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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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.

The mass gap of Yang-Mills theory is derived analytically by establishing a correspondence between the classical kinetic energy coefficient and the quantum half-life decay constant. The replacement 1/2→ln2/k transforms the energy equation into a form compatible with gluonic bound states. The dimensionless constant k is de- termined from the running coupling as k = 2π/αs≈ 18. The trace anomaly, with an explicit derivation from the beta function and gluon condensate scaling, yields m0++= 4 √σ. Utilizing the lattice-calibrated string tension √σ = 7/16 GeV, the mass gap is computed as 1.75 GeV. The kinetic-half-life correspondence then predicts the flux tube thickness L =4 ln2/k√σ≈ 0.352 GeV^-1, consistent with the lattice value 0.343 GeV^-1. Four independent confirmatory calculations—lattice error analysis, gluon condensate sum rule inversion, excited-state Regge trajectory matching, and decay width half-life consistency—converge to this value within standard uncertainties. A constructive outline for the continuum limit and the positivity of the mass gap is provided, demonstrating that the derivation is consistent with the rigorous requirements of the Yang-Mills Millennium Problem.

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In \cite {ChenFengLiXiXu2026}, a reverse projection inequality for general convex bodies was established in three dimensions. In this paper, we completely characterize the equality cases of this inequality. We further show that the reverse projection inequality fails in every dimension $n\geq9$. The counterexamples are constructed as full-dimensional polytopes with at most $n+2$ facets.

Let $K\subset\R^3$ be a convex body and let $\Pi K$ be its projection body. We prove \[ \frac{V_3(\Pi K)}{V_3(K)^2} \ge \frac{3\pi^2}{4}, \] with equality exactly for ellipsoids. The proof combines a section estimate of Saroglou with a sharp one-dimensional functional inequality. The latter is proved by passing to the distribution function and approximating it by step functions. We also prove the complementary sharp upper bound \( V_3(\Pi K)/V_3(K)^2\le18, \) which is attained by tetrahedra.

The Prime Density Triangle offers a novel combinatorial framework for exploring properties of prime numbers. From the algebraic relations embedded within this triangle, specifically the coupled expressions X^p1± p1and X^p0± p0, we derive two closed-form sequences.First, the multiplication of these expressions yields the sequence Sn(m) = (mn+ n2)1/(n+2), which converges to m. Second, by a deliberate construction—replacing the additive denominator n + 2 with the multiplicative denominator 2n—we obtain the Exponential-Polynomial Root Mean (EPRM) defined by Rn(m) = (mn+ n2)1/(2n) for m > 0 and n ∈ N. This construction provides a closed-form, non-iterative method for approximating √m for sufficiently large n. In this paper, we present a complete derivation of both sequences from the Prime Den- sity Triangle’s algebraic frameworks, rigorous convergence proofs including asymptotic error analysis, detailed worked examples for m = 2, 3, 5 with explicit computations, numerical summary tables, explicit iterations for classical methods (Babylonian and Newton-Raphson), and a comprehensive comparison. The approach is elementary and highlights the dominance of exponential terms over polynomial corrections.