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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2answers
77 views

In Mathematical Logic, What is a Language?

I've been reading about mathematical logic and computability theory, but I'm somewhat confused on one note: what exactly is a language? What does it mean when I am told "let $\langle 0, +, \leq ...
3
votes
1answer
73 views

Definition of intersection of a family of sets

The definition of an intersection of family($F$) is: $$\cap F=\bigl\{x\mid\forall A: (A\in F \implies x\in A)\bigr\} $$ If my understanding serves me correctly this notation means that all the $x$ ...
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1answer
47 views

Upper Triangular Matrix Definition

Is \begin{bmatrix} 1 & 1 & 2 & 3 \\ 0 & 0 & 4 & 4 \\ 0 & 0 & 4 & 4 \end{bmatrix} an upper triangular matrix? My linear algebra teacher says that the main ...
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0answers
10 views

what are the basics of the Divergence operator?

Can you please help me with the basics about the divergence operator $div$. I don't know how to use it ? where to use it ?
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0answers
48 views

“Filters” of associative rings

Does there exist some good notion of "filters" for arbitrary associative rings with unity that would generalize filters of Boolean rings? For me, if $R$ is an associative ring with unity $1$, it ...
2
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1answer
57 views

Kronecker product and outer product confusion

I have two column vectors: \begin{equation} u = \left[\matrix{ 1 \cr 2\cr }\right] \end{equation} \begin{equation} v = \left[\matrix{ 4 \cr 4\cr }\right] \end{equation} I'm trying to ...
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0answers
37 views

Binary operation (english) terminology

Foreword: I have read R.H. Bruck's A Survey of binary systems, where the notion of halfoperation is given. A halfoperation $\ast$ differs from a (binary) operation since $a\ast b$ may not be defined ...
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0answers
52 views

Definition of smooth (variety)

I don't understand the motivation for the definition of smoothness of a variety: A variety $V(f_1,...,f_m)$ in $n$-space is smooth iff $\mbox{rank}$ = $n-\mbox{dim} V$. Could you please give me ...
0
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1answer
28 views

Should I use supremum or maximum in the definition?

Let $U$ be a certain function on sets ($U(S)$ represents the 'value' of the set $S$). If a set $S$ is partitioned to two disjoint subsets $A$ and $B$ with $A \cup B=S$, then: $V(A,B):=U(A)+U(B)$ ($V$ ...
0
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1answer
32 views

Definition of Limit of a Function

I'm a little confused by the definition of the limit of a function-on one hand I feel the definition suggests that your limiting variable is shrinking into a little delta ball- on the other hand when ...
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0answers
8 views

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 ...
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0answers
79 views

Is it false that the complement of an open set is closed?

Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be a continuous function. Let $Z(f)$ be the zero of $f$. Prove that $Z(f)$ is closed. This is one of problems in my mid-term exam. I have used ...
6
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1answer
140 views

Stokes' Theorem Explanation

Can someone explain what Stokes' Theorem is measuring? What would taking the integral of a vector on a surface give you? When would you use it? This is the only definition I have and I don't really ...
0
votes
1answer
31 views

Does a bijection between two sets $A$ and $B$ implies $P(a \in C ) = P(b \in C)$, if $A,B \subset C$?

I'm just thinking about it. For example, a bijection between $\mathbb{Z^*_+}$ and $\mathbb{Q}$ implies that the probability of a random real number being rational or positive integer is the same (in ...
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0answers
25 views

What is a Complete Set of Weights of a Representation of a Lie Subalgebra?

In relation to Lie Group and Lie Algebra theory, I am studying about the weights of representations. I have come across the terminology "a complete string of weights" in my lecture course, but it is ...
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1answer
48 views

Why does $0/0$ have to be undefined? [duplicate]

Why can't it no be $\pm$ Infinity? If $x/1$ is $x$ then $x/0$ should be $\pm$ Infinity.
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0answers
38 views

Difference between a vector space and an algebra

I'm new to the subject of algebras and I would like to get a better understanding of what they are exactly. Am I right to say the follwing: ...
0
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2answers
67 views

Word for “openness”/“closedness” of an interval

What word properly completes the phrase the radius of convergence does not depend on the $\text{______}$ of the interval to mean that it doesn't matter whether $(a, b)$, $[a, b)$, $(a, b]$, or ...
0
votes
1answer
29 views

Convergence of vector spaces

I'm considering vector spaces over $\mathbb{R}$ or $\mathbb{C}$ : $(E_n)_n$ is a sequence of vector space (all included in $\mathbb{R}^p$ for example, with $p$ fixed). What meaning(s) do we usually ...
1
vote
1answer
49 views

Some troubles about topology and definition of a Vector Bundle

Disclaimer: It's heavily related to my old question : Visualizing the Topology of a Vector Bundle but I wanted to open a new question because the former had already got an answer and this time my ...
0
votes
1answer
19 views

Formal notation for finite intersection

How to state following sentence formally? Let $\tau$ be family of sets (possibly infinite). Any finite intersection of sets in $\tau$ is in $\tau$. My attempt is: $\tau$ is family of sets and ...
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0answers
35 views

A question about the names of the components of an interval

Suppose you have the interval $[n,10n]$. How do you call the $n$ in this interval? I think the $n$ can be called the "independent variable of the interval" (since the interval $[n,10n]$ can be written ...
0
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1answer
28 views

What is the name of the theorem that allows me to write : $f(x,y) = \frac{\partial F_1(x,y)}{\partial x} +\frac{\partial F_2(x,y) }{\partial y} $ ? [closed]

I would like to decompose $f$ into $F_1$ and $F_2$ : $f(x,y) = \frac{\partial F_1(x,y)}{\partial x} +\frac{\partial F_2(x,y) }{\partial y} $ What is the name of the theorem that allows me to do that ...
0
votes
2answers
30 views

What is the meaning of an “objective function” ?

I have studied Mathematics in french. Right now I am in front of this expression "objective function" and "objective functionals". I don't know its meaning, some one can help please ?
4
votes
3answers
182 views

The difference between hermitian, symmetric and self adjoint operators.

I am struggling with the concept of hermitian operators, symmetric operators and self adjoint operators. All of the relevant material seems quite self contradictory, and the only notes I have to go ...
6
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8answers
579 views

Positive and negative complex numbers?

Can there be such a thing as positive and negative complex numbers? Why or why not? What about positive or negative imaginary numbers? It seems very tempting to say $+5i$ is a positive number ...
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3answers
75 views

Why is $ |z|^2 = z z^* $?

I've been working with this identity but I never gave it much thought. Why is $ |z|^2 = z z^* $ ? Is this a definition or is there a formal proof?
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1answer
26 views

Is there any definition for homogeneous rotations?

Most of the geometric transformations can only be represented into square matrices via homogeneous coordinates, e.g., translation and 3D rotations with axes not through coordindate system origin. ...
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1answer
25 views

Reusing Variables First Order Logics

Assume we have a parametrized FO formula of this form: $$\varphi(x_1,x_2, y_1, \dots ,y_m) := \xi(x_1,x_2) \land \psi(y_1,\dots,y_m)$$ We want to use as few additional bound (quantified by $\exists$ ...
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0answers
10 views

Borel - Regular elements

In Borel's Linear Algebraic Groups (2ed) page 160 a regular element is defined in terms of its semisimple part, “thus $g$ is regular if and only if $g_s$ is regular.” A unipotent element $g$ has ...
0
votes
1answer
79 views

How is addition on N formally defined in textbooks on real analysis?

This is a follow-up question to Why does the definition of addition require proofs? In Landau's Foundations of Analysis, his definition of addition on the natural numbers seems a bit strange to me -- ...
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1answer
27 views

How would you define the square of the linear operator

If you define the linear operator norm of $A:X\to Y$ to be $$\|A\|_{op} = \inf\{C>0: \|Ax\|_Y \leq C\|x\|_X \text{ for all } x \in X \}$$ Then how would you define $\|A\|_{op}^2$? My guess is you ...
0
votes
1answer
57 views

Defining the integral of differential $1$-forms

Assume $M$ is a smooth manifold, $g:[0,1]\to M$ is a smooth curve on $M$, and $w$ is $1$-form on $M$. Definition: $$\int_gw=\lim_{\Delta\to 0}\sum_{i=1}^nw(x_i)$$ The tangent vectors $x_i$ are ...
0
votes
1answer
15 views

Equivalence between submodular function definitions.

I am trying to show that the definitions, given by wikipedia, of a submodular set function are equivalent. See section definition of: http://en.wikipedia.org/wiki/Submodular_set_function. Mainly I ...
0
votes
1answer
33 views

Graph Theory: What is the definition of the “Sorted Edge” algorithm?

I've been googling for a while and can't find a clear definition of the "sorted edge" algorithm--can anyone provide it please? A description would be helpful, but a simple statement of the algorithm ...
0
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0answers
29 views

Uniformly bounded function?

For all I know, the term "uniformly bounded" only makes sense when applied to a family of functions, sequences of functions, sets of functions with one parameter and such. But I've seen in several ...
2
votes
0answers
25 views

Connected sum of surfaces with boundary

The connected sum of closed surfaces (2-manifolds) is defined by removing a disk from each and gluing the exposed edges together. When defining the connected sum of surfaces with boundary, is the ...
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1answer
99 views

Characterization of lim sup, lim inf

If $(a_n)$ is a real sequence, in lecture we had: $\limsup_{n\to\infty} a_n=a \iff (i)\forall \epsilon >0 \,\exists n_0\in \mathbb{N} :a_n<a+\epsilon$ $ \forall n\ge n_0$ and $(ii) \forall ...
5
votes
1answer
81 views

What is the intuition behind the name “Flat modules”?

I am studying Atiyah and MacDonald's book "Introduction to Commutative Algebra" and I have just read the definition of a flat module. It seems to me that if they have called that kind of modules ...
6
votes
3answers
57 views

Codomain of a function

At high school we were told that a function has a domain and a range, the function maps from the domain to the range. Such that the domain contains all and only the possible inputs and the range ...
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1answer
33 views

Precise definition of oscillation behavior of functions like $\sin(\frac1{x})$

I tried today precisely defining the oscillation behavior present in functions like $\sin(\frac1{x})$ i.e: To do this, I started with the domain of function as in limits, and let $f(x)-L$ have ...
3
votes
1answer
76 views

Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP

I am still struggling quite a lot with the Picard-Lindelöf-Theorem (also known as the Cauchy-Lipschitz-Theorem). Problem: Consider the following IVP with $\alpha \neq 1$ $$\begin{cases} y'&= ...
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1answer
26 views

What is the Support of a permuation

Is this definition of the support of a permutation correct: let $\pi\in S_{\Omega}$ for $\Omega$ a finite set, and $S_\Omega$ the set of all permutations (bijections) on $\Omega$. Ie ...
1
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1answer
60 views

Is the function $f(x)=x$ on $\{\pm\frac1n:n\in\Bbb N\}$ differentiable at $0$?

This is really a question of definitions. If a function $f$ is not defined on an open set containing $x$, how do we define the derivative of $f$? Is it sufficient to be locally approximable by linear ...
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3answers
97 views

How to define the magnitude of a rotation in $\mathbb{R}^n$?

Is there some well established way on how to quantify rotations in $\mathbb{R}^n$? To say which rotation is greater and which is smaller? In $\mathbb{R}^2$ the rotation is characterized by a single ...
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0answers
44 views

If a series converges does its sequence of partial sums converge?

By Definition, a series $\sum_{n=1}^\infty a_n$ converges if it's sequence of partial sums $S_n = \sum_{k=1}^n a_k$ converges. My question is, is the converse true? If $\sum_{n=1}^\infty a_n$ ...
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1answer
27 views

Why is the length of one of the segments negative in the Menelaus' theorem? Aren't all distances by definition positive?

Why is the length of one of the segments negative in the Menelaus' theorem? Aren't all distances by definition positive? I think we all know the Menelaus' theorem, which claims that the following ...
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0answers
40 views

How is this function continuous?

The textbook stated that the following function whose domain is $\mathbb{R}$is continuous for every point in the domain: $g(x)=1, 0\le x\le1$ $=2, 2\le x \le3$, and it continues this patern. What ...
2
votes
2answers
48 views

A problem with the domain of function in the defintion of limits

My Stewart's Calculus gives the following definition of limit: $f(x)$ is defined on some open interval containing $a$, except at possibly $a$. So, $\lim_{x\to a} f(x) = L $ if and only if for ...
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1answer
29 views

Simple closed curve definition of genus

The genus of a connected surface can be defined as the maximum number of disjoint simple closed curves that can be removed from it without disconnecting it. Why must the simple closed curves be ...