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3 votes
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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0 votes

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 ...
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4 votes
Accepted

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 "...
2 votes
Accepted

What's the term for the coefficient that change any floating point number to its next or previous value?

There isn't a term for this constant because it doesn't exist: You can't perform this operation with a multiplication by a constant. For example, consider the double-precision numbers: $a = 1$ $b = ...
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0 votes

Meaning of 'argument of a function'

We can start from a natural language example, the expression "The capital of France is Paris". In it we have two names: "Paris" and "France". If we replace into this ...
1 vote

Is there a name for a vector space together with a bilinear form?

The closest thing I've seen is "semi-indefinite-inner product space" (https://arxiv.org/abs/0901.4872) which does not require the bilinear function to be symmetric or positive, but does ...
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0 votes

figure, number, digit and

Let's first define the $3$ terms that you refer to in your question, and then we can look at some examples to illustrate any differences between them. Number: a mathematical object used to count, ...
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1 vote
Accepted

Why is complex numbers considered numbers while vectors quantities?

More formally, a vector is a member of a vector space, a set $V$ with the associated operations of addition and scalar multiplication satisfying certain axioms. It can easily be shown that $\mathbb{C}...
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2 votes

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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3 votes

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 ...
-1 votes

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 ...
2 votes
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 ...
2 votes

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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0 votes

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 ...
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0 votes
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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 ...
1 vote
Accepted

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 ...
3 votes
Accepted

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 ...
0 votes

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, ...
-1 votes

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/...
-1 votes

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 ...
2 votes

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 ...
3 votes

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.
3 votes

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 ...
2 votes
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 $(3)$.
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3 votes
Accepted

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 ...
4 votes
Accepted

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 ...
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2 votes
Accepted

"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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3 votes
Accepted

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]&...
0 votes

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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2 votes

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 $\varphi(\cdot,t)(x) \doteq \varphi(x,t)$.
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1 vote
Accepted

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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