The Hartree-Fock equation which is the Euler-Lagrange equation corresponding to the Hartree-Fock energy functional is used in many-electron problems. Since the Hartree-Fock equation is a system of nonlinear eigenvalue problems, the study of structures of sets of all solutions needs new methods different from that for the set of eigenfunctions of linear operators. In this paper we prove that the sets of all solutions to the Hartree-Fock equation associated with critical values of the Hartree-Fock energy functional less than the first energy threshold are unions of a finite number of compact connected real-analytic spaces. The result would also be a basis for the study of approximation methods to solve the equation.

The purpose of the paper is to study the behavior of the classes of functions of bounded lower oscillation under the change of measure given by the weight $w$. More specifically, we provide sharp upper and lower bounds for the norm of the inclusion $BLO\hookrightarrow BLO(w)$ in terms of $A_\infty$ constant of the weight. The results hold in the general context of probability spaces equipped with a treelike structure.

We prove the existence of quadratic relations between periods of meromorphic flat bundles on complex manifolds with poles along a divisor with normal crossings under the assumption of “goodness”. In dimension one, for which goodness is always satisfied, we provide methods to compute the various pairings involved. In an appendix, we recall some classical results needed for the proofs.

We consider a mass critical nonlinear Schrödinger equation with a real-valued potential. In this work, we construct a minimal mass solution that blows up at finite time, under weaker assumptions on spatial dimensions and potentials than Banica, Carles, and Duyckaerts (2011). Moreover, we show that the blow-up solution converges to a blow-up profile. Furthermore, we improve some parts of the arguments in Raphaël and Szeftel (2011) and Le Coz, Martel, and Raphaël (2016).

A tangent developable surface is defined as a ruled developable surface by tangent lines to a space curve and it has singularities at least along the space curve, called the directrix or the edge of regression. The class of tangent developable surfaces is invariant under the parallel deformations. In this paper the notions of tangent developable surfaces and their parallels are naturally generalised for frontal curves in general in Euclidean spaces of arbitrary dimensions. The singularities appearing on parallels to tangent developable surfaces of frontal curves are studied and the classification of generic singularities on them for frontal curves in 3 or 4 dimensional Euclidean spaces are given.

We give a purely cubical argument for the localization theorem for the cubical version of higher Chow groups.

The class of warped product manifolds plays an important role in differential geometry and physics. In this paper, we shall study product manifolds $\mathbb{R} \times M$ with Finsler metrics arising from warped product structure. We give an equivalent condition for those metrics to be of quadratic Weyl curvature.