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.

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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This paper introduces a new method of finding the sum and product of roots of quadratic equations.The approach adapts the half-life concept from radioactive decay in Physics and Chemistry. During the derivation of the quadratic formula by completing the square, the step where we add half the coefficient of x to both sides is replaced by the half-life relation 1/2 = ln 2/k, leading to k = 2 ln 2. This substitution produces an auxiliary formula that yields two values x1and x2whose sum equals the Vieta sum.A second relation involving ln 2 then recovers the exact Vieta product. The method, called the Logarithmic Root-Sum-Product Extractor (or Half-Life Quadratic Formula), gives results identical to the conventional method while creating a strong link between mathematics and science.