arXiv subjects

Anatoly Eydelzon

Publications and source records attributed to Anatoly Eydelzon.

4 recordsLinked to original sources

arXiv2026

From Pythagorean Runs to Pell-Generated Cubic Runs

A classical Pythagorean run is an identity in which a block of consecutive squares is equal to the immediately following block of consecutive squares. Boardman's construction gives such a run for every prescribed length. We consider a cubic analogue in which the second block is allowed to have common difference 2. We prove that there are infinitely many positive integer triples $(m,A,B)$, with $B>A+2m-1$, such that $$ \sum_{j=0}^{2m-1}(A+j)^3 = \sum_{j=0}^{m-1}(B+2j)^3. $$ An explicit infinite family is obtained from the Pell-type equation $$ 48329z^2-156m^2=161, $$ which reduces to a generalized Pell equation in normalized form. The ordering condition ensures that the step-2 progression begins strictly after the consecutive block ends. The resulting sequence of half-lengths satisfies an explicit second-order linear recurrence.

math.NT↗

arXiv2026

From the Steiner Inellipse to the John Ellipsoid of a Simplex: A Corner-Volume Characterization

For a triangle of area $T$, a planar corner-area characterization states that an interior point $M$ lies on the Steiner inellipse precisely when the three corner triangles cut off by the lines through $M$ parallel to the sides have areas $T_1,T_2,T_3$ satisfying $$ T_1+T_2+T_3=\frac12 T. $$ We give the corresponding statement for a simplex in arbitrary dimension. If $S$ is a nondegenerate $n$-simplex of volume $V$ and $V_1(M),\ldots,V_{n+1}(M)$ are the volumes of the facet-parallel corner simplices determined by $M$, then $$ M\in\partial E_J(S) \quad\Longleftrightarrow\quad \sum_{i=1}^{n+1}V_i(M)^{2/n}=\frac1n V^{2/n}, $$ where $E_J(S)$ is the John ellipsoid of $S$. We also identify the entire corner-volume functional with the central second-moment quadratic of the uniform simplex. The novelty claimed here is limited to the corner-volume formulations and their connections with the planar Steiner-inellipse result; the underlying barycentric, covariance, and John-ellipsoid facts are classical.

math.MG↗

arXiv2017

On Finite Gauss Transform

We present an invariant density for the finite Gauss transformation of the unit interval and discuss some properties of this transformation.

math.DS↗

arXiv2014

On recoverability properties of fixed measurement matrices

The purpose of this paper is to extend a result by Donoho and Huo, Elad and Bruckstein, Gribnoval and Nielsen on sparse representations of signals in dictionaries to general matrices. We consider a general fixed measurement matrix, not necessarily a dictionary, and derive sufficient condition for having unique sparse representation of signals in this matrix. Currently, to the best of our knowledge, no such method exists. In particular, if matrix is a dictionary, our method is at least as good as the method proposed by Gribnoval and Nielsen.

cs.IT↗