In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical...In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical at all. Zeno’s paradoxes indirectly prove that distances are not composed of extensionless points and, in general, that a higher dimension cannot be completely composed of lower ones. Conversely, lower dimensions can be understood as special cases of higher dimensions. To illustrate this approach, I consider Cantor’s only apparent proof that the real numbers are uncountable. However, his widely accepted indirect proof has the disadvantage that it depends on whether there is another way to make the real numbers countable. Cantor rightly assumes that there can be no smallest number between 0 and 1, and therefore no beginning of counting. For this reason he arbitrarily lists the real numbers in order to show with his diagonal method that this list can never be complete. The situation is different if we start with the largest number between 0 and 1 (0.999…) and use the method of an inverted triangle, which can be understood as a special fractal form. Here we can construct a vertical and a horizontal stratification with which it is actually possible to construct all real numbers between 0 and 1 without exception. Each column is infinite, and each number in that column is the starting point of a new triangle, while each row is finite. Even in a simple sine curve, we experience finiteness with respect to the y-axis and infinity with respect to the x-axis. The first parts of this article show that Zeno’s assumptions contradict the concept of motion as such, so it is not surprising that this misconstruction leads to contradictions. In the last part, I discuss Cantor’s diagonal method and explain the method of an inverted triangle that is internally structured like a fractal by repeating this inverted triangle at each column. The consequence is that we encounter two very different methods of counting. Vertically it is continuous, horizontally it is discrete. While Frege, Tarski, Cantor, Gödel and the Vienna Circle tried to derive the higher dimension from the lower, a procedure that always leads to new contradictions and antinomies (Tarski, Russell), I take the opposite approach here, in which I derive the lower dimension from the higher. This perspective seems to fail because Tarski, Russell, Wittgenstein, and especially the Vienna Circle have shown that the completeness of the absolute itself is logically contradictory. For this reason, we agree with Hegel in assuming that we can never fully comprehend the Absolute, but only its particular manifestations—otherwise we would be putting ourselves in the place of the Absolute, or even God. Nevertheless, we can understand the Absolute in its particular expressions, as I will show with the modest example of the triangle proof of the combined horizontal and vertical countability of the real numbers, which I developed in rejection of Cantor’s diagonal proof. .展开更多
This note is mainly concerned with the creation of oppositely converging and alternatingly converging iterative methods that have the added advantage of providing ever tighter bounds on the targeted root. By a slight ...This note is mainly concerned with the creation of oppositely converging and alternatingly converging iterative methods that have the added advantage of providing ever tighter bounds on the targeted root. By a slight parametric perturbation of Newton’s method we create an oscillating super-linear method approaching the targeted root alternatingly from above and from below. Further extension of Newton’s method creates an oppositely converging quadratic counterpart to it. This new method requires a second derivative, but for it, the average of the two opposite methods rises to become a cubic method. This note examines also the creation of high order iterative methods by a repeated specification of undetermined coefficients.展开更多
This paper presents a thorough study of the effect of the Constant Eddy Viscosity(CEV)assumption on the optimization of a discrete adjoint-based design optimization system.First,the algorithms of the adjoint methods w...This paper presents a thorough study of the effect of the Constant Eddy Viscosity(CEV)assumption on the optimization of a discrete adjoint-based design optimization system.First,the algorithms of the adjoint methods with and without the CEV assumption are presented,followed by a discussion of the two methods’solution stability.Second,the sensitivity accuracy,adjoint solution stability,and Root Mean Square(RMS)residual convergence rates at both design and offdesign operating points are compared between the CEV and full viscosity adjoint methods in detail.Finally,a multi-point steady aerodynamic and a multi-objective unsteady aerodynamic and aeroelastic coupled design optimizations are performed to study the impact of the CEV assumption on optimization.Two gradient-based optimizers,the Sequential Least-Square Quadratic Programming(SLSQP)method and Steepest Descent Method(SDM)are respectively used to draw a firm conclusion.The results from the transonic NASA Rotor 67 show that the CEV assumption can deteriorate RMS residual convergence rates and even lead to solution instability,especially at a near stall point.Compared with the steady cases,the effect of the CEV assumption on unsteady sensitivity accuracy is much stronger.Nevertheless,the CEV adjoint solver is still capable of achieving optimization goals to some extent,particularly if the flow under consideration is benign.展开更多
文摘In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical at all. Zeno’s paradoxes indirectly prove that distances are not composed of extensionless points and, in general, that a higher dimension cannot be completely composed of lower ones. Conversely, lower dimensions can be understood as special cases of higher dimensions. To illustrate this approach, I consider Cantor’s only apparent proof that the real numbers are uncountable. However, his widely accepted indirect proof has the disadvantage that it depends on whether there is another way to make the real numbers countable. Cantor rightly assumes that there can be no smallest number between 0 and 1, and therefore no beginning of counting. For this reason he arbitrarily lists the real numbers in order to show with his diagonal method that this list can never be complete. The situation is different if we start with the largest number between 0 and 1 (0.999…) and use the method of an inverted triangle, which can be understood as a special fractal form. Here we can construct a vertical and a horizontal stratification with which it is actually possible to construct all real numbers between 0 and 1 without exception. Each column is infinite, and each number in that column is the starting point of a new triangle, while each row is finite. Even in a simple sine curve, we experience finiteness with respect to the y-axis and infinity with respect to the x-axis. The first parts of this article show that Zeno’s assumptions contradict the concept of motion as such, so it is not surprising that this misconstruction leads to contradictions. In the last part, I discuss Cantor’s diagonal method and explain the method of an inverted triangle that is internally structured like a fractal by repeating this inverted triangle at each column. The consequence is that we encounter two very different methods of counting. Vertically it is continuous, horizontally it is discrete. While Frege, Tarski, Cantor, Gödel and the Vienna Circle tried to derive the higher dimension from the lower, a procedure that always leads to new contradictions and antinomies (Tarski, Russell), I take the opposite approach here, in which I derive the lower dimension from the higher. This perspective seems to fail because Tarski, Russell, Wittgenstein, and especially the Vienna Circle have shown that the completeness of the absolute itself is logically contradictory. For this reason, we agree with Hegel in assuming that we can never fully comprehend the Absolute, but only its particular manifestations—otherwise we would be putting ourselves in the place of the Absolute, or even God. Nevertheless, we can understand the Absolute in its particular expressions, as I will show with the modest example of the triangle proof of the combined horizontal and vertical countability of the real numbers, which I developed in rejection of Cantor’s diagonal proof. .
文摘This note is mainly concerned with the creation of oppositely converging and alternatingly converging iterative methods that have the added advantage of providing ever tighter bounds on the targeted root. By a slight parametric perturbation of Newton’s method we create an oscillating super-linear method approaching the targeted root alternatingly from above and from below. Further extension of Newton’s method creates an oppositely converging quadratic counterpart to it. This new method requires a second derivative, but for it, the average of the two opposite methods rises to become a cubic method. This note examines also the creation of high order iterative methods by a repeated specification of undetermined coefficients.
基金supported by the National Science and Technology Major Project,China(No.2017-II-0009-0023)China’s 111 project(No.B17037)sponsored by Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University,China.
文摘This paper presents a thorough study of the effect of the Constant Eddy Viscosity(CEV)assumption on the optimization of a discrete adjoint-based design optimization system.First,the algorithms of the adjoint methods with and without the CEV assumption are presented,followed by a discussion of the two methods’solution stability.Second,the sensitivity accuracy,adjoint solution stability,and Root Mean Square(RMS)residual convergence rates at both design and offdesign operating points are compared between the CEV and full viscosity adjoint methods in detail.Finally,a multi-point steady aerodynamic and a multi-objective unsteady aerodynamic and aeroelastic coupled design optimizations are performed to study the impact of the CEV assumption on optimization.Two gradient-based optimizers,the Sequential Least-Square Quadratic Programming(SLSQP)method and Steepest Descent Method(SDM)are respectively used to draw a firm conclusion.The results from the transonic NASA Rotor 67 show that the CEV assumption can deteriorate RMS residual convergence rates and even lead to solution instability,especially at a near stall point.Compared with the steady cases,the effect of the CEV assumption on unsteady sensitivity accuracy is much stronger.Nevertheless,the CEV adjoint solver is still capable of achieving optimization goals to some extent,particularly if the flow under consideration is benign.
