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.

Disclaimer: This article does not constitute investment advice.

In an era marked by severe monetary expansion, intense geopolitical tensions, global instability, and the steady depletion of natural resources, where resources are increasingly becoming the ultimate source of value, gold is destined to rise substantially over the long term. The magnitude of that rise may well exceed anything people can judge from past experience. That broad trend is unlikely to change. In the short run, however, factors such as short squeezes triggered by higher futures margin requirements and deliberate market intervention by large capital can create enormous volatility and significant near-term risk. In short, gold remains a long-term bullish asset, but it also carries considerable short-term risk. We should therefore adhere to the basic principle of keeping positions light, investing for the long term, and investing with peace of mind.

Introduction

Markov Chain

Let $(\Omega,\mathcal{F})$ be a measurable space, $P:\Omega\times \mathcal{F}\to [0,1]$ be a transition kernel, that is,

• for every $x\in \Omega$, $P(x,\cdot)$ is a probability measure on $\mathcal{F}$;
• for every $A\in \mathcal{F}$, $x\mapsto P(x,A)$ is measurable.

This is the talk I presented in the group seminar, titled: Sampling and Diffusion Model.

This is the talk I presented in the group seminar, titled: Analogy between Sampling and Optimization.

In revisiting the proofs of many theorems in mathematics, we often need to use sequential compactness. However, the Riesz lemma in normed linear spaces tells us that every bounded set in an infinite-dimensional normed linear space is not sequentially compact, which brings many inconveniences. In fact, the essential reason for this phenomenon is that the topology induced by the norm in a normed linear space is “too strong.” For this reason, we need to introduce the weak topology, so that bounded sets are sequentially compact in the weak-topology sense. In this article, we mainly review local convex topological vector spaces and some main results on weak convergence.

The Wasserstein distance has wide applications in problems such as optimal transport. It characterizes the distance between two probability measures. In German, Wasser means water, and Stein means stone (although I have not carefully verified whether Wasserstein is a person’s name; after all, many German names are of this kind, for example: Einstein = one stone). This article will mainly answer the following two questions:

• On what set is the Wasserstein distance defined?
• Why is the Wasserstein distance a distance? That is, why does it satisfy the three axioms of a metric?

In this article, we only consider the Wasserstein distance on the $d$-dimensional Euclidean space $\mathbb{R}^d$. Since $\mathbb{R}^d$ is a special Polish space, the conclusions of this article can also be extended to general Polish spaces.

How can one define a product probability measure on infinitely many probability spaces? That is, given a family of probability spaces $(\Omega_t,\mathcal F_t)$, $t\in T$, suppose that for every nonempty finite subset $S\subset T$, a probability measure $\mathbb P_S$ has already been defined on $\prod_{t\in S}\mathcal F_t$. Then how can we define a “suitable” probability measure $\mathbb P$ on the product measurable space

$$
\left(\prod_{t\in T}\Omega_t,\prod_{t\in T}\mathcal F_t\right)
$$

such that its marginal distribution measures are exactly $\mathbb P_S$?

Unless otherwise specified, all classes of sets in this article refer to classes of subsets of $\Omega$.

The Monotone Class Theorem