Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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1answer
58 views

Restricted bilinear map

Suppose $\phi$ is a bilinear form on the vector space $V$. What does it mean (perhaps in matrix form) for $\phi|_X$ to be non-singular, where $X\leq V$? This question is probably elementary, but I ...
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2answers
2k views

Rank and signature of a quadratic form

Is the rank and signature of a quadratic form $x^TAx$ basically the rank and signature of $A$? Thanks.
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2answers
374 views

Exact sequence in a nonabelian category [previously: “Exact sequence for topological groups?”]

If $A$, $B$, and $C$ are topological groups, and $f: A \to B$ and $g: B \to C$ are two continuous group homomorphisms, what does it usually mean for $$1 \to A \stackrel{f}{\to} B \stackrel{g}{\to} C ...
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2answers
662 views

What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
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1answer
200 views

Differentiability of a matrix function

To prove the differentiability of a function defined on $M\in M_n$, how should I adapt the definition of differentiability? $h^{-1}[f(x+h)-f(x)-hL(x)]\to 0$  How do I take the reciprocal of a ...
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2answers
27 views

Definition clarifications on functions of vectors and their derivatives

Definition clarifications would be appreciated: How do I interpret the following ? For $f: R^n\to R^m$, $Df(\vec\xi)(\vec{x})$ in differentiation of a vector function. I know it as a function that ...
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1answer
91 views

Definition of a groupoid of fractions

The title sums it up : I am looking for a definition of "a groupoid of fractions for a category". I would be interested in any example someone might have as well...
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3answers
164 views

What is the proper term for a function where domain and codomain coincide?

What is the proper term for a function where domain and codomain coincide? E.g. in programming languages a function f : Int => Int or f : Double => Double. Thanks.
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1answer
543 views

Time-invariant IVP

How does one know that a system (of differential equation and initial value constraints) is time-invariant (perhaps by inspection...)? What are the implications of a system with this property (esp. ...
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1answer
97 views

Definition of $\ell_\infty$

What does the $\ell_\infty$ space stand for? Thank you.
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1answer
163 views

Definition of “or”

A quick definition clarification: Does the set $\{(x,y):x =0 \,\,\,\,\text{or} \,\,\,\,y=1 \}$ include the element $(0,1)$? (Sorry, English is not my first language, I get confused sometimes... Also ...
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2answers
653 views

Definition: Eigenspace of a matrix

If I am given a matrix and told to find a basis for its eigenspace, does that just mean find the eigenvectors of the matrix? In my understanding, an eigenspace of an eigenvalue $\lambda$ is the set of ...
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5answers
510 views

What is a Real Number?

I believe I'm over thinking it, but I want to be 100% sure. A Real Number is any number, correct? Whether it be an integer or something else. It's the set $\mathbb R$ from $(-\infty, +\infty)$ ...
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2answers
690 views

Definition: transient random walk

What exactly does a "transient random walk on a graph/binary tree" mean? Does it mean that we never return to the origin (assuming there is one as for the tree) or just any vertex of the graph or ...
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1answer
174 views

What does unique “minimal” partition mean (Context: Partitioning of Vertex-Sets)?

I am studying R. Diestel's Book Graph Theory and I encountered a formulation which I don't quiet understand. Mr. Diestel speaks in this proof on page 180 (Google Books Link) in the second last line of ...
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4answers
332 views

Definition of injective function

From wikipedia I obtain the following definition of an injective function : Let $f$ be a function whose domain is a set $A$. The function $f$ is injective if for all $a$ and $b$ in $A$, if $f(a) = ...
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1answer
452 views

Example of a simple pole

I was told that $\operatorname{sech} x$ has a simple pole. Could someone please explain what that means? I have looked up the definition but it involves too much jargon like holomorphic, etc. Is there ...
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0answers
99 views

question about definition Silverman's AEC

I am sure many of you have Silverman's book the arithmetic of elliptic curves (2nd edition). On page 417, proposition 1.2, he defines $\delta$. He also defines $\xi: G \rightarrow M:\ ...
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1answer
110 views

Clarification on Dirac notation

I am new to the Dirac notation, so would appreciate some clarification. Suppose $\Psi=\psi_1+\psi_2$ where $\Psi$ is normalized and $H$ is a linear operator such that $H\psi_1=E_1\psi_1$ and ...
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1answer
275 views

Definition of (formally) self-adjoint

Does formally self-adjoint imply self-adjoint and vice versa? Thanks.
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4answers
299 views

What is the name for a maximal convex set of points contained in another set of points?

What is the name for a maximal convex set of points contained in another set of points X? Maximal in terms of inclusion. For the desired set to be unique, X can be restricted to be a simple polygon ...
2
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2answers
1k views

What is Galois Field

When I am reading some algorithm related to error correction. To generate some polynomial it uses finite-field which is Galois field. I am not from mathematical background. Can anybody explain me in ...
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7answers
2k views

What is combinatorics?

I've tried to search the web and in books, but I haven't found a good definition or definitive explanation of what combinatorics is. Could anyone give me a definition/explanation of combinatorics, ...
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3answers
155 views

How can the geometry (and the reals) be motivated from the bottom up?

I'm really not sure that I know what I'm talking about, or if I should just go and learn more math before questioning such things, but I'd like to have answers to the following questions that don't ...
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3answers
187 views

After saw this piece of discussion, i ask myself what is the most rigorous definition of the circle, but i can't figure it out

Discussion: Is value of $\pi$ = 4? so what is the "real definition" of a circle? i think the original solution from wikipedia is too ambigous, i couldn't find why the circumference of the circle is ...
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0answers
174 views

Equinumerosity without ordered pairs

After some discouraging comments to this question let me ask straight ahead: Can the concept of equinumerosity be defined basically without the concept of ordered pairs (in any of its ...
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3answers
191 views

Equivalence relations and bijections without ordered pairs

Is it correct that — opposed to general relations and functions — equivalence relations and bijective functions can be defined without reference to ordered pairs? Especially, do the ...
3
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1answer
149 views

Complex conjugates?

Is $e^{1\over 1-ix}$ the complex conjugate of $e^{1\over 1+ix}$? Is there a simple rule to compute complex conjugates without having to find $a+ib$ form? Thanks.
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3answers
233 views

What is the definition of $\infty$

I saw in a note that say $\infty$ is not a real number, and there is no interval of the form $(a, \infty]$? So what is the definintion of $\infty$?
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1answer
229 views

Definition of Genus

Are genus necessarily toral -- as shown in the illustration on wiki what about a tube, does it qualify for having genus 1? What about this? Does this have genus 1 or 2? Thanks.
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0answers
245 views

why are curl and divergence defined the way they are?

Curl of a given vector field, $F$, at a given point is a vector, $C$, that gives the measure of amount of rotation the vector undergoes at that point. It has been defined using cross products and ...
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1answer
217 views

Does perfectly normal $\implies$ normal?

A question here on perfectly normal spaces got me into investigating the definition of such a space. The definition on wikipedia says A perfectly normal space is a topological space X in which ...
3
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4answers
433 views

what exactly is an open set?

Many, infact all the books on topology I have come across define open sets in the following way: "A set $A$ is said to be open if by moving in small amounts in any direction about any point we ...
6
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1answer
637 views

Why does the definition of an open subscheme / open immersion of schemes allow for an “extra” isomorphism?

After taking an algebraic geometry course last year, I've been reviewing the material this year, and I remembered something that struck me as odd, but which I'd neglected to ask about at the time: ...
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3answers
972 views

Embedding, immersion

Could someone please explain what "embedding" means? (Maybe a more intuitive definition) I read that the Klein bottle and real projective plane cannot be embedded in ${\mathbb R}^3$ but is embedded in ...
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2answers
830 views

Definition of a sphere

If there is no further specification (such as solid or hollow), does a "sphere" refer to the solid/filled form or the hollow shell? Thanks.
2
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2answers
2k views

What is the “Donkey Theorem”?

I was watching the Turkish version of Who Wants to Be a Millionaire? and they asked this question: What field is the Donkey Case (or I guess it can be translated as Donkey Theorem) related to? ...
2
votes
1answer
213 views

$A_{i,j}B_{i,j}$ is matrix dot-product in Einstein Notation?

Skim-read an engineering book stopped to this assertion "matrix dot product is $\sum_{i,j \in I} A_{i,j} B_{i,j} := A_{i,j} B_{i,j}$ in Einstein Notation". Sorry but what does it really mean? I have ...
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2answers
1k views

Interpolation, Extrapolation and Approximations rigorously

A foreign book mentioned that "when the Lagrange's interpolation formula fails (for example with large sample due to Runge's phenomenon), you should use approximation methods such as ...
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1answer
379 views

What is empirical accuracy?

Can anyone give a definition on empirical accuracy? I Googled this keyword but cannot find a satisfactory definition. It seems to be the accuracy rate of a specific sample. Any link or typed text ...
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2answers
852 views

Subexponential growing functions

What is the most common definition of a subexponential growing function ? It seems there are different notions in literature.
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3answers
222 views

How to define $-\infty$?

I think I understand the fundamental concept of infinity. Elementary mathematics define $\infty := \frac{x}{0}$, for every $x$. And also $\infty := \frac{-x}{0}$ for every $x$. I know only one ...
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0answers
165 views

Is the extra condition in this definition superfluous?

I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me ...
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1answer
193 views

“There is a natural [way/map/etc.]…” [duplicate]

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 ...
8
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3answers
352 views

Why is closure omitted in some group definitions?

In some texts, there are three group axioms and in some there are four. The difference is that one of the axioms, the closure ($a,b\in G$ then $a*b \in G$) is omitted. Why is this so?
2
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1answer
258 views

Partial derivative notation: is that a projection function?

Consider the following definition: Let $(U,\phi)$ be a chart and $f$ a $C^\infty$ function on a manifold $M$ of dimension $n$. As a function into $\mathbb{R}^n$, $\phi$ has $n$ components ...
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5answers
2k views

What is a universal property?

Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia. To make the following summary more readable I do equate "universal" with "initial" and omit the ...
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3answers
3k views

How can an ordered pair be expressed as a set?

My book says \begin{equation} (a,b)=\{\{a\},\{a,b\}\} \end{equation} I have been staring at this for a bit and it is not making since to me. I have read several others posts on this, but none made ...
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1answer
541 views

The definition of submanifold of a manifold with boundary

What is the exact definition of submanifold of a manifold with boundary? For example, when $H$ is the half space of the plane and S is a cycle which intersects with the origin in the half plane. Then ...
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5answers
1k views

Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...