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Differential Calculus
Published in Prem K. Kythe, Elements of Concave Analysis and Applications, 2018
Just imagine there is flooding in the terrain M. Then the region covered by water when it reaches an elevation of the point a is given by f-1(-∞,a] $ f^{-1}(-\infty , a] $ , i.e., water reaches the points with at most elevation a. When water passes the height of the point a, a critical point where the gradient ∇f=0 $ \nabla f=0 $ , then the water either (i) starts filling the terrain basin, (ii) covers a saddle point (a mountain pass), or (iii) submerges a peak. In each of these three types of critical points (basins, passes, and peaks), we have the case of minima, saddle points, and maxima, respectively. The safest place to escape flooding is either the front or back high elevation at a saddle point (known as the horn and the cantle of a saddle), or the highest peak (maximum elevation), and the worst places are the basins (minima) and the inflection points.
The Willamette and Vistula Rivers: Contrast and Comparison
Published in Antonius Laenen, David A. Dunnette, River Quality, 2018
Antonius Laenen, Jan R. Dojlido
The upper Willamette River is called the Middle Fork Willamette River. The river originates in a mountain pass at an elevation of about 2340 m and, after flowing for 196 km, reaches an elevation of 135 m at the confluence with the Coast Fork Willamette River near Eugene. The main stem of the Willamette River is formed by this confluence and flows another 301 km to the Columbia River. The main stem can be divided into four distinct reaches. The upper reach from Eugene to Corvallis is 110 km in length and is characterized by a meandering and braided channel with many islands and sloughs; the river is shallow, and the bed is composed almost entirely of cobbles and gravel. The middle reach from Corvallis to the confluence of the Yamhill River is 103 km in length and is characterized by a meandering channel deeply incised into the valley floor. The next reach from the Yamhill River to Oregon City is a 46 km long natural pool impounded behind the 15 m high Willamette Falls. Hydraulically, the deep, slow-moving pool can be characterized as a reservoir. The pool is a depositional area for small gravel- to silt-size material and provides a source of gravel, which is regularly mined from the streambed. The lower 42 km of river below the falls is affected by tides, and, during spring and early summer, by backwater from the Columbia River.
Bioremediation of artificially contaminated soil with petroleum using animal waste: cow and poultry dung
Published in Cogent Engineering, 2020
O. Olawale, K. S. Obayomi, S. O. Dahunsi, O. Folarin
The dimensional plots for the cow dung and poultry dung interaction are depicted in Figures 2–4. Figure 1 shows the effect of cow dung and poultry dung on the percentage removal of petroleum in the soil. The interaction as seen in the plots is minimal with F-value of 0.23. The poultry dung interaction is more as compared to the cow dung in terms of singular interaction. Figure 2 shows that the bioremediation effect is relatively flat as seen in the 3D contour plots; the 3D plot shows it actually as a saddle or minimal nature. The saddle is a surface that curves up in one direction, and curves down in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. This implies that the biodegradation is stable for most part of the values uses in the plot. As the cow dung composition increases, higher removal percentage was observed. The exit of the saddle is observed at the upper quadrant of the 3D contour plot at values of cow dung higher than 3.75 and cow dung/poultry dung ratio higher than 2.5. Hence, for effective bioremediation of petrol, the cow dung and cow dung/poultry dung ratio should be maintained with the ranges that allow for optimal removal of petrol. Figure 3 shows a similar saddle nature as Figure 2. However, the contour lines are more pronounced indicating the fact that with the poultry dung, the removal of petrol has a higher level of sensitivity to the low and extreme values of poultry dung and at all levels of the cow dung/poultry dung ratio.
Atomic scale study of the anti-vortex domain structure in polycrystalline ferroelectric
Published in Philosophical Magazine, 2018
Xiaobao Tian, Xiaoqiao He, Jian Lu
Based on the above analysis and contrasts to the vortex which is a flux-closer as shown in the left of Figure 5(a), the definition of the anti-vortex can be given as follows: the anti-vortex is formed by four quarter streamlines domains without domain wall and with a zero value at the core. The standard pattern of the anti-vortex is shown in the right of Figure 5(a). The anti-vortex can be segmented as two 90° domains The anti-vortex in two dimensions is a surface that curvers upward in one direction and, then, curves downward in a different direction, resembling a saddle or a mountain pass. In terms of contour lines, the anti-vortex core in two dimensions gives rise to a contour that appears to intersect itself. The anti-vortex core is an saddle point of the polarisation configuration. In mathematic definition, the saddle point is a stationary point but not a local extremum. In ferroelectric, the core of the anti-vortex is saddle point, but the polarisation of the anti-vortex is zero. According to the stable characteristic of stationary point or the saddle point, the anti-vortex core is stable based on the number of the cores found in this simulation. And the polarisation of the core of the anti-vortex is zero, which is unlike the normal saddle point mathematically. Hence, the mechanism of the stability and formation of anti-vortex is discovered.
A generalized mountain pass lemma with a closed subset for locally Lipschitz functionals
Published in Applicable Analysis, 2022
Fengying Li, Bingyu Li, Shiqing Zhang
Saddle points in the Mountain pass Lemma [1–28] are different from maximum points and minimum points. Maximum and Minimum problems in infinite dimensional space have a very long and prominent history [25] with‘isoperimetric problems’ and the ‘problem of the brachistochrone’ as two notable examples. In the 19th century Dirichlet principle we essentially encountered the problem of minimizing a functional; however, complete rigor was mostly lacking and we had to wait for Hilbert for satisfactory completion of the Dirichlet principle. Continuing in the 20th century, Italian mathematician Tonelli introduced the concept of a weakly lower semi-continuous(w.l.s.c) functional and proved that a w.l.s.c functional defined on a weakly closed subset of a reflexive Banach space can attain its infimum if it is coercive [25]. At times, the existence of a saddle point, which is neither a maximum nor minimum point, is of considerable importance. Minimax methods in the finite dimensional case [25,28] can be traced back to Birkhoff in 1917 and von Neumann's minimax theorem in 1928. We can also observe that the Mountain Pass Lemma of Ambrosseti-Rabinowitz [1] in 1973 is a type of minimax theorem, which can be traced back to Courant in 1950 for the finite dimensional case [25]. The Palais-Smale condition first appeared in connection with infinite dimensional problems, but it is necessary even in finite dimensional situations. Reference [4] gives an example of a polynomial in two variables that has exactly two non-degenerate critical points, both of which are global minimizers, and points out that the given polynomial does not satisfy the Palais-Smale condition. From the finite dimensional case to the infinite dimensional case, the key step in the proof of the Mountain Pass Lemma is the use of a Palais-Smale type compactness condition(PS) to drive Palais's Deformation Lemma. We should note the original proof of the Ambrosseti-Rabinowitz's Mountain Pass Lemma used Palais's Deformation Lemma [25]. In the 1970's, Ekeland discovered a very important principle for lower semi-continuous functions on a complete metric space. Until the middle of the 1980's, Aubin-Ekeland [3], Shi [24] discovered the relationship between the Mountain Pass Lemma of Ambrosseti-Rabinowitz and Ekeland's variational priciple. The Mountain Pass Lemma of Ambrosseti-Rabinowitiz has been intensively studied and has found numerous applications [1–28]. Of special note, it was generalized to the case of locally Lipschitz functionals by K. C. Chang [5] where he also obtained more minimax theorems by using a deformation lemma.