G+G^{+} Method in Action: New Classes of Nonnegative Matrices with Results

The $G^{+}$ method is a new method, a powerful one, for the study of (homogeneous and nonhomogeneous) products of nonnegative matrices -- for problems on the products of nonnegative matrices. To study such products, new classes of matrices are introduced: that of the sum-positive matrices, that of the $\left[ \Delta \right]$-positive matrices on partitions (of the column index sets), that of the $g_{k}^{+}$-matrices... On the other hand, the $g_{k}^{+}$-matrices lead to necessary and sufficient conditions for the $k$-connected graphs. Using the $G^{+}$ method, we prove old and new results (Wielandt Theorem and a generalization of it, etc.) on the products of nonnegative matrices -- mainly, sum-positive, $\left[ \Delta \right]$-positive on partitions, irreducible, primitive, reducible, fully indecomposable, scrambling, or Sarymsakov matrices, in some cases the matrices being, moreover, $g_{k}^{+}$-matrices (not only irreducible)

Partition and composition matrices

This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another. We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel. We show that composition matrices on X are in one-to-one correspondence with (2+2)-free posets on X. Also, composition matrices whose rows satisfy a column-ordering relation are shown to be in one-to-one correspondence with parking functions. Finally, we show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on {1,...,n}.Comment: 14 page

A Further Study of Vectorial Dual-Bent Functions

Vectorial dual-bent functions have recently attracted some researchers' interest as they play a significant role in constructing partial difference sets, association schemes, bent partitions and linear codes. In this paper, we further study vectorial dual-bent functions $F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}$, where $2\leq m \leq \frac{n}{2}$, $V_{n}^{(p)}$ denotes an $n$-dimensional vector space over the prime field $\mathbb{F}_{p}$. We give new characterizations of certain vectorial dual-bent functions (called vectorial dual-bent functions with Condition A) in terms of amorphic association schemes, linear codes and generalized Hadamard matrices, respectively. When $p=2$, we characterize vectorial dual-bent functions with Condition A in terms of bent partitions. Furthermore, we characterize certain bent partitions in terms of amorphic association schemes, linear codes and generalized Hadamard matrices, respectively. For general vectorial dual-bent functions $F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}$ with $F(0)=0, F(x)=F(-x)$ and $2\leq m \leq \frac{n}{2}$, we give a necessary and sufficient condition on constructing association schemes. Based on such a result, more association schemes are constructed from vectorial dual-bent functions

Crinkly curves, Markov partitions and dimension

We consider the relationship between fractals and dynamical systems. In particular we look at how the construction of fractals in (D1) can be interpreted-in a dynamical setting and additionally used as a simple method of describing the construction of invariant sets of dynamical systems. There is often a confusion between Hausdorff dimension and capacity -which is much easier to compute- and we show that simple examples of fractals, arising in dynamical systems, exist for which the two quantities differ. In Chapter One we outline the mathematical background required in the rest of the thesis. Chapter Two reviews the work of F. M. Dekking on generating 'recurrent sets', which are types of fractals. We show how to interpret this construction dynamically. This approach enables us to calculate Hausdorff dimension and describe Hausdorff measure for certain recurrent sets. We also prove a conjecture of Dekking about conditions under which the best general estimate of dimension actually equals dimension. In Section One of Chapter Three recurrent sets are used to construct special Markou partitions for expanding endomorphisms of T2 and hyperbolic automorphisms of T3. These partitions have transition matrices closely related to the covering maps. It is also shown that Markov partitions can be constructed for the same map whose boundaries have different capacities. Section Two looks at the problem of coding between two Markov partitions for the same expanding endomorphism of T2. It is shown that there is a relationship between mean coding time and the capacities of the boundaries. Section Three uses recurrent sets to construct fractal subsets of tori which have non-dense orbits under the above mappings. Finally, Chapter Four calculates capacity and Hausdorff dimension for a class of fractals (which are also recurrent sets) whose scaling maps are-not similitudes. Examples are given for which capacity and Hausdorff dimension give different answers