Minkowski type theorems for convex sets in cones

Acta Mathematica Hungarica - Minkowski’s classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface...     

A Neural Network-Based Policy Iteration Algorithm with Global $$H^2$$ -Superlinear Convergence for Stochastic Games on Domains

In this work, we propose a class of numerical schemes for solving semilinear Hamilton–Jacobi–Bellman–Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the \(H^2\)-norm and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to oblique derivative boundary conditions. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

     

Iodine(III)‐Catalyzed Rearrangements of Imides: A Versatile Route to α,α‐Dialkylated α‐Hydroxy Carboxylamides

A tertiary hydroxy group α to a carboxyl moiety comprises a key structural motif in many bioactive substances. With the herein presented metal‐free rearrangement of imides triggered by hypervalent λ³‐iodane, an easy and selective way to gain access to such a compound class, namely α,α‐disubstituted‐α‐hydroxy carboxylamides, was established. Their additional methylene bromide side chain constitutes a useful handle for rapid diversification, as demonstrated by a series of further functionalizations. Moreover, the in situ formation of an iodine(III) species under the reaction conditions was proven. Our findings clearly corroborate that hypervalent λ³‐benziodoxolones are involved in these organocatalytic reactions.     

Discovery of Novel Symmetrical 1,4-Dihydropyridines as Inhibitors of Multidrug-Resistant Protein (MRP4) Efflux Pump for Anticancer Therapy

Despite the development of targeted therapies in cancer, the problem of multidrug resistance (MDR) is still unsolved. Most patients with metastatic cancer die from MDR. Transmembrane efflux pumps as the main cause of MDR have been addressed by developed inhibitors, but early inhibitors of the most prominent and longest known efflux pump P-glycoprotein (P-gp) were disappointing. Those inhibitors have been used without knowledge about the expression of P-gp by the treated tumor. Therefore the use of inhibitors of transmembrane efflux pumps in clinical settings is reconsidered as a promising strategy in the case of the respective efflux pump expression. We discovered novel symmetric inhibitors of the symmetric efflux pump MRP4 encoded by the ABCC4 gene. MRP4 is involved in many kinds of cancer with resistance to anticancer drugs. All compounds showed better activities than the best known MRP4 inhibitor MK571 in an MRP4-overexpressing cell line assay, and the activities could be related to the various substitution patterns of aromatic residues within the symmetric molecular framework. One of the best compounds was demonstrated to overcome the MRP4-mediated resistance in the cell line model to restore the anticancer drug sensitivity as a proof of concept.     

The Effect of the Spacer of Bis(biurea) Ligands on the Structure of A2L3‐type (A=anion) Phosphate Complexes

By tuning the length and rigidity of the spacer of bis(biurea) ligands L, three structural motifs of the A₂L₃ complexes (A represents anion, here orthophosphate PO₄^3?), namely helicate, mesocate, and mono‐bridged motif, have been assembled by coordination of the ligand to phosphate anion. Crystal structure analysis indicated that in the three complexes, each of the phosphate ions is coordinated by twelve hydrogen bonds from six surrounding urea groups. The anion coordination properties in solution have also been studied. The results further demonstrate the coordination behavior of phosphate ion, which shows strong tendency for coordination saturation and geometrical preference, thus allowing for the assembly of novel anion coordination‐based structures as in transition‐metal complexes.     

Intramolecular Hydroamination of Selenoalkynes to 2-Selenylindoles in the Absence of Catalyst

In this work, a series of 2-chalcogenylindoles was synthesized by an efficient methodology, starting from chalcogenoalkynes, including a previously unreported tellurium indole derivative. For the first time, these 2-substituted chalcogenylindoles were obtained in the absence of metal catalyst or base, under thermal conditions only. In addition, the results described herein represent a methodology with inverse regioselectivity for the chalcogen functionalization of indoles.     

Self-similar sets 3. Constructions with sofic systems

Using sofic systems, modifications of the self-similar sets of Hutchinson are defined as solutions of systems of fixed-point equations. Their Hausdorff dimension is determined.     

Non-linear Stability of Modulated Fronts¶for the Swift–Hohenberg Equation

We consider front solutions of the Swift–Hohenberg equation ∂ _t u= -(1+ ∂ _x ²)² u + ɛ² u -u ³. These are traveling waves which leave in their wake a periodic pattern in the laboratory frame. Using renormalization techniques and a decomposition into Bloch waves, we show the non-linear stability of these solutions. It turns out that this problem is closely related to the question of stability of the trivial solution for the model problem ∂ _t u(x,t) = ∂ _x ² u (x,t)+(1+tanh(x-ct))u(x,t)+u(x,t) ^p with p>3. In particular, we show that the instability of the perturbation ahead of the front is entirely compensated by a diffusive stabilization which sets in once the perturbation has hit the bulk behind the front. Received: 23 February 2001 / Accepted: 27 August 2001