A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a containing line in each direction. We show that, when $N$ any square-free integer, the size of smallest $(\mathbb{Z}/N\mathbb{Z})^n$ at least $C_{n,\epsilon} N^{n - \epsilon}$ for $\epsilon$ -- resolving special case conjecture Hickman and Wright. Previously, such bounds were only known prime $N$. also that general can be reduced to lower...

We consider a chemotaxis system with singular sensitivity and logistic-type source: \begin{document}$ u_t = \Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+ru-\mu u^k $\end{document}, id="M2">\begin{document}$ v_t \epsilon\Delta v-v+u $\end{document} in smooth bounded domain id="...

Epsilon-toxin is produced by Clostridium perfringens types B and D and is responsible for a rapidly fatal enterotoxemia in animals, which is characterized by edema in several organs due to an increase in blood vessel permeability. The Madin-Darby canine kidney (MDCK) cell line has been found to be susceptible to epsilon-toxin (D. W. Payne, E. D. Williamson, H. Havard, N. Modi, and J. Brown, FEM...

Let P be a set of n points in d-dimensional Euclidean space. A (k,eps)-kernel is a subset of P that, for every direction, epsilon-approximates the directional width of P, when k “outliers’’ can be ignored in that direction. We prove that small (k,eps)-kernels exist and describe efficient algorithms for computing them. Such kernels are instrumental in solving shape-fitting problems with k outlie...

This paper develops a two-gene, single fitness peak model for determining the equilibrium distribution of genotypes in a unicellular population which is capable of genetic damage repair. The first gene, denoted by sigma(via), yields a viable organism with first-order growth rate constant k>1 if it is equal to some target "master" sequence sigma(via,0). The second gene, denoted by sigma(rep), yi...

Spectrally resolved infrared stimulated vibrational echo data were obtained for sperm whale carbonmonoxymyoglobin (MbCO) at 300 K. The measured dephasing dynamics of the CO ligand are in agreement with dephasing dynamics calculated with molecular dynamics (MD) simulations for MbCO with the residue histidine-64 (His64) having its imidazole epsilon nitrogen protonated (N(epsilon)-H). The two conf...

In the present study, flow characteristics of butterfly valves with different sizes DN 80 (nominal diameter: 76.2 mm), 262 254 400 406 mm) were numerically investigated under valve opening percentages. Representative two-equation turbulence models k-epsilon model Launder and Sharma, k-omega Wilcox, SST Menter selected. Flow examined to determine effects. It was determined that increasing effect...

We prove a surprising higher regularity for solutions to the nonlinear elliptic autonomous Beltrami equation in planar domain \(\Omega\), \[f_\overline{z} = \mathcal{A}(f_z)\ a.e.\ z\in\Omega,\] when \(\mathcal{A}\) is linear at \(\infty\). Namely \(W^{1,1}_{\operatorname{loc}}(\Omega)\) are \(W^{2,2+\epsilon}_{\operatorname{loc}}(\Omega)\). Here \(\epsilon>0\) depends explicitly on ellipticity...