Logarithmically Concave Sequence
المؤلف:
Levit, V. E. and Mandrescu, E.
المصدر:
"The Independence Polynomial of a Graph--A Survey." In Proceedings of the 1st International Conference on Algebraic Informatics. Held in Thessaloniki, October 20-23
الجزء والصفحة:
...
1-11-2020
1028
Logarithmically Concave Sequence
A finite sequence of real numbers ![<span style=]()
{a_k}_(k=1)^n" src="https://mathworld.wolfram.com/images/equations/LogarithmicallyConcaveSequence/Inline1.gif" style="height:17px; width:42px" /> is said to be logarithmically concave (or log-concave) if
holds for every 
with 
.
A logarithmically concave sequence of positive numbers is also unimodal.
If ![<span style=]()
{a_i}" src="https://mathworld.wolfram.com/images/equations/LogarithmicallyConcaveSequence/Inline4.gif" style="height:15px; width:21px" /> and ![<span style=]()
{b_i}" src="https://mathworld.wolfram.com/images/equations/LogarithmicallyConcaveSequence/Inline5.gif" style="height:15px; width:21px" /> are two positive log-concave sequences of the same length, then ![<span style=]()
{a_ib_i}" src="https://mathworld.wolfram.com/images/equations/LogarithmicallyConcaveSequence/Inline6.gif" style="height:15px; width:35px" /> is also log-concave. In addition, if the polynomial 
has all its zeros real, then the sequence ![<span style=]()
{p_i/(n; i)}" src="https://mathworld.wolfram.com/images/equations/LogarithmicallyConcaveSequence/Inline8.gif" style="height:34px; width:66px" /> is log-concave (Levit and Mandrescu 2005).
An example of a logarithmically concave sequence is the sequence of binomial coefficients 
for fixed 
and 
.
REFERENCES:
Levit, V. E. and Mandrescu, E. "The Independence Polynomial of a Graph--A Survey." In Proceedings of the 1st International Conference on Algebraic Informatics. Held in Thessaloniki, October 20-23, 2005 (Ed. S. Bozapalidis, A. Kalampakas, and G. Rahonis). Thessaloniki, Greece: Aristotle Univ., pp. 233-254, 2005.
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