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Raymond Cheng - One of the best experts on this subject based on the ideXlab platform.
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Weak Parallelogram laws on banach spaces and applications to prediction
Periodica Mathematica Hungarica, 2015Co-Authors: Raymond Cheng, William T. RossAbstract:This paper concerns a family of weak Parallelogram laws for Banach spaces. It is shown that the familiar Lebesgue spaces satisfy a range of these inequalities. Connections are made to basic geometric ideas, such as smoothness, convexity, and Pythagorean-type theorems. The results are applied to the linear prediction of random processes spanning a Banach space. In particular, the weak Parallelogram laws furnish coefficient growth estimates, Baxter-type inequalities, and criteria for regularity.
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Duality of the weak Parallelogram laws on Banach spaces
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Raymond Cheng, Charles B. HarrisAbstract:Abstract This paper explores a family of weak Parallelogram laws for Banach spaces. Some basic properties of such spaces are obtained. The main result is that a Banach space satisfies a lower weak Parallelogram law if and only if its dual satisfies an upper weak Parallelogram law, and vice versa. Connections are established between the weak Parallelogram laws and the following: subspaces, quotient spaces, Cartesian products, and the Rademacher type and co-type properties.
Yvan Le Borgne - One of the best experts on this subject based on the ideXlab platform.
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Parallelogram polyominoes the sandpile model on a complete bipartite graph and a q t narayana polynomial
Journal of Combinatorial Theory Series A, 2013Co-Authors: Mark Dukes, Yvan Le BorgneAbstract:We classify recurrent configurations of the sandpile model on the complete bipartite graph K"m","n in which one designated vertex is a sink. We present a bijection from these recurrent configurations to decorated Parallelogram polyominoes whose bounding box is an mxn rectangle. Several special types of recurrent configurations and their properties via this bijection are examined. For example, recurrent configurations whose sum of heights is minimal are shown to correspond to polyominoes of least area. Two other classes of recurrent configurations are shown to be related to bicomposition matrices, a matrix analogue of set partitions, and (2+2)-free partially ordered sets. A canonical toppling process for recurrent configurations gives rise to a path within the associated Parallelogram polyominoes. This path bounces off the external edges of the polyomino, and is reminiscent of [email protected]?s well-known bounce statistic for Dyck paths. We define a collection of polynomials that we call q,t-Narayana polynomials, defined to be the generating function of the bistatistic (area,parabounce) on the set of Parallelogram polyominoes, akin to the (area,hagbounce) bistatistic defined on Dyck paths in Haglund (2003). In doing so, we have extended a bistatistic of Egge et al. (2003) to the set of Parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the q,t-Narayana polynomials to be symmetric and prove this conjecture for numerous special cases. We also show a relationship between [email protected]?s (area,hagbounce) statistic on Dyck paths, and our bistatistic (area,parabounce) on a sub-collection of those Parallelogram polyominoes living in a (n+1)xn rectangle.
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Parallelogram polyominoes the sandpile model on a complete bipartite graph and a q t narayana polynomial
arXiv: Combinatorics, 2012Co-Authors: Mark Dukes, Yvan Le BorgneAbstract:We classify recurrent configurations of the sandpile model on the complete bipartite graph K_{m,n} in which one designated vertex is a sink. We present a bijection from these recurrent configurations to decorated Parallelogram polyominoes whose bounding box is a m*n rectangle. Several special types of recurrent configurations and their properties via this bijection are examined. For example, recurrent configurations whose sum of heights is minimal are shown to correspond to polyominoes of least area. Two other classes of recurrent configurations are shown to be related to bicomposition matrices, a matrix analogue of set partitions, and (2+2)-free partially ordered sets. A canonical toppling process for recurrent configurations gives rise to a path within the associated Parallelogram polyominoes. This path bounces off the external edges of the polyomino, and is reminiscent of Haglund's well-known bounce statistic for Dyck paths. We define a collection of polynomials that we call q,t-Narayana polynomials, defined to be the generating function of the bistatistic (area,parabounce) on the set of Parallelogram polyominoes, akin to the (area,hagbounce) bistatistic defined on Dyck paths in Haglund (2003). In doing so, we have extended a bistatistic of Egge, Haglund, Kremer and Killpatrick (2003) to the set of Parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the q,t-Narayana polynomials to be symmetric and prove this conjecture for numerous special cases. We also show a relationship between Haglund's (area,hagbounce) statistic on Dyck paths, and our bistatistic (area,parabounce) on a sub-collection of those Parallelogram polyominoes living in a (n+1)*n rectangle.
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The sandpile model on a bipartite graph, Parallelogram polyominoes, and a q,t-Narayana polynomial
2012Co-Authors: Mark Dukes, Yvan Le BorgneAbstract:We give a polyomino characterisation of recurrent configurations of the sandpile model on the complete bipartite graph Km,n in which one designated vertex is the sink. We present a bijection from these recurrent configurations to decorated Parallelogram polyominoes whose bounding box is a m×n rectangle. Other combinatorial structures appear in special cases of this correspondence: for example bicomposition matrices (a matrix analogue of set partitions), and (2+2)-free posets. A canonical toppling process for recurrent configurations gives rise to a path within the associated Parallelogram polyominoes. We define a collection of polynomials that we call q,t-Narayana polynomials, the generating functions of the bistatistic (area, bounceWeight) on the set of Parallelogram polyominoes, akin to Haglund's (area, hbounce) bistatistic on Dyck paths. In doing so, we have extended a bistatistic of Egge et al. to the set of Parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the q,t-Narayana polynomials to be symmetric and discuss the proofs for numerous special cases. We also show a relationship between the q,t-Catalan polynomials and our bistatistic (area, bounceWeight) on a subset of Parallelogram polyominoes.
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The sandpile model, polyominoes, and a $q,t$-Narayana polynomial
Discrete Mathematics and Theoretical Computer Science, 2012Co-Authors: Mark Dukes, Yvan Le BorgneAbstract:We give a polyomino characterisation of recurrent configurations of the sandpile model on the complete bipartite graph $K_{m,n}$ in which one designated vertex is the sink. We present a bijection from these recurrent configurations to decorated Parallelogram polyominoes whose bounding box is a $m×n$ rectangle. Other combinatorial structures appear in special cases of this correspondence: for example bicomposition matrices (a matrix analogue of set partitions), and (2+2)-free posets. A canonical toppling process for recurrent configurations gives rise to a path within the associated Parallelogram polyominoes. We define a collection of polynomials that we call $q,t$-Narayana polynomials, the generating functions of the bistatistic $(\mathsf{area ,parabounce} )$ on the set of Parallelogram polyominoes, akin to Haglund's $(\mathsf{area ,hagbounce} )$ bistatistic on Dyck paths. In doing so, we have extended a bistatistic of Egge et al. to the set of Parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the $q,t$-Narayana polynomials to be symmetric and discuss the proofs for numerous special cases. We also show a relationship between the $q,t$-Catalan polynomials and our bistatistic $(\mathsf{area ,parabounce}) $on a subset of Parallelogram polyominoes.
Charles B. Harris - One of the best experts on this subject based on the ideXlab platform.
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Duality of the weak Parallelogram laws on Banach spaces
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Raymond Cheng, Charles B. HarrisAbstract:Abstract This paper explores a family of weak Parallelogram laws for Banach spaces. Some basic properties of such spaces are obtained. The main result is that a Banach space satisfies a lower weak Parallelogram law if and only if its dual satisfies an upper weak Parallelogram law, and vice versa. Connections are established between the weak Parallelogram laws and the following: subspaces, quotient spaces, Cartesian products, and the Rademacher type and co-type properties.
Weixu Wang - One of the best experts on this subject based on the ideXlab platform.
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effects of Parallelogram shaped pavement markings on vehicle speed and safety of pedestrian crosswalks on urban roads in china
Accident Analysis & Prevention, 2016Co-Authors: Yanyong Guo, Pan Liu, Qiyu Liang, Weixu WangAbstract:The primary objective of this study was to evaluate the effects of Parallelogram-shaped pavement markings on vehicle speed and crashes in the vicinity of urban pedestrian crosswalks. The research team measured speed data at twelve sites, and crash data at eleven sites. Observational cross-sectional studies were conducted to identify if the effects of Parallelogram-shaped pavement markings on vehicle speeds and speed violations were statistically significant. The results showed that Parallelogram-shaped pavement markings significantly reduced vehicle speeds and speed violations in the vicinity of pedestrian crosswalks. More specifically, the speed reduction effects varied from 1.89km/h to 4.41km/h with an average of 3.79km/h. The reduction in the 85th percentile speed varied from 0.81km/h to 5.34km/h with an average of 4.19km/h. Odds ratios (OR) showed that the Parallelogram-shaped pavement markings had effects of a 7.1% reduction in the mean speed and a 6.9% reduction in the 85th percentile speed at the pedestrian crosswalks. The reduction of proportion of drivers exceeding the speed limit varied from 8.64% to 14.15% with an average of 11.03%. The results of the crash data analysis suggested that the use of Parallelogram-shaped pavement markings reduced both the frequency and severity of crashes at pedestrian crosswalks. The Parallelogram-shaped pavement markings had a significant effect on reducing the vehicle-pedestrian crashes. Two crash prediction models were developed for vehicle-pedestrian crashes and rear-end crashes. According to the crash models, the presence of Parallelogram-shaped pavement markings reduced vehicle-pedestrian crashes at pedestrian crosswalks by 24.87% with a 95% confidence interval of [10.06-30.78%]. However, the model results also showed that the presence of Parallelogram-shaped pavement markings increased rear-end crashes at pedestrian crosswalks by 5.4% with a 95% confidence interval of [0-11.2%]. Language: en
William T. Ross - One of the best experts on this subject based on the ideXlab platform.
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Weak Parallelogram laws on banach spaces and applications to prediction
Periodica Mathematica Hungarica, 2015Co-Authors: Raymond Cheng, William T. RossAbstract:This paper concerns a family of weak Parallelogram laws for Banach spaces. It is shown that the familiar Lebesgue spaces satisfy a range of these inequalities. Connections are made to basic geometric ideas, such as smoothness, convexity, and Pythagorean-type theorems. The results are applied to the linear prediction of random processes spanning a Banach space. In particular, the weak Parallelogram laws furnish coefficient growth estimates, Baxter-type inequalities, and criteria for regularity.
