1. [20 pts] Find an integrating factor and solve the equation y
... and determine the interval in which the solution exists. Solution: The differential equation can be written as (2y + 1)dy = 2xdx. Integrating the left side with respect to y and the right side with respect to x gives y 2 + y = x2 + C, where C is an arbitrary constant. To satisfy y(2) = 0 we must hav ...
... and determine the interval in which the solution exists. Solution: The differential equation can be written as (2y + 1)dy = 2xdx. Integrating the left side with respect to y and the right side with respect to x gives y 2 + y = x2 + C, where C is an arbitrary constant. To satisfy y(2) = 0 we must hav ...
Mathematical Physics Allowed material: No material is allowed. 1
... (a) What is saddle point approximation and when is it useful? (b) Describe at least three ways of summing infinite series. (c) What are advanced and retarded Green’s functions and when are they useful? (d) Consider an oil spill in the Baltic Sea. Assume that you know the density of oil in different ...
... (a) What is saddle point approximation and when is it useful? (b) Describe at least three ways of summing infinite series. (c) What are advanced and retarded Green’s functions and when are they useful? (d) Consider an oil spill in the Baltic Sea. Assume that you know the density of oil in different ...
Intersection Body of n–Cube, Siegel`s Lemma and Sum–Distinct
... and Siegel’s Lemma w. r. t. the maximum norm. We show that for any non–zero vector a ∈ Zn , n ≥ 5, there exist linearly independent vectors x1 , . . . , xn−1 ∈ Zn such that xi a = 0, i = 1, . . . , n − 1 and ¶n Z µ ||a||∞ 2 ∞ sin t 0 < ||x1 ||∞ · · · ||xn−1||∞ < , σn = dt . ...
... and Siegel’s Lemma w. r. t. the maximum norm. We show that for any non–zero vector a ∈ Zn , n ≥ 5, there exist linearly independent vectors x1 , . . . , xn−1 ∈ Zn such that xi a = 0, i = 1, . . . , n − 1 and ¶n Z µ ||a||∞ 2 ∞ sin t 0 < ||x1 ||∞ · · · ||xn−1||∞ < , σn = dt . ...
DEPARTMENT OF NON-METALLIC MATERIALS ENGINEERING
... called the objective function (cTx in this case). The inequalities Ax ≤ b and x ≥ 0 are the constraints which specify a convex polytope over which the objective function is to be optimized. In this context, two vectors are comparable when they have the same dimensions. If every entry in the first is ...
... called the objective function (cTx in this case). The inequalities Ax ≤ b and x ≥ 0 are the constraints which specify a convex polytope over which the objective function is to be optimized. In this context, two vectors are comparable when they have the same dimensions. If every entry in the first is ...
Homework 3
... Please use a relational schema (not a text format as in the book), which includes functional dependencies and referential integrity constraints, to demonstrate decomposed relations. Note that each of the four sub-problems is an independent problem. 3. (10 points) Problem 4-34 (all parts), pp.195-196 ...
... Please use a relational schema (not a text format as in the book), which includes functional dependencies and referential integrity constraints, to demonstrate decomposed relations. Note that each of the four sub-problems is an independent problem. 3. (10 points) Problem 4-34 (all parts), pp.195-196 ...
Exercise 4.1 True and False Statements about Simplex x1 x2
... x1 + x2 ≤ 1 becomes x1 + x2 + x3 = 1 with x3 ≥ 0 when transformed into standard form. Assume that the simplex algorithm starts at (x1 , x2 , x3 ) = (0, 0, 1). Taking either x1 or x2 into the basis decreases the objective value, wherefore both of them are candidate variables for entering the basis. N ...
... x1 + x2 ≤ 1 becomes x1 + x2 + x3 = 1 with x3 ≥ 0 when transformed into standard form. Assume that the simplex algorithm starts at (x1 , x2 , x3 ) = (0, 0, 1). Taking either x1 or x2 into the basis decreases the objective value, wherefore both of them are candidate variables for entering the basis. N ...
Multiple-criteria decision analysis
Multiple-criteria decision-making or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research that explicitly considers multiple criteria in decision-making environments. Whether in our daily lives or in professional settings, there are typically multiple conflicting criteria that need to be evaluated in making decisions. Cost or price is usually one of the main criteria. Some measure of quality is typically another criterion that is in conflict with the cost. In purchasing a car, cost, comfort, safety, and fuel economy may be some of the main criteria we consider. It is unusual that the cheapest car is the most comfortable and the safest one. In portfolio management, we are interested in getting high returns but at the same time reducing our risks. Again, the stocks that have the potential of bringing high returns typically also carry high risks of losing money. In a service industry, customer satisfaction and the cost of providing service are two conflicting criteria that would be useful to consider.In our daily lives, we usually weigh multiple criteria implicitly and we may be comfortable with the consequences of such decisions that are made based on only intuition. On the other hand, when stakes are high, it is important to properly structure the problem and explicitly evaluate multiple criteria. In making the decision of whether to build a nuclear power plant or not, and where to build it, there are not only very complex issues involving multiple criteria, but there are also multiple parties who are deeply affected from the consequences.Structuring complex problems well and considering multiple criteria explicitly leads to more informed and better decisions. There have been important advances in this field since the start of the modern multiple-criteria decision-making discipline in the early 1960s. A variety of approaches and methods, many implemented by specialized decision-making software, have been developed for their application in an array of disciplines, ranging from politics and business to the environment and energy.