We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Gröbner basis, is...We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Gröbner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.展开更多
Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investiga...Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investigate signed tilings of rectangles by T<sub>n</sub> and T<sup>+</sup><sub>n</sub> . We show that a rectangle has a signed tiling by T<sub>n</sub> if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by . We also show that a rectangle has a signed tiling by T<sup>+</sup><sub>n, </sub> n≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit GrÖbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.展开更多
We introduce the definition of q-Stancu operator and investigate its approximation and shape-preserving property. With the help of the sign changes of f（x） and Ln = f（f, q;x） the shape-preserving property of q-Sta...We introduce the definition of q-Stancu operator and investigate its approximation and shape-preserving property. With the help of the sign changes of f（x） and Ln = f（f, q;x） the shape-preserving property of q-Stancu operator is obtained.展开更多
The human placenta nourishes the growing fetus during pregnancy. The newly developing field of placenta analysis seeks to understand relationships between the health of a placenta and the health of the baby. Previous ...The human placenta nourishes the growing fetus during pregnancy. The newly developing field of placenta analysis seeks to understand relationships between the health of a placenta and the health of the baby. Previous studies have shown that the median placental chorionic shape at term is round, and deviation from such prototypical shape is related to a decreased placental functional efficiency. In this study, we propose the use of a nearly-continuous shape descriptor termed signed deviation vector to systematically study the relationship between various maternal and fetal characteristics and the shape of the placental surface. The proposed shape descriptor measures the amount of deviation along with the direction of the deviation a placental shape has away from the shape of normality. Using Linear Discriminant Analysis, we can independently examine how much of the placental shape is affected by maternal, newborn, and placental characteristics. The results allow us to understand how significantly various maternal and fetal conditions affect the overall shape of the placenta growth. Though the current study is largely exploratory, the initial findings indicate significant relationships between shape of the placental surface and newborn’s birth weight as well as their gestational age.展开更多
文摘We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Gröbner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.
文摘Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investigate signed tilings of rectangles by T<sub>n</sub> and T<sup>+</sup><sub>n</sub> . We show that a rectangle has a signed tiling by T<sub>n</sub> if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by . We also show that a rectangle has a signed tiling by T<sup>+</sup><sub>n, </sub> n≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit GrÖbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.
基金Supported by the Education Department of Zhejiang Province(20071078)
文摘We introduce the definition of q-Stancu operator and investigate its approximation and shape-preserving property. With the help of the sign changes of f（x） and Ln = f（f, q;x） the shape-preserving property of q-Stancu operator is obtained.
文摘The human placenta nourishes the growing fetus during pregnancy. The newly developing field of placenta analysis seeks to understand relationships between the health of a placenta and the health of the baby. Previous studies have shown that the median placental chorionic shape at term is round, and deviation from such prototypical shape is related to a decreased placental functional efficiency. In this study, we propose the use of a nearly-continuous shape descriptor termed signed deviation vector to systematically study the relationship between various maternal and fetal characteristics and the shape of the placental surface. The proposed shape descriptor measures the amount of deviation along with the direction of the deviation a placental shape has away from the shape of normality. Using Linear Discriminant Analysis, we can independently examine how much of the placental shape is affected by maternal, newborn, and placental characteristics. The results allow us to understand how significantly various maternal and fetal conditions affect the overall shape of the placenta growth. Though the current study is largely exploratory, the initial findings indicate significant relationships between shape of the placental surface and newborn’s birth weight as well as their gestational age.
