arXiv subjects

Naveen Kumar Kakumanu

Publications and source records attributed to Naveen Kumar Kakumanu.

3 recordsLinked to original sources

arXiv2026

Finite diagonalizable torsors and Kummer cohomology over semiring schemes

Classical descent treats a finite diagonalizable torsor as a higher-rank locally free object. Over semirings that route is incomplete: fpqc-local freeness is not known to imply Zariski-local freeness in arbitrary rank. We show that diagonalizable symmetry bypasses this obstruction. The coordinate semialgebra splits into character pieces, each piece descends in rank one, and the torsor identity makes their multiplication maps invertible. This gives a natural equivalence between fpqc $D_X(Q)$-torsors and Picard-strong $Q$-gradings, and represents every torsor by a finite locally free morphism of rank $|Q|$. The same viewpoint proves that finite-index monomial maps of split tori are finite locally free torsors and yields matrix Kummer classification sequences. A concrete example is the nontrivial Boolean torsor $\Spec\mathbb B[v^{\pm1}]\to\Spec\mathbb B[u^{\pm1}]$, $u\mapsto v^m$, whose unit-power classes contribute $\mathbb Z/m\mathbb Z$ to degree-one cohomology. A two-term resolution of the character group then gives a presentation-independent universal-coefficient filtration for diagonalizable fpqc cohomology. This formalism organizes, but does not compute, $\mathbf G_{\mathrm m}$-cohomology. In degree two it identifies the Kummer boundary of a line bundle with its root gerbe. On projective space over the Boolean or real tropical semifield all $μ_m$-torsors vanish, whereas the root gerbe of $\mathcal O(1)$ has exact order $m$. Thus torsors and gerbes retain information that can disappear under ordinary ring completion.

math.RA↗

arXiv2026

Central Splitting, Division-Stable Residuals, and Opposite-Ring Transfer for Strongly C4*-Rings

The known decomposition theorem for strongly $\Cfourstar$-modules gives a semisimple summand and a summand-square-free residual summand, with two-way Hom-orthogonality when the ambient module is projective. Applied to the regular module, the two cross-corners vanish, so the idempotent defining the decomposition is central. Thus every strongly right $\Cfourstar$-ring splits as $R\cong Σ\times T$, where $Σ$ is semisimple artinian and $T_T$ is summand-square-free. The splitting need not be unique. We organize all admissible central splittings into a join-semilattice and prove a comparison theorem: any two residual factors have a common direct factor, and their complementary factors are finite products of division rings. Consequently the right-to-left defect is independent of the chosen splitting. If the central idempotents satisfy the ascending chain condition, there is a unique greatest admissible idempotent; it captures every central semisimple artinian direct factor and hence yields a canonical residual with no further such factor. An infinite product of fields shows that this finiteness hypothesis cannot simply be omitted. We also prove anti-isomorphism transport for $\Cfour$, $\Cfourstar$, semi-weak-CS, and strongly $\Cfourstar$ modules. It yields transfer whenever a residual factor is a product of a semisimple summand-square-free ring and a self-opposite core. This criterion does not assume regularity, exchange, or a left-sided hypothesis. Skew Laurent rings provide noncommutative nonregular examples, while the known injective-nonsurjective skew-polynomial construction gives a sharp one-sided residual obstruction.

math.RA↗

arXiv2026

Certified Reduced-Order Surrogates and Stability Margins in Viscous Incompressible Flow and Fluid--Structure Interaction

Let $(u,p)$ solve the incompressible Navier--Stokes equations in a regime in which an energy inequality is available and each constant in that inequality is computable from declared data. We construct a reduced-order model $u_n$ constrained so that its discrete evolution satisfies a certified energy inequality. This certificate yields global-in-time boundedness of the ROM energy and a regime-of-validity test that fails when a stated hypothesis fails. It follows that one can attach a computable residual functional $\mathcal{R}_n$ to the ROM trajectory. We prove an a posteriori bound of the form \[ \norm{u-u_n}_{\mathsf{X}(0,T)} \le C(\text{declared data})\,\mathcal{R}_n, \] with $C$ explicit and with $\mathcal{R}_n$ computed from the ROM and the discretization operators. Conversely, if the certificate constraint is relaxed, the bound can fail even for stable full-order dynamics, by an explicit instability mechanism recorded in the text. We then derive transition indicators from rigorous energy and enstrophy budgets in simplified geometries. Each indicator is an inequality involving declared quantities such as forcing norms, viscosity, Poincaré-type constants, and a computable resolvent surrogate. These inequalities provide thresholds that preclude transition, or else certify the presence of transient growth beyond a stated level. Finally, for a class of fluid--structure interaction models, we identify a parameter regime that implies existence and uniqueness of weak solutions. We derive discrete coupled energy estimates that produce computable stability margins. These margins yield explicit constraints on time step and mesh parameters. They are stated as inequalities with constants determined by fluid viscosity, structure stiffness, density ratios, and interface trace bounds.

math.NA↗