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Water Resources Engineering
Published in P.K. Jayasree, K Balan, V Rani, Practical Civil Engineering, 2021
P.K. Jayasree, K Balan, V Rani
Natural channel sections are generally very irregular in shape, whereas artificial channels are usually designed with sections of regular geometry shapes. The trapezoid is a commonly used shape; the rectangle and triangle are special case of trapezoid. Since the rectangle has vertical sides, it is commonly used for channels built of stable materials, such as lined masonry, rocks, metal, or timber. The depth of flow y is the vertical distance from the water surface to the lowest point of the channel section. Stage is the elevation or vertical distance of the free surface above a datum. Top width T is the width of the natural section at the water surface. Water area A is the cross-sectional area of the flow measured normal to the direction of the flow. The wetted perimeter P is the length of the line of intersection of the channel wetted surface with the cross-sectional plane normal to the direction of the flow. Figure 14.11 shows the different geometric elements of channel section.
Mixed Integer Optimization
Published in Timothy R. Anderson, Optimization Modeling Using R, 2023
Binary variables are a special case of general integer variables. Rather than variables taking on values such as 46 and 973, acceptable values are limited to just 0 and 1. This greatly reduces the possible search space for branch and bound since you know that in any branch, you will never branch on a single variable more than once. On the other hand, with a large number of variables, the search space can still be very large. If the previous case with 1000 variables were binary, a full enumeration list would rely on a list of 21000 or one followed by about 300 zeros) possible solutions. While better than the case of integers, it is still vastly more than the number of atoms in the universe.
Mathematical preliminaries
Published in J.L. Meek, Computer Methods in Structural Analysis, 2017
For the special case of a rectangle, it is easily shown from eqn (1.104) that det J is constant over the rectangle, and thence I1=[1111][X]detJand,area=A=4detJ
Zero Initialised Unsupervised Active Learning by Optimally Balanced Entropy-Based Sampling for Imbalanced Problems
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
where is the upper rounding of X and . Note that unlike in Section 5.2, here is the entropy of case 1B before revealing the next item, so after the iteration is completed, but before the started. Moreover, in special case , where is an integer, then both and can be substituted with in Eq. 56, which yields the entropy of uniform distribution as the entropy of the optimal situation.
Comparing the concept images and hierarchical classification skills of students at different educational levels regarding parallelograms: a cross-sectional study
Published in International Journal of Mathematical Education in Science and Technology, 2022
In the Turkish Primary Mathematics Curriculum (1st–8th grade), the teaching of geometric shapes begins in the first grade of primary education with distinguishing and classifying the triangle, square, rectangle, and circle. In the second grade, the curriculum is focused on recognizing the cube, square prism, rectangular prism, triangular prism, cylinder, and sphere as well as distinguishing them. In the third grade, the sides of triangles, squares, and rectangles are classified, while in the fourth grade, the relationship between the perimeters of the square and rectangle along with the side lengths is explored. In the fifth grade, polygons are defined, constructed, and their basic elements are discovered. In addition, the basic elements of rectangle, parallelogram, rhombus, and trapezoid are defined, and hierarchical classifications are also created (MoNE, 2018). Also, in the Turkish mathematics curriculum (MoNE, 2018) the importance of hierarchical classification of the trapezoid and other special quadrilaterals is emphasized, for example, ‘a square is regarded as a special case of quadrilateral and rhombus. In addition, rectangle and rhombus are regarded as a special case of a parallelogram. Rectangle, rhombus, and parallelogram are also regarded as trapezium.’ (p. 8). Moreover, the properties of polygons and quadrilaterals are learned more deeply in the middle and high school curriculum.
Vibronic mean-field and perturbation theory for Jahn-Teller and pseudo-Jahn-Teller molecules
Published in Molecular Physics, 2021
Of course, the special case of accidental degeneracy will never be exactly fulfilled. For arbitrary values of the diabatic harmonic frequencies and , one of the linear coupling terms can be eliminated with the transformed coordinates where Equation (11) applies identically for and . The diabatic Hamiltonian in terms of the new coordinates has the same form as Equation (9), except for the addition of a new bilinear term along the diagonal of the matrix, where . The transformed coupling constants are The inequality of the diabatic harmonic frequencies spoils the otherwise exact separation into one vibronically active e mode and one spectator e mode. In the strong coupling limit where , however, the minimal vibronically correlated wavefunction, Equation (10), should be a good approximation.