Here, you’ve been given a function (x3 + x – 1) set to zero. So if the function has at least one solution, then that solution is a root (i.e. a zero). In order to apply Bolzano’s theorem, you need to find out two things:
Step 2: Locate the endpoints and see if they have opposite signs. Here, you’re given the function and the endpoints [0, 1], so plug the endpoints into the function and see what values come out:
The two values have opposite signs, and the function is continuous. Therefore, Bolzano’s theorem tells us that the equation does indeed have a real solution. A quick look at the graph of x3 + x – 1 can verify our finding:
Differential equations have many solutions and it’s usually impossible to find them all. To narrow down the set of answers from a family of functions to a particular solution, conditions are set. These conditions can be initial conditions (which define a starting point at the extreme of an interval) or boundary conditions (which define bounds that constrain the whole solution). Different types of boundary conditions can be imposed on the boundary of a domain.
One way to think of the difference between the two is that initial conditions deal with time, while boundary conditions deal with space. Boundaries can describe all manner of shapes: e.g. triangles, circles, polygons.
Dirichlet: Specifies the function’s value on the boundary. For example, you could specify Dirichlet boundary conditions for the interval domain [a, b], giving the unknown at the endpoints a and b. For two dimensions, the boundary conditions stretch along an entire curve; for three dimensions, they must cover a surface. This type of problem is called a Dirichlet Boundary Value Problem..
Neumann: Similar to the Dirichlet, except the boundary condition specifies the derivative of the unknown function. For example, we could specify u′(a) = α which imposes a Neumann boundary condition at the right endpoint of the interval domain [a, b].
Robin: A weighted combination of the function’s value and its derivative. For example, for unknown u(x) on the interval domain [a, b] we could specify the Robin condition u(a) −2u′(a) = 0.
Mixed: Similar to the Robin, except that parts of the boundary are specified by different conditions. For example, on the interval [a, b], the unknown u′(x) at x = a could be specified by a Neumann condition and the unkownn u(x) at x = b could be specified by a Dirichlet condition. [1]
As an example, let’s say you wanted to find the equation for a straight line on a curve-length function between two points (a, A) and (b, B). The function could be set up as with the points as boundary conditions [2]:
The Bray Curtis dissimilarity is used to quantify the differences in species populations between two different sites. It’s used primarily in ecology and biology, and can be calculated with the following formula:
To calculate Bray-Curtis, let’s first calculate Cij (the sum of only the lesser counts for each species found in both sites). Goldfish are found on both sites; the lesser count is 6. Guppies are only on one site, so they can’t be added in here. Rainbow fish, though, are on both, and the lesser count is 4.
To make it easy to work with, it’s often multiplied by 100, and then treated as a percentage. You may see a Bray Curtis dissimilarity of 0.21, for instance, being referred to as a Bray Curtis dissimilarity percent of 21%.
There’s another percentage which is often used to describe species counts; this one, though, tells you how similar two sites are rather than how different. It’s called the Bray Curtis index, and to calculate it you simply subtract the Bray Curtis dissimilarity (remember, a number between 0 and 1) from 1, then multiply by 100.
Let’s calculate this number for the fish example. The Bray Curtis dissimilarity was 0.39, and if we wanted it in terms of percentages we would have called it 39%. But the Bray Curtis index will be (1 – 0.39) · 100, or 61%. Notice this is, in a way, the opposite of the Bray Curtis dissimilarity. Identical sites have a Bray Curtis dissimilarity of 0, or 0%, and a Bray Curtis index of 100%. Sites which share no species would have a Bray Curtis dissimilarity of 1 (100%), and a Bray Curtis index of 0.
To calculate the Bray-Curtis dissimilarity between two sites you must assume that both sites are the same size, either in area or volume (as is relevant to species counts). This is because the equation doesn’t include any notion of space; it works only with the counts themselves.
Both the Levene and B-F tests transform dependent variables for use in an ANOVA test. The only difference between the two tests is in how those transformed variables are constructed. The Levene test uses deviations from group means, which usually results in a highly-skewed set of data; This violates the assumption of normality. The Brown-Forsythe test attempts to correct for this skewness by using deviations from group medians. The result is a test that’s more robust. In other words, the B-F test is less likely than the Levene test to incorrectly declare that the assumption of equal variances has been violated.
The test statistic used in a regular ANOVA is an F-statistic. The statistic used in an ANOVA with transformed variables is sometimes called a W-Statistic — but it’s really just an F-Statistic with a different name. It should not be confused with the coefficient of concordance W-statistic, which is used to assess agreement between raters.
For the most part, the B-F test is thought to perform as well as or better than other available tests for equal variances. However, Glass and Hopkins (1996 p. 436) state that the Levene and B-F tests are “fatally flawed”; It isn’t clear how robust they are when there is significant differences in variances and unequal sample sizes. Hill et. al (2006) advise repeating the test using a non-parametric method.
Best Statistics SiteCalculus is a unique branch of mathematics, and includes many symbols and equations that are also unique. Some are intuitive and make sense at a glance, but others can be very confusing if they are new to you. Here is a quick overview of some of the calculus symbols you will come across.
This is the format for writing a limit in calculus. When read aloud, it says “The limit of the function f of x, as x tends to 0.” (See: What is a limit?)
This is another symbol for a derivative. You can read it as “The derivative of y with respect to x.” Y is equivalent to f(x), as y is a function of x itself.
Both of these symbols represent the second derivative of the function, which means you take the derivative of the first derivative of the function. You would read it simply as “The second derivative of f of x.”
These symbols represent the nth derivative of f(x). Much like the second derivative, you would perform differentiation on the formula for n successive times. It reads as “The nth derivative of f of x.” If n were 4, it would be “The fourth derivative of x,” for example.
This symbol represents integration of the function. Integration of a function is the opposite of the differentiation. The variables a and b represent the lower limit and upper limit of the section of the graph the integral is being applied to. If there are no values for a and b, it represents the entire function. You would read it as “The integral of f of x with respect to x (over the domain of a to b.)”
A difference, or change, in a quantity: For example , where we say “delta x” we mean how much x changes. You will often come across delta in this context when working with values that characteristically change, such as velocity or acceleration. We also see this meaning when working with slope; The slope is the ratio of the vertical and horizontal changes between two points on a line. You’ll see the use of upper-case delta in the formula for slope: Slope = rise / run = Δy/Δx.
Lower-case δ is used when calculating limits. The epsilon-delta definition of a limit is a precise method of evaluating the limit of a function. Epsilon (ε) in calculus terms means a very small, positive number. The epsilon-delta definition tells us that:
Where f(x) is a function defined on an interval around x0, the limit of f(x) as x approaches x0 is L. For every ε > 0 there exists δ > 0 such that for all x:
This definition is particularly useful; It makes sure that values returned by the function f(x) are as close to the limit as possible by only using points in a small interval around x0. It gives us a useful measure regardless of how close to L we wish f(x) to be.
Case studies are in-depth studies of a phenomenon, like a person, group, or situation. The phenomenon is studied in detail, cases are analyzed and solutions or interpretations are presented. It can provide a deeper understanding of a complex topic or assist a person in gaining experience about a certain historical situation. Although case studies are used across a wide variety of disciplines, they are more frequently found in the social sciences.
Case studies are a type of qualitative research. This method does not involve statistical hypothesis testing. It has been criticized as being unreliable, too general, and open to bias. To avoid some of these problems, studies should be carefully planned and implemented. The University of Texas suggests the following six steps for case studies to ensure the best possible outcome:
Choose the cases and state how data is to be gathered and which techniques for analysis you’ll be using. Well designed studies consider all available options for cases and for ways to analyze those cases. Multiple sources and data analysis methods are recommended.
Prepare to collect the data. Consider how you will deal with large sets of data in order to avoid becoming overwhelmed once the study is underway. You should formulate good questions and anticipate how you will interpret answers. Multiple collection methods will strengthen the study.
Text books are including more real-life studies to veer away from the “clean” data sets that are found in traditional books. These data sets do little to prepare students for applying statistical concepts to their ultimate careers in industry or the social sciences.
Censoring in a study is when there is incomplete information about a study participant, observation or value of a measurement. In clinical trials, it’s when the event doesn’t happen while the subject is being monitored or because they drop out of the trial.
