On the number of symbols that forces a transversal

Akbari and Alipour conjectured that any Latin array of order $n$ with at least $n^2/2$ symbols contains a transversal. We confirm this conjecture for large $n$, and moreover, we show that $n^{399/200}$ symbols suffice.Comment: 6 pages, 1 figur

Latin transversals of rectangular arrays

Let m and n be integers, $2 \leq m \leq n$. An m by n array consists of mn cells, arranged in m rows and n columns, and each cell contains exactly one symbol. A transversal of an array consists of m cells, one from each row and no two from the same column. A latin transversal is a transversal in which no symbol appears more than once. We will establish a sufficient condition that a 3 by n array has a latin transversal.Comment: Theorem 4 has been added, which provides a lower bound on L(m,n

Parity of transversals of Latin squares

We introduce a notion of parity for transversals, and use it to show that in Latin squares of order $2 \bmod 4$, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving $E_1,\dots, E_n$, where $E_i$ is the number of diagonals of a given Latin square that contain exactly $i$ different symbols. Let $A(i\mid j)$ denote the matrix obtained by deleting row $i$ and column $j$ from a parent matrix $A$. Define $t_{ij}$ to be the number of transversals in $L(i\mid j)$, for some fixed Latin square $L$. We show that $t_{ab}\equiv t_{cd}\bmod2$ for all $a,b,c,d$ and $L$. Also, if $L$ has odd order then the number of transversals of $L$ equals $t_{ab}$ mod 2. We conjecture that $t_{ac} + t_{bc} + t_{ad} + t_{bd} \equiv 0 \bmod 4$ for all $a,b,c,d$. In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a $k$-regular bipartite graph on $2n$ vertices is divisible by $4$ when $n$ is odd and $k\equiv0\bmod 4$. We also show that $${\rm per}\, A(a \mid c)+{\rm per}\, A(b \mid c)+{\rm per}\, A(a \mid d)+{\rm per}\, A(b \mid d) \equiv 0 \bmod 4$$ for all $a,b,c,d$, when $A$ is an integer matrix of odd order with all row and columns sums equal to $k\equiv2\bmod4$

Racing car chassis

Cílem této bakalářské práce je analýza současných konceptů podvozků závodních okruhových aut. V první části práce je zpracován historický vývoj, charakteristika kol a pneumatik s reprezentací dobře známých produktů. V druhé části je popsán systém odpružení, pružné média a tlumící členy. Systémy odpružení je zde rozdělen na nezávisle a polozávislé zavěšení kol a odpružení pevných náprav. Následující oddíl této práce je zaměřený na standardní kontrolní systémy, jako jsou ABS, ESC a TSC. Závěr přináší rychlé shrnutí této problematiky.The aim of this bachelor thesis is to analyse contemporary concepts of circuit race car chassis. In the first part of the thesis, the historical evolution is described and then wheels and tires characteristic within some well-known brand products are represented. The second important part includes the suspension systems, springing medium and damping members. The suspension systems are further divided to independent and semi-independent solutions and rigid axle suspensions. The end of this thesis deals with the standard braking control systems, such as ABS, ESC and TCS. The conclusion brings the quick summary of this subject.