Magic square spectra

To study the eigenvalues of low order singular and non-singular magic squares we begin with some aspects of general square matrices. Additional properties follow for general semimagic squares (same row and column sums), with further properties for general magic squares (semimagic with same diagonal sums). Parameterizations of general magic squares for low orders are examined, including factorization of the linesum eigenvalue from the characteristic polynomial.For nth order natural magic squares with matrix elements 1,…,n² we find examples of some remarkably singular cases. All cases of the regular (or associative, or symmetric) type (antipodal pair sum of 1+n²) with n-1 zero eigenvalues have been found in the only complete sets of these squares (in fourth and fifth order). Both the Jordan form and singular value decomposition (SVD) have been useful in this study which examines examples up to 8th order.In fourth order these give examples illustrating a theorem by Mattingly that even order regular magic squares have a zero eigenvalue with odd algebraic multiplicity, m. We find 8 cases with m=3 which have a non-diagonal Jordan form. The regular group of 48 squares is completed by 40 squares with m=1, which are diagonable. A surprise finding is that the eigenvalues of 16 fourth order pandiagonal magic squares alternate between m=1, diagonable, and m=3, non-diagonable, on rotation by π/2. Two 8th order natural magic squares, one regular and the other pandiagonal, are also examined, found to have m=5, and to be diagonable.Mattingly also proved that odd order regular magic squares have a zero eigenvalue with even multiplicity, m=0,2,4,... Analyzing results for natural fifth order magic squares from exact backtracking calculations we find 652 with m=2, and four with m=4. There are also 20, 604 singular seventh order natural ultramagic (simultaneously regular and pandiagonal) squares with m=2, demonstrating that the co-existence of regularity and pandiagonality permits singularity. The singular odd order examples studied are all non-diagonable.     

Generalized latin square

LetX be a n-set and letA = [a_ij] be an xn matrix for whichaij ?X, for 1 ≤i, j ≤n. A is called a generalized Latin square onX, if the following conditions is satisfied: $ \cup _{i = 1}^n a_{ij} = X = \cup _{j = 1}^n a_{ij} $ . In this paper, we prove that every generalized Latin square has an orthogonal mate and introduce a H_v -structure on a set of generalized Latin squares. Finally, we prove that every generalized Latin square of ordern, has a transversal set.     

Stability of an orthotropic flexible square plate weakened by a square opening

The postbuckling behavior of square flexible orthotropic and isotropic plates weakened by a square opening is investigated. The load is uniformly distributed along two opposite hinged edges, the other two unloaded edges being free. The solution is obtained in finite differences by successive approximations. Calculations have been made for 1:1, 5:1, and 10:1 SVAM and steel using a Minsk-22 computer.Kazakh Polytechnic Institute, Alma-Ata. Translated from Mekhanika Polimerov, No. 3, pp. 482–488, May–June, 1971.     

Hyperinterpolation on the square

We show that hyperinterpolation at (near) minimal cubature points for the product Chebyshev measure, along with Xu compact formula for the corresponding reproducing kernel, provide a simple and powerful polynomial approximation formula in the uniform norm on the square. The Lebesgue constant of the hyperinterpolation operator grows like log² of the degree, as that of quasi-optimal interpolation sets recently proposed in the literature. Moreover, we give an accurate implementation of the hyperinterpolation formula with linear cost in the number of cubature points, and we compare it with interpolation formulas at the same set of points.     

Some unit square integrals

In this article, we prove some identities which allow us to evaluate some multiple unit square integrals. In our examples, we will give the value of some double and triple integrals. We then prove several classical integral formulas with the help of these identities and present others that seem to be new. Finally, we get double integrals for classical constants and different expressions for two Ramanujan’s integral formulas.     

Orthogonal latin square graphs

An orthogonal latin square graph (OLSG) is one in which the vertices are latin squares of the same order and on the same symbols, and two vertices are adjacent if and only if the latin squares are orthogonal. If G is an arbitrary finite graph, we say that G is realizable as an OLSG if there is an OLSG isomorphic to G. The spectrum of G [Spec(G)] is defined as the set of all integers n that there is a realization of G by latin squares of order n. The two basic theorems proved here are (1) every graph is realizable and (2) for any graph G, Spec G contains all but a finite set of integers. A number of examples are given that point to a number of wide open questions. An example of such a question is how to classify the graphs for which a given n lies in the spectrum.     

Mean square spline approximation

The approximation to a specified function on the real line by fitting a cubic in a piecewise fashion is achieved by minimizing the deviations in the mean square sense. The coefficients of the cubic are determined sequentially employing the method of dynamic programming. Employing this method a known function is approximated and the results of the computation are tabulated.     

All square chiliagonal numbers

A square chiliagonal number is a number which is simultaneously a chiliagonal number and a perfect square (just as the well-known square triangular number is both triangular and square). In this work, we determine which of the chiliagonal numbers are perfect squares and provide the indices of the corresponding chiliagonal numbers and square numbers. The study revealed that the determination of square chiliagonal numbers naturally leads to a generalized Pell equation x² ? Dy² = N with D = 1996 and N = 996², and has six fundamental solutions out of which only three yielded integer values for use as indices of chiliagonal numbers. The crossing/independent recurrence relations satisfied by each class of indices of the corresponding chiliagonal numbers and square numbers are obtained. Finally, the generating functions serve as a clothesline to hang up the indices of the corresponding chiliagonal numbers and square numbers for easy display and this was used to obtain the first few sequence of square chiliagonal numbers.     

The square lattice shuffle

We show that the operations of permuting columns and rows separately and independently mix a square matrix in constant time. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Random Latin square graphs

In this paper we introduce new models of random graphs, arising from Latin squares which include random Cayley graphs as a special case. We investigate some properties of these graphs including their clique, independence and chromatic numbers, their expansion properties as well as their connectivity and Hamiltonicity. The results obtained are compared with other models of random graphs and several similarities and differences are pointed out. For many properties our results for the general case are as strong as the known results for random Cayley graphs and sometimes improve the previously best results for the Cayley case. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Embedding an incomplete latin square in a latin square with a prescribed diagonal

We present necessary and sufficient conditions for the embedding of a given incomplete latin square R of side n,n?10 on the symbols σ₁,…,σ_t in a latin square T of side t on the symbols σ₁,…,σ_t for all t?2n+1 where the diagonal of T has been prescribed. The lower bound of 2n+1 for t is the best possible.     

Quadratic algebra of square groups

Square groups are quadratic analogues of abelian groups. Many properties of abelian groups are shown to hold for square groups. In particular, there is a symmetric monoidal tensor product of square groups generalizing the classical tensor product.     

On equilibria on the square

We construct a non-zero sum game on the square, with separately continuous payoff functions, and which has no correlated equilibrium. This answers a recent question of Nowak.     

Coefficients of symmetric square L-functions

Let λsym2f(n) be the n-th coefficient in the Dirichlet series of the symmetric square L-function associated with a holomorphic primitive cusp form f.We prove Ω± results for λsym2f(n) and evaluate the number of positive(resp.,negative) λsym2f(n) in some intervals.     

On dually flat square metrics

Square metrics arise from several classification problems in Finsler geometry. They are the rare Finsler metrics to be of excellent geometry properties. It is proved that every non-Riemannian dually flat square metric must be Minkowskian if the dimension ≥3. We also obtain a rigidity result in dually flat Matsumoto metrics.     

On a square haystack game

A haystack game is a hider-seeker zero-sum game of locating a needle in a haystack. Baston and Bostock have obtained partial solutions to this game for the case of a square haystack. This paper supplements their results, thus confirming their belief that a complete solution is difficult.The author would like to thank a referee for his helpful comments, in particular for suggesting Lemma 2.1 which has led to a more concise and transparent treatment of Section 2.