Given matrices $N\in C^{s\times s}$ and $S_0,\dots ,S_q\in C^{s\times s}$, we solve the linear differential equation

$$\begin{array}{r}\sum _{n=0}^q{T}_n(t)(d/dt{)}^{n}f(t)=g(t),\end{array}$$

where $t\in R$, $T_n(t)=e^{tN}{S}_n{e}^{-tN}$, and $f(t):R\to C^s$, using the roots of $d(\nu )=detD(\nu )$, where

$$\begin{array}{r}D(\nu )=\sum _{n=0}^q{S}_n{\left(\nu I_r+N\right)}^n.\end{array}$$

For example,

$$\begin{array}{r}N=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\end{array}$$

implies

$$\begin{array}{r}{e}^{tN}=\left(\begin{array}{cc}\cos t& -\sin t\\ \sin t& \cos t\end{array}\right),\end{array}$$

so that $T_n(t)$ are periodic, giving an explicit solution to a form of Floquet's theorem.

We used a new type of characteristics to solve a class of homogeneous linear multivariate partial differential equations on $C^p$. For $x$ in $R^p$ and $n$ in $Z^p$, set ${\mathrm{\partial }}_x^n=\prod _{j=1}^p{\left(\mathrm{\partial }/\mathrm{\partial }{x}_j\right)}^{n_j}$. Given square matrices $\left\{N_j\right\}$ and $\left\{S_n\right\}$ in $C^{s\times s}$, set

$$\begin{array}{r}Y(x)=\exp \left(\sum _{j=1}^p{x}_j{N}_j\right)\text{in}{C}^{s\times s}\end{array}$$

and $T_n(x)=Y(x)S_{n}Y(-x)$ in $C^{s\times s}$. When $\left\{N_j\right\}$ commute, we show that the linear partial differential equation

$$\begin{array}{r}\sum _{n=0_p}^q{T}_n(x){\mathrm{\partial }}_x^{n}f(x)=0_s\text{for}f(x)\text{in}{C}^s\end{array}$$

has solutions $f(x)=f_n(x,\nu )$ for each admissible $n\le sq$ and any $\nu$ in $C^p$ such that $d(\nu )=0$, where

$$\begin{array}{r}d(\nu )=\mathrm{d}\mathrm{e}\mathrm{t}D(\nu ),D(\nu )=\sum _{n=0_p}^q{S}_n\prod _{j=1}^p{\left({\nu }_j{I}_s+N_j\right)}^{n_j}.\end{array}$$

The research aims to develop a new method, based on a novel type of characteristics, for solving a broad class of multivariate homogeneous linear partial differential equations with matrix coefficients of a specific exponential-conjugate form, extending classical Cauchy characteristic techniques beyond the univariate case and providing explicit basis solutions parameterized over complex surfaces.

Copulas are important because they allow for modeling and analyzing the dependence structure between random variables, providing insights into complex relationships beyond linear correlations. In this paper, we produced a compendium of expressions for four of the most popular dependence measures of bivariate copulas. Over twenty-five families of bivariate copulas were considered.

Unit continuous probability distributions play a fundamental role in modeling variables bounded within the interval $[0,1]$, such as proportions and probabilities. In recent decades, there has been a significant increase in the development of new parametric families of these distributions. In this work, we present a comprehensive and up-to-date review of more than one hundred unit continuous distributions, including classical models, such as the beta and Kumaraswamy distributions, along with their various extensions. We examined key statistical properties such as moments and demonstrated the practical effectiveness of twelve selected distributions through applications to nine distinct datasets, thereby highlighting their flexibility in modeling a wide range of data types. To the best of our knowledge, this is the most extensive review focused specifically on unit distributions and is a valuable reference for researchers and practitioners.