arXiv subjects

Gautam Neelakantan Memana

Publications and source records attributed to Gautam Neelakantan Memana.

5 recordsLinked to original sources

arXiv2026

Fractional elementary symmetric functions and Hessian estimates in the plane

We prove an endpoint Pucci maximum principle for domains in $\mathbb R^2$, formulated using a generalization of elementary symmetric polynomials to noninteger orders. As a result, we sharpen an earlier result of Astala, Iwaniec and Martin, and deduce a new Sobolev inequality for $q$-Hessians where $1 < q < 2$. The fractional elementary symmetric functions we use are derived through fractional differentiation. Their signed singular value extension coincides with the Burkholder functional introduced by Burkholder and gives us access to a local Burkholder inequality due to Astala, Iwaniec, Prause and Saksman, leading to a different proof of the Pucci maximum principle from the one in that used quasiconformal maps. We include an elementary real-variable proof of this local Burkholder inequality, obtained by combining a few known techniques from existing literature.

math.AP↗

arXiv2026

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition

We give a negative solution to MAIS-O60. We first construct an example in which an initially active ReLU neuron becomes completely inactive in finite time and thereafter remains frozen at a limit whose Fourier energy is equally distributed among all nonzero real frequency classes. The counterexample holds on an open set of initial conditions and therefore occurs with positive probability under Gaussian initialization. An appendix prepared by GPT-5.6 Sol strengthens the counterexample by showing that the same failure can occur for every Clarke trajectory from an open set of initial conditions, under the convention $\mathrm{ReLU}'(0)=0$, for smooth dead-zone approximations of ReLU, and for fixed-step full-batch gradient descent. Thus, single-frequency alignment is not a general consequence of training a single neuron on modular addition.

math.OC↗

arXiv2026

Comparison theorems for weak solutions of nonlinear maximally sub-elliptic PDEs

We establish a comparison principle for viscosity subsolutions and supersolutions of a broad class of second-order quasilinear, maximally subelliptic PDEs on general manifolds. In fact, we prove the comparison theorem for a larger class of degenerate subelliptic PDEs. Our result strengthens a recent theorem of Manfredi-Mukherjee, which was established in the setting of Carnot groups. Our main aim is to highlight that maximal subellipticity allows one to obtain a comparison principle for weak solutions in close analogy with the classical elliptic theory.

math.AP↗

arXiv2024

A regularity theorem for fully nonlinear maximally subelliptic PDE

We prove a sharp interior regularity theorem for fully nonlinear maximally subelliptic PDE in non-isotropic Sobolev spaces adapted to maximally subelliptic operators. This result complements the regularity theorem proven by Street for adapted Zygmund-Hölder spaces. This also recovers the classical regularity theorem for fully nonlinear elliptic differential operators for classical Sobolev spaces. We also obtain a sharp interior regularity theorem for fully nonlinear maximally subelliptic PDE in isotropic Sobolev spaces.

math.AP↗

arXiv2022

Uniform Poincaré inequalities on measured metric spaces

Consider a proper geodesic metric space $(X,d)$ equipped with a Borel measure $μ.$ We establish a family of uniform Poincaré inequalities on $(X,d,μ)$ if it satisfies a local Poincaré inequality ($P_{loc}$) and a condition on growth of volume. Consequently if $μ$ is doubling and supports $(P_{loc})$ then it satisfies a $(σ,β,σ)$-Poincaré inequality. If $(X,d,μ)$ is a $δ$-hyperbolic space then using the volume comparison theorem in \cite{BCS} we obtain a uniform Poincaré inequality with exponential growth of the Poincaré constant. If $X$ is the universal cover of a compact $CD(K,\infty)$ space then it supports a uniform Poincaré inequality and the Poincaré constant depends on the growth of the fundamental group.

math.MG↗