Main Points:
If you have a constant that must be used to satisfy two equations which is impossible, a constant can be found that is the best, but not perfect for both equations using a vector that is perpindicular to the other vector from the original. Mathematically this constant is represented by x dot y divided by x dot x (x is the equation in vector form).
Challenges:
I forgot how to find the perpendicular vector. I realized now that it is the vector that when dotted with the original it =0.
Reflections:
I realize this is a very shitty blog, but I just got done with my consulting project a little while ago and realized I still had to blog.
Monday, November 24, 2008
Wednesday, November 19, 2008
Linear Algebra 4.1 - 4.2
Main Points:
Dot products can be used to find the lengths of vectors and the angle between them. The length of a vector can be calculated by taking the square root of the sum of the squares of the components of the vector. The cosine of the angle between two vectors is given by dividing the dot product of the two by the product of their lengths. When a vector isn't a multiple of another, one can figure out the closest the vector can get to the intended point. To do this, one must utilize two equations. The first is that the closest the vector can get (which is represented by another vector r) is the point perpendicular to the intended point, so r dotted with the vector trying to be reached equals zero. The second equation is formed from the fact that a multiple times the vector plus the vector r between the two must equal the vector trying to be reached.
Challenges
The biggest hang up of this reading for me was the section describing how to find if two vectors are perpendicular. I understand that the product of the slopes of the vectors equals negative one when the vectors are perpendicular, but the way the reading went about showing this was confusing.
Reflections
This concept of finding an answer that is the best possible, rather than flat out the best brings about some useful applications. We've been finding optimized solutions for problems given restraints, and this seems like it could be used in a similar way.
Dot products can be used to find the lengths of vectors and the angle between them. The length of a vector can be calculated by taking the square root of the sum of the squares of the components of the vector. The cosine of the angle between two vectors is given by dividing the dot product of the two by the product of their lengths. When a vector isn't a multiple of another, one can figure out the closest the vector can get to the intended point. To do this, one must utilize two equations. The first is that the closest the vector can get (which is represented by another vector r) is the point perpendicular to the intended point, so r dotted with the vector trying to be reached equals zero. The second equation is formed from the fact that a multiple times the vector plus the vector r between the two must equal the vector trying to be reached.
Challenges
The biggest hang up of this reading for me was the section describing how to find if two vectors are perpendicular. I understand that the product of the slopes of the vectors equals negative one when the vectors are perpendicular, but the way the reading went about showing this was confusing.
Reflections
This concept of finding an answer that is the best possible, rather than flat out the best brings about some useful applications. We've been finding optimized solutions for problems given restraints, and this seems like it could be used in a similar way.
Monday, November 17, 2008
Linear Algebra 1.3, 2.0 -- 2.3, 3.0 -- 3.3
Main Points
Vectors can be added together by adding the all the x components together and all the y components together. Multiples of vectors can be added together to match another vector. The multiple used can be found by changing the i and j values into constants being multiplied by the variable to equal the final vector value. If the vectors are parallel, then there is no solution. This same method can also be applied in three dimensions and with many different vectors. The span of a vector, or sum of vectors, is represented by all possible multiples of the vector. A subspace is a list of vectors that is the span of another list of vectors. The dimension of a subspace is the minimum number of vectors it takes to span that subspace.
Challenges:
The biggest challenge for me is keeping all of the columns and rows and how they multiply straight in my head. Also I thought some of the applications of spanning were quite confusing. The example using a span to solve for an equation was hard to follow. Hearing the explanation in a human voice will help.
Reflections
I had no idea vectors could be used for so damn much! I guess my only reflection is that I really hope that these concepts have some translation into the real world because vectors aren't my strong point.
Vectors can be added together by adding the all the x components together and all the y components together. Multiples of vectors can be added together to match another vector. The multiple used can be found by changing the i and j values into constants being multiplied by the variable to equal the final vector value. If the vectors are parallel, then there is no solution. This same method can also be applied in three dimensions and with many different vectors. The span of a vector, or sum of vectors, is represented by all possible multiples of the vector. A subspace is a list of vectors that is the span of another list of vectors. The dimension of a subspace is the minimum number of vectors it takes to span that subspace.
Challenges:
The biggest challenge for me is keeping all of the columns and rows and how they multiply straight in my head. Also I thought some of the applications of spanning were quite confusing. The example using a span to solve for an equation was hard to follow. Hearing the explanation in a human voice will help.
Reflections
I had no idea vectors could be used for so damn much! I guess my only reflection is that I really hope that these concepts have some translation into the real world because vectors aren't my strong point.
Monday, November 10, 2008
10.7 Modeling the Spread of a Disease
Main Points:
Differential equations can be used to describe disease outbreaks and are helpful in finding the necessary vaccination to stop the spread of a disease. The SIR model is used to model this. S stands for the number of people capable of becoming sick. I represents the the number of infected people, and R represents the number of recovered or people who are unable to get sick. The number of people susceptible decreases, so the slope that represents this (dS/dt) = -aSI. The rate of infected increases at the same rate as the rate of susceptible category minus those who have recovered OR DIED. This slope can be represented as (dI/dt)= aSI - bI. The recovered group OR DEAD GROUP has the rate of increase of (dR/dt)= bI. a represents how infectious a disease is, so if you know the initial conditions, you can roughly calculate this constant. b represents the rate at which the infected are removed from the infected population. If you consider vaccination, the vaccinated represent people removed from the infected population. The threshold is the point when the number of infecteds starts to decrease. If enough people are vaccinated that the disease can only infect as many as are represented by the threshold, then it will decrease rapidly just like the graph shows.
Challenges
I honestly think the book explains how to graph this very poorly, I do understand it, but I feel it is a challenging thing to look at this section in the text book and figure out through their jargon how the heck to graph it.
Reflections
This is really really useful stuff. Learning how and when to stop an epidemic or at least an outbreak with calculus is pretty amazing.
Differential equations can be used to describe disease outbreaks and are helpful in finding the necessary vaccination to stop the spread of a disease. The SIR model is used to model this. S stands for the number of people capable of becoming sick. I represents the the number of infected people, and R represents the number of recovered or people who are unable to get sick. The number of people susceptible decreases, so the slope that represents this (dS/dt) = -aSI. The rate of infected increases at the same rate as the rate of susceptible category minus those who have recovered OR DIED. This slope can be represented as (dI/dt)= aSI - bI. The recovered group OR DEAD GROUP has the rate of increase of (dR/dt)= bI. a represents how infectious a disease is, so if you know the initial conditions, you can roughly calculate this constant. b represents the rate at which the infected are removed from the infected population. If you consider vaccination, the vaccinated represent people removed from the infected population. The threshold is the point when the number of infecteds starts to decrease. If enough people are vaccinated that the disease can only infect as many as are represented by the threshold, then it will decrease rapidly just like the graph shows.
Challenges
I honestly think the book explains how to graph this very poorly, I do understand it, but I feel it is a challenging thing to look at this section in the text book and figure out through their jargon how the heck to graph it.
Reflections
This is really really useful stuff. Learning how and when to stop an epidemic or at least an outbreak with calculus is pretty amazing.
Wednesday, November 5, 2008
10.6 Modeling the Interaction of two Populations
Main Points:
When considering the interaction of two populations, one needs a system of two differential equations. In the predator prey model, the idea that the prey will grow exponentially without the predator can be used to form a definition of the slope. Other ideas that should be considered are that the predator's population would decrease if there is too little prey, and the robins benefit from the worms' presence, but the worms do not. Also the number of prey being preyed upon should be directly proportional to the population of the predator. To form a phase plane, the two slope values for the populations must be multiplied. The chart will usually show a equilibrium at the origin and somewhere else that the slope points flow around. The slope points show a trajectory of the populations. As you move a little farther away from one point the slope changes slightly and you can trace the approximate path starting at one point. You can see that with this predator - prey model, the populations oscillate at regular intervals that are separated by one quarter of a period.
Challenges:
The hardest part for me was to figure out how they came up with the equation used to plot the phase plane. It looks like they just treated the slopes like functions and multiplied them together. I need some clarification though.
Reflections:
IT'S THE CIRCLE OF LIFE! It really represents that saying perfectly. The slopes of the population trace a circle around the equilibrium, and since we don't live in a perfect world of equilibrium, this graphic representation is really usefull.
When considering the interaction of two populations, one needs a system of two differential equations. In the predator prey model, the idea that the prey will grow exponentially without the predator can be used to form a definition of the slope. Other ideas that should be considered are that the predator's population would decrease if there is too little prey, and the robins benefit from the worms' presence, but the worms do not. Also the number of prey being preyed upon should be directly proportional to the population of the predator. To form a phase plane, the two slope values for the populations must be multiplied. The chart will usually show a equilibrium at the origin and somewhere else that the slope points flow around. The slope points show a trajectory of the populations. As you move a little farther away from one point the slope changes slightly and you can trace the approximate path starting at one point. You can see that with this predator - prey model, the populations oscillate at regular intervals that are separated by one quarter of a period.
Challenges:
The hardest part for me was to figure out how they came up with the equation used to plot the phase plane. It looks like they just treated the slopes like functions and multiplied them together. I need some clarification though.
Reflections:
IT'S THE CIRCLE OF LIFE! It really represents that saying perfectly. The slopes of the population trace a circle around the equilibrium, and since we don't live in a perfect world of equilibrium, this graphic representation is really usefull.
Sunday, November 2, 2008
10.4 Exponential Growth and Decay & 10.5 Applications and Modeling
Main Points:
The general solution to the differential equation dy/dt = ky is y = Ce^kt (C is a constant). When k is greater than 0, the function is exponentially growing, if k is less than 0, the function is exponentially decaying. Population growth, pollution concentration, and interest are all situations that can be solved by this model. Another common model for a function used in a differential equation is dy/dt = k(y - A). In this form k and A are constants. The general solution for this differential equation is y = A+Ce^kt. Equilibrium solutions are ones that give a constant for all values of the independent variable. This yields a horizontal line. An equilibrium solution is stable if a small change in the initial conditions produces solution that approaches the equilibrium as the independent variable approaches infinity. An equilibrium solution is unstable if the new solution veers away from the equilibrium as the variable approaches infinity.
Challenges:
I think I have a decent grip on the subjects talked about in these chapters. The most challenging thing for me was following book's description of solving the differential equation dy/dt = k(y - A). Ce^kt justs pops in there unexpectedly, so it's a little confusing at first.
Reflections:
Well now I got to see how to find the differential equation, which is good since it was bothering me that we couldn't necessarily solve to find the equation. This new tactic is nice, but it only works in some situations. The situations are pretty broad, so it is usefull.
The general solution to the differential equation dy/dt = ky is y = Ce^kt (C is a constant). When k is greater than 0, the function is exponentially growing, if k is less than 0, the function is exponentially decaying. Population growth, pollution concentration, and interest are all situations that can be solved by this model. Another common model for a function used in a differential equation is dy/dt = k(y - A). In this form k and A are constants. The general solution for this differential equation is y = A+Ce^kt. Equilibrium solutions are ones that give a constant for all values of the independent variable. This yields a horizontal line. An equilibrium solution is stable if a small change in the initial conditions produces solution that approaches the equilibrium as the independent variable approaches infinity. An equilibrium solution is unstable if the new solution veers away from the equilibrium as the variable approaches infinity.
Challenges:
I think I have a decent grip on the subjects talked about in these chapters. The most challenging thing for me was following book's description of solving the differential equation dy/dt = k(y - A). Ce^kt justs pops in there unexpectedly, so it's a little confusing at first.
Reflections:
Well now I got to see how to find the differential equation, which is good since it was bothering me that we couldn't necessarily solve to find the equation. This new tactic is nice, but it only works in some situations. The situations are pretty broad, so it is usefull.
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