We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A1 structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given embeddings with isomorphic normal bundles of a closed manifold B into manifolds Q1 and Q2, we construct a differential graded category from the singular cochains of these spaces, and prove that it is equivalent to the A1 category obtained by considering exact Lagrangian embeddings of Q1 and Q2 which intersect cleanly along B.

We give a new characterization of the Orlicz–Sobolev space$W$^{1,Ψ}(R^{$n$}) in terms of a pointwise inequality connected to the Young function Ψ. We also study different Poincaré inequalities in the metric measure space.

This article focuses on numerically studying the eigenstructure behavior of generalized eigenvalue problems (GEPs) arising in three dimensional (3D) source-free Maxwell's equations with magnetoelectric coupling effects which model 3D reciprocal chiral media. It is challenging to solve such a large-scale GEP efficiently. We combine the null-space free method with the inexact shift-invert residual Arnoldi method and MINRES linear solver to solve the GEP with a matrix dimension as large as 5,308,416. The eigenstructure is heavily determined by the chirality parameter $\gamma$. We show that all the eigenvalues are real and finite for a small chirality $\gamma$. For a critical value $\gamma = \gamma^*$, the GEP has $2 \times 2$ Jordan blocks at infinity eigenvalues. Numerical results demonstrate that when $\gamma$ increases from $\gamma^*$, the $2 \times 2$ Jordan block will first split into a complex conjugate eigenpair, then rapidly collide \tr{with} the real axis and bifurcate into \tr{positive (resonance) and negative eigenvalues with modulus} smaller than the other existing positive eigenvalues. The resonance band also exhibits an anticrossing interaction. Moreover, the electric and magnetic fields of the resonance modes are localized inside the structure, with only a slight amount of field leaking into the background (dielectric) material.

We show that closed hypersurfaces in Euclidean space with nonnegative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the kth mean curvature, for k greater than 2, as we construct the counterexamples for all k greater than 2. Our proof relies on a new geometric argument which relates the scalar curvature and mean curvature of a hypersurface to the mean curvature of the level sets of a height function. By extending the argument, we show that complete noncompact asymptotically flat hypersurfaces with nonnegative scalar curvature are weakly mean convex and prove the positive mass theorem for such hypersurfaces in all dimensions.

This paper gives a new and direct construction of the multi-prime big de Rham–Witt complex, which is defined for every commutative and unital ring; the original construction by Madsen and myself relied on the adjoint functor theorem and accordingly was very indirect. The construction given here also corrects the 2-torsion which was not quite correct in the original version. The new construction is based on the theory of modules and derivations over a$λ$-ring which is developed first. The main result in this first part of the paper is that the universal derivation of a$λ$-ring is given by the universal derivation of the underlying ring together with an additional structure depending directly on the$λ$-ring structure in question. In the case of the ring of big Witt vectors, this additional structure gives rise to divided Frobenius operators on the module of Kähler differentials. It is the existence of these divided Frobenius operators that makes the new construction of the big de Rham–Witt complex possible. It is further shown that the big de Rham–Witt complex behaves well with respect to étale maps, and finally, the big de Rham–Witt complex of the ring of integers is explicitly evaluated.