Harmonic maps with torsion

In this article we introduce a natural extension of the well-studied equation for harmonic maps between Riemannian manifolds by assuming that the target manifold is equipped with a connection that is metric but has non-vanishing torsion. Such connections have already been classified in the work of Cartan(1924).The maps under consideration do not arise as critical points of an energy functional leading to interesting mathematical challenges. We will perform a first mathematical analysis of these maps which we will call harmonic maps with torsion.     

Spatiotemporal Studies of the One-Dimensional Coordination Polymer [Fe(ebtz)2(C2H5CN)2](BF4)2: Tug of War between the Nitrile Reorientation Versus Crystal Lattice as a Tool for Tuning the Spin Crossover Properties**

Reaction of 1,2-di(tetrazol-2-yl)ethane (ebtz) with Fe(BF₄)₂⋅6 H₂O in different nitriles yields one-dimensional coordination polymers [Fe(ebtz)₂(RCN)₂](BF₄)₂⋅nRCN (n=2 for R=CH₃ ( 1 ) and n=0 for R=C₂H₅ ( 2 ) C₃H₇ ( 3 ), C₃H₅ ( 4 ), CH₂Cl ( 5 )) exhibiting spin crossover (SCO). SCO in 1 and 3 – 5 is complete and occurs above 160 K. In 2 , it is shifted to lower temperatures and is accompanied by wide hysteresis (T_1/2^↓=78 K, T_1/2^↑=123 K) and proceeds extremely slowly. Isothermal (80 K) time-resolved single-crystal X-ray diffraction studies revealed a complex nature for the HS→LS transition in 2 . An initial, slow stage is associated with shrinkage of polymeric chains and with reduction of volume at 77 % (in relation to the difference between cell volumes V_HS−V_LS) whereas only 16 % of iron(II) ions change spin state. In the second stage, an abrupt SCO occurs, associated with breathing of the crystal lattice along the direction of the Fe–nitrile bonds, while the nitriles reorient. HS→LS switching triggered by light (808 nm) reveals the coupling of spin state and nitrile orientation. The importance of this coupling was confirmed by studies of [Fe(ebtz)₂(C₂H₅CN/C₃H₇CN)₂](BF₄)₂ mixed crystals ( 2 a , 2 b ), showing a shift of T_1/2 to higher values and narrowing of the hysteresis loop concomitant with an increase of the fraction of butyronitrile. This increase reduces the capability of nitrile molecules to reorient. Density functional theory (DFT) studies of models of 1 – 5 suggest a particular possibility of 2 to adopt a low (140–145°) value of its Fe-N-C(propionitrile) angle.     

Half-a-century of gamma-ray astrophysics at the Max-Planck Institute for Extraterrestrial Physics

The European Physical Journal H - This is a tutorial for the many-worlds theory by Everett, which includes some of my personal views. It has two main parts. The first main part shows the emergence...     

Tensor method for optimal control problems constrained by fractional three-dimensional elliptic operator with variable coefficients

We introduce the tensor numerical method for solving optimal control problems that are constrained by fractional two- (2D) and three-dimensional (3D) elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D $n\times n\times n$ spacial grids. Using the diagonalization of the arising matrix-valued functions in the eigenbasis of the one-dimensional Sturm–Liouville operators, we construct the rank-structured tensor approximation with controllable precision for the discretized fractional elliptic operators and the respective preconditioner. The right-hand side in the constraining equation (the optimal design function) is supposed to be represented in a form of a low-rank canonical tensor. Then the equation for the control function is solved in a tensor structured format by using preconditioned CG iteration with the adaptive rank truncation procedure that also ensures the accuracy of calculations, given an $\epsilon$-threshold. This method reduces the numerical cost for solving the control problem to $O(n\log n)$ (plus the quadratic term $O(n^2)$ with a small weight), which outperforms traditional approaches with $O(n^3\log n)$ complexity in the 3D case. The storage for the representation of all 3D nonlocal operators and functions involved is also estimated by $O(n\log n)$. This essentially outperforms the traditional methods operating with fully populated $n^3\times n^3$ matrices and vectors in ${?}^{n^3}$. Numerical tests for 2D/3D control problems indicate the almost linear complexity scaling of the rank truncated preconditioned conjugate gradient iteration in the univariate grid size n.