# Tag Info

Accepted

### A logic with no axioms or no inference rules

(1) Where did $\Gamma$ come from? Ultimately it must have come from an axiom (if it is not an axiom itself). (2) You are using a rule of inference: from $\Phi$ deduce $\neg(\neg\Phi)$. And BTW, "...
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### Which one is the $width$ and $height$ of a rectangle?

In English , Side : A line segment forming part of the perimeter of a plane figure Width : The extent of something from side to side Height : The vertical dimension of extension; distance from the ...
• 1,112
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### Is there a formal term for a "subset connected by comparability" in a poset?

Every poset is a directed graph with a directed edge $a \to b$ whenever $a \le b$, and in a directed graph the concept you're looking for is called being weakly connected (as opposed to "...
• 377k
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• 10.7k

### Why is complex numbers considered numbers while vectors quantities?

I don't think there is a precise mathematical definition or reason - it's more a question of English language than mathematics. But usage suggests we commonly use the word "number" when ...
1 vote
Accepted

### What do we call a sequence with a finite limit superior?

You would call it bounded above. A sequence of real numbers $(s_n)$ has a finite $\limsup$ if and only if there is an absolute bound $M$ with $s_n \leq M$ for all $n$. It's clear that if $s_n$ is ...
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### What does normalizing a real valued function mean?

This depends on which norm you're using. Every continuous real function with compact domain assumes both its maximum and minima at least once, so on a compact domain $D$ it's natural to define the ...
• 6,087

### Can every group be called a symmetry group?

Before even mentioning what set your group elements act as symmetries of? Please, no. If you're starting with the group, just call them "(group) elements". If you really want to pay ...
Accepted

### Question about terminology regarding "Grothendieck Group," "Grothendieck Ring," perhaps "Grothendieck field"?

"Grothendieck group" also refers to a different but related construction which takes as input a category $C$ of some sort, typically abelian, and returns as output the free abelian group on ...
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### Synonym for "corollary" in English?

This is Soft Question, with no Exact "Correct" Answer. With that comment, I suggest : illation : The reasoning involved in drawing a conclusion or making a logical judgment on the basis of ...
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### Synonym for "corollary" in English?

In Sanskrit we use the word प्रमेयम् (Prameyam) for a theorem and उपप्रमेयम्(UpPrameyam) for a corollary. For example sometimes the Pythagoras Theorem is called the Baudhayan Prameyam as he ...
• 23
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### Terminology: Isomorphism to mean bijection

Is this true, does mathematical literature use "isomorphism" (perhaps with some qualifiers) to mean bijection? Should I be prepared to face such term in the wild (in non-CT context)? As for ...
• 184k
1 vote
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### Expressing $\!\lim\limits_{n\to\infty} (n,n)\!"$ without losing information?

Losing the information that two elements are the same, or more generally are not independent, is indeed a common problem. E.g. when taking into account computation error, or observation error, adding ...
• 2,297
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### Are mathematical theorems entailments, or merely implications?

Your question is indeed interesting. A theorem may or may not be in the form of $P \Rightarrow Q$. When it is $P \Rightarrow Q$, $P$ is usually not a tautology (i.e $P$ is not always true), otherwise ...
• 2,297

### Are mathematical theorems entailments, or merely implications?

I understand that many (if not all) of the theorems I've seen are in the form of implication: $P \implies Q$ What if both $P$ and $Q$ are true, but irrelevant/independent? In classical logic anyway, ...
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### Does the concept of "dynamic average" makes any sense?

Below link to the paper Dynamic programming: Arithmetic, Geometric, Harmonic, and Power Means. https://www.researchgate.net/publication/...

### What is the opposite to "discretization"?

It seems to be an embedding as you already wrote. Check this Wikipedia entry: https://en.m.wikipedia.org/wiki/Embedding The function or process could be called embedding or maybe continuity ...

### Why are exact differential equations called so?

An exact differential has the property that its integral over a path is path-independent: it does not depend upon the taken path, but only upon the origin and end of the path. In mechanics, the work ...
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### Name for subgroup of elements commuting with another group action

It's the intersection of $G$ and $\text{Aut}_H(X)$. That's also what your natural transformation computation says.
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### Terminology: how do we call a vector space equipped with a symmetric bilinear form over a general field?

I'm not aware of any terminology for this. The term quadratic space is used for a pair $(V, q)$ of a vector space $V$ and a quadratic form $q : V \to K$ on it; this is equivalent to specifying a ...
• 377k
Accepted

### What is the opposite of relaxation

If $(1)$ is a relaxation of $(2)$, then $(2)$ is a restriction of $(1)$. Also, you have the roles of $C$ and $F$ reversed in $(3)$.
• 35.6k
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### Proper meaning of 'algebra'.

A $K$-algebra $A$ is a vector space over a field $K$ equipped with a bilinear product, see here. If it is not necessarily associative or commutative it is often called a non-associative algebra. For ...
• 120k
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### Number of self-inverses in a group

An element of order $2$ in a group is often called an involution. In geometric groups reflections are involutions, as are half turns and central involutions, for example. Involutions are normally ...
• 97.1k
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### "Node-independent path"?

Node-independent, also vertex-independent, means the two paths do not share any nodes (vertices). That is, the two paths are completely separate except for their start and end nodes. Also equivalently,...
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1 vote

### Why addition/multiplication are not considered a mathematical functions?

Addition, multiplication, and exponentiation are "functions". Furthermore, they are all hyperoperations belonging to the hyperoperation sequence, which has no upper limit. Basically, we can ...
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### What is a "proper face" of a graph?

You may rest assured that: This is not common terminology, and Neither of the two textbooks cited in "We assume the reader is familiar with basic graph theory appearing for example in [11, 26]&...
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### Complement versus Negation

Suppose $P(C)=0.2.$ Its complement is $0.8,$ i.e., $P(C)^\complement=0.8.$ But what does $P(¬C)$ mean? I think I am mixing up the term complement and negation? $$P(C)=0.2$$ is a statement. Its ...
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### What is this "φ(·, t)" means?

The symbol $\varphi(\cdot, t)$, for fixed $t$, is a shorthand for the function that maps $x$ to $\varphi(x,t)$. Being pedantic, it would be $\varphi(\cdot,t)(x) \doteq \varphi(x,t)$.
• 72.1k
1 vote
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### What is a pointed connected groupoid?

You've mixed up category levels; the definition of pointedness is about the terminal object of the $2$-category of connected groupoids itself, which is the point $1$. Then the $2$-category of pointed ...
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