Joseph Emerson,1 Robert Alicki,2 and Karol Życzkowski1, 3, 4 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Wita Stwosza 57, PL 80-952 Gdańsk, Poland Institute of Physics, Jagiellonian University, ul. Reymonta 4, 31-059 Cracow, Poland Center for Thoretical Physics, Polish Academy of Sciences, Al Lot...

There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant. This solves a problem of Nandakumar.

The author presents a simple technique for teaching the Cournot model to first-year students. The approach involves demonstrating to the students that out of all rectangles with a common perimeter, the square has the greatest area. No use is made of derivatives. The same approach can be used to understand some other market forms.

spherical with an indented base and of volume less than the parent and this was not varied. Other work has shown (ref. [3]) that sphere. When they are large the dhTerence in shape between two parameters describing the bubbles do vary with bed thickness and and three-dimensinal bubbles is considerable. can be sensitive to its value when it is smal]. Further treatment of al1 the data referred to ...

We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions. This is extended variant of an extended abstract accepted to Eurocomb 2017 which contains proofs of the main statements.

The site-perimeter of a polyomino is the number of empty cells connected to the polyomino by an edge. A formula for the minimum site-perimeter with a given cell size is found. This formula is used to show the effectiveness of a simple random strategy in polyomino set achievement games.

We show that if the unit square is covered by n rectangles, then at least one must have perimeter at least 4(2m + 1)/(n + m(m + 1)), where m is the largest integer whose square is at most n. This result is exact for n of the form m(m + 1) (or m2).

We deal with the asymptotic behavior of the s-perimeter of a set E inside a domain Ω as s ↘ 0. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of E and Ω. Moreover, we construct examples of sets for which the limit does not exist.

An approach is presented for the enumeration of directed-convex polyominos that are not parallelogram polyominoes and we establish that there are ( 2n n−2 ) with a perimeter of 2n + 4. Finally using known results we prove that there are ( 2n n ) directed-convex polyominos with a perimeter of 2n + 4.

In the Heisenberg group, we prove that the boundary of sets with finite H-perimeter and having a bound on the measure theoretic normal is an H-Lipschitz graph. Then we show that if the normal is, on the boundary, the restriction of a continuous mapping, then the boundary is an H-regular surface.