687 research outputs found
Wavelet analysis on symbolic sequences and two-fold de Bruijn sequences
The concept of symbolic sequences play important role in study of complex
systems. In the work we are interested in ultrametric structure of the set of
cyclic sequences naturally arising in theory of dynamical systems. Aimed at
construction of analytic and numerical methods for investigation of clusters we
introduce operator language on the space of symbolic sequences and propose an
approach based on wavelet analysis for study of the cluster hierarchy. The
analytic power of the approach is demonstrated by derivation of a formula for
counting of {\it two-fold de Bruijn sequences}, the extension of the notion of
de Bruijn sequences. Possible advantages of the developed description is also
discussed in context of applied
Non-Euclidean geometry in nature
I describe the manifestation of the non-Euclidean geometry in the behavior of
collective observables of some complex physical systems. Specifically, I
consider the formation of equilibrium shapes of plants and statistics of sparse
random graphs. For these systems I discuss the following interlinked questions:
(i) the optimal embedding of plants leaves in the three-dimensional space, (ii)
the spectral statistics of sparse random matrix ensembles.Comment: 52 pages, 21 figures, last section is rewritten, a reference to
chaotic Hamiltonian systems is adde
The Roots and Links in a Class of -Matrices
In this paper, we discuss exiting roots of sub-kernel transient matrices
associated with a class of matrices which are related to generalized
ultrametric matrices. Then the results are used to describe completely all
links of the class of matrices in terms of structure of the supporting tree.Comment: 11 pages, 1 figur
Geometric aspects of the symmetric inverse M-matrix problem
We investigate the symmetric inverse M-matrix problem from a geometric
perspective. The central question in this geometric context is, which
conditions on the k-dimensional facets of an n-simplex S guarantee that S has
no obtuse dihedral angles. First we study the properties of an n-simplex S
whose k-facets are all nonobtuse, and generalize some classical results by
Fiedler. We prove that if all (n-1)-facets of an n-simplex S are nonobtuse,
each makes at most one obtuse dihedral angle with another facet. This helps to
identify a special type of tetrahedron, which we will call sub-orthocentric,
with the property that if all tetrahedral facets of S are sub-orthocentric,
then S is nonobtuse. Rephrased in the language of linear algebra, this
constitutes a purely geometric proof of the fact that each symmetric
ultrametric matrix is the inverse of a weakly diagonally dominant M-matrix.
Review papers support our belief that the linear algebraic perspective on the
inverse M-matrix problem dominates the literature. The geometric perspective
however connects sign properties of entries of inverses of a symmetric positive
definite matrix to the dihedral angle properties of an underlying simplex, and
enables an explicit visualization of how these angles and signs can be
manipulated. This will serve to formulate purely geometric conditions on the
k-facets of an n-simplex S that may render S nonobtuse also for k>3. For this,
we generalize the class of sub-orthocentric tetrahedra that gives rise to the
class of ultrametric matrices, to sub-orthocentric simplices that define
symmetric positive definite matrices A with special types of k x k principal
submatrices for k>3. Each sub-orthocentric simplices is nonobtuse, and we
conjecture that any simplex with sub-orthocentric facets only, is
sub-orthocentric itself.Comment: 42 pages, 20 figure
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