Related papers: Some remarks on the midrange crossing constant

For the quadratic Lagrange interpolation function, an algorithm is proposed to provide explicit and verified bound for the interpolation error constant that appears in the interpolation error estimation. The upper bound for the…

We prove that a suitably adjusted version of Peter Jones' formula for interpolation by bounded holomorphic functions gives a sharp upper bound for what is known as the constant of interpolation. We show how this leads to precise and…

The crossing number of a graph $G$ denotes the minimum number of crossings in any planar drawing of $G$. In this short note, we confirm a long-standing conjecture posed by Pach, Spencer, and T\'oth over 25 years ago, establishing an optimal…

Baader, J\"org, and Parlier recently established an upper bound for the crossing number of curve systems of size $m\asymp g^{1+\alpha}$ on a genus $g$ surface, obtaining a leading coefficient of $9/4=2.25$. Their construction relies on…

The classical Crossing Lemma by Ajtai et al.~and Leighton from 1982 gave an important lower bound of $c \frac{m^3}{n^2}$ for the number of crossings in any drawing of a given graph of $n$ vertices and $m$ edges. The original value was $c=…

For all $s \geq 1$ and $N \geq 1$ there exist sequences $(z_1,\ldots,z_N)$ in $[0,1]^s$ such that the star-discrepancy of these points can be bounded by $$D_N^*(z_1,\ldots,z_N) \leq c \frac{\sqrt{s}}{\sqrt{N}}.$$ The best known value for…

We show an upper bound of $\frac{ \sin\left(\frac{3\pi}{10}\right) }{ \sin\left(\frac{2\pi}{5}\right)-\sin\left(\frac{3\pi}{10}\right) } <5.70$ on the spanning ratio of $\Theta_5$-graphs, improving on the previous best known upper bound of…

We improve the constant $\frac{\pi}{2}$ in $L^1$-Poincar\'e inequality on Hamming cube. For Gaussian space the sharp constant in $L^1$ inequality is known, and it is $\sqrt{\frac{\pi}{2}}$. For Hamming cube the sharp constant is not known,…

Determining whether there exists a graph such that its crossing number and pair crossing number are distinct is an important open problem in geometric graph theory. We show that $\textit{cr}(G)=O(\mathop{\mathrm{pcr}}(G)^{3/2})$ for every…

Concepts and results of determinations of the strong coupling constant in hadron collisions are discussed. A recent alpha_s result from the inclusive jet cross section in pp-bar collisions at sqrt(s)=1.96 TeV is presented which is based on…

We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds, the outgoing resolvent satisfies $\|\chi R(k) \chi\|_{L^2\to L^2}\leq C{k}^{-1}$ for ${k}>1$, but the constant $C$…

The momentum-differential invariant cross sections of ${\pi^{0}}$ and $\eta$ mesons are reported for pp collisions at $\sqrt{s}$ = 13 TeV at midrapidity ($|y|<0.8$). The measurement is performed in a broad transverse-momentum range of…

The cross section of the $p(e,e'\pi^+)n$ reaction has been measured for five kinematic settings at an invariant mass of $W = 1094$ MeV and for a four-momentum transfer of $Q^2 = 0.078$ (GeV/$c$)$^2$. The measurement has been performed at…

We introduce remarkable upper bounds for the interpolation error constants on triangles, which are sharp and given by simple formulas. These constants are crucial in analyzing interpolation errors, particularly those associated with the…

We study the correlation of edges, vectors or elements to be in a randomly chosen spanning tree or a basis, respectively. Here we follow the guideline of Huh and Wang and introduce as a measure an invariant that is called the correlation…

Let $P$ be a set of points in general position in the plane. Join all pairs of points in $P$ with straight line segments. The number of segment-crossings in such a drawing, denoted by $\crg(P)$, is the \emph{rectilinear crossing number} of…

Motivated by an approximation problem from mathematical finance, we analyse the stability of the boundary crossing probability for the multivariate Brownian motion process, with respect to small changes of the boundary. Under broad…

The inclusive cross section for top quark pair production is measured in proton-proton collisions at sqrt(s) = 7 and 8 TeV, corresponding to 5.0 and 19.7 invers-femtobarns, respectively, with the CMS experiment at the LHC. The cross…

Within the conventional QCD sum rules, we calculate the $\pi NN$ coupling constant, $g_{\pi N}$, beyond the chiral limit using two-point correlation function with a pion. We consider the Dirac structure, $i\gamma_5$, at $m_\pi^2$ order,…

We formulate and discuss the affine-invariant matrix midrange problem on the cone of $n\times n$ positive definite Hermitian matrices $\mathbb{P}(n)$, which is based on the Thompson metric. A particular computationally efficient midpoint of…