The number of islands on a rectangular sea

Host Institution:

La Trobe University, AGR Room.

Title of Seminar:

The number of islands on a rectangular sea

Speaker's Name:

Csaba Szabό

Speaker's Institution:

Eőtvős Loránd University, Budapest

Time and Date:

2:00pm Friday 13 June 2008

Seminar Abstract:

The talk is intended for a general mathematical audience. Let a rectangle of size m × n be divided into mn many unit squares.

For every square of this rectangular grid a positive real number is given, its height. We think of this grid as an island in the sea. The height is constant on each square. Now, a rectangle R on our grid is called an island if the height of each cell of R is greater than the heights of the squares adjacent to R. That is if there is a possible water level such that the rectangle is an island of the sea in the usual sense. We look for the maximum number of rectangular islands.

There are other examples requiring only m × n cells; for example, aij may mean a colour on a gray-scale (before we convert the picture to black and white) or transparency (against X-rays). For higher dimensions let us imagine that, for example a three dimensional table consists of cube-shaped unit cells, and the whole table is subject to some homogeneous (physical, chemical, etc.) effect E. Each cell (h, I, j) has a threshold value A(h, I, j), and this cell disappears when E > A(h, I, j). In this model, R is a brick island according to our definition iff it is a “real island” in the obvious sense at some value of E. A possible example is the case when E resp. A(h, I, j) means temperature resp. melting temperature.

The primary motivation of the research comes from coding theory. The number of islands on the 1 × n rectangle gives the lexicographical lengths of binary instantaneous codes. The talk is based on works of Gábor Czédli (University of Szeged, Szeged) and Gabriella Pluhár (Eőtvős Loránd University, Budapest).

Seminar Convenor:

This email address is being protected from spambots. You need JavaScript enabled to view it.

AGR IT support:

This email address is being protected from spambots. You need JavaScript enabled to view it.