91Ó°ÊÓ

Give a Latin square of order 6 .

Let \(A=\\{a\\}, G=\left\\{g_{0}\right\\}, \rightarrow=\left\\{g_{0} \rightarrow a a, g_{0} \rightarrow g_{0} a\right\\}, G=\left(A, G, \rightarrow, g_{0}\right) .\) Find \(L(G) .\)

Is \(S=\left\\{x^{2}+1, y^{2}+1\right\\}\) solvable in \(R ?\) In some extension of \(\mathbb{R}\) ?

Prove that if Hadamard matrices of orders \(m\) and \(n\) exist, then there is also a Hadamard matrix of order \(m n\).

(1) Find an isomorphism from \((\mathcal{P}(M), \cap)\) onto \((\mathcal{P}(N), \cap)\) if \(|M|=|N|\). (ii) For \(M \subseteq N\) find an embedding of \((\mathcal{P}(M), \cap\) ) into \((\mathcal{P}(N), \cap)\).

Let \(n=4\) and \(\omega=i \in \mathbb{C}\). Compute the Fourier transform of \((1,1,1,1)\), \((1,0,0,0)\), and \((1, i, i,-i)\).

Are the sisters of daughters in \(32.3\) considered to be "the same" as the daughters of sisters?

Compute \(e^{\mathbf{A}}\) for \(\mathbf{A}=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right), \mathbf{A}=\left(\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right), \mathbf{A}=\left(\begin{array}{cc}0 & \omega \\ -\omega & 0\end{array}\right) .\)

Give a term over some \(\Omega=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)\) of type \((2,1,0)\) which is not a term over the type \(\Omega^{\prime}=\left(\omega_{1}\right)\).

Let \(A=\\{a\\}, G=\left\\{g_{0}\right\\}, \rightarrow=\left[g_{0} \rightarrow a g_{0}, g_{0} \rightarrow g_{0} a\right\\}, G=\left(A, G, \rightarrow, g_{0}\right) .\) Find \(L(G) .\)