Multi-latin squares

A multi-latin square of order $n$ and index $k$ is an $n\times n$ array of multisets, each of cardinality $k$, such that each symbol from a fixed set of size $n$ occurs $k$ times in each row and $k$ times in each column. A multi-latin square of index $k$ is also referred to as a $k$-latin square. A $1$-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in $k$-latin square of order $m$ embeds in a $k$-latin square of order $n$, for each $n\geq 2m$, thus generalizing Evans' Theorem. Exploiting this result, we show that there exist non-separable $k$-latin squares of order $n$ for each $n\geq k+2$. We also show that for each $n\geq 1$, there exists some finite value $g(n)$ such that for all $k\geq g(n)$, every $k$-latin square of order $n$ is separable. We discuss the connection between $k$-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and $k$-latin trades. We also enumerate and classify $k$-latin squares of small orders.Comment: Final version as sent to journa

Generalized Multi-latin Squares

The research explores properties of generalized multi-latin squares and proposes ways to construct them. Much like a Sudoku puzzle, generalized multi-latin squares have parameters restricting the symbols in an array. A (n, t, m, p, q)-generalized multi-latin square is an array consisting of n rows and n columns, where each cell is filled with m symbols from a collection consisting of t different symbols, any symbol appears in each row and in each column p times, and any pair of different symbols occur together q times. Understanding trivial examples, the properties, and the math behind the problem reveals multiple examples and a systematic way to build generalized multi-latin squares.https://ecommons.udayton.edu/stander_posters/1280/thumbnail.jp

How sensitive are Latin American exports to Chinese competition in the U.S. market ?

This paper estimates the elasticity of substitution of U.S. imports using detailed trade data over the 1990-2003 period. The authors use a two-stage least squares framework in order to identify the elasticity parameter of interest. The authors use the elasticity estimates to assess the extent to which Latin American and Chinese goods compete in the U.S. market by providing forecasts of how alternative policy scenarios may affect exports to the United States. The analysis considers the following scenarios: (i) currency revaluation in China; (ii) elimination of U.S. tariffs on Latin American exports under a hemispheric free trade agreement; and (iii) the elimination of quotas on apparel and textile exports under the Multi-Fiber Agreement. The findings show that a 20-percent appreciation of the renminbi reduces Chinese exports to the United States by a fifth, although since other regions increase sales to that market (0.5 percent for Latin America), U.S. imports decline by only 1.7 percent. Hemispheric free trade would increase Latin America's exports to the United States by around 3 percent. The removal of the quotas would lead to a sharp increase in Chinese sales to the United States (40 percent), but Latin America would see its share of the U.S. market decline by around 2 percent (2.5 percentage points). China's gains would come mainly at the expense of other regions of the world.Economic Theory&Research,Free Trade,Markets and Market Access,Trade Policy,Debt Markets

K-Space at TRECVID 2008

In this paper we describe K-Space’s participation in TRECVid 2008 in the interactive search task. For 2008 the K-Space group performed one of the largest interactive video information retrieval experiments conducted in a laboratory setting. We had three institutions participating in a multi-site multi-system experiment. In total 36 users participated, 12 each from Dublin City University (DCU, Ireland), University of Glasgow (GU, Scotland) and Centrum Wiskunde and Informatica (CWI, the Netherlands). Three user interfaces were developed, two from DCU which were also used in 2007 as well as an interface from GU. All interfaces leveraged the same search service. Using a latin squares arrangement, each user conducted 12 topics, leading in total to 6 runs per site, 18 in total. We officially submitted for evaluation 3 of these runs to NIST with an additional expert run using a 4th system. Our submitted runs performed around the median. In this paper we will present an overview of the search system utilized, the experimental setup and a preliminary analysis of our results

K-Space at TRECVid 2008

In this paper we describe K-Space’s participation in TRECVid 2008 in the interactive search task. For 2008 the K-Space group performed one of the largest interactive video information retrieval experiments conducted in a laboratory setting. We had three institutions participating in a multi-site multi-system experiment. In total 36 users participated, 12 each from Dublin City University (DCU, Ireland), University of Glasgow (GU, Scotland) and Centrum Wiskunde & Informatica (CWI, the Netherlands). Three user interfaces were developed, two from DCU which were also used in 2007 as well as an interface from GU. All interfaces leveraged the same search service. Using a latin squares arrangement, each user conducted 12 topics, leading in total to 6 runs per site, 18 in total. We officially submitted for evaluation 3 of these runs to NIST with an additional expert run using a 4th system. Our submitted runs performed around the median. In this paper we will present an overview of the search system utilized, the experimental setup and a preliminary analysis of our results

On the completability of incomplete orthogonal Latin rectangles

We address the problem of completability for 2-row orthogonal Latin rectangles (OLR2). Our approach is to identify all pairs of incomplete 2-row Latin rectangles that are not com- pletable to an OLR2 and are minimal with respect to this property; i.e., we characterize all circuits of the independence system associated with OLR2. Since there can be no poly- time algorithm generating the clutter of circuits of an arbitrary independence system, our work adds to the few independence systems for which that clutter is fully described. The result has a direct polyhedral implication; it gives rise to inequalities that are valid for the polytope associated with orthogonal Latin squares and thus planar multi-dimensional assign- ment. A complexity result is also at hand: completing a set of (n - 1) incomplete MOLR2 is NP-complete

Urban squares morphologies, contributes of a multidimensional analysis

The word Square and the Latin – platea – derived terms (piazza, plaza, praça, piaţă) are used to identify a public space of an exceptional character that is morphologically distinct in the urban morphology. The study of urban morphology seeks to understand the spatial structure and character of the city by identifying the patterns of its elements and the process of its development. The characterizing traits of the urban square are diverse and their origin twofold: global properties, referred to its relationships within the whole grid, and local properties, depending on the intrinsic morphologic features of its space; what requires a multi-dimensional and multi-scale approach. This paper will present a multidimensional analysis of two Italian Tuscan historic squares and two Portuguese historic squares. The squares will be analysed from a simultaneous view of their attributes. Thus, it is proposed, in an ongoing joint research project, to address the limitations of traditional-descriptive urban morphology in dealing with this simultaneity. Developing the relations between formal attributes and intangible spatial properties, their identity and closeness may be disclosed by multivariate statistical analysis and computational techniques.info:eu-repo/semantics/acceptedVersio

Entanglement and quantum combinatorial designs

We introduce several classes of quantum combinatorial designs, namely quantum Latin squares, cubes, hypercubes and a notion of orthogonality between them. A further introduced notion, quantum orthogonal arrays, generalizes all previous classes of designs. We show that mutually orthogonal quantum Latin arrangements can be entangled in the same way than quantum states are entangled. Furthermore, we show that such designs naturally define a remarkable class of genuinely multipartite highly entangled states called $k$-uniform, i.e. multipartite pure states such that every reduction to $k$ parties is maximally mixed. We derive infinitely many classes of mutually orthogonal quantum Latin arrangements and quantum orthogonal arrays having an arbitrary large number of columns. The corresponding multipartite $k$-uniform states exhibit a high persistency of entanglement, which makes them ideal candidates to develop multipartite quantum information protocols.Comment: 14 pages, 3 figures. Comments are very welcome