Definition 1. A Lie algebra $L$ is abelian if $[L,L]=0$, which means for all $x,y\in L,\ [x,y]=0$.

Definition 1. A mapping $f$ from the Borel sets of a metrizable space $(X,\tau)$ to $[0,\infty]$ is tight if $f(K)<\infty$ for each compact $K$ in $X$ and
$$
f(A)=\sup\{f(K):K \text{ compact},\ K\subseteq A\},
\quad \text{for each } A\in \mathcal{B}(X).
$$

Throughout this article $(X,d)$ will be a given complete metric space.

Representations

Definition 1. Let $M_n(k)$ be the algebra of $n\times n$ matrices over $K$ and let
$$
\mathfrak{gl}_n(K):=[M_n(K)]
$$

be the corresponding Lie algebra, which is also called the general linear Lie algebra.

Lie Algebra

Definition 1. A Lie algebra is a vector space $L$ over a field $K$ with a Lie bracket
$$
\begin{aligned}
\left[ \cdot,\cdot \right] &: L \times L \to L\\\
(x,y) &\mapsto [x,y],
\end{aligned}
$$
such that

Definition 1. If $(X,\tau)$ is a topological space, then the Borel $\sigma$-field $\mathcal{B}$ of $X$ is the $\sigma$-field generated by the open sets of $X$ and we say a measure $\mu$ defined on $(X,\mathcal{B})$ is a Borel measure.

Let $\mathcal{P}_2(\mathbb{R}^d)$ denote the space of probability measures on $\mathbb{R}^d$ with finite second moment and $F: \mathcal{P}_2(\mathbb{R}^d)\to \mathbb{R}$ be the potential functional. We are interested in the following optimization problem

$$
\min_{m\in \mathcal{P}_2(\mathbb{R}^d)} F(m).
$$

Motivation

Let $(X,d)$ be a metric space, $F, F_n: X\to \overline{\mathbb{R}}, n=1,2,\cdots$ be functionals, suppose that $x_n\in X$ minimizes $F_n$ for each $n=1,2,\cdots$, does $\lim\limits_{n\to\infty} x_n$ (if it exists) minimize any functional $F$? And in what sense does $F_n$ converge to $F$ ensure the minimizer of $F_n$ converges to minimizer of $F$?

On January 19, 2026, data released by the National Bureau of Statistics showed that, according to preliminary estimates, China’s gross domestic product (GDP) in 2025 was approximately 140.2 trillion CNY. In constant-price terms, this represented year-on-year growth of 5.0%, meeting the annual growth target of around 5% set at the beginning of the year and matching the growth rate of the previous year. A major reason China was able to achieve the 5% target was the substantial contribution from exports. According to data from the General Administration of Customs, China’s total merchandise trade in 2025 amounted to 45.47 trillion CNY, of which exports totaled 26.99 trillion CNY and imports 18.48 trillion CNY. In U.S. dollar terms, China’s full-year trade surplus in 2025 reached 1.19 trillion USD, setting a new historical record. In addition, National Bureau of Statistics data showed that net exports of goods and services contributed 1.6 percentage points to GDP growth in 2025, accounting for approximately 32% of the full-year 5.0% economic growth rate, or nearly one-third.

Disclaimer: This article does not constitute any investment advice.

In this era of severe monetary overissuance, extreme geopolitical tensions, global instability, and the gradual depletion of resources, where resources are paramount, the future price of silver is bound to rise substantially. Such an increase may far exceed anything within people’s memory. Silver and gold are both precious metals. Both have financial attributes, both serve as stores of value, and both possess collectible value. However, the biggest difference between silver and gold is that silver has a very pronounced industrial character. In particular, silver is widely used in many high-tech industries. From a longer-term perspective, silver may have stronger support than gold, which also means that its future price fluctuations may be more intense. In short, silver remains promising in the long run but carries high short-term risk. Therefore, we should adhere to the basic principle of maintaining light positions for the long term and investing with peace of mind.