Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

Diane M. Donovan; Mike Grannell; Emine Ş. Yazıcı

Displaying similar documents to “Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results”

Equivalence classes of Latin squares and nets in ${ℂ P}^{2}$

Corey Dunn, Matthew Miller, Max Wakefield, Sebastian Zwicknagl (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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The fundamental combinatorial structure of a net in $ℂ P^{2}$ is its associated set of mutually orthogonal Latin squares. We define equivalence classes of sets of orthogonal Latin squares by label equivalences of the lines of the corresponding net in $ℂ P^{2}$ . Then we count these equivalence classes for small cases. Finally we prove that the realization spaces of these classes in $ℂ P^{2}$ are empty to show some non-existence results for 4-nets in $ℂ P^{2}$ .

On Sums of Squares in $Q^{\frac{1}{2}} (X)$ etc

W. J. Ellison (1970-1971)

Séminaire de théorie des nombres de Bordeaux

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The number of squares and $B_{h} [g]$ sets

Ben Green (2001)

Acta Arithmetica

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Exceptional sets in Waring's problem: two squares and s biquadrates

Lilu Zhao (2014)

Acta Arithmetica

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Let $R_{s} (n)$ denote the number of representations of the positive number n as the sum of two squares and s biquadrates. When $s = 3$ or 4, it is established that the anticipated asymptotic formula for $R_{s} (n)$ holds for all $n \leq X$ with at most $O (X^{(9 - 2 s) / 8 + ε})$ exceptions.

Solvability classes for core problems in matrix total least squares minimization

Iveta Hnětynková, Martin Plešinger, Jana Žáková (2019)

Applications of Mathematics

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Linear matrix approximation problems $A X \approx B$ are often solved by the total least squares minimization (TLS). Unfortunately, the TLS solution may not exist in general. The so-called core problem theory brought an insight into this effect. Moreover, it simplified the solvability analysis if $B$ is of column rank one by extracting a core problem having always a unique TLS solution. However, if the rank of $B$ is larger, the core problem may stay unsolvable in the TLS sense, as shown for the first...

A hybrid method for nonlinear least squares that uses quasi-Newton updates applied to an approximation of the Jacobian matrix

Lukšan, Ladislav, Vlček, Jan

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In this contribution, we propose a new hybrid method for minimization of nonlinear least squares. This method is based on quasi-Newton updates, applied to an approximation $A$ of the Jacobian matrix $J$ , such that $A^{T} f = J^{T} f$ . This property allows us to solve a linear least squares problem, minimizing $∥ A d + f ∥$ instead of solving the normal equation $A^{T} A d + J^{T} f = 0$ , where $d \in R^{n}$ is the required direction vector. Computational experiments confirm the efficiency of the new method.

On orthogonal Latin $p$ -dimensional cubes

Marián Trenkler (2005)

Czechoslovak Mathematical Journal

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We give a construction of $p$ orthogonal Latin $p$ -dimensional cubes (or Latin hypercubes) of order $n$ for every natural number $n \neq 2, 6$ and $p \geq 2$ . Our result generalizes the well known result about orthogonal Latin squares published in 1960 by R. C. Bose, S. S. Shikhande and E. T. Parker.

A discrepancy principle for Tikhonov regularization with approximately specified data

M. Thamban Nair, Eberhard Schock (1998)

Annales Polonici Mathematici

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Many discrepancy principles are known for choosing the parameter α in the regularized operator equation $(T * T + α I) x_{α}^{δ} = T * y^{δ}$ , $| y - y^{δ} | \leq δ$ , in order to approximate the minimal norm least-squares solution of the operator equation Tx = y. We consider a class of discrepancy principles for choosing the regularization parameter when T*T and $T * y^{δ}$ are approximated by Aₙ and $z ₙ^{δ}$ respectively with Aₙ not necessarily self-adjoint. This procedure generalizes the work of Engl and Neubauer (1985), and particular cases of the results...

Sum of squares and the Łojasiewicz exponent at infinity

Krzysztof Kurdyka, Beata Osińska-Ulrych, Grzegorz Skalski, Stanisław Spodzieja (2014)

Annales Polonici Mathematici

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Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h ₁ (x) = \dots = h_{r} (x) = 0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then ${f |}_{V}$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h (x) = \sum_{i = 1}^{r} h ²_{i} (x) σ_{i} (x)$ , where $σ_{i}$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) >...

Overlapping latin subsquares and full products

Joshua M. Browning, Petr Vojtěchovský, Ian M. Wanless (2010)

Commentationes Mathematicae Universitatis Carolinae

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We derive necessary and sufficient conditions for there to exist a latin square of order $n$ containing two subsquares of order $a$ and $b$ that intersect in a subsquare of order $c$ . We also solve the case of two disjoint subsquares. We use these results to show that: (a) A latin square of order $n$ cannot have more than $\frac{n}{m} (\binom{n}{h}) / (\binom{m}{h})$ subsquares of order $m$ , where $h = ⌈ (m + 1) / 2 ⌉$ . Indeed, the number of subsquares of order $m$ is bounded by a polynomial of degree at most $\sqrt{2 m} + 2$ in $n$ . (b) For all $n \geq 5$ there exists a loop of order...

On sets of polynomials whose difference set contains no squares

Thái Hoàng Lê, Yu-Ru Liu (2013)

Acta Arithmetica

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Let $_{q} [t]$ be the polynomial ring over the finite field $_{q}$ , and let $_{N}$ be the subset of $_{q} [t]$ containing all polynomials of degree strictly less than N. Define D(N) to be the maximal cardinality of a set $A \subseteq_{N}$ for which A-A contains no squares of polynomials. By combining the polynomial Hardy-Littlewood circle method with the density increment technology developed by Pintz, Steiger and Szemerédi, we prove that $D (N) ≪ q^{N} {(l o g N)}^{7} / N$ .