Groups as Unions of Proper Subgroups
By: Mira Bhargava
When is a group the union of two of its proper subgroups? The answer, as is now well known, is never.
There are several natural variants of the above question, however, that have positive and, in many cases, very pretty and surprising answers. For example, when is a group the union of three, four, or five proper subgroups? When is it the union of n proper subgroups? What about n proper normal subgroups?
In this article, we discuss some of the many fascinating recent results in this direction, and also describe some interesting related questions that remain open.
Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results
By: Rick Mabry and Paul Deiermann
Two friends, Gray and White, dine out to share a pizza. Their server brings a pizza that has been sliced using N straight, concurrent, equiangular cuts. This should enable Gray and White to share the pizza equally, presuming they both take the same number (N) of slices. The trouble is, the pizza-cutter missed the mark — the point of concurrency is not the center! Questions now arise. In particular, How can the pizza be equally shared? And, Who says they want to share equally?
It has been known since the 1960s that when N is even and greater than 2, an answer to the first question is for Gray and White to choose alternate slices about the point P of concurrency. (The figures below show the situation for N=6 and N=7.) This alternation scheme also results in equal shares for any N, quite obviously, if the center O lies on one of the cuts. But if N is odd, and if O does not lie on a cut, then, as has been known since the 1990s, this method of alternating slices does not result in equal sharing. Those more inclined to the second question above now want to know, Who gets the most pizza?
It was conjectured by Stan Wagon and others, that for N=3,7,11,15,..., whoever gets the center gets the most pizza, while for N=5,9,13,17,..., whoever gets the center gets the least. We prove this Pizza Conjecture by first showing its equivalence to a (pretty wild) trigonometric inequality. This inequality is proved with the aid of a theorem that counts lattice paths. Our main theorem is sufficiently general that, as a bonus, results concerning the equiangular slicing of other dishes are obtained.
Parallel Transport and Thomas-Wigner Rotation
By: Nieves Alamo and Carlos Criado
The Thomas-Wigner effect consists in the fact that, after a gyroscope has completed a circular trip, the initial and final directions of its spin axis do not coincide. In this article we give, in an elementary way, a geometric explanation of this effect as an application of the concept of parallelism on a hyperbolic space. For that we prove that the vector field determined by the directions of the gyroscope’s spin axis in the hyperbolic space of relativistic velocities is parallel. By using only elementary calculus, we calculate the angle by which this vector is twisted after a complete circular trip.
Symmetric and Alternating Groups Generated by a Full Cycle and Another Element
By: Daniel Heath, I. M. Isaacs, John Kiltinen, and Jessica Sklar
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Let σ be a permutation of the integers 1,2, ..., k. In this note, we establish a necessary and sufficient condition on σ such that for all sufficiently large integers n, the permutation group generated by σ and the n-cycle (1,2, ..., n) will be either the full symmetric group or the alternating group on the set of integers 1,2, ..., n.
Revisiting a Century-Old Characterization of Baire Class One, Darboux Functions
By: Michael J. Evans and Paul D. Humke
In most every first course in real analysis one proves that derivatives have the intermediate value property (that is, are Darboux) and also are the pointwise limits of sequences of continuous functions (that is, are Baire class one). The problem of characterizing derivatives is old and celebrated, but has also been quite vexing. Characterizations have been neither transparent nor particularly useful. However, there are many quite elegant and very useful characterizations of real-valued functions of one variable that are both Darboux and Baire class one.
The situation in higher dimensions is considerably more delicate. First, there are several ``natural'' ways to generalize the notion of Darboux to functions of several variables, but these are no longer equivalent. To confound matters further, each of the many equivalences of Darboux, Baire class one functions of one variable gives rise to one or more equally ``natural'' generalizations to functions of several variables, and these too are not equivalent, in many cases not even comparable.
In 1997, Jan Malý showed that partial derivatives of differentiable functions map closed convex sets with nonempty interior to intervals.
In this paper we give an elementary proof that Malý's condition is equivalent to a natural generalization to the several variable situation of a celebrated characterization, due to W. H. Young, of Darboux, Baire one functions of one real variable.
A Proof of the Cayley Hamilton Theorem
By: Chris Bernhardt
We use formal power series to give a simple proof of the Cayley Hamilton Theorem.
A New Constructive Proof of the Malgrange-Ehrenpreis Theorem
By: Peter Wagner
It is very well known that -1/(4π|x|) is a fundamental solution of the 3-dimensional Laplacean. The famous Malgrange-Ehrenpreis theorem states that every linear partial differential operator with constant coefficients has a fundamental solution in the space of distributions. In this article, this theorem is proven constructively by means of a very elementary formula.
Understanding and Using Linear Programming
By: Jiří Matouek and Bernd Gärtner
Introduction to Optimization
By: Pablo Pedregal
Reviewed by: Allen Holder