We, the students of Professor Bart Goddard’s Differential Equations class (M 427J), are writing to you about a recent incident for which he is at risk of losing his position in the university. During the 12:00 PM class on Wednesday, February 22, 2017, Professor Goddard told a cautionary tale about how, in the Holocaust, Nazi engineers used lime to prevent blood from ruining the train tracks of the trains carrying the Jews to the camps, calling on the engineering students in the audience to understand the implications of their work and to avoid evildoing. As a result, he was relieved of his teaching duties on Friday morning, for two class sessions and is currently in a state of limbo pending the decision of the dean for his case. However, we believe that the professor in no way attacked, demeaned, or otherwise discredited a student, race or religious affiliation (nor intended to do so) with his story. Furthermore, the story he told does not reflect his ability to effectively teach the course at all, and it is not representative of his highly positive relationship with his classes as a whole. Professor…
investors will not pay for risk premiums, and defined by following stochastic differential equations: (1) (2) Where: S(t) represents the price process of an underlying assets. V(t) is the variance of the corresponding instantaneous returns. And the initial condition S(0), V(0) should be strictly positive. r is the risk-free return rate (non-negative constant). k is the speed of the mean reversion. θ is the mean value of the variance. σ is the volatility of the variance. These three are positive…
machines were unexplainable with wasan. For example, to understand the locomotive, one must apprehend the mechanics of a steam engine, which ultimately boils down to thermodynamics. Many thermodynamic equations are differential, so a grasp on calculus, considered yôsan, would be necessary to implement the steam locomotive. While some may argue that enri could have been used, applying this to differential equations would have been difficult as values were extrapolated rather than calculated…
Fundamental Theorem of Calculus The Fundamental Theorem of Calculus evaluate an antiderivative at the upper and lower limits of integration and take the difference. This theorem is separated into two parts. The first part is called the first fundamental theorem of calculus and states that one of the antiderivatives of some function may be obtained as the integral of the function with a variable bound of integration. The second part of the theorem, called the second fundamental theorem of…
The fundamental theorem of Calculus: The fundamental theorem of calculus asserts the interrelated properties of integration and differentiation. It says that a function when differentiated, can be brought back by integrating (anti-derivative) or a function when integrated, can be brought back by differentiation. First theorem: Let f be a function that is integrable on [a,x] for each x in [a,b], then let c be such that a≤c≤b and define a new function A as follows, A(x)=∫_c^x▒f(x)dt, if a≤x≤b.…
1. What are the least, and most, amount of distinct zeroes of a 7th degree polynomial, given that at least one root is a complex number? Answer: If the equation is 7th degree then it has 7 roots. Those roots can be complex or real. Complex roots always come in pairs, so if it has one, then it has 2, the other one being the conjugate of the first one. This in other words, if one complex root is a + bi, then the other complex root is a – bi. If at least one root were complex, then we would have a…
To give an another definition of complex number, we have to show z= ew for w, where z is any non-zero complex number. If we assume w = u + iv then eueiv = |z|e^i arg z and this means that |z| = e^u , v = arg z . The equation |z| = e^u is a real equation, so we can write u = ln |z|, where ln |z| is the ordinary logarithm with positive real numbers. Hence, w = u + iv = ln |z| + i arg z = ln |z| + i(Arg z + 2n) , n = 0 , ±1 , ±2 , ±3 , . . .6 We profoundly examined the connection between…
Title: Application of Self-Organization Neural Network Technique (SOM) to Optimize Finite- Element Partial Differential Equation (PDEs) Results in Square-Shaped Structures Analysis. The finite-element method (FEM) is a computationally method for solving partial differential equations (PDEs) with specific boundary conditions over a domain. When we applying the FEM to a domain, it has to divide to a finite number of elements and nodes. The collections of the elements and nodes form the…
(34) Substituting , and into (31), and calculating the derivatives with respect to , we obtain (35) where Since , we have , and . Hence there is a Hopf bifurcation at . We have the following result: Theorem 13. Suppose holds. Then the system (29) undergoes a Hopf bifurcation when passes through emanating from the steady state leading to periodic solutions trajectories for either (super-critical bifurcation), or (sub-critical-bifurcation) or at . Next,…
Differential Equations was the course that made me question my decision in taking up Chemical Engineering. I actually had no idea it was this challenging. All I had in mind was that it’s just Calculus 1 and 2 combined. Before we officially started the class, Engr. Cabatingan introduced us the syllabus and made us study it. The topics were “bearable”. However, I really had no idea what was Modeling all about. We even joked about it, more so when we knew that there’s going to be a gallery walk. In…