# Danylo RadchenkoETH Zurich | ETH Zürich · Department of Mathematics

Danylo Radchenko

PhD

## About

34

Publications

3,240

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479

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Introduction

**Skills and Expertise**

## Publications

Publications (34)

Let $\chi(\mathbb{E}^n)$ denote the chromatic number of the Euclidean space $\mathbb{E}^n$, i.e., the smallest number of colors that can be used to color $\mathbb{E}^n$ so that no two points unit distance apart are of the same color. We present explicit constructions of colorings of $\mathbb{E}^n$ based on sublattice coloring schemes that establish...

Bezdek and Kiss showed that existence of origin-symmetric coverings of unit sphere in ${\mathbb {E}}^n$ by at most $2^n$ congruent spherical caps with radius not exceeding $\arccos \sqrt {\frac {n-1}{2n}}$ implies the X -ray conjecture and the illumination conjecture for convex bodies of constant width in ${\mathbb {E}}^n$ , and constructed such co...

We answer a question of Brass about vertex degrees in unit distance graphs of finitely generated additive subgroups of \(\mathbb {R}^2\).

Significance
We show an interrelation between the uniqueness aspect of the recent Fourier interpolation formula of D.R. and M.V. and the lattice-cross uniqueness set for the Klein–Gordon equation studied by H.H. and A.M.-R. With appropriate modifications, the approach applies in any even dimension ≥ 4 and is based on a sophisticated analysis of the...

We present new functional equations in weights 5, 6 and 7 and use them for explicit depth reduction of multiple polylogarithms. These identities generalize the crucial identity $\mathbf{Q}_4$ from the recent work of Goncharov and Rudenko that was used in their proof of the weight 4 case of Zagier's Polylogarithm Conjecture.

We show an interrelation between the uniqueness aspect of the recent Fourier interpolation formula of Radchenko and Viazovska and the Heisenberg uniqueness study for the Klein-Gordon equation and the lattice-cross of critical density, studied by Hedenmalm and Montes-Rodriguez. This has been known since 2017.

We prove that, for a tropical rational map if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative)....

We construct a large family of Fourier interpolation bases for functions analytic in a strip symmetric about the real line. Interesting examples involve the nontrivial zeros of the Riemann zeta function and other $L$-functions. We establish a duality principle for Fourier interpolation bases in terms of certain kernels of general Dirichlet series w...

We give a new explicit formula for Grassmannian polylogarithms in terms of iterated integrals. We also explicitly reduce the Grassmannian polylogarithm in weight 4 and in weight 5 each to depth 2. Furthermore, using this reduction in weight 4 we obtain an explicit, albeit complicated, form of the so-called 4-ratio, which gives an expression for the...

Let $\Gamma$ be a simple graph and $I_\Gamma(x)$ its multivariate independence polynomial. The main result of this paper is the characterization of chordal graphs as the only $\Gamma$ for which the power series expansion of $I_\Gamma^{-1}(x)$ is Horn hypergeometric.

We derive new functional equations for Nielsen polylogarithms. We show that, when viewed modulo $\mathrm{Li}_5$ and products of lower weight functions, the weight $5$ Nielsen polylogarithm $S_{3,2}$ satisfies the dilogarithm five-term relation. We also give some functional equations and evaluations for Nielsen polylogarithms in weights up to 8, and...

We prove for a tropical rational map that if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative)....

We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), whic...

We study the general properties of certain rank four rigid local systems considered by Goursat. We analyze when they are irreducible, give an explicit integral description as well as the invariant Hermitian form when it exists. By a computer search we find what we expect are all irreducible such systems all whose solutions are algebraic functions a...

We answer a question of Brass about vertex degrees in unit distance graphs of finitely generated additive subgroups of $\mathbb{R}^2$.

We prove that there is no strongly regular graph (SRG) with parameters (460, 153, 32, 60). The proof is based on a recent lower bound on the number of 4-cliques in a SRG and some applications of Euclidean representation of SRGs.

We prove the non-existence of strongly regular graph with parameters . We use Euclidean representation of a strongly regular graph together with a new lower bound on the number of 4-cliques to derive strong structural properties of the graph, and then use these properties to show that the graph cannot exist.

In this paper we construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set {0,±1,±2,±3,…}. The functions in the interpolating basis are constructed in a closed form as an integral tr...

Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions, and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an anal...

We prove that there is no strongly regular graph (SRG) with parameters
(460,153,32,60). The proof is based on a recent lower bound on the number of
4-cliques in a SRG and some applications of Euclidean representation of SRGs.

Our main result is the non-existence of strongly regular graph with
parameters (76,30,8,14). We heavily use Euclidean representation of a strongly
regular graph, and develop a number of tools that allow to establish certain
structural properties of the graph. In particular, we give a new lower bound
for the number of 4-cliques in a strongly regular...

Let S d be a unit sphere in ℝ d+1, and let α be a positive real number. For pairwise different points x 1,x 2, . . . ,x N ∈ S d , we consider a functional E α (x 1,x 2, . . . ,x N ) = Σ i≠j ||x i − x j ||−α . The following theorem is proved: for α ≥ d − 2, the functional E α (x 1,x 2, . . . ,x N ) does not have local maxima.

Pippenger (1977) [3] showed the existence of (6m,4m,3m,6)(6m,4m,3m,6)-concentrator for each positive integer mm using a probabilistic method. We generalize his approach and prove existence of (6m,4m,3m,5.05)(6m,4m,3m,5.05)-concentrator (which is no longer regular, but has fewer edges). We apply this result to improve the constant of approximation o...

We describe, up to unitary equivalence, all k-tuples (A
1, A
2, . . ., A
k
) of unitary operators such that
$ A_i^{{{n_i}}}=I\;\mathrm{for}\;i=\overline{1,k} $
and A
1A
2 . . . A
k
= λI, where the parameters (n
1, . . . , n
k
) correspond to one of the extended Dynkin diagrams
$ {{\tilde{D}}_4} $
,
$ {{\tilde{E}}_6} $
,
$ {{\tilde{E}}_7} $...

For each $N\ge C_dt^d$ we prove the existence of a well separated spherical
$t$-design in the sphere $S^d$ consisting of $N$ points, where $C_d$ is a
constant depending only on $d$.

In this paper we prove the conjecture of Korevaar and Meyers: for each N≥c_d t^d, there exists a spherical t-design in the sphere S^d consisting of N points, where c_d is a constant depending only on d.

The problem of approximating continuous maps by smooth maps with nonnegative Jacobian is considered.

We estimate the degree of comonotone polynomial approximation of continuous functions f, on [−1,1], that change monotonicity s≥1 times in the interval, when the degree of unconstrained polynomial approximation En(f)≤n^−α, n≥1. We ask whether the degree of comonotone approximation is necessarily ≤c(α,s)n^−α, n≥1, and if not, what can be said. It tur...

In this paper, we give a complete description of strongly regular graphs with
parameters ((n^2+3n-1)^2,n^2(n+3),1,n(n+1)). All possible such graphs are: the
lattice graph $L_{3,3}$ with parameters (9,4,1,2), the Brouwer-Haemers graph
with parameters (81,20,1,6), and the Games graph with parameters
(729,112,1,20).

In this paper we investigate a problem of approximation of continuous mappings by smooth mappings with nonnegative Jacobian.

In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge
c_dt^d$ there exists a spherical $t$-design in the sphere $S^d$ consisting of
$N$ points, where $c_d$ is a constant depending only on $d$.

An $N$-dimensional parallelepiped will be called a bar if and only if there are no more than $k$ different numbers among the lengths of its sides (the definition of bar depends on $k$). We prove that a parallelepiped can be dissected into finite number of bars iff the lengths of sides of the parallelepiped span a linear space of dimension no more t...