Related papers: Equipartitions of measures in $\mathbb{R}^4$

We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over the complex numbers all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When…

In this paper, we address the problem of prescribing non-constant $Q$ and boundary $T$ curvatures on the upper hemisphere $\mathbb{S}^4_+\subset \mathbb{R}^5$, via a conformal change of the background metric. This is equivalent to solve a…

We show that the space $\mathcal{R}^{\mathrm{pRc}}(W_g^{2n})$ of metrics with positive Ricci curvature on the manifold $W^{2n}_g := \sharp^g (S^n \times S^n)$ has nontrivial rational homology if $n \not \equiv 3 \pmod 4$ and $g$ are both…

We prove several results of the following type: any $d$ measures in $\mathbb R^d$ can be partitioned simultaneously into $k$ equal parts by a convex partition (this particular result is proved independently by Pablo Sober\'on). Another…

For planar graphs, it is well known that high connectivity implies a Hamiltonian cycle and hence any 4-connected planar graph has a near-perfect matching. Nevertheless, whether 6-connected 1-planar graphs admit near-perfect matchings…

Considered is the problem of local equivalence of generic four-dimensional metrics possessing two commuting and orthogonally transitive Killing vector fields. A sufficient set of eight differential invariants is explicitly constructed,…

A torsion-free G_2 structure admitting an infinitesimal isometry is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations.…

We prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly. This solves a…

This work introduces a new class of symmetric matrix structures, called harmonic structures, which enable the generation of all possible directed transitions $(x_i, x_{i+1})$ over a set of $n$ symbols, without internal repetitions. Unlike…

Using a Mathematica code, we present a straightforward numerical analysis of the 384-dimensional solution space of signed permutation 4x4 matrices, which in sets of four provide representations of the GR(4,4) algebra, closely related to the…

We perform a complete integration of the Einstein-dilaton-antisymmetric form action describing black p-branes in arbitrary dimensions assuming the transverse space to be homogeneous and possessing spherical, toroidal or hyperbolic topology.…

The form of realistic space-time supersymmetry is fixed, by Haag-Lopuszanski-Sohnius theorem, either to the familiar form of Poincare supersymmetry or, in massless case, to that of conformal supersymmetry. We question necessity for such…

A path-integral method effective beyond the perturbation expansion approach is suggested to consider the quartic anharmonicity in different spatial dimensions. Due to an optimal representation of the partition function, the leading term has…

We computationally resolve an open problem concerning the expressibility of $4 \times 4$ full-rank matrices as Hadamard products of two rank-2 matrices. Through exhaustive search over $\mathbb{F}_2$, we identify 5,304 counterexamples among…

With no criteria of the index type, it is proved the existence of a solution for the Riemann-Hilbert problem in the fairly general setting of arbitrary Jordan domains, measurable coefficients and measurable boundary data. The theorem is…

The planar scattering amplitudes of $\mathcal{N} = 4$ super-Yang--Mills theory display symmetries and structures which underlie their relatively simple analytic properties such as having only logarithmic singularities and no poles at…

A variant of the classical optimal transportation problem is: among all joint measures with fixed marginals and which are dominated by a given density, find the optimal one. Existence and uniqueness of solutions to this variant were…

The famous $4n^2$-inequality is extended to generic complete intersection singularities: it is shown that the multiplicity of the self-intersection of a mobile linear system with a maximal singularity is higher than $4n^2\mu$, where $\mu$…

For a domain $G$ in the one-point compactification $\overline{\mathbb{R}}^n = \mathbb{R}^n \cup \{ \infty\}$ of $\mathbb{R}^n, n \ge 2$, we characterize the completeness of the modulus metric $\mu_G$ in terms of a potential-theoretic…

Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context…