Related papers: A note on a perfect Euler cuboid
We compute the dynamical degrees of certain compositions of reflections in points on a smooth cubic fourfold. Our interest in these computations stems from the irrationality problem for cubic fourfolds. Namely, we hope that they will…
We find an algorithm to compute the quadratic Euler characteristic of a smooth projective complete intersection of hypersurfaces of the same degree. As an example, we compute the quadratic Euler characteristic of a smooth projective…
Building on the genus-3 reduction $C_A : w^2 = \lambda^8 + A \lambda^4 + 1$ established in our companion paper (arXiv:2604.09328), we give an unconditional proof of the perfect-cuboid conjecture ("Conjecture B") on $1{,}072$ explicit…
Following Lortz, we construct a family of smooth steady states of the ideal, incompressible Euler equation in three dimensions that possess no continuous Euclidean symmetry. As in Lortz, they do possess a planar reflection symmetry and, as…
In this paper we obtain four new parametric ideal solutions of the Tarry-Escott problem of degree 7, that is, of the simultaneous diophantine equations, $\sum_{i=1}^8x_i^r=\sum_{i=1}^8y_i^r,\;r=1,\,2,\,\dots,\,7$. While all the known…
A remarkably simple Diophantine quadratic equation is known to generate all Apollonian integral gaskets (disk packings). A new derivation of this formula is presented here based on inversive geometry. Also occurrences of Pythagorean triples…
Optimally convergent (with respect to the regularity) quadratic finite element method for two dimensional obstacle problem on simplicial meshes is studied in (Brezzi, Hager, Raviart, Numer. Math, 28:431--443, 1977). There was no analogue of…
The Euler-Bernoulli equation describing the deflection of a beam is a vital tool in structural and mechanical engineering. However, its derivation usually entails a number of intermediate steps that may confuse engineering or science…
We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such…
In 2009, Luca and Nicolae proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are $1,2$, and $3$. In 2015, Faye and Luca proved that the only Pell numbers whose Euler totient function is another…
We construct a graded Lie algebra in which a solution to the vacuum Einstein equations is any element of degree 1 whose bracket with itself is zero. Each solution generates a cochain complex, whose first cohomology is linearized gravity…
In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space $L_2^{(m)}(0,1)$is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first…
In this paper we obtain an explicit formula for the number of degree d curves in two dimensional complex projective space, passing through (d(d+3)/2 -k) generic points and having a codimension k singularity, where k is at most 7. In the…
In this note we provide a direct proof of the complete classification of conformally flat isoparametric submanifolds of Euclidean space.
We give a formula for the Euler characteristic of a triangulated manifold of even dimension in terms of the numbers of even-dimensional faces only. The coefficients in this formula are universal (they do not depend on the dimension of the…
We show that for any positive integer $N$, there are only finitely many holomorphic eta quotients of level $N$, none of which is a product of two holomorphic eta quotients other than 1 and itself. This result is an analog of Zagier's…
In this paper we present a conjecture on the construction of generalised elliptic units above number fields with exactly one complex place. These elliptic units obtained as values of multiple elliptic Gamma functions. These form a…
We determine the $166\,104$ extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with $D$-equivalence classes of regular triangulations of the $3$-dilated tetrahedron. We describe how to compute these…
Based on a PML for the advective wave equation, we propose two PML models for the linearized Euler equations. The derivation of the first model can be applied to other physical models. The second model was implemented. Numerical results are…
We study integrable Euler equations on the Lie algebra $\mathfrak{gl}(3,\mathbb{R})$ by interpreting them as evolutions on the space of hexagons inscribed in a real cubic curve.