Mutually Orthogonal Binary Frequency Squares

A frequency square is a matrix in which each row and column is a permutation of the same multiset of symbols. We consider only binary frequency squares of order $n$ with $n/2$ zeros and $n/2$ ones in each row and column. Two such frequency squares are orthogonal if, when superimposed, each of the 4 possible ordered pairs of entries occurs equally often. In this context we say that a set of $k$-MOFS$(n)$ is a set of $k$ binary frequency squares of order $n$ in which each pair of squares is orthogonal. A set of $k$-MOFS$(n)$ must satisfy $k\le(n-1)^2$, and any set of MOFS achieving this bound is said to be complete. For any $n$ for which there exists a Hadamard matrix of order $n$ we show that there exists at least $2^{n^2/4-O(n\log n)}$ isomorphism classes of complete sets of MOFS$(n)$. For $2<n\equiv2\pmod4$ we show that there exists a set of $17$-MOFS$(n)$ but no complete set of MOFS$(n)$. A set of $k$-maxMOFS$(n)$ is a set of $k$-MOFS$(n)$ that is not contained in any set of $(k+1)$-MOFS$(n)$. By computer enumeration, we establish that there exists a set of $k$-maxMOFS$(6)$ if and only if $k\in\{1,17\}$ or $5\le k\le 15$. We show that up to isomorphism there is a unique $1$-maxMOFS$(n)$ if $n\equiv2\pmod4$, whereas no $1$-maxMOFS$(n)$ exists for $n\equiv0\pmod4$. We also prove that there exists a set of $5$-maxMOFS$(n)$ for each order $n\equiv 2\pmod{4}$ where $n\geq 6$.

Free Vibration of the Triple-Walled Carbon Nanotubes

Natural frequencies of the triple-walled carbon nanotubes (TWCNTs) are determined both exactly and approximately. For the case of TWCNT which is simply supported at it ends closed-form solutions are obtained. It is shown that there are three series of natural frequencies corresponding to the cubic polynomial equation for natural frequency squares. For the TWCNT that has other boundary conditions approximate Bubnov-Galerkin and Petrov-Galerkin methods are applied. Simple polynomial coordinate functions are utilized. Each of these methods yields three natural frequencies corresponding to the lower and of each frequency spectra.

Frequency Squares and Affine Designs

The known methods for constructing complete sets of mutually orthogonal frequency squares all yield one of two parameter sets. We show that almost all these constructions can be derived from one basic design theory construction.

A Construction Method for Complete Sets of Mutually Orthogonal Frequency Squares

A construction is described for combining affine designs with complete sets of mutually orthogonal frequency squares to produce other complete sets.