Questions tagged [definition]
For requesting, clarifying, and comparing definitions of mathematical terms.
8,223 questions
0 votes
4 answers
565 views
Are theorems the same, semantically, as definitions? Or axioms?
Okay, I know about this question and have been looking for others that might speak directly to this. When I ask this question to Google AI (I don't use AI much at all), Are all valid, proven ...
0 votes
2 answers
108 views
Definition of one sided limit
I am currently taking a real analysis class, and we've learned some notions from topology (nothing deep just key concepts). While I was proving a theorem, I came across something that really confused ...
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1 vote
0 answers
45 views
generalizing space diagonals to all (or most) geometric solids
While writing about diagonals of shapes, I defined a space diagonal as a diagonal of a 3D shape connecting vertices that are not on the same face of the shape. I then realized that this definition ...
- 279
3 votes
1 answer
169 views
"Universality" of terminology associated with quotient spaces
Originally this question came to mind due to a similar question I asked a couple of days ago. Briefly speaking, in the earlier question I asked for help regarding the terminology associated with ...
- 187
4 votes
2 answers
467 views
Robinson's word vs Rotman's word: can the second be derived from the first?
In "A Course in the Theory of Groups" written by Derek J.S. Robinson it is done as follows. Let be $X$ a set. Choose a set disjoint from $X$ with the same cardinality (see here for details):...
- 5,410
2 votes
1 answer
116 views
Definition of locally connected topological space - multiple (seemingly) different definitions
At the moment I am learning about connected, path connected and locally connected topological spaces. I came across several (seemingly) different definitions for locally connected spaces and would ...
- 187
0 votes
1 answer
67 views
Case where infeasibility comes from $Ax=b$ unmet in standard polyhedron not handled in this proof?
In linear optimization, a solution that is not feasible can be so because it violates an equality constraint, isn't it? Be it an equality constraint from a non-standard polyhedron or an equality ...
- 1,371
3 votes
1 answer
355 views
Definition of tightness
Recently, I read some materials on tightness of random variables and probability measures. There are two definitions: Definition 1. A sequence of random variables $(X_n)_{n \ge 1}$ is tight if for ...
- 826
0 votes
0 answers
100 views
What is the mathematical definition of rationality in game theory
From Wikipedia https://en.wikipedia.org/wiki/Rationality#In_various_fields Rationality is a core assumption of game theory: it is assumed that each player chooses rationally based on what is most ...
1 vote
1 answer
116 views
A question about the formal definition of a function graph.
With some friends I am currently reading and trying to understand Category Theory by Steve Awodey. As I am no trained mathematician, even simple issues can halt my progress. One occurred when I tried ...
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2 votes
0 answers
25 views
What is the relationship between the two different definitions of Concave Function? [duplicate]
Some articles indicate the definition of a concave function $f(x)$ as follows: $$\forall x_1,x_2\in D_f, \forall\lambda\in(0,1): f\left((1-\lambda)x_1+\lambda x_2\right) > (1-\lambda)f(x_1) + \...
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0 votes
0 answers
118 views
What is the most concise complete definition of a rigid framework?
From what I've seen, the key characteristic of a rigid framework in a polygon is that the sides of the polygon, once set, force the distance between every pair of vertices to remain constant. Is "...
- 279
1 vote
1 answer
141 views
Terminology confusion regarding quotient topological spaces
At the moment I am learning about quotients of topological spaces and am struggling with the notion related to them. In the class we have defined the following: Let $\left( X, \tau \right)$ be a ...
- 187
-1 votes
0 answers
38 views
When does an optimization problem that is "worth solving" also satisfy strong duality?
I want some minimalistic/easy to remember statements that relate someone's willingness to solve an optimization problem with the strong duality property of that problem. Of course, you would only want ...
- 8,711
-1 votes
1 answer
54 views
Justification for Definition of Directed and Undirected Graph
It is commonly known that directed graphs are defined as a double $G_d:=(V,E)$ such that $E \subseteq V^2$, and that undirected graphs $G_u:=(V,E)$ such that $E \subseteq \left\{ \{a,b\}\Big\vert a \...
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