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

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1answer
339 views

Why do some authors use terms “non-ascending” and “non-descending” instead of ascending and descending? [duplicate]

In my math book, everywhere the author has used "non-ascending" instead of descending and "non-descending" instead of ascending. I was wondering if there is some special meaning or use associated with ...
0
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1answer
46 views

Definition of monomial

I thought the definition of a monomial is an algebraic term that has no subtraction or addition. I saw on my online college homework that 2/x is not a monomial. Why?
3
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2answers
25 views

What's the name of a solid that results from extruding an area straight along an axis?

If you have any kind of 2D shape and move it up into the third dimension, what do you call it, because it seems like prism is used only if the base is a polygon. It also seems like extrusion is a ...
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1answer
74 views

What is the connection between slant/oblique asymptote to the polynomial part of the function and polynomial division?

What is the connection between slant/oblique asymptote calculation to the polynomial part of the function and polynomial division? To find the slant asymptote $y=mx+n$ we can can calculate it in two ...
3
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1answer
86 views

Understanding the condition of $\diamondsuit^+$

I try to understand $\diamondsuit^+$, a variant of the diamond principle. It says that there is a sequence $\langle \mathcal{A}_\alpha\rangle_{\alpha<\omega_1}$ of collection of subsets which ...
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2answers
54 views

Proving rigorously that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$ using divisibility definition

Let $a,b\in \mathbb Z$. Prove rigorously using divisibility definition that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$ After a bit of algebra I get that $$3\overset{?}{\mid}(a+b)^3-3ab(a+b)$$ So now how ...
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1answer
54 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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0answers
54 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
540 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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1answer
225 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
51 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
28 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
575 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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0answers
42 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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0answers
2k 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
69 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
64 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
78 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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1answer
66 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
54 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 pre-...
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1answer
25 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
523 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
184 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
49 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
68 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
71 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?
0
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1answer
38 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 ...
2
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2answers
110 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
147 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
143 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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2answers
51 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 = \{\{a\},...
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1answer
113 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., $\textbf{...
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2answers
28 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
38 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 U_x\...
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1answer
130 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
48 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
533 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
455 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
25 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
56 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
152 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
38 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
20 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
60 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
54 views

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

$$\mathcal{P}(x)=\{y\mid y\subseteq x\}$$ $$\mathcal{P}(\varnothing)=\{\varnothing\}$$ $$\mathcal{P}(\{\varnothing\})=\{\varnothing,\{\varnothing\}\}$$ $$\mathcal{P}(\{a,b\})=\{\varnothing,\{a\},\{...
0
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1answer
56 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
126 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
893 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 $f$...
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1answer
99 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 "...
2
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2answers
72 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" ...