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Register a new userOn monotonicity of Ramanujan function for binomial random variables
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For a binomial random variable $xi$ with parameters $n$ and $b/n$, it is well known that the median equals $b$ when $b$ is an integer. In 1968, Jogdeo and Samuels studied the behaviour of the relative difference between ${sf P}(xi=b)$ and $1/2-{sf P}(xi<b)$. They proved its monotonicity in $n$ and posed a question about its monotonicity in $b$. This question is motivated by the solved problem proposed by Ramanujan in 1911 on the monotonicity of the same quantity but for a Poisson random variable with an integer parameter $b$. In the paper, we answer this question and introduce a simple way to analyse the monotonicity of similar functions.
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The behavior of a certain random growth process is analyzed on arbitrary regular and non-regular graphs. Our argument is based on the Expander Mixing Lemma, which entails that the results are strongest for Ramanujan graphs, which asymptotically maximize the spectral gap. Further, we consider ErdH{o}s--Renyi random graphs and compare our theoretical results with computational experiments on flip graphs of point configurations. The latter is relevant for enumerating triangulations.
Suppose you and your friend both do $n$ tosses of an unfair coin with probability of heads equal to $alpha$. What is the behavior of the probability that you obtain at least $d$ more heads than your friend if you make $r$ additional tosses? We obtain asymptotic and monotonicity/convexity properties for this competing probability as a function of $n$, and demonstrate surprising phase transition phenomenons as parameters $ d, r$ and $alpha$ vary. Our main tools are integral representations based on Fourier analysis.
The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group $G$ over a local field $F$. We show that if $T$ is any $k$-regular $G$-equivariant operator on the Bruhat-Tits building with a simple combinatorial property (collision-free), the associated random walk on the $n$-vertex Ramanujan complex has cutoff at time $log_k n$. The high dimensional case, unlike that of graphs, requires tools from non-commutative harmonic analysis and the infinite-dimensional representation theory of $G$. Via these, we show that operators $T$ as above on Ramanujan complexes give rise to Ramanujan digraphs with a special property ($r$-normal), implying cutoff. Applications include geodesic flow operators, geometric implications, and a confirmation of the Riemann Hypothesis for the associated zeta functions over every group $G$, previously known for groups of type $widetilde A_n$ and $widetilde C_2$.
The distribution of sum of independent non-identical binomial random variables is frequently encountered in areas such as genomics, healthcare, and operations research. Analytical solutions to the density and distribution are usually cumbersome to find and difficult to compute. Several methods have been developed to approximate the distribution, and among these is the saddlepoint approximation. However, implementation of the saddlepoint approximation is non-trivial and, to our knowledge, an R package is still lacking. In this paper, we implemented the saddlepoint approximation in the textbf{sinib} package. We provide two examples to illustrate its usage. One example uses simulated data while the other uses real-world healthcare data. The textbf{sinib} package addresses the gap between the theory and the implementation of approximating the sum of independent non-identical binomials.
The Hardy hypothesis, as an analogue to the Riemann hypothesis for the Riemann zeta function, is a conjecture proposed by Hardy in 1940, that all of the nontrivial zeros for the Ramanujan zeta function have a real part equal to 6. In this paper, we propose the power series expansion for the entire Ramanujan zeta function using the work of Mordell. Then, we suggest an alternative infinite product for the entire Ramanujan zeta function derived from the work of Conrey and Ghosh. We also establish the class of the entire Ramanujan zeta function related to the functional equation coming from Wilton. Motivated by the work of Lekkerkerker, we prove an conjecture due to Bruijn that all of the zeros of the Ramanujan Xi function are nonzero real numbers. From theory of the entire functions, we also prove that the Hardy hypothesis is true.
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