Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics

In this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn–Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, and weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution – non-negativity, conservation of the total mass and dissipation of the energy – are automatically guaranteed by the construction from minimizing movements in the energy landscape.     

Multi-dimensional smoothing transformations: Existence,regularity and stability of fixed points

We analyze a class of smoothing transformations on probability measures in multiple space dimensions. Applying a synthesis of probabilistic methods and Fourier analysis, we prove existence and uniqueness of a fixed point inside the class of probability measures of finite second moment, characterize it as a scale mixture of Gaussians, and discuss its regularity. We also classify its tail, which might be of Pareto type. As an application, we study the stability of stationary solutions in a Kac-type kinetic model. In particular, we prove that the domain of attraction is precisely the probability measures of finite second moment.     

Variation on a theme by Bobylëv and Villani

It is shown that the collisional gain operator for a Maxwell gas does not increase the Fisher information. Our proof is a variant of the one given by Villani in 1998 [2], but it is shorter and based on Fourier techniques rather than direct estimates. The method we use also applies to general (non-symmetric) Wild convolutions.     

The Series of Rare Earth Complexes [Ln2Cl6(μ‐4,4′‐bipy)(py)6], Ln=Y,Pr, Nd,Sm‐Yb: A Molecular Model System for Luminescence Properties in MOFs Based on LnCl3 and 4,4′‐Bipyridine

A series of 12 dinuclear complexes [Ln₂Cl₆(μ‐4,4′‐bipy)(py)₆], Ln=Y, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, ( 1 – 12 , respectively) was synthesized by an anhydrous solvothermal reaction in pyridine. The complexes contain a 4,4′‐bipyridine bridge and exhibit a coordination sphere closely related to luminescent lanthanide MOFs based on LnCl₃ and 4,4‐bipyridine. The dinuclear complexes therefore function as a molecular model system to provide a better understanding of the luminescence mechanisms in the Ln‐N‐MOFs ${\hbox{}{{\hfill 2\atop \hfill \infty }}}$ [Ln₂Cl₆(4,4′‐bipy)₃] ? 2(4,4′‐bipy). Accordingly, the luminescence properties of the complexes with Ln=Y, Sm, Eu, Gd, Tb, Dy, ( 1 , 4 – 8 ) were determined, showing an antenna effect through a ligand–metal energy transfer. The highest efficiency of luminescence is observed for the terbium‐based compound 7 displaying a high quantum yield (QY of 86 %). Excitation with UV light reveals typical emission colors of lanthanide‐dependent intra 4f–4f‐transition emissions in the visible range (Tb^III: green, Eu^III: red, Sm^III: salmon red, Dy^III: yellow). For the Gd^III‐ and Y^III‐containing compounds 6 and 1 , blue emission based on triplet phosphorescence is observed. Furthermore, ligand‐to‐metal charge‐transfer (LMCT) states, based on the interaction of Cl^? with Eu^III, were observed for the Eu^III compound 5 including energy‐transfer processes to the Eu^III ion. Altogether, the model complexes give further insights into the luminescence of the related MOFs, for example, rationalization of Ln‐independent quantum yields in the related MOFs.     

The K-Theory of Heegaard-Type Quantum 3-Spheres

We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C^*-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C^*-algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts. Dedicated to the memory of Olaf Richter An erratum to this article is available at .     

Short proofs of normalization for the simply- typed λ-calculus,permutative conversions and Gödel's T

 Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simply-typed λ-calculus and for an extension by sum types with permutative conversions. The analogous treatment of a new system with generalized applications inspired by generalized elimination rules in natural deduction, advocated by von Plato, shows the flexibility of the approach which does not use the strong computability/candidate style à la Tait and Girard. It is also shown that the extension of the system with permutative conversions by η-rules is still strongly normalizing, and likewise for an extension of the system of generalized applications by a rule of ``immediate simplification'. By introducing an infinitely branching inductive rule the method even extends to G?del's T. Received 12 October 1998 Published online: 19 December 2002 Mathematics Subject Classification (2000): 03B40 Keywords or phrases: λ-calculus – Permutative/commuting conversions – Sum types – Generalized application – G?del's T_q-ω-rule – Inductive characterization     

Existence of Weak Solutions to a Class of Fourth Order Partial Differential Equations with Wasserstein Gradient Structure

We prove the global-in-time existence of nonnegative weak solutions to a class of fourth order partial differential equations on a convex bounded domain in arbitrary spatial dimensions. Our proof relies on the formal gradient flow structure of the equation with respect to the L²-Wasserstein distance on the space of probability measures. We construct a weak solution by approximation via the time-discrete minimizing movement scheme; necessary compactness estimates are derived by entropy-dissipation methods. Our theory essentially comprises the thin film and Derrida-Lebowitz-Speer-Spohn equations.     

1,3,4-Oxadiazole-containing hybrids as potential anticancer agents: Recent developments,mechanism of action and structure-activity relationships

Chemotherapy is an important therapeutic approach for the treatment of cancer. Currently, many anticancer drugs are available in the market that plays an important role in cancer treatment, but concerns such as, drug resistance and side effects create an urgent need for the development of new anti-tumor drugs with high potency and less side effects. Heterocycles are of great interest due to their fascinating anticancer activity. Among them, 1,3,4-oxadiazoles showed attracting anti-tumor activity and its derivatives are under clinical trials for the treatment of cancer. Hybridization of 1,3,4-oxadiazole moiety with other heterocyclic pharmacophoresis a promising approach to overcome various disadvantages of current anticancer drugs such as drug resistance, toxicity, and other side effects. Thus, 1,3,4-oxadiazole-heterocycle hybrids occupy a significant position in the discovery of anti-tumor drugs. Among the reported oxadiazole-based hybrids reviewed here, compounds 45i, 59j, and 62x showed the highest anticancer activity with IC₅₀ values in the nanomolar range. This review summarizes the recent developments in the anticancer potential, structure–activity relationships, and mechanisms of actions of 1,3,4-oxadiazole-heterocycle hybrids.