Many sequences from number theory, such as the primes, are defined by recursive procedures, often leading to complex local behavior, but also to graphical similarity on different scales — a property that can be analyzed by fractal dimension. This paper computes sample fractal dimensions from the graphs of some number-theoretic functions. It argues for the usefulness of empirical fractal dimensi...

We construct a Π1-class X that has classical packing dimension 0 and effective packing dimension 1. This implies that, unlike in the case of effective Hausdorff dimension, there is no natural correspondence principle (as defined by Lutz) for effective packing dimension. We also examine the relationship between upper box dimension and packing dimension for Π1-classes.

Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C∗ is prime. This definition, however, is not intrinsic it strongly depends on the base ring being a field...

Using the theory of non-negative intersections, duality of the Schubert varieties, and Pieri-type formula for a maximal orthogonal grassmannian, we get an upper bound for the canonical dimension cd(Spinn) of the spinor group Spinn. A lower bound is given by the canonical 2-dimension cd2(Spinn), computed in [8]. If n or n + 1 is a power of 2, no space is left between these two bounds; therefore ...

We show that the classical discrete logarithm problem over prime fields can be reduced to that of solving a system of linear modular equations.

Assuming the axiom of constructibility, points in closed discrete subspaces of certain normal spaces can be simultaneously separated. This is a partial result towards the normal Moore space conjecture. The normal Moore space conjecture states that every normal Moore space is metrizable. This is known to be not provable from the usual axioms of set theory, since Silver [4] shows that Martin's ax...

We analyze properties of a map f sending a unitary matrix U of size N into a doubly stochastic matrix B = f (U) defined by Bi,j = |Ui,j| 2. For any U we define its defect, determined by the dimension of the image Df (TU U) of the space TU U tangent to the manifold of unitary matrices U at U under the tangent map Df corresponding to f. The defect, equal to zero for a generic unitary matrix, give...

Explicit formulas are obtained for the number of periodic points and maximum tail length split polynomial maps over finite fields affine projective space. This work includes a detailed analysis structure directed graph Chebyshev polynomials non-prime degree in dimension 1 powering map any dimension. The results applied to an algorithm determining type given through its cycle statistics modulo p...