Holes or Empty Pseudo-Triangles in Planar Point Sets

Let $E(k, \ell)$ denote the smallest integer such that any set of at least $E(k, \ell)$ points in the plane, no three on a line, contains either an empty convex polygon with $k$ vertices or an empty pseudo-triangle with $\ell$ vertices. The existence of $E(k, \ell)$ for positive integers $k, \ell\geq 3$, is the consequence of a result proved by Valtr [Discrete and Computational Geometry, Vol. 37, 565--576, 2007]. In this paper, following a series of new results about the existence of empty pseudo-triangles in point sets with triangular convex hulls, we determine the exact values of $E(k, 5)$ and $E(5, \ell)$, and prove bounds on $E(k, 6)$ and $E(6, \ell)$, for $k, \ell\geq 3$. By dropping the emptiness condition, we define another related quantity $F(k, \ell)$, which is the smallest integer such that any set of at least $F(k, \ell)$ points in the plane, no three on a line, contains a convex polygon with $k$ vertices or a pseudo-triangle with $\ell$ vertices. Extending a result of Bisztriczky and T\'oth [Discrete Geometry, Marcel Dekker, 49--58, 2003], we obtain the exact values of $F(k, 5)$ and $F(k, 6)$, and obtain non-trivial bounds on $F(k, 7)$.Comment: A minor error in the proof of Theorem 2 fixed. Typos corrected. 19 pages, 11 figure

Effect of Salt Concentration on the Electrophoretic Speed of a Polyelectrolyte through a Nanopore

In a previous paper [S. Ghosal, Phys. Rev. E 74, 041901 (2006)] a hydrodynamic model for determining the electrophoretic speed of a polyelectrolyte through an axially symmetric slowly varying nanopore was presented in the limit of a vanishingly small Debye length. Here the case of a finite Debye layer thickness is considered while restricting the pore geometry to that of a cylinder of length much larger than the diameter. Further, the possibility of a uniform surface charge on the walls of the nanopore is taken into account. It is thereby shown that the calculated transit times are consistent with recent measurements in silicon nanopores.Comment: 4 pages, 2 figure

Properties of Accretion Shocks in Viscous Flows with Cooling Effects

Low angular momentum accretion flows can have standing and oscillating shock waves. We study the region of the parameter space in which multiple sonic points occur in viscous flows in presence of various cooling effects such as bremsstrahlung and Comptonization. We also quantify the parameter space in which shocks are steady or oscillating. We find that cooling induces effects opposite to heating by viscosity even in modifying the topology of the solutions, though one can never be exactly balanced by the other due to their dissimilar dependence on dynamic and thermodynamic parameters. We show that beyond a critical value of cooling, the flow ceases to contain a shock wave.Comment: 18 pages, 12 figures, Accepted for Publication in Int. J. Mod. Phys.

Disjoint Empty Convex Pentagons in Planar Point Sets

In this paper we obtain the first non-trivial lower bound on the number of disjoint empty convex pentagons in planar points sets. We show that the number of disjoint empty convex pentagons in any set of n points in the plane, no three on a line, is at least ⌊5n/47⌋. This bound can be further improved to (3n−1)/28 for infinitely many n

On the Minimum Size of a Point Set Containing a 5-Hole and a Disjoint 4-Hole

Let H(k; l), k ≤ l denote the smallest integer such that any set of H(k; l) points in the plane, no three on a line, contains an empty convex k-gon and an empty convex l-gon, which are disjoint, that is, their convex hulls do not intersect. Hosono and Urabe [JCDCG, LNCS 3742, 117–122, 2004] proved that 12 ≤ H(4, 5) ≤ 14. Very recently, using a Ramseytype result for disjoint empty convex polygons proved by Aichholzer et al. [Graphs and Combinatorics, Vol. 23, 481–507, 2007], Hosono and Urabe [Kyoto CGGT, LNCS 4535, 90–100, 2008] improve the upper bound to 13. In this paper, with the help of the same Ramsey-type result, we prove that H(4; 5) = 12

Disjoint Empty Convex Pentagons in Planar Point Sets

Harborth [{\it Elemente der Mathematik}, Vol. 33 (5), 116--118, 1978] proved that every set of 10 points in the plane, no three on a line, contains an empty convex pentagon. From this it follows that the number of disjoint empty convex pentagons in any set of $n$ points in the plane is least $\lfloor\frac{n}{10}\rfloor$. In this paper we prove that every set of 19 points in the plane, no three on a line, contains two disjoint empty convex pentagons. We also show that any set of $2m+9$ points in the plane, where $m$ is a positive integer, can be subdivided into three disjoint convex regions, two of which contains $m$ points each, and another contains a set of 9 points containing an empty convex pentagon. Combining these two results, we obtain non-trivial lower bounds on the number of disjoint empty convex pentagons in planar points sets. We show that the number of disjoint empty convex pentagons in any set of $n$ points in the plane, no three on a line, is at least $\lfloor\frac{5n}{47}\rfloor$. This bound has been further improved to $\frac{3n-1}{28}$ for infinitely many $n$.Comment: 23 pages, 28 figure

INHIBITION OF CALCIUM OXALATE CRYSTALLIZATION BY AN EXTRACT OF OCIMUM BASILICUM SEEDS: AN IN-VITRO STUDY

Objective: Ocimum basilicum (OB) has been used to treat diverse illnesses which include urinary stone disorder for a reason that historical time in India. We investigated OB seeds for antiurolithic activity. Methods: Calcium oxalate crystallization becomes triggered by the addition of 0.01 M sodium oxalate answers in normal human urine and nucleation was done. Results: OB seeds were discovered to be robust and promising antiurolithiatic agents which are in accordance with its use in traditional medication. Conclusion: An extract of the traditional herb OB has super inhibitory activity on crystalluria and therefore might be useful in dissolving urinary stone; however, in addition, a study in animal fashions of urolithiasis is needed to assess its capability antiurolithiatic interest

Improving the process performance of magnetic abrasive finishing of SS304 material using multi-objective artificial bee colony algorithm

Magnetic abrasive finishing is a super finishing process in which the magnetic field is applied in the finishing area and the material is removed from the workpiece by magnetic abrasive particles in the form of microchips. The performance of this process is decided by its two important quality characteristics, material removal rate and surface roughness. Significant process variables affecting these two characteristics are rotational speed of tool, working gap, weight of abrasive, and feed rate. However, material removal rate and surface roughness being conflicting in nature, a compromise has to be made between these two objective to improve the overall performance of the process. Hence, a multi-objective optimization using an artificial bee colony algorithm coupled with response surface methodology for mathematical modeling is attempted in this work. The set of Pareto-optimal solutions obtained by multi-objective optimization offers a ready reference to process planners to decide appropriate process parameters for a particular scenario