Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [definition]

For requesting, clarifying, and comparing definitions of mathematical terms.

Definitions are at the core of mathematical precision; they answer the question "What is X?" in mathematics. Into this category fit questions regarding equivalence of definitions, clarification of complicated definitions, or proposed new definitions for mathematical notions, with requests for improvements or comments.

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What is a sparse subset?

In a work about fully homomorphic encryption I found usage of the expression: "sparse subset", as in: Our hint will consist of a set of vectors that has a (secret) sparse subset of vectors whose sum is v$_J^{sk^*}$ How exactly is such sparseness…
Golob
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What does it mean for two conjectures to be incompatible?

What does it mean for two conjectures to be incompatible? I read about Incompatibility of two Hardy-Littlewood Conjectures. http://mathworld.wolfram.com/Hardy-LittlewoodConjectures.html What does it mean?
ppthegreat
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acclivity of differential equation not defined

the differential equation y'*y=cos(x) has the acclivity of y'=cos(x)/y. Obviously the acclivity is not defined for points (x=r;y=0). r= real numbers Is there a way to give a short but logic argument for this? Does this mean there is no kurve within…
Jln
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Homomorphisms, Linear Transformations, and Distributivity

What are the differences between homomorphisms, linear transformations, and distributive operations? To me, they all seem essentially the same, they just are different names for the same phenomenon based on context.
kevin
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A question about the span of functions.

I am having trouble understanding the span of functions, my problem is: What is $\operatorname{span}\{a\sin(x),b\}$, and what is its dimension? I understand this in terms of vectors, but not in terms of functions, so any enlightenment would be very…
Ellya
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"There is a natural [way/map/etc.]..."

Possible Duplicate: What is a natural isomorphism? I have often encountered the phrase "There is a natural [way/map/etc.]..." when describing say isomorphisms, maps, etc. What exactly does "natural" mean?
elsa
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Upside-down triangle symbol on function

I came across the symbol that looks like an upside-down triangle, and coming in front of a function $f(x,y)$. What does that mean?
JJ Beck
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What does "(additional) features" mean in the definition of a structure?

I am reading the Wikipedia article on mathematical structure, and it says "a structure is a set endowed with some additional features". Features can encompass a metric, a topology, etc... I'd like to know if there exists a rigorous definition of…
niobium
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nomenclature concerning inverse maps

Let $X$ and $Y$ be sets with map $\psi: X \to Y$. If $\psi$ is bijective, then the inverse map $\psi^{-1}: Y \to X$ exists. Denote $2^X$ and $2^Y$ as power sets of $X$ and $Y$, respectively. For any map $\psi: X \to Y$ we can always define the map…
Moritz
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Intuitively, why are two integers $(a,b)$ and $(c,d)$ equal iff $a+d=c+b$?

I am going off the construction of integers from natural numbers as stated in this wikipedia article: https://en.m.wikipedia.org/wiki/Integer#construction I have heard that the first integer in the question would mean $a-b$ and the second one $c-d$,…
Dark Rebellion
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Assigning quantitative 'velocity' to the growth of the function

I already know that we can compare the 'growth of the function' by little O or big O notation. The meaning of 'growth of the function' I mean like $\log x
one potato two potato
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What does this definition of a function mean?

I've never seen before the following function definition. Image taken from Ch. 2 of the book "Principles of Eventual Consistency" What does it try to convey in this specific form and how does it differ from the commonly used notation? (The book is…
Curious
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Can a formal sum be 'infinite'?

I'm reading Abstract Algebra (by Doomit and Foote), where a polynomial is defined as the formal sum $$a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$$ with $n \geq 0$. Is it valid to define a polynomial as $\sum_{j=0}^{\infty} a_jx^j$, for which exists an $n$…
John Mars
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What is the formal definition of a mathematical expression?

I am interested in what the definition of a mathematical expression is. For example $x_1 + x_2$ is a different expression than $x_2 + x_1$, even though they both evaluate to the same function on the reals. Has anyone rigorously defined what a…
user107952
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Definition of sigmoid function

Wikipedia gives the following definition for a sigmoid function: A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a non-negative derivative at each point.[1] A sigmoid "function" and a…
mic
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