Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis

Phase field models recently gained a lot of interest in the context of tumour growth models. Typically Darcy-type flow models are coupled to Cahn–Hilliard equations. However, often Stokes or Brinkman flows are more appropriate flow models. We introduce and mathematically analyse a new Cahn–Hilliard–Brinkman model for tumour growth allowing for chemotaxis. Outflow boundary conditions are considered in order not to influence tumour growth by artificial boundary conditions. Existence of global-in-time weak solutions is shown in a very general setting.     

Synthesis,spectral investigation and development of tetrahedral copper(II) complexes as artificial metallonucleases and antimalarial agents

Seven new copper(II) complexes of type [Cu(A)(L)]?H₂O (A = sparfloxacin, ciprofloxacin, levofloxacin, gatifloxacin, pefloxacin, ofloxacin, norfloxacin; L = 5‐[(3‐chlorophenyl)diazenyl]‐4‐hydroxy‐1,3‐thiazole‐2(3H)‐thione) were synthesized and characterized using elemental and thermogravimetric analyses, and electronic, electron paramagnetic resonance (EPR), Fourier transform infrared and liquid chromatography–mass spectroscopies. Tetrahedral geometry around copper is assigned in all complexes using EPR and electronic spectral analyses. All complexes were investigated for their interaction with herring sperm DNA utilizing absorption titration (K_b = 1.27–3.13 × 10⁵ M^?1) and hydrodynamic volume measurement studies. The studies suggest the classical intercalative mode of DNA binding. The cleavage reaction on pUC19 DNA was monitored by agarose gel electrophoresis. The results indicate that the Cu(II) complexes can more effectively promote the cleavage of plasmid DNA. The superoxide dismutase mimic activity of the complexes was evaluated by nitroblue tetrazolium assay, and the complexes catalysed the dismutation of superoxide at pH = 7.8 with IC₅₀ values in the range 0.597–0.900 μM. The complexes were screened for their in vitro antibacterial activity against five pathogenic bacteria. All the complexes are good cytotoxic agents and show LC₅₀ values ranging from 5.559 to 11.912 µg ml^?1. All newly synthesized Cu(II) complexes were also evaluated for their in vitro antimalarial activity against Plasmodium falciparum strain (IC₅₀ = 0.62–2.0 µg ml^?1). Copyright © 2015 John Wiley & Sons, Ltd.     

On actions of Drinfel'd doubles on finite dimensional algebras

Let q be an nth root of unity for $n>2$ and let $T_n(q)$ be the Taft (Hopf) algebra of dimension $n^2$. In 2001, Susan Montgomery and Hans-Jürgen Schneider classified all non-trivial $T_n(q)$-module algebra structures on an n-dimensional associative algebra A. They further showed that each such module structure extends uniquely to make A a module algebra over the Drinfel'd double of $T_n(q)$. We explore what it is about the Taft algebras that leads to this uniqueness, by examining actions of (the Drinfel'd double of) Hopf algebras H “close” to the Taft algebras on finite-dimensional algebras analogous to A above. Such Hopf algebras H include the Sweedler (Hopf) algebra of dimension 4, bosonizations of quantum linear spaces, and the Frobenius–Lusztig kernel $u_q({\mathfrak{sl}}_2)$.     

Bounded sets structure of CpX and quasi‐(DF)‐spaces

For wide classes of locally convex spaces, in particular, for the space $C_p(X)$ of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of $(DF)$‐spaces have led us to introduce quasi‐ $(DF)$‐spaces, a class of locally convex spaces containing $(DF)$‐spaces that preserves subspaces, countable direct sums and countable products. Regular $(LM)$‐spaces as well as their strong duals are quasi‐ $(DF)$‐spaces. Hence the space of distributions $D^{\prime }(\mathrm{\Omega })$ provides a concrete example of a quasi‐ $(DF)$‐space not being a $(DF)$‐space. We show that $C_p(X)$ has a fundamental bounded resolution if and only if $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if the strong dual of $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if X is countable. If X is metrizable, then $C_k(X)$ is a quasi‐ $(DF)$‐space if and only if X is a σ‐compact Polish space.     

Automorphisms of Hilbert schemes of points on a generic projective K3 surface

We study automorphisms of the Hilbert scheme of n points on a generic projective K3 surface S, for any . We show that is either trivial or generated by a non‐symplectic involution and we determine numerical and divisorial conditions which allow us to distinguish between the two cases. As an application of these results we prove that, for any , there exist infinitely many admissible degrees for the polarization of the K3 surface S such that admits a non‐natural involution. This provides a generalization of the results of [7] for .     

K3 surfaces with an order 50 automorphism

In any characteristic p different from 2 and 5, Kondō gave an example of a K3 surface with a purely non-symplectic automorphism of order 50. The surface was explicitly given as a double plane branched along a smooth sextic curve. In this note we show that, in any characteristic $p\ne 2,5$, a K3 surface with a cyclic action of order 50 is isomorphic to the example of Kondō.     

Errata and addendum to “On the prime geodesic theorem for hyperbolic ‐manifolds” Math. Nachr. 291 (2018), no. 14–15, 2160–2167

We correct the exponent in the error term of the prime geodesic theorem for hyperbolic 3‐manifolds 1 and in Park's theorem for higher dimensions [ 3 , 2 ].