On influence of collimating slit imperfections on the quality of experimental data in high-resolution x-ray diffraction

High-resolution x-ray diffraction and imaging techniques commonly assume a well-defined plane wave incident on the sample. Experimentally, the wave-front is limited by a collimating slit. Slit imperfections, such as surface roughness on the edges, may significantly contribute to the formation of the diffraction pattern from a specimen placed behind the slit. These effects become more profound when imaging at the nano-scale. This Letter presents experimental and simulated x-ray diffraction data quantitatively demonstrating the influence of slit edge imperfections on the formation of the diffraction pattern in the far-field regime.     

New derivation of the Waller–Hartree–Fock spatial wave function

A new derivation is given for the Waller–Hartree–Fock double-determinantal spatial wave function. One starts from the single-determinant wave function in which a orbitals are doubly occupied, and decomposes it into a sum of products of spatial and spin functions. The spatial product of the first genealogical spin eigenfunction is a double-determinantal function. The derivation is based on the simple form of U_1?(P) when the representation matrix is obtained from the genealogical spin eigenfunction.     

The spin degeneracy problem in the AMO method

The use of different spin functions in the AMO method was investigated for benzene, the ring of six H atoms, and fulvene. The additional improvement in the energy obtained by the use of a linear combination of different spin functions is quite small for the singlet ground state. It is pointed out that there are great differences in energy between functions having the same spatial function but different spin functions.     

Short-Time Existence of the Möbius-Invariant Willmore Flow

In this article the author proves existence and uniqueness of a smooth short-time solution of the “Möbius-invariant Willmore flow” Eq. (9) starting in a \(C^{\infty }\)-smooth immersion \(F_0\) of a fixed smooth compact torus \({\varSigma }\) into \(\mathbb {R}^n\) without umbilic points. Hence, for some sufficiently small \(T^*>0\) there exists a unique smooth family \(\{f_t\}\) of smooth immersions of the torus \({\varSigma }\) into \(\mathbb {R}^n\), with \(f_0=F_0\), which solve the evolution Eq. (9) for \(t \in [0,T^*]\) and whose tracefree parts \(A^{0}_{f_t}(x)\) of their second fundamental forms do not vanish in any \((x,t) \in {\varSigma }\times [0,T^*]\). The right-hand side of Eq. (9) has the specific property that any family \(\{f_t\}\) of umbilic-free \(C^4\)-immersions \(f_t:{\varSigma }\longrightarrow \mathbb {R}^n\) solves Eq. (9) if and only if its composition \({\varPhi }(f_t)\) with any applicable Möbius-transformation \({\varPhi }\) of \(\mathbb {R}^n\) solves Eq. (9) as well.     

On purely non-free finite actions of abelian groups on compact surfaces

A finite group of self-homeomorphisms of a closed orientable surface is said to act on it purely non-freely if each of its elements has a fixed point; we also call it a gpnf-action. In this paper we observe that gpnf-actions exist for an arbitrary finite group and we discuss the minimum genus problem for such actions. We solve it for abelian groups. In the cyclic case we prove that the minimal gpnf-action genus coincides with Harvey’s minimal genus.     

Equations for the Missing Boundary Values in the Hamiltonian Formulation of Optimal Control Problems

Partial differential equations for the unknown final state and initial costate arising in the Hamiltonian formulation of regular optimal control problems with a quadratic final penalty are found. It is shown that the missing boundary conditions for Hamilton’s canonical ordinary differential equations satisfy a system of first-order quasilinear vector partial differential equations (PDEs), when the functional dependence of the H-optimal control in phase-space variables is explicitly known. Their solutions are computed in the context of nonlinear systems with ℝ ⁿ -valued states. No special restrictions are imposed on the form of the Lagrangian cost term. Having calculated the initial values of the costates, the optimal control can then be constructed from on-line integration of the corresponding 2n-dimensional Hamilton ordinary differential equations (ODEs). The off-line procedure requires finding two auxiliary n×n matrices that generalize those appearing in the solution of the differential Riccati equation (DRE) associated with the linear-quadratic regulator (LQR) problem. In all equations, the independent variables are the finite time-horizon duration T and the final-penalty matrix coefficient S, so their solutions give information on a whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage from the symplectic structure of the Hamiltonian formalism, which allows one to reformulate Bellman’s conjectures concerning the “invariant-embedding” methodology for two-point boundary-value problems. Results for LQR problems are tested against solutions of the associated differential Riccati equation, and the attributes of the two approaches are illustrated and discussed. Also, nonlinear problems are numerically solved and compared against those obtained by using shooting techniques.