Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects

In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first adopt the notion of weak solutions introduced in [C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321--359]. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques

Higher Homotopies in a Hierarchy of Univalent Universes

For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of univalent universes, we show that U(n) is not an n-type. Our construction also solves the problem of finding a type that strictly has some high truncation level without using higher inductive types. In particular, U(n) is such a type if we restrict it to n-types. We have fully formalized and verified our results within the dependently typed language and proof assistant Agda.Comment: v1: 30 pages, main results and a connectedness construction; v2: 14 pages, only main results, improved presentation, final journal version, ancillary files with electronic appendix; v3: content unchanged, different documentclass reduced the number of pages to 1

Complete damage in linear elastic materials - Modeling, weak formulation and existence results

In this work, we introduce a degenerating PDE system with a time-depending domain for complete damage processes under time-varying Dirichlet boundary conditions. The evolution of the system is described by a doubly nonlinear differential inclusion for the damage process and a quasi-static balance equation for the displacement field which are strongly nonlinearly coupled. In our proposed model, the material may completely disintegrate which is indispensable for a realistic modeling of damage processes in elastic materials. Complete damage theories lead to several mathematical problems since for instance coercivity properties of the free energy are lost and, therefore, several difficulties arise. For the introduced complete damage model, we propose a classical formulation and a corresponding suitable weak formulation in an $SBV$-framework. The main aim is to prove existence of weak solutions for the introduced degenerating model. In addition, we show that the classical differential inclusion can be regained from the notion of weak solutions under certain regularity assumptions which is a novelty in the theory of complete damage models of this type

Renormalization of the Minimal Supersymmetric Standard Model

The renormalization of the Minimal Supersymmetric Standard Model (MSSM) is presented. We describe symmetry identities that constitute a framework in which the MSSM is completely characterized and renormalizability can be proven. Furthermore, we discuss applications of this framework for the determination of symmetry-restoring counterterms, the gauge dependence of tan(beta) and the derivation of non-renormalization theorems.Comment: Talk given by D.S. at the RADCOR/Loops and Legs in Quantum Field Theory 2002, September 8--13, Kloster Banz, Germany. 5 pages. Layout improve

Supercurrent and Local Coupling in the Wess-Zumino Model

We study the Wess-Zumino model with the coupling extended to a chiral superfield. In order to incorporate the renormalization effects a further external real field has to be introduced. It is then possible to derive a Callan-Symanzik equation and to prove renormalizability. By constructing the supercurrent in this context the whole machinery for describing the superconformal symmetries becomes available. The presence of the external fields allows also to define multiple insertions of all relevant composite operators. Interesting relations to the curved superspace treatment show up.Comment: 16 page

Influence of the pruning system on the fungal community of grapevine (Vitis vinifera)

„Minimalschnitt im Spalier“ (MMS) ist ein neues Reben-Erziehungssystem, das den Winzern die Möglichkeit eröffnet, Einsparungen bei den Produktionskoste vorzunehmen. Bei diesem Erziehungssystem wird der zeit- und arbeitsaufwändige Winterschnitt, der normalerweise per Hand erfolgt, maschinell, mit Hilfe eines Laubschneiders, durchgeführt. Aufgrund dieses ökonomischen Vorteils erlebte der MMS eine erhöhte Popularität unter den Winzern in den letzten Jahren, Tendenz steigend. Jedoch fehlen wichtige Informationen zu einige Kultivierungsaspekten, da es kaum detaillierte Feldstudien zu diesem Erziehungssystem gibt. Einer dieser Aspekte betrifft das Vorkommen von pilzlichen Krankheiten und den entsprechenden Schutz der Reben. Aus diesem Grund war es Ziel dieser Arbeit, basierend auf Langzeitversuchen (2015–2018), den Einfluss des Schnittsystems auf die Pilzgemeinschaft der Rebe zu untersuchen, mit besonderem Hinblick auf phytopathogene Pilze. Ein Teil dieser Arbeit verglich die Laubwandarchitektur und das Mikroklima von MMS-Reben mit Reben in der klassischen Bogenerziehung (BE). Zudem wurde das Auftreten der wichtigsten Pilzkrankheiten, nämlich Falscher Mehltau (FM), Echter Mehltau (EM) und Graufäule (GF), vier Jahre hintereinander in den beiden Erziehungssystemen untersucht. In diesem Zeitraum wurde festgestellt, dass MMS-Reben im Vergleich zu BE-Reben anfälliger sind gegenüber FM und EM. Es wird vermutet, dass das vergleichsweise langsamere Abtrocknen der MSS-Laubwand ein Mikroklima erzeugt, welches das Wachstum und die Verbreitung der Krankheiten begünstigt. GF betreffend waren MMS-Trauben weniger befallen als BE-Trauben, was vermutlich mit der lockeren Traubenarchitektur in MMS-Reben zusammenhängt. Eine Analyse der zeitlichen Entwicklung der Pilzgemeinschaft in äußerlich gesunden Rebtriebe, von zwei Monate bis acht Jahre, ergab, dass schnell wachsende Pilze wie Alternaria spp., Aureobasidium pullulans, Botrytis cinerea, Cladosporium spp., Diaporthe spp., Epicoccum nigrum, Penicillium spp. und Truncatella angustata charakteristisch sind für junge Triebe (zwei Monate bis ein Jahr). In älteren Trieben (> ein Jahr) allerdings ist die Pilzgemeinschaft mehr durch grapevine trunk disease (GTD)-assoziierte Pilze geprägt, mit Cadophora luteo-olivacea, Diplodia seriata und Eutypa lata als vorherrschende Arten. Diese Daten zeigen, dass Reben mit älteren Trieben, wie bei MSS-Reben zu finden, stärker durch GTDs gefährdet sind als BE-Reben mit ausschließlich einjährigen Trieben. In vier aufeinanderfolgenden Jahren wurde das Vorkommen von externen Esca-Blattsymptomen in zwölf Weinbergen, unterteilt in MMS und BE, untersucht, um herauszufinden, ob und inwieweit, dass Schnittsystem das Auftreten der Blattsymptome beeinflusst. Signifikante Unterschiede zwischen den zwei Erziehungssystemen hinsichtlich der Häufigkeit von Esca-Blattsymptomen wurden in drei von vier Jahren gefunden, jedoch sind die Ergebnisse widersprüchlich. Während in 2016 MMS-Reben häufiger Blattsymptome zeigten als BE-Reben, war es in den Jahren 2017 und 2018 das Gegenteil der Fall. Sporenfallen wurden in einem Weinberg an Trieben von MMS- und BE-Reben befestigt, um das Vorkommen des mit Esca assoziierten Pilzes Phaeomoniella chlamydospora im Jahresverlauf zu erfassen. Neben P. chlamydospora wurden dabei auch andere Arten der Phaeomoniellales isoliert; eine bereits bekannte Art, Neophaeomoniella zymoides, und sechs unbekannte Arten, die neu beschrieben wurden. Die Anzahl an Pilzstämmen und Arten von Phaeomoniellales war tendenziell höher für Sporenfallen, die an Ästen von MMS-Reben befestigt waren. Ein Pathogenitätstest mit Topfpflanzen im Gewächshaus ergab, dass von den acht isolierten Arten der Phaeomoniellales nur P. chlamydospora signifikante Läsionen im Holz verursachen kann. Die Daten, welche in dieser Arbeit generiert wurden, sollen helfen, ein an MMS-Reben angepasstes Pflanzenschutzregime zu entwickeln.The novel grapevine training system “semi minimal pruned hedge” (SMPH), enables farmers to reduce their production costs by replacing the time and labour intensive winter pruning, usually done by hand, with a mechanical pruning approach. Due to its economic benefits, SMPH became more popular in Germany over the last decade and the number of SMPH vineyards is still increasing. However, due to the lack of extended monitoring in the field, some cultivation aspects of SMPH are poorly understood. One important issue is related to the incidence of fungal grapevine diseases and the respective protection of the vines. Therefore, it was the aim of this study to investigate on a long term basis (2015–2018) the influence of the pruning system on the fungal community of grapevine, with special regard to phytopathogenic fungi. One part of this study compared the canopy architecture and microclimate of SMPH and “vertical shoot positioning” (VSP) trained vines. The incidence of major fungal grapevine diseases, i.e. Downy Mildew (DM), Powdery Mildew (PM) and Botrytis Bunch Rot (BR) was surveyed over four consecutive years. Within this time frame it was noted that SMPH trained vines were more susceptible to DM and PM compared to VSP trained vines. It is assumed that the comparatively slower drying process of SMPH canopies produces a microclimate that favours growth and spread of these diseases. By contrast, SMPH bunches were less susceptible to BR compared to VSP bunches, which is probably linked to the more loose bunch architecture. An analysis of the temporal development of the fungal community in healthy grapevine branches covering two months to eight years old branches revealed that fast growing fungi like Alternaria spp., Aureobasidium pullulans, Botrytis cinerea, Cladosporium spp., Diaporthe spp., Epicoccum nigrum, Penicillium spp., and Truncatella angustata are characteristic for young branches (two months to one year old). However, in older branches (> one year) the fungal community is more characterized by grapevine trunk disease (GTD) associated fungi, with Cadophora luteo-olivacea, Diplodia seriata and Eutypa lata as predominant species. These data show that grapevines with older branches, as found in SMPH trained vines, are more threatened by GTDs than grapevines with only annual branches, as in VSP trained vines. In four consecutive years a monitoring for Esca related foliar symptoms was performed in twelve vineyards each subdivided in SMPH and VSP trained sections to investigate the influence of the training system on symptom occurrence. Significant differences between the two training systems regarding the incidence of Esca foliar symptoms were found in three out of the four years. However, overall the results are inconsistent. While in 2016 SMPH trained vines expressed more symptoms than VSP trained vines, the opposite was the case in 2017 and 2018. Spore traps were placed in a vineyard subdivided into SMPH and VSP trained sections to monitor the incidence of the Esca associated fungus Phaeomoniella chlamydospora over the year. Besides P. chlamydospora also other species of the Phaeomoniellales were isolated, including one known species, Neophaeomoniella zymoides, and six unknown species, which were newly described. The number of fungal strains and species of the Phaeomoniellales tend to be higher for spore traps attached to SMPH trained vines. A pathogenicity test made with potted plants in the greenhouse revealed that of the eight isolated species of the Phaeomoniellales only P. chlamydospora was able to induce significant lesions in the wood. Data collected in this study shall help to developing a plant protection regime adapted to SMPH

Two-Level Type Theory and Applications

We define and develop two-level type theory (2LTT), a version of Martin-L\"of type theory which combines two different type theories. We refer to them as the inner and the outer type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level $n$ can be constructed in HoTT for any externally fixed natural number $n$. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where $n$ will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the inner and outer natural numbers to be isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of (infinity,1)-category and give some examples.Comment: 53 page

Limit on sterile neutrino contribution from the Mainz Neutrino Mass Experiment

The recent analysis of the normalization of reactor antineutrino data, the calibration data of solar neutrino experiments using gallium targets, and the results from the neutrino oscillation experiment MiniBooNE suggest the existence of a fourth light neutrino mass state with a mass of O(eV), which contributes to the electron neutrino with a sizable mixing angle. Since we know from measurements of the width of the Z0 resonance that there are only three active neutrinos, a fourth neutrino should be sterile (i.e., interact only via gravity). The corresponding fourth neutrino mass state should be visible as an additional kink in beta-decay spectra. In this work the phase II data of the Mainz Neutrino Mass Experiment have been analyzed searching for a possible contribution of a fourth light neutrino mass state. No signature of such a fourth mass state has been found and limits on the mass and the mixing of this fourth mass states are derived

Modeling and analysis of a phase field system for damage and phase separation processes in solids

In this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced model, we start with an approximation system. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.Comment: Keywords: Cahn-Hilliard system, phase separation, elliptic-parabolic systems, doubly nonlinear differential inclusions, complete damage, existence results, energetic solutions, weak solutions, linear elasticity, rate-dependent system