Let $f(x) in mathbb{Z}[x]$; for each integer $alpha$ it is interesting to consider the number of iterates $n_{alpha}$, if possible, needed to satisfy $f^{n_{alpha}}(alpha) = alpha$. The sets ${alpha, f(alpha), ldots, f^{n_{alpha} - 1}(alpha), alpha}$
generated by the iterates of $f$ are called cycles. For $mathbb{Z}[x]$ it is known that cycles of length 1 and 2 occur, and no others. While much is known for extensions to number fields, we concentrate on extending $mathbb{Z}$ by adjoining reciprocals of primes. Let $mathbb{Z}[1/p_1, ldots, 1/p_n]$ denote $mathbb{Z}$ extended by adding in the reciprocals of the $n$ primes $p_1, ldots, p_n$ and all their products and powers with each other and the elements of $mathbb{Z}$. Interestingly, cycles of length 4, called 4-cycles, emerge for polynomials in $mathbb{Z}left[1/p_1, ldots, 1/p_nright][x]$ under the appropriate conditions. The problem of finding criteria under which 4-cycles emerge is equivalent to determining how often a sum of four terms is zero, where the terms are $pm 1$ times a product of elements from the list of $n$ primes. We investigate conditions on sets of primes under which 4-cycles emerge. We characterize when 4-cycles emerge if the set has one or two primes, and (assuming a generalization of the ABC conjecture) find conditions on sets of primes guaranteed not to cause 4-cycles to emerge.
We generalize a construction of families of moderate rank elliptic curves over $mathbb{Q}$ to number fields $K/mathbb{Q}$. The construction, originally due to Steven J. Miller, Alvaro Lozano-Robledo and Scott Arms, invokes a theorem of Rosen and Silv
erman to show that computing the rank of these curves can be done by controlling the average of the traces of Frobenius, the construction for number fields proceeds in essentially the same way. One novelty of this method is that we can construct families of moderate rank without having to explicitly determine points and calculating determinants of height matrices.
A statistical model for the fragmentation of a conserved quantity is analyzed, using the principle of maximum entropy and the theory of partitions. Upper and lower bounds for the restricted partitioning problem are derived and applied to the distribu
tion of fragments. The resulting power law directly leads to Benfords law for the first digits of the parts.
The spherical orthogonal, unitary, and symplectic ensembles (SOE/SUE/SSE) $S_beta(N,r)$ consist of $N times N$ real symmetric, complex hermitian, and quaternionic self-adjoint matrices of Frobenius norm $r$, made into a probability space with the uni
form measure on the sphere. For each of these ensembles, we determine the joint eigenvalue distribution for each $N$, and we prove the empirical spectral measures rapidly converge to the semicircular distribution as $N to infty$. In the unitary case ($beta=2$), we also find an explicit formula for the empirical spectral density for each $N$.
In this note we show how the irrationality measure of $zeta(s) = pi^2/6$ can be used to obtain explicit lower bounds for $pi(x)$. We analyze the key ingredients of the proof of the finiteness of the irrationality measure, and show how to obtain good
lower bounds for $pi(x)$ from these arguments as well. Whi
Given an ensemble of NxN random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N --> oo. While this has been proved for many thin patterned ensemble
s sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view m as a dial we can turn from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities f_m show a visually stunning convergence to the semi-circle as m tends to infinity, which we prove. In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that f_m is the product of a Gaussian and a degree 2m-2 polynomial; the formula equals that of the m x m Gaussian Unitary Ensemble (GUE). The proof is by the moments. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the m-block circulant pattern, dropping the assumption that the m random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending on not only the frequency at which each element appears, but also the way the elements are arranged.
