Systemic magnesium homeostasis in mammals is primarily governed by the activities of the TRPM6 and TRPM7 cation channels, which mediate both uptake by the intestinal epithelial cells and reabsorption by the distal convoluted tubule cells in the kidney. In the nematode, C. elegans, intestinal magnesium uptake is dependent on the activities of the TRPM channel proteins, GON-2 and GTL-1. In this p...

We study a generalization of the von Koch Curve, which has two parameters, an integer n and a real number c on the interval (0, 1). This von Koch type curve is constructed as the limit of a recursive process that starts with a regular n-gon (or line segment) and repeatedly replaces the middle c portion of an interval by the n− 1 other sides of a regular n-gon placed contiguous to the interval. ...

Trigonometric and trigonometric-algebraic entropies are introduced. Regularity increases the entropy and the maximal entropy is shown to result when a regular n-gon is inscribed in a circle. A regular n-gon circumscribing a circle gives the largest entropy reduction, or the smallest change in entropy from the state of maximum entropy which occurs in the asymptotic infinite n limit. EOM are show...

We present a solution for the largest regular m-gon contained in a regular n-gon. We find that the answer depends critically on the coprimality of m and n. We show that the optimal polygons are concentric if and only if gcd(m,n) > 1. Our principal result is a complete solution for the case where m and n share a common divisor. For the case of coprime m and n, we present partial results and a co...

The present paper expands details and confirms the transition mechanism between two subsequent polygonal patterns of the hollow-core vortex. Using power spectral analysis, we confirm in this work that the transition from any N-gon to (N+1)-gon pattern observed within a hollow-core vortex of shallow rotating flows occurs in two steps. The regime was quasi-periodic before the frequencies lock (sy...

In 1978 Erdős asked if every sufficiently large set of points in general position in the plane contains the vertices of a convex k-gon, with the additional property that no other point of the set lies in its interior. Shortly after, Horton provided a construction—which is now called the Horton set—with no such 7-gon. In this paper we show that the Horton set of n points can be realized with int...

We show that every heptagon is a section of a 3-polytope with 6 vertices. This implies that every n-gon with n > 7 can be obtained as a section of a (2 + ⌊ n 7 ⌋ )dimensional polytope with at most ⌈ 6n 7 ⌉ vertices; and provides a geometric proof of the fact that every nonnegative n ×m matrix of rank 3 has nonnegative rank not larger than ⌈ 6min(n,m) 7 ⌉ . This result has been independently pro...

Given any convex n-gon, in this article, we: (i) prove that its vertices can form at most n2/2 + Θ(n log n) isosceles trianges with two sides of unit length and show that this bound is optimal in the first order, (ii) conjecture that its vertices can form at most 3n2/4 + o(n2) isosceles triangles and prove this conjecture for a special group of convex n-gons, (iii) prove that its vertices can f...

We present a short proof of Füredi’s theorem [F] stated in the title.