I like these pictures. They were taken in the final fading days of autumn, a few weeks ago. The colours are quite lovely, and the pictures were tastefully composed. Maybe this gives you a little insight into myself.
Monday, November 17, 2008
Myself in Pictures
While I appreciate the semi-anonymity I have in the blogosphere, here are a few pictures taken while I was out with my sweet fiancee.
I like these pictures. They were taken in the final fading days of autumn, a few weeks ago. The colours are quite lovely, and the pictures were tastefully composed. Maybe this gives you a little insight into myself.
I like these pictures. They were taken in the final fading days of autumn, a few weeks ago. The colours are quite lovely, and the pictures were tastefully composed. Maybe this gives you a little insight into myself.
Being a Good Potluck Guest
As promised, I am continuing my posts on potluck etiquette, offering some advice based on my humble experiences.
Let's start with being a good guest, since most people will be a guest at a potluck before they are a host (at least the way I think of it). Being invited to a potluck is a bit unlike other social events since you have a clear responsibility of bringing something. One does not simply show up to a potluck, as one would be outcast immediately. Or just receive some nasty glares from other guests. Either way, the obvious first piece of advice is to make sure you bring something to a potluck.
Now the question becomes what to bring. In my experience, this is difficult for most people, mainly because they "don't" cook or "can't make anything good". If you, dear reader, can relate to this line of reasoning, do not fret. There a few avenues you can take to remedy this problem, which usually involves talking to the host or to some other guests. First, you should ask them what the expectations are for this potluck. If it's a relatively informal gathering where the food is merely a means to an end (of having a great time), then perhaps bringing snack foods, alcohol, or a ready-made, store-bought dessert is a possibility. These are easy things to bring, as you just have pick them up from your favourite grocer and remember to bring them. If the event is somewhat more formal, and food is a higher order of the day, then you may want to bring something you have prepared yourself or a classier store-bought item, such a fine wine or more decadent dessert. The last case is where the host has something particular in mind for some or all guests to bring, in which case you may have already been volunteered to make something. This is either a blessing or a curse depending on your esteemed host. That delicious three bean dip or blueberry fruitcake may taste good and impress, but it could also take time, resources and money to prepare that you just don't feel you have.
In any event, make sure to let the host know about your comfort level in preparing something. You may be able to bring utility items such as soft drinks or paper plates in lieu of actual food. You may also be able to avoid getting locked into preparing something you do not want to. As with all social aspects of humans, communication is key, so talk to the host if you're sure of the expectations. Also make sure you know of any dietary restrictions, so you don't bring bacon-wrapped shrimp to a kosher party. This could be awkward for you and other guests, so be mindful. On the other side of the coin, make sure the host and possibly other guests know about dietary restrictions you have.
The only aspect to being a gracious guest is how you actually arrive and interact at the event. Make sure you're punctual, which should be an obvious point. Also remember to let the host know of any special equipment or preparations that should be done before you arrive. There are instances where it makes more sense to prepare or finish a dish on-site at the event. If this is the case, make sure the host knows about it! The last point is to be polite and courteous to other guest and dishes. Don't turn your nose up at other prepared dishes, as this is rude. Nor should you offer nonconstructive criticism of other dishes; if you have nothing nice to say, say nothing at all. In my experiences, most of the dishes prepared are actually really good. Potlucks are a perfect chance to experience new foods and perspectives, so don't ruin things by being a snob or an ass. Be yourself and enjoy!
In my next post, I'll give some tips for preparing good dishes and provide some potluck-friendly recipes.
Let's start with being a good guest, since most people will be a guest at a potluck before they are a host (at least the way I think of it). Being invited to a potluck is a bit unlike other social events since you have a clear responsibility of bringing something. One does not simply show up to a potluck, as one would be outcast immediately. Or just receive some nasty glares from other guests. Either way, the obvious first piece of advice is to make sure you bring something to a potluck.
Now the question becomes what to bring. In my experience, this is difficult for most people, mainly because they "don't" cook or "can't make anything good". If you, dear reader, can relate to this line of reasoning, do not fret. There a few avenues you can take to remedy this problem, which usually involves talking to the host or to some other guests. First, you should ask them what the expectations are for this potluck. If it's a relatively informal gathering where the food is merely a means to an end (of having a great time), then perhaps bringing snack foods, alcohol, or a ready-made, store-bought dessert is a possibility. These are easy things to bring, as you just have pick them up from your favourite grocer and remember to bring them. If the event is somewhat more formal, and food is a higher order of the day, then you may want to bring something you have prepared yourself or a classier store-bought item, such a fine wine or more decadent dessert. The last case is where the host has something particular in mind for some or all guests to bring, in which case you may have already been volunteered to make something. This is either a blessing or a curse depending on your esteemed host. That delicious three bean dip or blueberry fruitcake may taste good and impress, but it could also take time, resources and money to prepare that you just don't feel you have.
In any event, make sure to let the host know about your comfort level in preparing something. You may be able to bring utility items such as soft drinks or paper plates in lieu of actual food. You may also be able to avoid getting locked into preparing something you do not want to. As with all social aspects of humans, communication is key, so talk to the host if you're sure of the expectations. Also make sure you know of any dietary restrictions, so you don't bring bacon-wrapped shrimp to a kosher party. This could be awkward for you and other guests, so be mindful. On the other side of the coin, make sure the host and possibly other guests know about dietary restrictions you have.
The only aspect to being a gracious guest is how you actually arrive and interact at the event. Make sure you're punctual, which should be an obvious point. Also remember to let the host know of any special equipment or preparations that should be done before you arrive. There are instances where it makes more sense to prepare or finish a dish on-site at the event. If this is the case, make sure the host knows about it! The last point is to be polite and courteous to other guest and dishes. Don't turn your nose up at other prepared dishes, as this is rude. Nor should you offer nonconstructive criticism of other dishes; if you have nothing nice to say, say nothing at all. In my experiences, most of the dishes prepared are actually really good. Potlucks are a perfect chance to experience new foods and perspectives, so don't ruin things by being a snob or an ass. Be yourself and enjoy!
In my next post, I'll give some tips for preparing good dishes and provide some potluck-friendly recipes.
Tuesday, November 11, 2008
"You can use math to do that?" Part 2
In the previous post of this series, I introduced the idea of mathematical modelling and gave a simple model of population growth of bacteria. We ended with the expression
P(t+1) = P(t) + bP(t) - dP(t) ,
This "new" form of our population equation can tell us something about what will happen to our population in the long-run (that is, as we let t get larger and larger). The above expression is an example of a geometric sequence. A sequence is just a list of numbers, following some pattern. In a geometric sequence, each number in the sequence is related to the previous one by multiplying it by a constant value. Using another mathematical trick, let's make a substitution, and let
k = 1 + b - d ,
so that our equation really is
where P(t) is the population at (discrete) time t and b and d are the positive birth and death rates, respectively. Now that we've used our biological knowledge to come up with this equation, let's use some math knowledge to learn more about it and what it can tell us about bacteria growth.
First, we can collect like terms, and make the expression simpler. This is like using the fact that x + 2x = 3x, since we have like terms of x and 2x. Since most people become physically ill when algebra is mentioned, I'll just write down the collected terms in our P(t) formula, and get
P(t+1) = (1 + b - d) P(t) .
First, we can collect like terms, and make the expression simpler. This is like using the fact that x + 2x = 3x, since we have like terms of x and 2x. Since most people become physically ill when algebra is mentioned, I'll just write down the collected terms in our P(t) formula, and get
This "new" form of our population equation can tell us something about what will happen to our population in the long-run (that is, as we let t get larger and larger). The above expression is an example of a geometric sequence. A sequence is just a list of numbers, following some pattern. In a geometric sequence, each number in the sequence is related to the previous one by multiplying it by a constant value. Using another mathematical trick, let's make a substitution, and let
P(t+1) = k P(t) .
In this form, we can really see what will happen to our population. If we start with some initial population size, such as P(0) = 100, then using our simplified equation above, we can find P(1), which is
Then
P(1) = k P(0) = 100k.
Then
P(2) = k P(1) = k (100k) = 100 k^2
and
P(3) = k P(2) = 100k^3
and so on. If we look at the pattern here, we see that there is a general equation for
P(t) = P(0) k^t
if the initial population is P(0) and k = 1+ b - d. If you're still with me at this point, you might be asking "so what does this get us, exactly?". Well, if we know what k is, it tells us a lot. The growth in the population depends completely on what k is, which depends on what b and d are. This makes sense: the birth and death rates determine what will happen to the little bacteria culture we have growing. Let's look at an example to see how.
Suppose that b = 0.2 and d = 0.1, meaning that k = 1 + 0.2 - 0.1 = 1.1 in the equation above. Then if we look at powers of k, we get (1.1)^2 = 1.21, (1.1)^3 = 1.331, (1.1)^4 = 1.4641, and so on. You can check for yourself that each power gets larger, and is larger than the previous one. That means as t increases, (1.1)^t is also increases, and in turn, P(t) is increasing. This makes some intuitive sense, since this is a result of the birth rate being higher than the death rate. Since at each time step there are more births and deaths, the population will increase indefinitely. This is a little bit unrealistic, since there are limits to growth (the amount of food the bacteria have available to them, having no more space in the petri dish, etc). This is due to the simplicity of the model, by the strong assumption that birth and death rates are constant in time. Still, the model does tell us something that makes sense and is realistic.
Let's look at the opposite case now. Suppose that b = 0.1 and d = 0.2 so that k = 1 + 0.1 - 0.2 = 0.9. If we look at powers of this k, we get (0.9)^2 = 0.81, (0.9)^3 = 0.729, (0.9)^4 = 0.6561 and so on. Here, as t increases, (0.9)^t is decreasing, and since it can never be negative, it will shrink smaller and smaller until it is almost zero. Biologically this means the bacteria will eventually go extinct if there more deaths than births at each time step. Again, this is exactly what you'd expect. The model, while pretty simple, makes biological sense, and so if our assumptions are not too strict, it would be a good model of bacteria growing in a laboratory.
So, there's a simple model of population growth. Of course, there more complicated models, like the logistic equation, which has rich mathematical properties as well. Personally, I think this area of study is really cool. Math is as useful as it is fun!
Suppose that b = 0.2 and d = 0.1, meaning that k = 1 + 0.2 - 0.1 = 1.1 in the equation above. Then if we look at powers of k, we get (1.1)^2 = 1.21, (1.1)^3 = 1.331, (1.1)^4 = 1.4641, and so on. You can check for yourself that each power gets larger, and is larger than the previous one. That means as t increases, (1.1)^t is also increases, and in turn, P(t) is increasing. This makes some intuitive sense, since this is a result of the birth rate being higher than the death rate. Since at each time step there are more births and deaths, the population will increase indefinitely. This is a little bit unrealistic, since there are limits to growth (the amount of food the bacteria have available to them, having no more space in the petri dish, etc). This is due to the simplicity of the model, by the strong assumption that birth and death rates are constant in time. Still, the model does tell us something that makes sense and is realistic.
Let's look at the opposite case now. Suppose that b = 0.1 and d = 0.2 so that k = 1 + 0.1 - 0.2 = 0.9. If we look at powers of this k, we get (0.9)^2 = 0.81, (0.9)^3 = 0.729, (0.9)^4 = 0.6561 and so on. Here, as t increases, (0.9)^t is decreasing, and since it can never be negative, it will shrink smaller and smaller until it is almost zero. Biologically this means the bacteria will eventually go extinct if there more deaths than births at each time step. Again, this is exactly what you'd expect. The model, while pretty simple, makes biological sense, and so if our assumptions are not too strict, it would be a good model of bacteria growing in a laboratory.
So, there's a simple model of population growth. Of course, there more complicated models, like the logistic equation, which has rich mathematical properties as well. Personally, I think this area of study is really cool. Math is as useful as it is fun!
Wednesday, November 5, 2008
Potluck 101: The Basics
I love entertaining and social events in general, particularly when they come together well. Your friends come, they bring other cool people, you mix and mingle, eat, drink and be merry. Frankly, I can't think of a better way to spend an evening but with good people at a good party/get-together. There are lots of options you can take when it comes to organizing such an evening, from the ridiculous to the sublime. When you are a student and so are your friends, a potluck is a good way to go. As a person who has been to some good potlucks, I thought I'd have a little series of posts on the in's and out's of this form of entertaining, as I have some experience in such things. I'll go over the basics, as well how to be a good host, a good guest, and some other tips along the way. Who knows, maybe fun will be had. But let's start at the beginning.
Essentially, a potluck is a type of party where every person invited brings an item of food or drink of their choice to share collectively with other guests. It's exactly what it sounds like, where you have to see what "luck" you have as to what's in the "pot". There are a few advantages to having potlucks. One is that you do not have to prepare an extensive menu for food and beverages while having potentially several friends over for a gathering. Another is the cost of having refreshments is lower on the host (which is good if you are a cash-strapped student). And of course, potlucks tend to be less formal events, which can bring an air of relaxation and mirth (especially when that really cute girl you told your friend to invite brings a spectacular dessert). Among the downsides are guests sometimes feeling obligated to make something "good" or "homemade", which can be testing for those culinarily-challenged, and making sure that not everyone brings the same or similar item. Nobody wants a meal of a dozen Caesar salads.
In general, potlucks are good event to plan if 1) cost is an issue, 2) you may be expecting a relatively large crowd and do not have the resources to prepare for them, 3) guests you invite enjoy cooking/baking, or 4) individuals who you would invite to such an event would actually enjoy it, as in, for example, no outrageously picky eaters. They are also a good basis for themed parties, where everyone brings a theme food item (eg East Asian cuisine, dessert). Or you can turn some other party into a potluck just by asking beforehand that guests bring particular items, which eases the obligation of the host. The biggest question you should ask yourself before organizing a potluck-style event is "Will this go over well with the guests I intend to invite?". If the answer is a no, perhaps consider something else.
I'll conclude with some general tips for potluck or potluck-esque events.
Essentially, a potluck is a type of party where every person invited brings an item of food or drink of their choice to share collectively with other guests. It's exactly what it sounds like, where you have to see what "luck" you have as to what's in the "pot". There are a few advantages to having potlucks. One is that you do not have to prepare an extensive menu for food and beverages while having potentially several friends over for a gathering. Another is the cost of having refreshments is lower on the host (which is good if you are a cash-strapped student). And of course, potlucks tend to be less formal events, which can bring an air of relaxation and mirth (especially when that really cute girl you told your friend to invite brings a spectacular dessert). Among the downsides are guests sometimes feeling obligated to make something "good" or "homemade", which can be testing for those culinarily-challenged, and making sure that not everyone brings the same or similar item. Nobody wants a meal of a dozen Caesar salads.
In general, potlucks are good event to plan if 1) cost is an issue, 2) you may be expecting a relatively large crowd and do not have the resources to prepare for them, 3) guests you invite enjoy cooking/baking, or 4) individuals who you would invite to such an event would actually enjoy it, as in, for example, no outrageously picky eaters. They are also a good basis for themed parties, where everyone brings a theme food item (eg East Asian cuisine, dessert). Or you can turn some other party into a potluck just by asking beforehand that guests bring particular items, which eases the obligation of the host. The biggest question you should ask yourself before organizing a potluck-style event is "Will this go over well with the guests I intend to invite?". If the answer is a no, perhaps consider something else.
I'll conclude with some general tips for potluck or potluck-esque events.
- Make sure you ask guests about allergies or dietary restrictions beforehand so guests know not to bring particular dishes. Don't invite a vegan to a "meatluck".
- Ask guests ahead what they plan on preparing. This is to avoid having multiples of the same dish, and to add some variety and balance to the meal.
- With any party or event, invite your guests carefully. Inviting people who have no interest in cooking or enjoying other people's food will likely lead to a downer of a time, so make sure you invite individuals with whom you will have a good time with.
- On the other side of the coin, do not be too draconian when asking what individuals bring. If they do not like cooking or really don't want to prepare a certain dish, don't force them to. This could put a strain on your enjoyment and theirs at the actual event.
- If necessary, establish who will bring needed non-food items (cutlery, cups, corkscrews, etc).
- Make sure there's enough food and drink to for the guests invited. Usually, in my experiences, this is not a problem, but it could potentially be if people cop out and bring nothing but napkins or a bag of chips.
- If necessary, establish beforehand what happens to leftovers when everyone has eaten. This can lead to surprisingly tricky situations, so make sure that people know what the deal is.
- Be polite, courteous and have fun!
"You can use math to do that?"
So, after a short hiatus (due to sketchy internet connections), I thought I'd continue along with some posts relating to my current position as a math grad student. Now that you've had a small taste of mathematical biology, you might be asking yourself "That sounds really interesting and stuff, but how do you actually use math for that?". This question comes up a surprising amount when it comes up that I'm an applied math person, and have done or seen projects on epidemic spreads or why individuals cooperate with one another. Most people just don't see the connection between mathematical modelling and "real life" problems. So, as a primer on this, let's talk about mathematical modelling.
The main idea behind mathematical modelling is to try to describe a problem or situation using mathematics and then gain insight from the maths that you've come up with. Think of it as translating the problem into another language, a language that you can work with to find a solution. Once you have this solution, you can just translate it back and see what it says. The goal is to make sure that the translation in either direction is good, and that nothing gets lost or distorted. Usually, this is the toughest part: coming up with a good model to begin with (and when I say model, I mean a set of expressions describing the problem, not Tyra Banks or Christine Brinkley). Once you get a solid model, usually you can analyze it shed some light onto the problem at hand. Perhaps this is a good time to lead by example.
Suppose I'm a biologist, and I am studying bacteria in laboratory settings. In between the lavish late nights I enjoy in such a profession, I want to understand what influences bacteria population growth or extinction. Here, there's at least two ways I can look how bacteria grows
First, I'll need some variables to work with. Suppose that P(t) is the population of the bacteria at time t. I want to find an expression for what P(t) for any general time t, so the goal is to find a mathematical model that represents this.
Now, I'll need to make some assumptions.
So, with these assumptions in place, I can now develop a model. The simplest model has the form
P(t+1) = P(t) + births - deaths
which means that my population at the next time step is just the population at the previous time step plus the increase from births and less the decrease from deaths. In population dynamics, many, many models start out with "births - deaths".
If assume that the birth and death rates are just a constant proportion of the population at time t, then I can turn this into
P(t+1) = P(t) + bP(t) - dP(t)
where b is the constant birth rate, and d is a constant death rate, and both are fixed constants. What these mean is that at each time step a proportion b of the population reproduces (remember, these are bacteria, so let's just say that they divide to reproduce), and another proportion d die.
So, there is it, I've come up with a mathematical model of bacteria growing. I've now translated the problem of how bacteria grow into mathematics. But that's a lot of work for one day for mathematical biologist, so in my next post, I'll look at working with this model to get some results out of it, and translating those back into real answers.
The main idea behind mathematical modelling is to try to describe a problem or situation using mathematics and then gain insight from the maths that you've come up with. Think of it as translating the problem into another language, a language that you can work with to find a solution. Once you have this solution, you can just translate it back and see what it says. The goal is to make sure that the translation in either direction is good, and that nothing gets lost or distorted. Usually, this is the toughest part: coming up with a good model to begin with (and when I say model, I mean a set of expressions describing the problem, not Tyra Banks or Christine Brinkley). Once you get a solid model, usually you can analyze it shed some light onto the problem at hand. Perhaps this is a good time to lead by example.
Suppose I'm a biologist, and I am studying bacteria in laboratory settings. In between the lavish late nights I enjoy in such a profession, I want to understand what influences bacteria population growth or extinction. Here, there's at least two ways I can look how bacteria grows
- Collect data from the bacteria in petri dishes, and make some plots with it.
- Create a mathematical model based on what I know about bacteria.
First, I'll need some variables to work with. Suppose that P(t) is the population of the bacteria at time t. I want to find an expression for what P(t) for any general time t, so the goal is to find a mathematical model that represents this.
Now, I'll need to make some assumptions.
- Assume the population changes at regular, discrete time intervals (for example, the population only changes every hour, and not all the time). This may seem a bit strong ("bacteria reproduce all the time, like a million times a minute!" you say) but this is part of translating the bacteria problem into math. I have to make some kind of simplification or else I won't have anything to work with.
- Also, I assume that the only way the population can change is by bacteria reproducing or dying. This means that I can't just "add in" bacteria once I've started, nor can they just "appear" out of nowhere. They have to be the result of reproduction.
- Finally, I'll assume that bacteria reproduce and die directly proportional to the population size. In layman's terms, this means that the more bacteria there are, the more will reproduce or die at any time step.
So, with these assumptions in place, I can now develop a model. The simplest model has the form
which means that my population at the next time step is just the population at the previous time step plus the increase from births and less the decrease from deaths. In population dynamics, many, many models start out with "births - deaths".
If assume that the birth and death rates are just a constant proportion of the population at time t, then I can turn this into
where b is the constant birth rate, and d is a constant death rate, and both are fixed constants. What these mean is that at each time step a proportion b of the population reproduces (remember, these are bacteria, so let's just say that they divide to reproduce), and another proportion d die.
So, there is it, I've come up with a mathematical model of bacteria growing. I've now translated the problem of how bacteria grow into mathematics. But that's a lot of work for one day for mathematical biologist, so in my next post, I'll look at working with this model to get some results out of it, and translating those back into real answers.
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