China
Africa
Explore chapters and articles related to this topic
Fourier Series and Orthogonal Functions
Published in George F. Simmons, Differential Equations with Applications and Historical Notes, 2016
At the age of 18 the Russian-German mathematician Hermann Minkowski (1864–1909) won the Grand Prize of the Academy of Sciences in Paris for his brilliant research on quadratic forms, starting from a problem about the representation of an integer as the sum of five squares. This work later led to the creation of a whole new branch of number theory now called the Geometry of Numbers, which in turn is based on his highly original ideas about the properties of convex bodies in n-dimensional space. In this connection he introduced the abstract concept of distance, analyzed the notions of volume and surface, and established the important inequality that bears his name. In the years 1907–1908 Minkowski became the mathematician of relativity by geometrizing the new subject. He created the concept of four-dimensional space -time as the proper mathematical setting for Einstein’s essentially physical (and nonmathematical) way of thinking about special relativity. In a now-famous lecture of 1908 he began with a sentence that is not easily forgotten: “From now on space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will retain an independent existence.”
Preparatory Tools of Lattice Point Theory
Published in Willi Freeden, M. Zuhair Nashed, Lattice Point Identities and Shannon-Type Sampling, 2019
Willi Freeden, M. Zuhair Nashed
With these preparations we are able to formulate Minkowski’s lattice point theorem, that is a classical result in the geometry of numbers. For more details the reader is referred, e.g., to L.J. Mordell (1935), E. Hlawka (1954), E. Hlawka et al. (1991), and the references therein).
Numerical study of the intensification of single-phase heat transfer in a sandwich-like channel using staggered miniature-pin fins
Published in Numerical Heat Transfer, Part A: Applications, 2023
In Figure 27, the variation of the average Nusselt number with the Reynolds number for each clearance distance is given. Nusselt numbers for different Prandtl numbers (5.12, 6.98, and 9.39) and heating modes (TQBQ and 2BQ) can be seen in these figures. As the Reynolds number increases, the difference between the Nusselt numbers for each heating mode and Prandtl number becomes larger. The highest Nusselt number is 346.06 when there is no clearance with a Prandtl number of 9.39 and Reynolds number of 20000 (Figure 27a). In this case, the heat transfer coefficients are almost the same for the two considered heating modes. However, in the case of only heating from the bottom surface, the Nusselt number (329.9) is higher with tip clearance of 0.1 than that for equal heating from the top and bottom (303.8) (Figure 27b). These values begin to approach each other at 0.3 (320.78, 316.24) (Figure 27c), are almost the same at 0.6 (311.8, 313.33) (Figure 27d), and both decrease at 1 mm (298.62, 308.83) (Figure 27e). For a better understanding of the increase in heat transfer with finned geometry, Nusselt numbers for finless geometry at Pr = 6.78 are also given. It can be seen that as the Prandtl number increases, the Nusselt number becomes higher at the same Reynolds number.
Causality and interpretation: a new design model inspired by the Aristotelian legacy
Published in Construction Management and Economics, 2022
Ergo Pikas, Lauri Koskela, Olli Seppänen
Both Platonic and Aristotelian epistemologies have come to underlie much of design research and practice (Booth 1996). Platonic epistemology has dominated design since the beginning of the Renaissance. It entered the design domain through several well-known architects and engineers (Lefèvre and Buchwald 2004, Murphy 2017). Booth (1996) argued that Leon Battista Alberti, influenced by Platonic epistemology, assigned mathematics (geometry and numbers) “a central role in architectural design”. The first engineering schools in Europe (Channell 2009) and the engineering science by Rankine (1872) had been founded on the recognition that different disciplines (military and civil) had their basis in the principle that engineering solutions are deduced from scientific knowledge (Koskela et al.2019).
3D numerical analysis of piled raft foundation in stone column improved soft soil
Published in International Journal of Geotechnical Engineering, 2019
The behaviour of piled raft foundation on stone column-improved ground has been investigated using three-dimensional finite element analysis. The stone columns were used to strengthen the shallow soft soil strata, and thus improve the overall behaviour of piled raft foundation. Based on the parametric study undertaken, the following conclusions were drawn:An increase in length of piles (slenderness ratio of piles) increased the percentage load shared by piles, while subsequently decreased the load shared by stone columns. It also played a significant role in reduction of settlements of the foundation, especially under higher vertical loads.An increase in diameter of stone columns (area replacement ratio) resulted in higher proportion of load shared by stone columns while the corresponding load shared by piles decreased. However, it had insignificant effect on reducing the settlement of the foundation system due to its low compression modulus.Increasing the length of stone columns (slenderness ratio of stone columns) was found effective in reducing settlements of the foundation system, but had no effect on the load shared by stone columns or piles.From the point of economy, an optimum balance between geometry and numbers of piles and stone columns can be obtained, which can result in balanced share of load among rigid and flexible piles and also reduce the overall cost of the foundation system as cost of construction of flexible piles is lower than rigid piles. For the considered geometry in the present study, almost equal load was shared among stone columns and piles when the area replacement ratio was 25 and pile slenderness ratio was 40.Increase in angle of friction of constituent stone column material was found to enhance the proportion of load shared by stone columns and reduce the settlement in the foundation.Increase in raft thickness not only enhanced the proportion of load shared by stone columns, but also drastically reduced the total and differential settlements in the foundation.