Let $G$ be the Lie group $SO\left(n,ℝ\right)\times SO\left(n,ℝ\right)$ and let $V$ be the vector space of $n\times n$ real matrices. An action of $G$ on $V$ is given by

$$\left(g,h\right).v:=g^{-1}vh,\left(g,h\right)\in G,v\in V.$$

We consider the orbits of this group action and demonstrate a cross-section to the orbits. We then determine the stabilizer for a typical element in this cross-section and completely describe the fundamental group of an orbit of maximal dimension.

We use elementary linear algebra to explicitly calculate a basis for, and the dimension of, the space of degree-three covariants for a binary form of arbitrary degree. We also give an explicit basis for the subspace of covariants complementary to the space of degree-three reducible covariants.

Norms of certain composition operators are given in terms of their symbols in some finite-dimensional setting. Then a family of isometric composition operators acting on certain vector spaces is identified.

We show that if $f$ is a diffeomorphism of a manifold to itself, $\Lambda$ is a mixing (or transitive) hyperbolic set, and $V$ is a neighborhood of $\Lambda$, then there exists a mixing (or transitive) hyperbolic set $\tilde{\Lambda }$ with a Markov partition such that $\Lambda \subset \tilde{\Lambda }\subset V$. We also show that in the topologically mixing case the set $\tilde{\Lambda }$ will have a unique measure of maximal entropy.

An elastica is a bendable one-dimensional continuum, or idealized elastic rod. If such a rod is subjected to compression while its ends are constrained to remain tangent to a single straight line, buckling can occur: the elastic material gives way at a certain point, snapping to a lower-energy configuration.

The bifurcation diagram for the buckling of a planar elastica under a load $\lambda$ is made up of a trivial branch of unbuckled configurations for all $\lambda$ and a sequence of branches of buckled configurations that are connected to the trivial branch at pitchfork bifurcation points. We use several perturbation expansions to determine how this diagram perturbs with the addition of a small intrinsic shape in the elastica, focusing in particular on the effect near the bifurcation points.

We find that for almost all intrinsic shapes $ϵf\left(s\right)$, the difference between the buckled solution and the trivial solution is $O\left({ϵ}^{1∕3}\right)$, but for some ineffective $f$, this difference is $O\left(ϵ\right)$, and we find functions $u_j\left(s\right)$ so that $f$ is ineffective at bifurcation point number $j$ when $\langle f,u_j\rangle =0$. These ineffective perturbations have important consequences in numerical simulations, in that the perturbed bifurcation diagram has sharper corners near the former bifurcation points, and there is a higher risk of a numerical simulation inadvertently hopping between branches near these corners.

In 1998 Filipponi and Hart introduced a number of Fibonacci product identities. This paper provides a combinatorial proof for such identities via tilings. The methods used in the proof are then further used to produce some new Zeckendorf representations and a known Fibonacci identity.

We develop the theory of frames and Parseval frames for finite-dimensional vector spaces over the binary numbers. This includes characterizations which are similar to frames and Parseval frames for real or complex Hilbert spaces, and the discussion of conceptual differences caused by the lack of a proper inner product on binary vector spaces. We also define switching equivalence for binary frames, and list all equivalence classes of binary Parseval frames in lowest dimensions, excluding cases of trivial redundancy.

Let $A$ and $B$ be finite subsets of positive real numbers. Solymosi gave the sum-product estimate $max\left(|A+A|,|A\cdot A|\right)\ge {\left(4\lceil log\left|A\right|\rceil \right)}^{-1∕3}|A{|}^{4∕3}$, where $\lceil \rceil$ is the ceiling function. We use a variant of his argument to give the bound

$$max\left(|A+B|,|A\cdot B|\right)\ge {\left(4\lceil log\left|A\right|\rceil \lceil log\left|B\right|\rceil \right)}^{-1∕3}\left|A{|}^{2∕3}\right|B{|}^{2∕3}.$$

(This isn’t quite a generalization since the logarithmic losses are worse here than in Solymosi’s bound.)

Suppose that $A$ is a finite subset of real numbers. We show that there exists an $a\in A$ such that $|aA+A|\ge c|A{|}^{4∕3}$ for some absolute constant $c$.

We prove the double bubble conjecture in ${ℝ}^2$: that the standard double bubble in ${ℝ}^2$ is boundary length-minimizing among all figures that separately enclose the same areas. Our independent proof is given using the new method of metacalibration, a generalization of traditional calibration methods useful in minimization problems with fixed volume constraints.