arXiv subjects

Pavel Grinfeld

Publications and source records attributed to Pavel Grinfeld.

6 recordsLinked to original sources

arXiv2026

Power sums of critical points of gap polynomials for symmetric and pseudo-symmetric numerical semigroups

For a numerical semigroup $S$ with Frobenius number $F$ and gaps $G$, a polynomial $P\left(z\right)=\sum_{g\in G}c_{g}z^{g}$ of degree $F$ is called a gap polynomial. We show that for a symmetric numerical semigroup, the power sums of the critical points of a gap polynomial vanish for every power $g\in G$. For a pseudo-symmetric numerical semigroup, the same result holds if the term $z^{F/2}$ is omitted from $P\left( z\right) $. We give several corollaries, including that the sum of the critical values of a gap polynomial vanishes.

math.AC↗

arXiv2026

An Elementary Proof of Hopf's Curvatura Integra Theorem

We provide an elementary proof of the key aspect of Hopf's curvatura integra theorem. Namely, we show that the total curvature, i.e. the surface integral of the Gauss-Kronecker curvature B, is independent of the shape of the hypersurface. We accomplish this task by using the Calculus of Moving Surfaces to show that the rate of change of the total curvature vanishes under smooth changes in shape.

math.DG↗

arXiv2026

A Method for Preserving Geometric Meaning in the Calculus of Variations

We discuss the advantages of the Calculus of Moving Surfaces over the Euler-Lagrange equation for optimization problems originating in Geometry. An extension of Tensor Calculus, it provides tools for analyzing geometric quantities directly rather than their coordinate representations. This allows us to avoid the many difficulties associated with the use of coordinates, from the untenable complexity of analytical expressions to the virtual impossibility of recovering the geometric interpretation of the final result. As an illustration, we analyze the brachistochrone and give its geometric characterization in terms of its curvature.

math.DG↗

arXiv2021

The Use of Geometric Quantities in the Tensor Description of a Euclidean Space

We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors. We demonstrate the effectiveness of the approach by proving of a number of integral identities with vector integrands.

math.DG↗

arXiv2012

Accurate Computation of Laplace Eigenvalues by an Analytical Level Set Method

This purpose of this write-up is to share an idea for accurate computation of Laplace eigenvalues on a broad class of smooth domains. We represent the eigenfunction $u$ as a linear combination of eigenfunctions corresponding to the common eigenvalue $\rho ^{2}$:\EQN{6}{1}{}{0}{\RD{\CELL{u(r,\theta) =\sum_{n=0}^{N}P_{n}J_{n}(\rho) \cos n\theta,}}{1}{}{}{}}We adjust the coefficients $P_{n}$ and the parameter $\rho $ so that the zero level set of $u$ approximates the domain of interest. For some domains, such as ellipses of modest eccentricity, the coefficients $P_{n}$ decay exponentially and the proposed method can be used to compute eigenvalues with arbitrarily high accuracy.

math.DG↗

arXiv2011

A Geometric Construction for the Evaluation of Mean Curvature

We give a relationship that yields an effective geometric way of evaluating mean curvature of surfaces. The approach is reminiscent of the Gauss's contour based evaluation of intrinsic curvature. The presented formula may have a number of potential applications including estimating the normal vector and mean curvature on triangulated surfaces. Given how brief is its derivation, it is truly surprising that this formula does not appear in the existing literature on differential geometry -- at least according to the author's search. We hope to learn about a reference containing this result.

math.NA↗