109 research outputs found
Extremal Lp-norms of linear operators and self-similar functions
AbstractWe prove that for any p∈[1,+∞] a finite irreducible family of linear operators possesses an extremal norm corresponding to the p-radius of these operators. As a corollary, we derive a criterion for the Lp-contractibility property of linear operators and estimate the asymptotic growth of orbits for any point. These results are applied to the study of functional difference equations with linear contractions of the argument (self-similarity equations). We obtain a sharp criterion for the existence and uniqueness of solutions in various functional spaces, compute the exponents of regularity, and estimate moduli of continuity. This, in particular, gives a geometric interpretation of the p-radius in terms of spectral radii of certain operators in the space Lp[0,1]
Cyclic classes and attraction cones in max algebra
In max algebra it is well-known that the sequence A^k, with A an irreducible
square matrix, becomes periodic at sufficiently large k. This raises a number
of questions on the periodic regime of A^k and A^k x, for a given vector x.
Also, this leads to the concept of attraction cones in max algebra, by which we
mean sets of vectors whose ultimate orbit period does not exceed a given
number. This paper shows that some of these questions can be solved by matrix
squaring (A,A^2,A^4, ...), analogously to recent findings concerning the orbit
period in max-min algebra. Hence the computational complexity of such problems
is of the order O(n^3 log n). The main idea is to apply an appropriate diagonal
similarity scaling A -> X^{-1}AX, called visualization scaling, and to study
the role of cyclic classes of the critical graph. For powers of a visualized
matrix in the periodic regime, we observe remarkable symmetry described by
circulants and their rectangular generalizations. We exploit this symmetry to
derive a concise system of equations for attraction cpne, and we present an
algorithm which computes the coefficients of the system.Comment: 38 page
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