The structure of DGA resolutions of monomial ideals

Let $I?\mathbb{k}[x_1,\dots ,x_n]$ be a squarefree monomial ideal in a polynomial ring. In this paper we study multiplications on the minimal free resolution $\mathbb{F}$ of $\mathbb{k}[x_1,\dots ,x_n]/I$. In particular, we characterize the possible vectors of total Betti numbers for such ideals which admit a differential graded algebra (DGA) structure on $\mathbb{F}$. We also show that under these assumptions the maximal shifts of the graded Betti numbers are subadditive.On the other hand, we present an example of a strongly generic monomial ideal which does not admit a DGA structure on its minimal free resolution. In particular, this demonstrates that the Hull resolution and the Lyubeznik resolution do not admit DGA structures in general.Finally, we show that it is enough to modify the last map of $\mathbb{F}$ to ensure that it admits the structure of a DG algebra.     

Timing and eco(nomic) efficiency of climate-friendly investments in supply chains

Emission trading schemes such as the European Union Emissions Trading System (EUETS) attempt to reconcile economic efficiency with ecological efficiency by creating financial incentives for companies to invest in climate-friendly innovations. Using real options methodology, we demonstrate that under uncertainty, economic and ecological efficiency continue to be mutually exclusive. This problem is even worse if a climate-friendly project depends on investing in of a whole supply chain. We model a sequential bargaining game in a supply chain where the parties negotiate over implementation of a carbon dioxide (CO₂) saving investment project. We show that the outcome of their bargaining is not economically efficient and even less ecologically efficient. Furthermore, we show that a supply chain becomes less economically efficient and less ecologically efficient with every additional chain link. Finally, we make recommendations for how managers or politicians can improve the situation and thereby increase economic as well as ecological efficiency and thus also the eco-efficiency of supply chains.     

Minimization- and Saddle-Point-Based Modeling of Diffusion-Deformation-Processes in Hydrogels

Hydrogels have gained importance during the last years due to their wide range of synthetically fabricable elastic properties as well their increasing meaning in biomedical applications. Future exploitation of the vast prospects of hydrogels is however only feasible by establishing reliable material models that precisely capture their behavior in different environments. To this end, we propose a consistent variational framework for deformation-diffusion processes, offering a canonically compact approach to the chemo-mechanical coupling of hydrogels via a saddle-point as well as a new minimization formulation. The work depicts the construction of rate-type potentials for the chemo-mechanical evolution problem and their transformation into time-discrete incremental potentials. In terms of spatial discretization, the finite element method is employed, benefiting from the intrinsic symmetric structure of the variational foundation. While the saddle-point formulation yields the well-known LBB condition as a constraint for finite element interpolations, on the part of its minimizing counterpart H(Div, ℬ︁)-conforming elements have to be chosen. We illustrate appropriate solutions to both challenges, using mixed Taylor-Hood for the saddle-point and Raviart-Thomas elements for the minimization formulation and discuss advantages of the new approach. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Practical use of robust GCV and modified GCV for spline smoothing

Generalized cross-validation (GCV) is a popular parameter selection criterion for spline smoothing of noisy data, but it sometimes yields a severely undersmoothed estimate, especially if the sample size is small. Robust GCV (RGCV) and modified GCV are stable extensions of GCV, with the degree of stabilization depending on a parameter \(\gamma \in (0,1)\) for RGCV and on a parameter \(\rho >1\) for modified GCV. While there are favorable asymptotic results about the performance of RGCV and modified GCV, little is known for finite samples. In a large simulation study with cubic splines, we investigate the behavior of the optimal values of \(\gamma \) and \(\rho \), and identify simple practical rules to choose them that are close to optimal. With these rules, both RGCV and modified GCV perform significantly better than GCV. The performance is defined in terms of the Sobolev error, which is shown by example to be more consistent with a visual assessment of the fit than the prediction error (average squared error). The results are consistent with known asymptotic results.     

A hyperbolic surface with a square grid net

We prove the existence of a hyperbolic surface spread over the sphere for which the projection map has all its singular values on the extended real line and such that the preimage of the extended real line under the projection map is homeomorphic to the square grid in the plane. This answers a question raised by è. B. Vinberg. Supported by the Alexander von Humboldt Foundation. Supported by NSF grants DMS-0400636, DMS-0244421, and DMS-0244547.     

A canonical module characterization of Serre’s (R1)

In this short note, we give a characterization of domains satisfying Serre’s condition (R₁) in terms of their canonical modules. In the special case of toric rings, this generalizes a result of the second author [9 Yanagawa, K. (2015). Dualizing complexes of seminormal a?ne semigroup rings and toric face rings. J. Algebra 425:367–391.[Crossref], [Web of Science ®] , [Google Scholar]] where the normality is described in terms of the “shape” of the canonical module.     

Robert Leslie Ellisʼs work on philosophy of science and the foundations of probability theory

The goal of this paper is to provide an extensive account of Robert Leslie Ellis?s largely forgotten work on philosophy of science and probability theory. On the one hand, it is suggested that both his ‘idealist’ renovation of the Baconian theory of induction and a ‘realism’ vis-à-vis natural kinds were the result of a complex dialogue with the work of William Whewell. On the other hand, it is shown to what extent the combining of these two positions contributed to Ellis?s reformulation of the metaphysical foundations of traditional probability theory. This parallel is assessed with reference to the disagreement between Ellis and Whewell on the nature of (pure) mathematics and its relation to scientific knowledge.     

On Configurational-Force-Driven Crack Propagation and Adaptive Mesh Refinement in Finite Inelasticity

We introduce a consistent variational framework for inelasticity at finite strains, yielding dual balances in physical and material space as the Euler equations. The formulation is employed for the simultaneous usage of configurational forces as both driving forces for crack propagation as well as h-adaptive mesh refinement. The theoretical basis builds upon a global balance of internal and external power, where the mechanical response is exclusively governed by two scalar functions, the free energy function and a dissipation potential. The resulting variational structure is exploited in the context of fracture mechanics and yields evolution equations for internal variables. In the discrete setting, we present a geometry model fully separated from the finite element mesh structure that represents structural changes of the material configuration due to crack propagation. Advanced meshing algorithms provide an optimal discretization at the crack tip. Local and global criteria are obtained via error estimators based on configurational forces being interpreted as indicators of an energetic misfit due to an insufficient discretization. The numerical handling is decomposed into a staggered algorithm scheme for the dual set of equilibrium equations in material and physical space and efficient mesh generation tools. Exemplary numerical examples are considered to illustrate the method and to underline the effects of inelastic material behaviour in the presented context. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

High-resolution non-invasive 3D imaging of paint microstructure by synchrotron-based X-ray laminography

The characterisation of the microstructure and micromechanical behaviour of paint is key to a range of problems related to the conservation or technical art history of paintings. Synchrotron-based X-ray laminography is demonstrated in this paper to image the local sub-surface microstructure in paintings in a non-invasive and non-destructive way. Based on absorption and phase contrast, the method can provide high-resolution 3D maps of the paint stratigraphy, including the substrate, and visualise small features, such as pigment particles, voids, cracks, wood cells, canvas fibres etc. Reconstructions may be indicative of local density or chemical composition due to increased attenuation of X-rays by elements of higher atomic number. The paint layers and their interfaces can be distinguished via variations in morphology or composition. Results of feasibility tests on a painting mockup (oak panel, chalk ground, vermilion and lead white paint) are shown, where lateral and depth resolution of up to a few micrometres is demonstrated. The method is well adapted to study the temporal evolution of the stratigraphy in test specimens and offers an alternative to destructive sampling of original works of art.