We introduce the roots of supercritical Leung--Yau--Zaslow pairs and prove pointwise strong positivity of the associated Schur forms. The resulting LYZ cones contain the pliant cone on smooth projective threefolds, with strict inclusion on a Fujino--Sato threefold. We also establish Hard Lefschetz and Hodge--Riemann relations for the associated Chern classes on compact K\"ahler manifolds in bidegrees $1\le p+q\le\dim X$ with $\min\{p,q\}\le1$.

We study an accelerated Langevin equation in which the classical overdamped Langevin drift is supplemented by a nonlocal gradient velocity determined through a target-weighted Poisson equation. The target \(p\) is only assumed to be strictly positive and need not be normalized. Writing the dynamics in terms of \(\omega=\log(p/q)\) yields a coupled parabolic--elliptic system whose parabolic component contains the usual Langevin diffusion, whereas the elliptic component generates a global transport. The system is invariant under multiplication of \(p\) by a positive constant. On a bounded smooth domain with homogeneous Neumann boundary conditions, we establish global existence and uniqueness of classical solutions for strictly positive \(C^{2+\alpha}\) target weights and compatible \(C^{2+\alpha}\) initial data, with full \(\alpha\)-H\"older regularity up to the initial time. For long-time behavior, the oscillation of \(\omega\) decays at the explicit rate \(e^{-t}\). For normalized initial densities, the solution density converges exponentially to the normalized target. Moreover, an exact entropy-dissipation identity holds relative to the normalized target, in which the relative Fisher information and the Jeffreys divergence give the two dissipation mechanisms associated with the Langevin and weighted-Poisson components. In particular, KL divergence also converges to zero with rate $e^{-t}$ without a convexity assumption on the target.

Consistent orientation of surface normals is a global problem in computational geometry. Orientation is intrinsically global: local geometry determines a normal line, but not which of its two directions is outward. We study the orientation problem at the continuum level for a smooth closed surface $\Gamma=\partial\Omega$. Starting from the Newtonian single-layer potential, we consider its averaged boundary divergence and curl traces, denoted by $B$ and $C$. The classical scalar Gauss constraint $B\varphi=1/2$ alone does not determine a free vector density. We show that adding the curl condition $C\varphi=0$ makes the system complete. More precisely, for every $s\in\R$, \[ \|\varphi\|_{H^s(\Gamma;\R^3)} \le C_{\Gamma,s} \bigl(\|B\varphi\|_{H^s(\Gamma)}^2+ \|C\varphi\|_{H^s(\Gamma;\R^3)}^2\bigr)^{1/2}. \] $P=(B,C)$ is therefore injective with closed range, and $P\varphi=(1/2,0)$ has the unique solution $\varphi=n_\Omega$. The Clifford Cauchy involution $T^2=I/4$ is classical. The new result proved here is the same-order lower bound for $P=(B,C)$; its problem-specific proof step is the factorization $TM=JP$, which identifies $B$ and $C$ with the scalar and bivector parts of the same Cauchy singular integral. This factorization also yields a compatibility condition for general boundary data. If the samples are distributed according to a positive density $\dd\mu=\rho\,\dd S$, the weighted system recovers $q=n_\Omega/\rho$ uniquely, so normalization removes the unknown sampling density and returns the outward orientation. Regularized point-cloud experiments illustrate the continuum result.

We establish coefficientwise equality between the refined supersymmetric indices of ABJM theory and the BMN matrix model in the plane-wave limit at every fixed positive Chern--Simons level $k$. For multi-membrane families with fixed multiplicities---and hence fixed finite ABJM gauge rank---the equality holds as all longitudinal momenta tend to infinity with their pairwise differences fixed. Scalar-dressed equal-flux ABJM monopoles correspond to BMN vacua of fuzzy-sphere membranes: multiplicities count coincident membranes, whereas flux magnitudes give the momentum per membrane along the M-theory circle. At the index level, the associated Higgs residues yield a diagonal-holonomy integral for the ABJM plane-wave contribution, whose monopole-harmonic kernel agrees with the BMN matrix-harmonic kernel up to the finite BMN upper-spin cutoff. At finite momentum, the full index difference splits into the non-plane-wave remainder and the harmonic tail above this cutoff. Increasing $k$ shifts the former to higher fugacity grade without changing the surviving kernel or the cutoff contribution. This gives a coefficientwise, multi-membrane extension of the earlier single-membrane index relation.

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