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 $D (N) ≪ q^{N} {(l o g N)}^{7} / N$ .

Orthogonal series regression estimation under long-range dependent errors

Waldemar Popiński (2001)

Applicationes Mathematicae

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This paper is concerned with general conditions for convergence rates of nonparametric orthogonal series estimators of the regression function. The estimators are obtained by the least squares method on the basis of an observation sample $Y_{i} = f (X_{i}) + η_{i}$ , i=1,...,n, where $X_{i} \in A \subset ℝ^{d}$ are independently chosen from a distribution with density ϱ ∈ L¹(A) and $η_{i}$ are zero mean stationary errors with long-range dependence. Convergence rates of the error $n^{- 1} \sum_{i = 1}^{n} (f (X_{i}) - f ̂_{N} (X_{i})) ²$ for the estimator $f ̂_{N} (x) = \sum_{k = 1}^{N} c ̂_{k} e_{k} (x)$ , constructed using an orthonormal system...

Strongly elliptic linear operators without coercive quadratic forms I. Constant coefficient operators and forms

Gregory C. Verchota (2014)

Journal of the European Mathematical Society

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A family of linear homogeneous 4th order elliptic differential operators $L$ with real constant coefficients, and bounded nonsmooth convex domains $Ω$ are constructed in $ℝ^{6}$ so that the $L$ have no constant coefficient coercive integro-differential quadratic forms over the Sobolev spaces $W^{2, 2} (Ω)$ .