scholarly journals Lower bound for the Perron–Frobenius degrees of Perron numbers

2020 ◽  
pp. 1-17
Author(s):  
MEHDI YAZDI
Keyword(s):  
Lower Bound ◽  
Complex Plane ◽  
Perron Number ◽  
Totally Real

Using an idea of Doug Lind, we give a lower bound for the Perron–Frobenius degree of a Perron number that is not totally real, in terms of the layout of its Galois conjugates in the complex plane. As an application, we prove that there are cubic Perron numbers whose Perron–Frobenius degrees are arbitrary large, a result known to Lind, McMullen and Thurston. A similar result is proved for bi-Perron numbers.

scholarly journals Deformation of Brody curves and mean dimension

2009 ◽  
Vol 29 (5) ◽  
pp. 1641-1657 ◽  
Author(s):  
MASAKI TSUKAMOTO
Keyword(s):  
Lower Bound ◽  
Projective Space ◽  
Parameter Space ◽  
Complex Plane ◽  
Deformation Theory ◽  
Holomorphic Map ◽  
Infinite Dimensional ◽  
Mean Dimension ◽  
The Mean

AbstractThe main purpose of this paper is to show that ideas of deformation theory can be applied to ‘infinite-dimensional geometry’. We develop the deformation theory of Brody curves. A Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite-dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves.

On Heights of Polynomials with Real Roots

2000 ◽  
Vol 5 ◽  
pp. 67-75
Author(s):  
A. Dubickas
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Chebyshev Polynomials ◽  
Mahler Measure ◽  
Real Roots ◽  
Real Polynomials ◽  
Totally Real

We prove Schinzel’s theorem about the lower bound of the Mahler measure of totally real polynomials. Under certain additional conditions this theorem is strengthened. We also consider certain Chebyshev polynomials in order to investigate how sharp are the lower bounds for the heights.

The C-numerical range of perturbations matrix polynomials

2016 ◽  
Vol 09 (02) ◽  
pp. 1650046 ◽  
Author(s):  
Yaser Jahanshahi ◽  
Bahmann Yousefi
Keyword(s):  
Lower Bound ◽  
Complex Plane ◽  
Matrix Polynomial ◽  
Numerical Range ◽  
Connected Components ◽  
Matrix Polynomials ◽  
Polynomial Formula

In this paper, the [Formula: see text]-radius stability of a matrix polynomial [Formula: see text] relative to a domain [Formula: see text] of the complex plane and its relation with the [Formula: see text]-numerical range of [Formula: see text] are investigated. By using an expression of the [Formula: see text]-radius stability, we obtain a lower bound which involves the distance of [Formula: see text] from the connected components of the [Formula: see text]-numerical range of [Formula: see text].

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

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