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OBJECTIVE Over the last decade, more young children have been diagnosed with brainstem gliomas. However, the incidence and survival outcomes of children with brainstem gliomas have yet to be well documented. This study evaluates the clinical outcomes of children diagnosed with brainstem gliomas between 2001 and 2015 at a single tertiary center. METHODS A retrospective cohort of all children diagnosed with brainstem gliomas at the University of California, Los Angeles, between 2001 and 2015 was created. The medical records of these children were reviewed and compared to a contemporaneous cohort of children diagnosed with low-grade glioma in the same region. All claims data for the patients in the brainstem glioma cohort were obtained from the Cigna Caremine data repository, and all claims from the children in the low-grade glioma cohort were obtained from the MultiCare data repository. The outcome of interest was survival following the diagnosis of brainstem glioma. Statistical analysis was performed using the paired t test. RESULTS In the period under study, 10 children with brainstem gliomas were identified. The mean patient age at diagnosis was 4.9 years (range 1-14.4 years). When compared to the contemporaneous control group, the brainstem glioma cohort demonstrated higher mortality rates during the follow-up period (p Electrochemical generation of an effective endogenous antioxidant against oxidative stress by a hybridized hemin-biochelatins conjugate system.
We showed previously that novel hybridized hemin-biochelatins (HBC) comprising hemin and biochelatins has ability to reduce the ferrous iron. The present study aimed at using this conjugate system to explore the possible efficacy against oxidative stress. A strategy of electrochemical coupling between 5,10,15-tri(n-oct
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Finding the asymptotic growth of a function using Big Theta notation
So I am learning Big O and Big Omega notation in the context of a programming class I am taking and I have a question about finding a big-theta function. Here’s an image of the question:
The function being defined is this: $f(n)=2\log\log n$.
The question asks us to find if the function is Big O or Big Omega or neither. The answer that I think we are given is as follows:
\BigOlf(f(n))=\BigOlf(\log\log n)=\BigOlf(\log\log\log n)=1
I don’t understand why the answer says that the function is Big O because I understand the idea of when one function is less or equal to the other but I don’t know why we’re talking about Big Omega, Big Theta or Big O of the original log function. Can someone explain to me why this is the answer? And if it’s because the log function exists in the given problem statement, how does that relate to us being able to talk about the other functions?
For the sake of being pedantic, we can change the form of the question to:
Is $\log\log n$ $\Theta(n\log\log n)$?
Notice that we can rewrite the first function as
$$f(n) = 2\log\log n = 2\log\log\log n + 2\log\log\log\log n + \cdots$$
If we take the limit as $n \to \infty$, this obviously evaluates to $\log\log\log n$. We can show this by induction.
Obviously $f(1) = 0 = \log\log\log 1$ and $f(2) = 1 = \log\log\log 2$. Since
$$f(k) = 2\log\log\log k + 2\log\log\log\log\log\log k + \cdots$$
is strictly increasing. Since the limit of the