Papers by Wacharin Wichiramala
We prove that each simple polygonal arc {\gamma} attains at most two pairs of support lines of gi... more We prove that each simple polygonal arc {\gamma} attains at most two pairs of support lines of given angle difference such that each pair has s1 < s2 < s3 that {\gamma}(s1) and {\gamma}(s3) are on one such line and {\gamma}(s2) is on the other line.
Proceedings of the American Mathematical Society
We prove that the only equilibrium double bubble in R 2 which is stable for fixed areas is the st... more We prove that the only equilibrium double bubble in R 2 which is stable for fixed areas is the standard double bubble. This uniqueness result also holds for small stable double bubbles in surfaces, where it is new even for perimeter-minimizing double bubbles.
Wetzel’s sector covers unit arcs
Periodica Mathematica Hungarica
Covers for Angleworms
The American Mathematical Monthly
A set in the plane is a cover for a family of planararcs if it is convex and if it contains a con... more A set in the plane is a cover for a family of planararcs if it is convex and if it contains a congruent copy of each arc of the family. For a given family of arcs, one is typically interested in the cover of least area. Although many such problems, commonly called "worm" problems, have ...
Weak Approach to Planar Soap Bubble Clusters
Missouri Journal of Mathematical Sciences
ABSTRACT The planar soap bubble problem seeks the least perimeter way to enclose and separate $m$... more ABSTRACT The planar soap bubble problem seeks the least perimeter way to enclose and separate $m$ regions of $m$ given areas. We discuss a useful approach, especially for $m\le 8$.
Sectorial Covers for Unit Arcs
Mathematics Magazine
Topological Methods in Nonlinear Analysis
Miskolc Mathematical Notes
Forty years ago Schaer and Wetzel showed that a 1 π × 1 2π √ π 2 − 4 rectangle, whose area is abo... more Forty years ago Schaer and Wetzel showed that a 1 π × 1 2π √ π 2 − 4 rectangle, whose area is about 0.122 74, is the smallest rectangle that is a cover for the family of all closed unit arcs. More recently Füredi and Wetzel showed that one corner of this rectangle can be clipped to form a pentagonal cover having area 0.11224 for this family of curves. Here we show that then the opposite corner can be clipped to form a hexagonal cover of area less than 0.11023 for this same family. This irregular hexagon is the smallest cover currently known for this family of arcs.
ScienceAsia, 2016
Optimal partitioning of a square is the search for the least-diameter way to partition a unit squ... more Optimal partitioning of a square is the search for the least-diameter way to partition a unit square into n pieces. The problem is here solved for some small n values. Although this problem has recently been approached by transforming the problem into a graphical enumeration, the algorithm had too large a computational cost for cases of n 7. In this paper, the existence of solutions in a more general sense is established and the graphical transformation method is improved by generating dual graphs of the combinatorial patterns. In particular, combinatorial patterns were generated using the triangulation of planar graphs. Theorems to eliminate some unnecessary partitions are presented and numerical optimization by convex programming is used to find the minimum diameters. Our results confirm the earlier reported cases for n = 9 and 10 and the predictions made for the case of n = 11.
Small convex covers for convex unit arcs
Chiang Mai Journal of Science, 2010
... These covers have area 0.39270, 0.34501 and 0.3214 respectively. The smallest Page 2. 186 Chi... more ... These covers have area 0.39270, 0.34501 and 0.3214 respectively. The smallest Page 2. 186 Chiang Mai J. Sci. 2010; 37(2) cover known with area 0.260437 is by Norwood andPoole [4]. One interesting aspect is that this cover is not convex. ...
Efficient cut for a subset of prescribed area
We discuss an interesting isoperimetric problem on the plane. Given a set S ‰R2 of flnite area A ... more We discuss an interesting isoperimetric problem on the plane. Given a set S ‰R2 of flnite area A and a real number 0 < a < A, we conjecture that there exists a set E such that S nE has connected components of area a and length(E) is less than or equal to the shortest length needed to enclose a
The planar soap bubble problem with equal pressure regions
The planar soap bubble problem seeks the least-perimeter way to enclose and separate regions of m... more The planar soap bubble problem seeks the least-perimeter way to enclose and separate regions of m given areas in R2. We study the possible con- flgurations for perimeter minimizing enclosures for more than three regions. For four and flve regions, we prove that a perimeter minimizing enclosure with equal pressure regions must have connected.
A smaller cover for convex unit arcs
Optimal partitioning of a square problem is the search for least-diameter way to partition a unit... more Optimal partitioning of a square problem is the search for least-diameter way to partition a unit square into n pieces. The problem has been settled for some cases. Unfortunately, the proof for each n has its own unique trick with no relation to the proof for the other cases. In this work, we introduce a new approach to this problem. We construct an algorithm to generate all of the combinatorial patterns of the effective partitions. Then, we use the numerical optimization to find the minimum diameter. The results agree with the previous works for n = 4 to 7.
Planar soap bubbles on a half plane for three and four areas with equal pressure regions
The planar soap bubble is the search for the least-perimeter way to enclose and separate m region... more The planar soap bubble is the search for the least-perimeter way to enclose and separate m regions of given areas on the plane. In this work, we study the possible configurations for the perimeter minimizing enclosures on a half plane for m≥3. For three and four regions, we prove that the perimeter minimizing enclosures with equal pressure regions and without empty chambers must have connected regions.
Planar m-bubbles with m-1 equal highest pressures
The planar soap bubble problem asks for the least-perimeter way to enclose and separate open regi... more The planar soap bubble problem asks for the least-perimeter way to enclose and separate open regions R 1 , R 2 ,⋯,R m of m given areas on the plane. In this work, we study properties for minimizing bubbles in case that the pressure of R m is lower than the equal pressures of R 1 ,R 2 ,⋯ and R m-1 . For m=4, we show that a minimizing bubble with nonnegative pressures and without empty chambers has at most one internal component of the region R 4 .
ScienceAsia
We show that the least-perimeter way to enclose and separate two regions of prescribed area outsi... more We show that the least-perimeter way to enclose and separate two regions of prescribed area outside a disc is a truncated standard double bubble.
We flnd minimal enclosures by rectangles for two and three regions of given areas. We show that e... more We flnd minimal enclosures by rectangles for two and three regions of given areas. We show that each minimizer has connected regions and has shape depending on ratio of areas.
Periodica Mathematica Hungarica, 2007
We describe the broadest three-segment unit arc in the plane, and we conclude with some conjectur... more We describe the broadest three-segment unit arc in the plane, and we conclude with some conjectures about the broadest n-segment unit arc for n > 3.
A covering theorem for families of sets in $\mbb{R}^{d}$
Journal of Combinatorics, 2010
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Papers by Wacharin Wichiramala