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

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13 views

Probability density function definition

The definition above is given in my lecture notes. However there is no further reference/explanation given for what $o(h)$ represents. Can anyone explain this in this case?
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1answer
19 views

Clarification of some doubts on the definition of submatrix

I don't fully understand how I can choose a submatrix in a matrix. Judging from this definition and picture (http://mathworld.wolfram.com/Submatrix.html), I would assume that you can't pick as a ...
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20 views

What abstract structures allows us to describe “nets that converge toward each other”?

Topological spaces are equipped with a bare minimum of structure to allow for a formalization of the statement "the net $a$ converges to the point $x$." Actually this isn't strictly true, but its true ...
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1answer
25 views

Definition of complex conjugate in complex vector space

I am starting reading about Hodge theory and while reading the definition of abstract Hodge structure a very basic question came to my mind... What is the definition of the conjugate of a subspace of ...
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18 views

Definition of a vector field

Reading Wikipedia, I see that a vector field is defined as a mapping $X: S \rightarrow \mathbb R^n$ where $S \subseteq R^n$. However, I sometimes see mappings $X: S \subseteq R^m \rightarrow R^n$ ...
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0answers
16 views

Elliptic boundary value problems and elliptic partial differential equations

I am interested in the relation between the definition of an 'elliptic boundary value problem' and an 'elliptic partial differential equation'. From the wiki entries it seems that 'elliptic boundary ...
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1answer
15 views

Is there a term for something equally distributed around zero?

Let's say X is uniformly distributed in [-1 , 1] Then what can we call the distribution of X³ ? It is not uniform, but it "mirrors" around 0 as well. Is there a simple word describing X³ that would ...
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1answer
18 views

Why are the two definitions of covariance equal? [duplicate]

For example in Wolfram mathworld, you get these two definitions of covariance. http://mathworld.wolfram.com/Covariance.html cov(X,Y) = E[ (X - E[X])(Y - E[Y]) ] = E[XY] - E[X]E[Y] ...
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3answers
320 views

How can we think and/or write rigorously about integration by substitution?

Define a function $I:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ as follows. $$I(a,b)=\int_a^b \sin t \cos t \,d t$$ Then we can find a more explicit description of $I$ using integration by ...
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11 views

What is the correct term for the equations which comprise a mathematical model?

I have a mathematical model constructed by myself and my supervisor. In writing my report do I refer to the equations that make up this model as "constitutive equations", or is there another term ...
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17 views

What is the difference between assuming for $\forall k< n$ and proving for $n$ than assuming for $n$ and proving for $n+1$

What is the difference between assuming for $\forall k< n$ and proving for $n$ than assuming for $n$ and proving for $n+1$ (or $n-1$ and proving for $n$)? both with induction. The first one is ...
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27 views

The definition of continuously differentiable functions

When we say $f \in C^1$, we mean that f is continuously differentiable. Isn't the continuity a redundant word? I mean, we have a theorem that says if $f$ is differentiable then it is continuous. So ...
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1answer
26 views

Definition of gradient?

Reference: A primer of nonlinear analysis - Antonio & Giovanni Let $H$ be a hilbert space over $\mathbb{K}$ and $U$ be open in $H$ and $p\in U$ and $f:U\rightarrow \mathbb{K}$ be a functional ...
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0answers
32 views

What does rigoruous but non-technical mean?

Hi I find the above expression a bit confusing. I am considering buying a book and it says that it's a rigorous but non-technical introduction to optimal stochastic control. Could someone explain ? ...
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1answer
15 views

What is the definition of differentiation in normed space?

I'm trying to generalize implicit&inverse function theorems in Euclidean spaces to the context of Banach spaces. I'm wondering what would be the definition of differentiation in Banach space and ...
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0answers
7 views

$k$-ary labeled trees with distinct labels

Classical definition of $k$-ary labeled trees doesn't restrict somehow the uniqueness of tree labels inside its branches. My question: Is any special definition (name) for such trees? To clarify ...
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0answers
30 views

What is an omega model?

I went to a seminar and a side question was if a theory had an omega model, however from the context I could not deduce the exact meaning. Does an omega model have a general meaning in mathematical ...
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0answers
22 views

Proper usage for term, addend, factor, multiplicand, expression, formula

The definitions and usage of the following words seem to vary, depending on the source text: term addend factor multiplicand expression formula The words are being used in the context of ...
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1answer
18 views

Clarification on definition of a basis

Quick question; lets say that $S$ is a basis of $V$. I understand that this means all vectors in $S$ are linearly independent, and that every vector in $V$ is an element of $\text{span} \ S$. Is it ...
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1answer
45 views

Is this relation transitive? $R=\{(1,2),(1,1),(2,1),(2,2)\}$ over $A=\{1,2,3\}$

Is this relation $R$ over $A$ transitive?$$A=\{1,2,3\}$$ $$R=\{(1,2),(1,1),(2,1),(2,2)\}$$ Since from the definition a relation is transitive if $\forall x,y,z\in A (xRy,yRz\to xRz)$, so since $3$ ...
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0answers
82 views

What is the “correct” way of making $\mathcal{P}(X)$ into a topological space?

If $X$ is a topological space, I want to know the "correct" way of making the powerset $\mathcal{P}(X)$ into a topological space. So let $\mathsf{SupLat}$ denote the infinitary Lawvere theory whose ...
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1answer
48 views

Are 'vectors' vectors?

Let us say I have a 'vector' $\vec v$ for which I can do the following operation on $A\vec v$ where $A$ is a matrix. Now most people (i think) would say that $\vec v \in R^n$ however $\vec v$ is not a ...
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1answer
48 views

Given a field extension $K\colon F$, $K$ is an $F$-vector space

I'm having a hard time understanding fields. Could someone help with the following I need to show that if $F$ $\subseteq$ $K$ are both fields and addition and multiplication on F are the ...
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3answers
40 views

If $f\circ g = g \circ f$ does that mean that both functions are to and from the same set and both are bijections? Does it tell us anything else?

If $f\circ g = g \circ f$ does that mean that both functions are to and from the same set and both are bijections? Does it tell us anything else?
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1answer
29 views

Question about defintion of inner product space

While practising I came across the following easy question: "Is the space $B(0,1):=\{f:[0,1]\rightarrow\mathbb{R}$ bounded$\}$ an inner product space?" But I'm not quite sure what the correct answer ...
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2answers
47 views

If the empty set is a subset of every set, why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$?

If the empty set is a subset of every set, why it isn't written with the elements of a set? like so $\{1,2,3,\emptyset\}$ Or why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$? I know one has two elements ...
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2answers
53 views

Find the right cosets of $H$ in $G$ simple example

Question: Let $G$ be a group and $H<G$ a subgroup with $|G:H|=2$ Show that the right cosets of $H$ in $G$ are $H$ and $G\backslash H$ Answer given: There are two right cosets, they are disjoint ...
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2answers
32 views

Give an example of a set which is not transitive

Transitive set: set $x$ is transitive if $\forall y\in x(y\subseteq x)$ I think $\{\varnothing\}$ is not transitive since $\varnothing\in\{\varnothing\}$ but $\varnothing\not\subseteq\{\varnothing\}$ ...
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1answer
37 views

slightly different definition of an ordered pair

In a paper I was reading an ordered pair had a slightly different definition $\langle a,b \rangle = \{a,\{a,b\}\}$ instead the normal Kuratowski definition which is that $\langle a,b \rangle = ...
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0answers
20 views

The maximum notation as regards the absolute value?

We know that $\max(\textbf{A})$ gives the maximum element of the array $\textbf{A}$. What is the notation, or a short formula, if we seek the element having the largest absolute value? e.g., ...
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2answers
24 views

Why can we write a uncontinual function continual?

Let's consider $$f(x) = \frac{(x-1)(x-2)(x-3)}{x^2-3x+2}$$ with definition $D_{f} = \mathbb{R} \setminus \lbrace 1, 2 \rbrace$. This means we are allowed to set $x$ to every value of $\mathbb{R}$ ...
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1answer
27 views

Local Homeomorphisms: Characterization

Problem Consider for simplicity a surjection $F:X\to Y$. Are these characterizations of local homeomorphisms equivalent: $$\forall x\in X:\exists U_x\in\mathcal{T}_X, V_y\in\mathcal{T}_Y:\quad ...
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1answer
31 views

Can every function be a composite to itself and how to know if a composite between two functions is defined?

Can every function be a composite to itself? like we have $f:A\to B$ is $f \circ f$ always defined? Can we say that if $f$ is a injection/surjection/bijection then so is $f\circ f$? Also, how do ...
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2answers
33 views

Determinant of Polynomial

I was reading some paper and it says 'Let $\Delta$ denote the determinant of the polynomials $P,Q$ and $R$ with respect to the basis $1,X,X^2$' ($P,Q$ and $R$ are degree 2 polynomials). And then I ...
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1answer
46 views

definition of a $\Delta$ - complex

I've been given this image from hatchers algebraic topology as an example of a $\Delta$ complex but with the explicit definition as follows A $\Delta$-complex structure on a space X is a collection ...
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1answer
71 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
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0answers
23 views

Holomorphs and split extensions

The notion holomorph was introduced in Maria S. Voloshina's Ph.D. thesis On the Holomorph of a Discrete Group. It is defined as follows: Let $G$ be a group and let $\mathrm{Aut}(G)$ be the ...
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0answers
27 views

Difference between upper bound, maximal element, and maximum

In order to learn the difference between upper bound, maximum, and maximal element (of a set), I wrote down the following. Is it correct? Upper bound not necessarily element (of set) greater than ...
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0answers
15 views

What is meant by “commutes with spatial translations”?

Let $f\colon(X,\mathcal{B},m_1)\to(X,\mathcal{B},m_2)$, where $(X,\mathcal{B},m_i), i=1,2$ are measure spaces. What is meant when saying that f commutes with all spatial translations?
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0answers
14 views

$k$-cube definition clarrification

If $c:I^k\rightarrow\mathbb{R}^n$ is a singular $k$-cube. What will $\partial c$ be? Will it be a singular $(k-1)$-cube or a $(k+1)$-cube or something else? Definition - A singular $k$-cube on $U$ ...
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1answer
88 views

Is there a name for a regular semigroup with zero in which the product of any two different idempotents is zero?

As the title says, my question is: is there a name for a regular semigroup with zero in which the product of any two different idempotents is zero? Note that any such semigroup is necessarily an ...
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0answers
37 views

Is there a name for a function that returns only binary values?

Is there a name for a function that returns only binary values (e.g., $f(x) : X \rightarrow \{0,1\}$)?
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1answer
14 views

Is $(-\infty,0)\times S$ for a compact closed manifold $S$ a “manifold with boundary and cylindrical ends”?

I read the following definition from this paper. Definition: Let $N$ be a Riemannian manifold with boundary $\partial N$. We say $N$ is a manifold with boundary and cylindrical ends if there ...
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1answer
34 views

What does “be the inclusion” mean?

Can anyone explain what the phrase means? To be specific, my notes has the phrase "let $f:A \rightarrow B $ be the inclusion". Does this mean the identity map?
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2answers
48 views

Power set of $\{\{\varnothing\}\}$

$$\mathcal{P}(x)=\{y\mid y\subseteq x\}$$ $$\mathcal{P}(\varnothing)=\{\varnothing\}$$ $$\mathcal{P}(\{\varnothing\})=\{\varnothing,\{\varnothing\}\}$$ ...
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6 views

What does bimeasurable mean? Is an invertible transformation bimeasurable?

What exactly is the meaning of a bimeasurable transformation? I did not find a very clear answer to that. As far as I see it means that Borelsets are maped to Borelsets. So an invertible ...
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2answers
23 views

Corresponding partition in equivalence relation

The relation $R$ on the set $A=\{2,4,6,8,10\}$ is defined by $$R=\{(2,2),(2,6),(2,10),(4,4),(4,8),(6,2),(6,6),(6,10),(8,4),(8,8),(10,2),(10,6),(10,10)\}$$ Question 1 Verify if $R$ is an ...
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2answers
169 views

Is a function defined at a single point continuous?

Is a function defined at a single point continuous? For example $f:\{0\}\to\{0\}$ defined by $f(x)=\sqrt{x}+\sqrt{-x}$ is a sum of two continuous functions and is therefore continuous, however for ...
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1answer
45 views

What is a regular homotopy?

The definition of regular homotopy from Wikipedia says that two immersion $f,g:M\to N$ are regularly homotopic if they represent points in the same path-component of $\text{Imm}(M,N)$. What does ...
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2answers
69 views

Questions which have false conditions

There are many "questions" on the internet like If $$1=5$$ $$2=6$$ $$3=7$$ $$4=8$$ then how many is $5$? With one "logic" answer is $9$ because $n=n+4$, then $5=9$. With other "logic" ...