𝕋he pure \(\lambda\)-calculus contains only functions as values. It is not exactly easy to write large or interesting programs in the pure \(\lambda\)-calculus. We can however encode objects, such as … Read More →

𝕎e have now seen two operational models for programming languages: small-step and large-step. In this nnote, we consider a different semantic model, called denotational semantics. The idea in … Read More →

𝕋he small-step and large-step semantics are equivalent as captured by the following theorem. Theorem. For all commands $c$ and stores $\sigma$ and $\sigma'$ we have […] The proof is left as an … Read More →

𝕎e will now consider a more realistic programming language, one where we can assign values to variables and execute control constructs such as if and while. The syntax for this imperative language, … Read More →

𝕀n the last lecture we defined a semantics for our language of arithmetic expressions using a small-step evaluation relation $\rightarrow \subseteq \mathbf{Config}\times \mathbf{Config}$ (and its … Read More →

𝕀n this note, we will use the semantics of our simple language of arithmetic expressions, […] to express useful program properties, and we will prove these properties by induction. […] … Read More →

𝕎hat is the meaning of a program? When we write a program, we represent it using sequences of characters. But these strings are just concrete syntax—they do not tell us what the program actually … Read More →

ℂatalan数，即卡特兰数（英语：Catalan number），又称明安图数，是组合数学中一种常出现于各种级数问题中的数列。以比利时的数学家欧仁·查理·卡特兰的名字来命名。 它是指满足以下递推关系的数列： […] Catalan数的递推关系很简单，乍看之下没什么特别的，但是Catalan数却是许多计数问题的最终形式。Stanley的Enumerative … Read More →

若$S_n$为前$n$项平方和： […] 它的通项公式我们在中学阶段已经学习过，即 […] 但是它的推导过程却不算简单。本文主要从组合数学的角度罗列一些此级数通项公式的推导方法。要理解本文的内容，读者应当学习过一些组合数学或离散数学的知识。 […] 考虑 […] 上式为常系数非线性递推关系。若我们尝试直接求解，则需要先求通解，再求特解，然后将 … Read More →

雅可比行列式在多元函数微积分中的应用非常广泛，我们在很多地方都能见到它的身影。合理运用雅可比行列式，可以简化我们的计算。 本文主要总结了一些雅可比行列式在多元函数微积分的计算方面的应用。 […] 雅可比行列式通常称为雅可比式(Jacobian)，它是以n个n元函数的偏导数为元素的行列式 。事实上，在函数都连续可微（即偏导数都连续）的前提之下，它就是函数组的微分形式下的系数矩阵（即雅可 … Read More →