Completing partial latin squares with prescribed diagonals

AbstractThis paper deals with completion of partial latin squares L=(lij) of order n with k cyclically generated diagonals (li+t,j+t=lij+t if lij is not empty; with calculations modulo n). There is special emphasis on cyclic completion. Here, we present results for k=2,…,7 and odd n⩽21, and we describe the computational method used (hill climbing). Noncyclic completion is investigated in the cases k=2,3 or 4 and n⩽21

The Intricacy of Avoiding Arrays

Abstract. Let A be any n × n array on the symbols [n], with at most m symbols in each cell. An n × n Latin square L avoids A if no entry in L is present in the corresponding cell in A. If m = 1 and A is split into two arrays B and C in a special way, there are Latin squares L B and L C avoiding B and C respectively. In other words, the intricacy of avoiding arrays is 2, the number of arrays into which A has to be split. We also investigate the case m > 1, derive some upper and lower bounds, and propose a conjecture on the exact value of the intricacy for the general case

Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups

An $r \times s$ partial Latin rectangle $(l_{ij})$ is an $r \times s$ matrix containing elements of $\{1,2,\ldots,n\} \cup \{\cdot\}$ such that each row and each column contain at most one copy of any symbol in $\{1,2,\ldots,n\}$. An entry is a triple $(i,j,l_{ij})$ with $l_{ij} \neq \cdot$. Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively. We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of $m$-entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) $r=s=n$, i.e., partial Latin squares, (b) $r=2$ and $s=n$, and (c) $r=2$ and $s \neq n$

Partially Filled Latin Squares

In this thesis, we analyze various types of Latin squares, their solvability and embeddings. We examine the results by M. Hall, P. Hall, Ryser and Evans first, and apply our understandings to develop an algorithm that the determines the minimum possible embedding of an unsolvable Latin square. We also study Latin squares with missing diagonals in detail

How Ordinary Elimination Became Gaussian Elimination

Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method that Euler did not recommend, that Legendre called "ordinary," and that Gauss called "common" - is now named after Gauss: "Gaussian" elimination. Gauss's name became associated with elimination through the adoption, by professional computers, of a specialized notation that Gauss devised for his own least squares calculations. The notation allowed elimination to be viewed as a sequence of arithmetic operations that were repeatedly optimized for hand computing and eventually were described by matrices.Comment: 56 pages, 21 figures, 1 tabl

Nelikulmion modulin numeerinen laskenta

The module of a quadrilateral is a positive real number which divides quadrilaterals into conformal equivalence classes. This is an introductory text to the module of a quadrilateral with some historical background and some numerical aspects. This work discusses the following topics: 1. Preliminaries 2. The module of a quadrilateral 3. The Schwarz-Christoffel Mapping 4. Symmetry properties of the module 5. Computational results 6. Other numerical methods Appendices include: Numerical evaluation of the elliptic integrals of the first kind. Matlab programs and scripts and possible topics for future research. Numerical results section covers additive quadrilaterals and the module of a quadrilateral under the movement of one of its vertex.Nelikulmion moduli on positiivinen reaaliluku, joka jakaa nelikulmiot konformisiin ekvivalenssi luokkiin. Tämä on johdanto teksti nelikulmion moduliin ja sen numeeriseen laskentaan. Lisäksi työssä on näiden alojen historiaa. Työssä käsitellään mm. seuraavia asioita: 1. Esitiedot 2. Nelikulmion modulin määritelmä 3. Schwarz-Christoffel kuvaus 4. Nelikulmion modulin symmetriaominaisuuksia 5. Laskennallisia tuloksia 6. Muita numeerisia menetelmiä Liitteet sisältävät: Elliptisten, ensimmäisen luokan, integraalien numeerinen laskeminen. Matlab ohjelmia, joita on käytetty työssä ja ehdotuksia tutkimuskohteiksi. Laskennallisissa tuloksissa osiossa tutkitaan summautuvia nelikulmioita ja nelikulmion modulia. Lisäksi tutkitaan miten nelikulmion moduli muuttuu kun yksi sen kärkipiste liikkuu