This study generalized the best copula to characterize the joint probability distribution between rainfall severity and duration in Peninsular Malaysia using two dimensional copulas. Specifically, to construct copulas, Inference Function for Margins (IFM) and Canonical Maximum Likelihood (CML) methods were specially exploited. For the purpose of achieving copula fitting, the derived rainfall variables by making use of the Standardized Precipitation Index (SPI) were fitted into several distributions. Five copulas, namely Gaussian, Clayton, Frank, Joe and Gumbel were put to the tests to establish the best data fitted copula. The tests produced acknowledged and satisfactory results of copula fitting for rainfall severity and duration. Surveying the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), only three copulas produced a better fit for parametric and semi parametric approaches. Finally, two consistency tests were conducted and the results had shown that Frank Copula produced consistent results.

Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying \begin{align*} \left|\sum_{j=1}^{n}\lambda_j^2\right|=\max_{1\leq j \leq n}|\lambda_j|^2, \quad \forall \lambda_j \in \mathbb{K}, 1\leq j \leq n, \forall n \in \mathbb{N}. \end{align*} For $d\in \mathbb{N}$, let $\mathbb{K}^d$ be the standard $d$-dimensional non-Archimedean Hilbert space. Let $m \in \mathbb{N}$ and $\text{Sym}^m(\mathbb{K}^d)$ be the non-Archimedean Hilbert space of symmetric m-tensors. We prove the following result. If $\{\tau_j\}_{j=1}^n$ is a collection in $\mathbb{K}^d$ satisfying $\langle \tau_j, \tau_j\rangle =1$ for all $1\leq j \leq n$ and the operator $\text{Sym}^m(\mathbb{K}^d)\ni x \mapsto \sum_{j=1}^n\langle x, \tau_j^{\otimes m}\rangle \tau_j^{\otimes m} \in \text{Sym}^m(\mathbb{K}^d)$ is diagonalizable, then \begin{align}\label{WELCHNONABSTRACT} \max_{1\leq j,k \leq n, j \neq k}\{|n|, |\langle \tau_j, \tau_k\rangle|^{2m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{align} We call Inequality (\ref{WELCHNONABSTRACT}) as the non-Archimedean version of Welch bounds obtained by Welch [\textit{IEEE Transactions on Information Theory, 1974}]. We formulate non-Archimedean Zauner conjecture.

We prove the continuous non-Archimedean (resp. p-adic) Banach space and Hilbert space versions of non-Archimedean (resp. p-adic) Welch bounds proved by M. Krishna. We formulate continuous non-Archimedean and p-adic functional Zauner conjectures.

We prove the non-Archimedean (resp. p-adic) Banach space version of non-Archimedean (resp. p-adic) Welch bounds recently obtained by M. Krishna. More precisely, we prove following results. \begin{enumerate}[\upshape(i)] \item Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying $\left|\sum_{j=1}^{n}\lambda_j^2\right|=\max_{1\leq j \leq n}|\lambda_j|^2$ for all $ \lambda_j \in \mathbb{K}, 1\leq j \leq n$, for all $n \in \mathbb{N}.$ Let $\mathcal{X}$ be a $d$-dimensional non-Archimedean Banach space over $\mathbb{K}$. If $\{\tau_j\}_{j=1}^n$ is any collection in $\mathcal{X}$ and $\{f_j\}_{j=1}^n$ is any collection in $\mathcal{X}^*$ (dual of $\mathcal{X}$) satisfying $f_j(\tau_j) =1$ for all $1\leq j \leq n$ and the operator $S_{f, \tau} : \text{Sym}^m(\mathcal{X})\ni x \mapsto \sum_{j=1}^nf_j^{\otimes m}(x)\tau_j^{\otimes m} \in \text{Sym}^m(\mathcal{X})$, is diagonalizable, then \begin{align}\label{NONFUNCTIONALWELCH} \max_{1\leq j,k \leq n, j \neq k}\{|n|, |f_j(\tau_k)f_k(\tau_j)|^{m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{align} We call Inequality (\ref{NONFUNCTIONALWELCH}) as non-Archimedean functional Welch bounds. \item For a prime $p$, let $\mathbb{Q}_p$ be the p-adic number field. Let $\mathcal{X}$ be a $d$-dimensional p-adic Banach space over $\mathbb{Q}_p$. If $\{\tau_j\}_{j=1}^n$ is any collection in $\mathcal{X}$ and $\{f_j\}_{j=1}^n$ is any collection in $\mathcal{X}^*$ (dual of $\mathcal{X}$) satisfying $f_j(\tau_j) =1$ for all $1\leq j \leq n$ and there exists $b \in \mathbb{Q}_p$ such that $ \sum_{j=1}^{n}f_j^{\otimes m}(x) \tau_j^{\otimes m} =bx$ for all $ x \in \text{Sym}^m(\mathcal{X}),$ then \begin{align}\label{PADICFUNCTIONALWELCH} \max_{1\leq j,k \leq n, j \neq k}\{|n|, |f_j(\tau_k)f_k(\tau_j)|^{m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{align} We call Inequality (\ref{PADICFUNCTIONALWELCH}) as p-adic functional Welch bounds. \end{enumerate} We formulate non-Archimedean functional and p-adic functional Zauner conjectures.

We introduce the notion of p-adic magic contraction on p-adic Hilbert space. We derive p-adic Halmos dilation, p-adic Egervary N-dilation, p-adic von Neumann inequality and p-adic Sz.-Nagy dilation for p-adic magic contraction.