scholarly journals MAHLER MEASURE OF ‘ALMOST’ RECIPROCAL POLYNOMIALS

2018 ◽  
Vol 98 (1) ◽  
pp. 70-76
Author(s):  
J. C. SAUNDERS
Keyword(s):  
Lower Bound ◽  
Mahler Measure

We give a lower bound of the Mahler measure on a set of polynomials that are ‘almost’ reciprocal. Here ‘almost’ reciprocal means that the outermost coefficients of each polynomial mirror each other in proportion, while this pattern may break down for the innermost coefficients.

scholarly journals On the Zhang–Zagier measure

2018 ◽  
Vol 14 (10) ◽  
pp. 2663-2671
Author(s):  
V. Flammang
Keyword(s):  
Lower Bound ◽  
Mahler Measure ◽  
Algebraic Integers ◽  
Totally Positive ◽  
The Absolute

We first improve the known lower bound for the absolute Zhang–Zagier measure in the general case. Then we restrict our study to totally positive algebraic integers. In this case, we are able to find six points for the related spectrum. At last, we give inequalities involving the Zhang–Zagier measure, the Mahler measure and the length of such integers.

scholarly journals Lower bounds for Mahler measure that depend on the number of monomials

2019 ◽  
Vol 15 (07) ◽  
pp. 1425-1436
Author(s):  
Shabnam Akhtari ◽  
Jeffrey D. Vaaler
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Mahler Measure ◽  
Ordered Group ◽  
Classical Inequality ◽  
Several Variables ◽  
Complex Coefficients

We prove a new lower bound for the Mahler measure of a polynomial in one and in several variables that depends on the complex coefficients and the number of monomials. In one variable, our result generalizes a classical inequality of Mahler. In [Formula: see text] variables, our result depends on [Formula: see text] as an ordered group, and in general, our lower bound depends on the choice of ordering.

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 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.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

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