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We study the canonical heat flow (H ) ≥0 on the cotangent module 2 ( * ) over an RCD( , ∞) space ( , d, m), ∈ ℝ. We show Hess-Schrader-Uhlenbrock's inequality and, if ( , d, m) is also an RCD * ( , ) space, ∈ (1, ∞), Bakry-Ledoux's inequality for (H ) ≥0 w.r.t. the heat flow (P ) ≥0 on 2 ( ). A crucial tool is that the dimensional vector 2-Bochner inequality is self-improving, entailing a dimensional vector 1-Bochner inequality-a version of which is also available in the dimension-free case-as a byproduct. Variable versions of these estimates are discussed as well. In conjunction with a study of logarithmic Sobolev inequalities for 1-forms, the previous inequalities yield various -properties of (H ) ≥0 , ∈ [1, ∞].Then we establish explicit inclusions between the spectrum of its generator, the Hodge Laplacian ⃗ Δ, of the negative functional Laplacian −Δ, and of the Schrödinger operator −Δ + .In the RCD * ( , ) case, we prove compactness of ⃗ Δ −1 if is compact, and the independence of the -spectrum of ⃗ Δ on ∈ [1, ∞] under a volume growth condition.We terminate by giving an appropriate interpretation of a heat kernel for (H ) ≥0 . We show its existence in full generality without any local compactness or doubling assumptions, and derive fundamental estimates and properties of it.

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We prove that if the Ricci curvature of a geodesically complete Riemannian manifold X, endowed with the Riemannian distance ρ and the Riemannian volume measure m, is bounded from below by a Dynkin decomposable function k : X → R, then X is stochastically complete. This assumption on k is satisfied if its negative part k − belongs to the Kato class of X. In addition, given f ∈ L p (X) for sufficiently large p in a range depending only on k − , we derive a global Bismut derivative formula for ∇P t f for every t > 0 along the heat flow (P t ) t≥0 , whose L ∞ -Lip-regularization we obtain as a corollary.Moreover, for such functions k, we show that the Ricci curvature of X is bounded from below by k if and only if X supports the L 1 -gradient estimate w.r.t. k, i.e. for every f ∈ W 1,2 (X) and t ≥ 0, |∇P t f | ≤ P k t |∇f | holds m-a.e., (P k t ) t≥0 being the Schrödinger semigroup in L 2 (X) generated by −(∆− k)/2. If k is additionally lower semicontinuous, another equivalent characterization of lower Ricci bounds by k is proven to be the existence, given any x, y ∈ X, of a Markovian coupling (b x , b y ) of Brownian motions on X starting in (x, y) such that a.s. for every s, t ∈ [0, ∞) with s ≤ t, one has the pathwise estimate

In the language of L ∞ -modules proposed by Gigli, we introduce a first order calculus on a topological Lusin measure space (M , m) carrying a quasi-regular, strongly local Dirichlet form E. Furthermore, we develop a second order calculus if (M , E, m) is tamed by a signed measure in the extended Kato class in the sense of Erbar, Rigoni, Sturm and Tamanini. This allows us to define e.g. Hessians, covariant and exterior derivatives, Ricci curvature, and second fundamental form.

Abstract. The calculation of hydraulic state variables for a network is an important task in managing the distribution of potable water. Over the years the mathematical modeling process has been improved by numerous researchers for utilization in new computer applications and the more realistic modeling of water distribution networks. But, in spite of these continuous advances, there are still a number of physical phenomena that may not be tackled correctly by current models. This paper will take a closer look at the two modeling paradigms given by demand-and pressure-driven modeling. The basic equations are introduced and parallels are drawn with the optimization formulations from electrical engineering. These formulations guarantee the existence and uniqueness of the solution. One of the central questions of the French and German research project ResiWater is the investigation of the network resilience in the case of extreme events or disasters. Under such extraordinary conditions where models are pushed beyond their limits, we talk about deficient network models. Examples of deficient networks are given by highly regulated flow, leakage or pipe bursts and cases where pressure falls below the vapor pressure of water. These examples will be presented and analyzed on the solvability and physical correctness of the solution with respect to demand-and pressure-driven models.

Mit dem Ziel, Slim‐Floor‐Träger möglichst wirtschaftlich als Verbundträger auszubilden, wurden zur Übertragung der Längsschubkräfte in der Verbundfuge tiefliegende Betondübel untersucht. Pilotversuche im Jahre 2009 belegten das große Potenzial dieser neuen Bauweise, machten aber auch deutlich, dass bestehende Bemessungsmodelle für Betondübel hier nicht ohne weiteres verwendet werden können. Im Jahr 2011 wurden weitere Versuche durchgeführt, um den bereits untersuchten Anwendungsbereich der Betondübel bei Slim‐Floor‐Trägern zu erweitern und eine allgemeine bauaufsichtliche Zulassung für die Verwendung von tiefliegenden Betondübeln zu erwirken. Die Versuche bestätigten das duktile Verhalten der Betondübel und die Analyse der Versuchsergebnisse führte zu der Erkenntnis, dass sich die Traglast im Wesentlichen aus den Anteilen einer in den Dübel laufenden Betondruckstrebe, der Tragfähigkeit der Dübelbewehrung und aus Reibung zusammensetzt. Im vorliegenden Beitrag wird speziell der Einfluss der Betondruckfestigkeit auf die Traglast erläutert, da in den Versuchen eine höhere Betondruckfestigkeit nicht zwangsläufig auch zu einer höheren Traglast geführt hat. Aus den Versuchsergebnissen wurden Empfehlungen für die charakteristische Längsschubtragfähigkeit der tiefliegenden Betondübel im untersuchten Anwendungsbereich abgeleitet. Im Beitrag gegebene Hinweise zum Entwurf fördern die wirtschaftliche Anwendung der entwickelten Verbund‐Slim‐Floor‐Träger (CoSFB), deren Potenzial durch Projektbeispiele demonstriert wird. Analysis of the composite action of concrete dowels – Application of concrete dowels for slim‐floor construction (CoSFB). With the aim to design slim‐floor beams more economically, acting as composite beams, concrete dowels have been investigated to transfer the longitudinal shear force. First pilot tests executed in 2009 have demonstrated a great potential of this new construction system. It has also shown that the existing models for concrete dowels cannot be directly applied. Consequently, further test campaign has been carried out in 2011 to extend the already examined application range of the concrete dowels with slim‐floor beams and to obtain General Technical Approval from the Deutschen Institut für Bautechnik (DIBt ‐ German Institute for Civil Engineering). Through out all the tests, a ductile behaviour of concrete dowels has been confirmed. The analysis of the test results leads further to the conclusion that the ultimate load bearing capacity is mainly composed of three bearing parts: a concrete compression strut running into the dowel, resistance of the reinforcement bar and friction. In this paper, an influence of the concrete compressive strength on the load bearing capacity of the shear connection is specifically explained, since the experiments showed that a higher concrete compressive strength has not necessarily led to a higher load bearing capacity of the dowels. From the experimental results recommendations for the characteristic longitudinal shear capacity of the concrete dowels in t...