论⽂中的定理（Theorem）、引理（Lemma）、推论（Corollary）

名词解释

- Theorem：就是定理，⽐较重要的，简称是 Thm。
- Lemma：⼩⼩的定理，通常是为了证明后⾯的定理，如果证明的篇幅很长時，可能會把证明拆成⼏个部分來论述，虽然篇幅可能变多，但

派络却很清楚。 - Corollary：推论。由定理⽴即可推知的結果。 - Property：性质，结果虽然值得⼀記，卻沒定理來的深刻。 - Proposition：有⼈翻译为命題，有些作者喜欢⽤，⼤概也可以算是⽐较简单的定理的⼀种称呼。 - Claim：证明时先论述⼀个结果，再作证明。看的⼈⽐较轻松。 - Note：通常只是⼀个注解。 - Remark：涉及⼀些结论，⽐较起來 "Note" ⽐较像说明， "remark" 則常是⾮正式的定理。

⾸先，定义和公理是任何理论的基础，定义解决了概念的范畴，公理使得理论能够被⼈的理性所接受。其次，定理和命题就是在定义和公理的基础上通过理性的加⼯使得理论的再延伸，我认为它们的区别主要在于，定理的理论⾼度⽐命题⾼些，定理主要是描述各定义（范畴）间的逻辑关系，命题⼀般描述的是某种对应关系（⾮范畴性的）。⽽推论就是某⼀定理的附属品，是该定理的简单应⽤。最后，引理就是在证明某⼀定理时所必须⽤到的其它定理。⽽在⼀般情况下，就像前⾯所提到的定理的证明是依赖于定义和公理的。

1.引理和定理应该是根据⽂章⽬的不同⽽区分的，同样的论点在这篇⽂章可以是引理，在那篇⽂章可以是定理。 2.如果为了说明⼀个问题进⾏论证，但是在论证前需要证明若⼲个⼩问题，那么这些若⼲个⼩问题的结论就是引理，⽽这个问题的论证将会需要引⽤到前⾯的引理，该问题的结论就是定理。 3.引理是为定理作准备的。⽂章中的定理才是需要说明的主要问题或者⽬的。

就如Doppler说的， "Theorem" 本⾝是⼀个⼤result，"Lemma" 是 prove “Theorem“ 之前⽤的⼀个 result。"Corollary" 是可以从 "Theorem" ⾥直接 deduce/prove 出来的 result。" Proposition" 是⼀个还⽆法⼤到变成 "Theorem" 的⼀个 result (当作⼩ theorem )。 (1) Definition（定义）------a precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true. (2) Theorem（定理）----a mathematical statement that is proved using rigorous mathemat-ical reasoning. In a mathematical paper, the term theorem is often reserved for the most important results. (3) Lemma（引理）----a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem. Very occasionally lemmas can take on a life of their own (Zorn's lemma, Urysohn's lemma, Burnside's lemma,Sperner's lemma). (4) Corollary（推论）-----a result in which the (usually short) proof relies heavily on a given theorem (we often say that is a corollary of Theorem A"). (5) Proposition（命题）-----a proved and often interesting result, but generally less important than a theorem. (6) Conjecture（推测，猜想）----a statement that is unproved, but is believed to be true (Collatz conjecture, Goldbach conjecture, twin prime conjecture). (7) Claim（断⾔）-----an assertion that is then proved. It is often used like an informal lemma. (8) Axiom/Postulate------（公理/假定）a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved (Eu-clid's ve postulates, Zermelo-Frankel axioms, Peano axioms). (9) Identity（恒等式）-----a mathematical expression giving the equality of two (often variable) quantities (trigonometric identities, Euler's identity). (10) Paradox（悖论）----a statement that can be shown, using a given set of axioms and de nitions, to be both true and false. Paradoxes are often used to show the inconsistencies in a awed theory (Russell's paradox). The term paradox is often used informally to describe a surprising or counterintuitive result that follows from a given set of rules (Banach- Tarski paradox, Alabama paradox, Gabriel's horn).

参考

英文论文中Remark所放位置及其作用

2023-06-20 22:47:59 Written by Fu, Jian

# 多模式、多模态

multi-mode 多模式

multimodal = multiple “modes” 多模态

multi-mode optical fiber 多模光纤 Multi-mode fiber has a fairly large core diameter that enables multiple light modes to be propagated and limits the maximum length of a transmission link because of modal dispersion. 多模光纤中的多模应该指的是光纤中存在多种光的模式。模式就是波导传输的时候电磁波的空间分布情况。

single-mode optical fiber 单模光纤

多模式多模态

什么是模态？模间色散，模式色散，模态色散模内色散三种光纤色散：模式色散、材料色散、波导色散。