success stories, promises such as the increase in production speed and the improvement in stakeholders' collaboration have contributed to making agile a transformation in the software industry in which many companies want to take part. However, driven either by a natural and expected evolution or by contextual factors that challenge the adoption of agile methods as prescribed by their creator(s), software processes in practice mutate into hybrids over time. Are these still agile? In this article, we investigate the question: what makes a software development method agile? We present an empirical study grounded in a large-scale international survey that aims to identify software development methods and practices that improve or tame agility. Based on 556 data points, we analyze the perceived degree of agility in the implementation of standard project disciplines and its relation to used development methods and practices. Our findings suggest that only a small number of participants operate their projects in a purely traditional or agile manner (under 15%). That said, most project disciplines and most practices show a clear trend towards increasing degrees of agility. Compared to the methods used to develop software, the selection of practices has a stronger effect on the degree of agility of a given discipline. Finally, there are no methods or practices that explicitly guarantee or prevent agility. We conclude that agility cannot be defined solely at the process level. Additional factors need to be taken into account when trying to implement or improve agility in a software company. Finally, we discuss the field of software process-related research in the light of our findings and present a roadmap for future research., Comment: Journal paper, 17 pages, 14 figures

A common way to avoid divergent integrals in homogeneous spatially non-compact gravitational systems is to introduce a fiducial cell by cutting-off the spatial slice at a finite region $V_o$. This is usually considered as an auxiliary regulator to be removed after computations by sending $V_o\to\infty$. In this paper, we analyse the dependence of the classical and quantum theory of homogeneous, isotropic and spatially flat cosmology on $V_o$. We show that each fixed $V_o$ regularisation leads to a different canonically independent theory. At the classical level, the dynamics of observables is not affected by the regularisation on-shell. For the quantum theory, however, this leads to a family of regulator dependent quantum representations and the limit $V_o\to\infty$ becomes then more subtle. First, we construct a novel isomorphism between different $V_o$-regularisations, which allows us to identify states in the different $V_o$-labelled Hilbert spaces to ensure equivalent dynamics for any value of $V_o$. The $V_o\to\infty$ limit would then correspond to choosing a state for which the volume assigned to the fiducial cell becomes infinite as appropriate in the late-time regime. As second main result of our analysis, quantum fluctuations of observables smeared over subregions $V\subset V_o$, unlike those smeared over the full $V_o$, explicitly depend on the size of the fiducial cell through the ratio $V/V_o$ interpreted as the (inverse) number of subcells $V$ homogeneously patched together into $V_o$. Physically relevant fluctuations for a finite region, as e.g. in the early-time regime, which would be unreasonably suppressed in a na\"ive $V_o\to\infty$ limit, become appreciable at small volumes. Our results suggest that the fiducial cell is not playing the role of a mere regularisation but is physically relevant at the quantum level and complement previous statements in the literature., Comment: Published version, 13 pages (REVTeX format), 1 figure. Minor clarifications added