Tuesday, April 30, 2013

The Arctic and North Atlantic Oscillations

The Arctic Oscillation (AO) and North Atlantic Oscillation (NAO) are two climatological phenomena that characterize the changes in atmospheric pressure over their respective regions, the Arctic, and the North Atlantic.

Based on anomalies in the pressure of the regions from their long-term averages (computed over a period of over 100 years), the oscillations are assigned parameters, called the AO index and NAO index, respectively, which change with time. The sign of the parameter (whether it is positive or negative) can predict certain features of the climate of much of the northern hemisphere, and are particularly important in the winter.

The Arctic Oscillation index is computed from the pressures of the subtropical and subarctic regions. The pressure gradient between the two latitudes determines the sign of the AO index. If pressures are higher in the subtropics than normal and lower in the subarctic, the AO index is positive, and the AO is said to be in its positive phase, while if subtropical pressures are anomalously low and subarctic pressures anomalously high, the AO index is negative, and the AO is said to be in its negative phase.

The above figure shows the general shape of the path of the jet stream during positive and negative phases of the AO. During positive AO, the jet stream tends to be stronger and more linear in its path during the winter, locking cold air in the Arctic regions and generally leading to warmer winters in the subtropical regions. Since the position of the jet stream allows tropical moisture to venture farther north, the subtropics are also generally wetter during these periods.

When the AO index is negative, the jet stream becomes more sinusoidal, with the amplitude of the variations in the jet stream's latitude generally proportional to the magnitude of the negative phase. Where the jet stream dips south, large masses of cold air can engulf regions for days or weeks, generally resulting in colder, snowier winters. At the same time, however, the upswings in the jet stream can bring warm air to generally cold areas. Winters in the northern hemisphere with a negative AO index generally tend to be more volatile.

The NAO is closely related to the AO, except that the index is determined only by the pressure gradient between latitudes only in a very specific region: the North Atlantic near 30°W longitude, or the subtropical region near the Azores Islands, and the Arctic region near Iceland. The effects of the NAO on the jet stream are similar to the AO, but they are not the same. The NAO index is generally a very good indicator of the winter temperature anomaly in the eastern U.S. and Europe, and though its sign usually agrees with that of the AO, this is not always the case:

The graphs of the AO index (top) and the NAO index (bottom) over the winters of roughly the same time period, from the late 1800's to the early 2000's. The indices clearly are related; both are predominantly negative in the period 1960-1980 and predominantly positive from 1980 to 2000, but on some years, they disagree. For example, if the NAO index is positive and the AO index negative, the jet stream may be straight over the Atlantic, bringing warm air to the eastern U.S. and western Europe, but sinusoidal elsewhere. This happened, for example, in the winter of 2011-2012. Conversely, if the NAO index is negative and the AO positive, there may be a large dip in the jet stream over the U.S. and a weak pressure gradient over the Atlantic, but cold air masses may be fairly well confined to the Arctic at other longitudes. This situation occurred in the winter of 2008-2009.

These oscillations, relative to El Nino and La Nina, are notoriously hard to predict. In addition, while the weather of any given winter is influenced by the El Nino/La Nina, the AO, and the NAO, many other factors also come into play. However, the fairly consistent accuracy with which the AO and NAO have predicted the winter weather of North America and Europe serve to exemplify the importance of atmospheric phenomena even in determining the climate of a region thousands of miles away.

Sources: The Winters of Our Discontent from The Scientific American, December 2012, AO and NAO on Wikipedia

Monday, April 15, 2013

More About Constructible Numbers and Figures

This series of posts deals with determining which geometric figures are "constructible", that is, can be formed using only a compass and straightedge.

The set of constructible real numbers, or those numbers whose absolute value is equal to the length of a line segment constructible with compass and straightedge, has already been shown to be of the type called a field. In addition, this field contains all of the rational numbers. But what others, if any, does it contain?

It may be noted that the circle has not featured in any of the constructions thus far. In fact, by use of a circle, one can construct a segment whose length is the (positive) square root of that of a given segment. The construction is illustrated below.

A segment of length a1/2 is constructed by first drawing segments of length 1 and a on end (1). Then, a circle is drawn with the combined segment (of length 1 + a) as a diameter (2). Finally, a perpendicular is erected from the diameter at the point of intersection of the two segments to the circle, and two other segments are drawn connecting the point of intersection of the perpendicular and the circle to the endpoints of the diameter (3). The resulting figure has three triangles, the largest partitioned into the two smaller by the perpendicular. One need only observe that the large triangle is a right triangle, as one of its angles subtends a semicircular arc, and since this large triangle shares a side and an angle with each of the two smaller right triangles, it is similar to each. The two smaller right triangles are then also perpendicular to each other, and so the ratio of 1 to the length of the perpendicular must be equivalent to the ratio of the same length to a. The length of the perpendicular is thus the square root of a.

Thus the field of constructible numbers includes any number that can be derived from a finite sequence of additions, subtractions, multiplications, divisions, and square roots from the unit length 1. In fact, these are all of the constructible numbers. To see this, note first that the construction of segments in the plane involves only the intersections of lines and/or circles. The general equation for a line is ax + by = c, a linear equation, and the general equation for a circle is (x - h)2 + (y - k)2 = r2. It is clear that in solving for the intersection of these two types of functions, the highest degree one could encounter for the intersection points to satisfy is 2, i.e., a quadratic. Finally, by the quadratic formula and the distance formula, which each involve only square roots, the most general type of number one can construct can be seen to be one involving nested square roots, a conclusion in agreement with the previous result.

To illustrate the power of this new concept, we know turn to some applications. First, we relate our result to the geometric construction problems posed at the beginning of this series of posts:

Example 1:
The problem of "squaring the circle" was shown to require the constructibility of the length π1/2. This condition is equivalent to the constructibility of π and is therefore impossible, as π is what is called a transcendental irrational number; there is no polynomial with integer (or rational, by extension) coefficients with π as a root.
Example 2:
"Doubling the cube" was shown to require the constructibility of a segment of length 21/3. This problem is impossible as well, because though there is a polynomial with this number as a root, namely x3 - 2 = 0, this polynomial is of degree 3, not 2, and cannot be factored in any way to reduce its degree. Please note that even though a number such as 21/4 is the root of the degree 4 polynomial x4 - 2 = 0, the substitution y = x2 reduces it to two polynomials of degree 2, and this number is thus constructible.
Example 3: To illustrate the applicability of this concept to figures that actually are constructible, consider the equilateral triangle. Since all three sides are of the same (arbitrary) length, the ability to draw an equilateral triangle depends on its angles, all of which are 60°. To equate the construction of an angle to the construction of a segment, we use trigonometry:

The above figure illustrates that the constructbility of the angle 60° is equivalent to that of the segments of lengths cos(60°) = 1/2 and sin(60°) = 31/2/2. If they are given, a right triangle can be drawn with legs of these lengths, thereby giving the angle. Since 31/2/2 involves only a square root, it is constructible, and 1/2 obviously is. Thus the equilateral triangle can be drawn with compass and straightedge as well.

The problem of constructibility played a greater role in ancient times than it does today. The standards that constitute "existence" for a mathematical object, though still debated, are much looser than in the time of the Ancient Greek mathematicians. For example, we now accept cubic curves, for example, as perfectly reasonable mathematical objects, even though they cannot be constructed with compass and straightedge (in fact, an arbitary point on one of these curves may not be constructible). The problem now mainly serves as a mathematical curiosity, and as an example of how one can calculate the power, in this case the constructing power, of certain systems in mathematics.

Sources: A First Course in Abstract Algebra by John B. Fraleigh, Constructible Number on Wikipedia

Sunday, April 7, 2013

Constructible Numbers and Figures

The problem of constructible figures—determining which geometrical objects can be constructed using only the straightedge and compass—was a longstanding problem of mathematics, resolved in the early 19th century. It is closely related to the famous problems of squaring the circle (constructing a square of equal area to a given circle) doubling the cube (finding a cube with double the volume of a given cube), and trisecting the angle (constructing an angle with measure exactly a third of a given one), and actually encompasses these problems, as we shall see below.

The Ancient Greeks did not merely focus on the problem of determining which figures were constructible to categorize geometric objects—their standards of rigor were such that, if a curve or figure could not be constructed, they did not consider it to exist!

But, as we shall show, the problem of constructible figures can be reduced to determining what length line segments can be constructed. Thus the set of constructible figures is determined by a set of real numbers that corresponds to a set of line segments with these numbers as lengths. To illustrate this concept, we will reduce a few of the problems mentioned above to the problem of constructing a line segment of a given length.

The problem of squaring the circle can be reduced to finding a line segment that is one side of the square, i.e., constructing a line segment of length π1/2. (More precisely, the construction requires the ability to construct a line segment whose length forms a ratio of π1/2 to the radius of the circle. However, assuming the base segment to be of length 1, the problem reduces to the one above)

The problem of doubling the cube is technically in three-dimensions, but it depends on the ability, in plane geometry, to construct a line segment of length 21/3 (again, this is actually the ratio of the side of the larger cube to that of the smaller).

Many other construction problems can similarly be translated into the language of lengths of line segments or ratios of such. Now, the problem is to find what lengths can be constructed. First, we determine what sort of set the set of constructible lengths is. To do this, consider two lengths a and b that are given to be constructible. In other words, if one were forming geometric objects on a piece of paper, one would have, in addition to a compass and straightedge, objects of length a and b from which things can be measured. What other lengths can be obtained from these?

Clearly, given lengths a and b, one can place two lines of these respective lengths end to end, giving a line of length a + b.

Similarly, the length b - a can be constructed from lines of respective lengths a and b. Some care must be taken here, as this construction only yields a positive number for the length of b - a if the length b is greater than the length a. Alternatively, one could consider oriented line segments, or vectors, with initial and terminal points that can be "negative". Here we shall limit ourselves to regular line segments, but allow negative values to be members of the set of constructible lengths. Hence a real number x is a member of the set if its absolute value is the length of a constructible line segment.

Constructing a line segment whose length is the product of two given numbers is a little trickier. In the above figure, we assume a and b to be positive. First, mark a line segment of length a on a given ray, beginning at the endpoint of the ray. Then, draw any other ray out of the endpoint not coincident with the first ray (1). On this second ray, mark two segments beginning from the endpoint of lengths 1 and b. Draw a line segment, l, connecting the the other end of the line segment of length 1 to the end of the segment of length a (2). Finally, draw a line, k, through the end of the segment of length b parallel to l. The intersection of this line with the initial ray demarcates a line segment of length ab from the endpoint. This conclusion follows from the similar triangle law, as the ratio of 1 to a is the same as the ratio of b to ab.

A similar method, again using similar triangles and ratios involving their sides, brings about a segment of length a/b for positive a and b. Begin by marking a line segment of length a along a ray, and draw another ray sharing its endpoint with the first (1). On the second ray, draw a line segment of length b from the endpoint, and connect the opposite end of that segment with that of the segment of length a, forming l (2). Finally, draw the line segment k, beginning at the point on the second ray one unit away from the endpoint (3). The intersection of k with the first ray will define a segment of length a/b, as the ratio of b to a is the same as that of 1 to a/b.

Therefore, given lengths a and b, one can construct a + b, a - b, ab, and a/b. In other words, performing any of these operations on two members of the set of constructible lengths creates another constructible length. The set is said to be closed under these operations. In addition, it is interesting to note that the unit length 1 is needed to compute the product and quotient of lengths. Therefore, we assume all lengths to be in terms of the unit 1, which is also given. Finally, we assume the existence of the length 0, which is simply a point. (It is easy to confirm that the algebraic properties of 0 are satisfied by its geometric counterpart. If one takes b to be 0 in the diagram for division, the line l coincides with the first ray, and the line k, being parallel to l but through a different point, never intersects the first ray. This is consistent with division by zero being undefined.)

A set of this type, closed under addition, subtraction, multiplication, and division, and including 0 and 1, (though division by 0 is undefined) is called a field. Thus the set of constructible real numbers is a field. Any rational function of a and b is a member of this field, where a rational function of a and b is a quotient of polynomials f(a,b)/g(a,b) where both are of finite degree and g(a,b) does not equal 0. Thus the set of constructible real numbers contains all rational numbers, a rather intuitive conclusion. The question of what other numbers the set contains, and its consequences on the motivating problems discussed above, is addressed in the next post, coming April 15.

Sources: History of Mathematical Thought from Ancient to Modern Times, vol. 2, by Morris Kline, A First Course in Abstract Algebra by John B. Fraleigh