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

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29 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 ...
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115 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 ...
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27 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 ...
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61 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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41 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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53 views

A question on Holder spaces

A function $f$ is said to belong to the Holder space if Holder condition is satisfied, i.e. $\exists \beta,L\geq0$ such that $$|f(x)-f(x')|\leq L|x-x'|^\beta$$ for all $x,x'$ in the domain of $f$. ...
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46 views

What can you tell me about integrable functions and riemann integrals?

Define the concepts of integrable function and Riemann integral for functions of two variables (across a rectangle and over an arbitrary area). I know how to define for a rectangle but not an ...
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1answer
20 views

'Union' of maps

Let $f : A \to Y$, $g : B \to Y$. Suppose that $f(x) = g(x)$ whenever $x \in A \cap B$. Define $$ h : A \cup B \to Y, \\ h(x) = \begin{cases} f(x) & \text{ if $x \in A$} \\ g(x) & \text{ if ...
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45 views

Solvable Group, which Quotients need to be Abelian?

In Wikipedia it says a group $G$is solvable if it has a subnormal series $\{e\}=G_1\lhd G_2,\dots \lhd G_n=G$ where $G_i$ is a normal subgroup of $G_{i+1}$ and all the factor groups are abelian. My ...
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145 views

What is the most accurate definition of the hyperboloid model of hyperbolic geometry?

For simplicity, let's focus on the two-dimensional case (the hyperbolic plane in 3-space). I have seen the hyperboloid model defined as variations on the following i) The positive sheet of a ...
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62 views

What does “weakly compact” mean when applied to subsets $X \subset Y$?

Let $X$ be a subset of a Banach space $Y$. Please can you give me a definition of what "$X$ is weakly compact" means? I want one which is in terms of sequences and boundedness, as opposed to one with ...
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56 views

About definition of recursive sequence

Can I define the recursive sequence in the following? Let $f: \Bbb{N} \to \Bbb{R}$ a sequence and $k \in \Bbb{N}$ and $\forall x \in \Bbb{N}(f_x:=f(x))$ , $f$ is recursive sequence if ...
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365 views

Definition of “the surface measure”?

Let $\mu_n$ be the $n$-dimensional Lebesgue measure. Let $||\cdot||$ be a norm on $\mathbb{R}^n$. Define $S^{n-1}=\{x\in\mathbb{R}:||x||=1\}$. I have proven that $\forall A\in\mathscr{B}_{S^{n-1}}, ...
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54 views

Definition of “point” and “vector” in $\Bbb{R}^n$, and a model for $\Bbb{A}^n(\Bbb{R})$..

Can I use the following definitions? let $(a_1,a_2,..,a_{n+1}) \in \Bbb{R}^{n+1}$, $(a_1,a_2,..,a_{n+1})$ is a point of $\Bbb{R}^n$ if $$a_1=1 \wedge \forall i \in \{2,...,n+1\} (a_i \in \Bbb{R})$$ ...
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62 views

One definition of the determinant of a matrix

Suppose you define as follows : for $(a,b,c,d)\in \mathbb{R}^4$, $\det \begin{pmatrix} a & b \\ c & d\end{pmatrix} = ad-bc$. for $A$ a square matrix of size $n$, you define $\det A$ ...
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55 views

Definition of a binary operation is the same as definition of a closed binary operation?

I'm reading Wikipedia about operations and binary operations . Intuitively I always thought that a binary operation is a operation that takes two arguments. But Wikipedia defines a binary operation as ...
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101 views

Questions on Killing form: its definition and a root space decomposition.

I have a question on Killing form. Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Consider the adjoint representation $(\mathrm{ad},\mathfrak{g})$ of $\mathfrak g$, i.e. $$ \mathrm{ad}: ...
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1answer
52 views

Big-$O$ notation definition.

I have come across the notation $$...+\ O_R(x)$$ where $R$ is any positive real number, and $R<2$. What is the term big $O_R$ of $x$ referring to?
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37 views

Quick question about chain homotopies.

In the definition of a chain homotopy (say $h$) between two chain maps (say $f$ and $g$), are the maps $h_i$ comprising the chain homotopy required to commute with all other maps involved (the $f_i$s, ...
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37 views

On (absolute) convergence of $f_c:= c + \sum_{n=0}^{+ \infty} \frac{a_n}{n+1}x^{n+1}$

Let $R> 0$ and let $g: (-R,R) \longrightarrow \mathbb{R}$ be given by the convergent power series $$g(x):= \sum_{n=0}^{+ \infty} a_nx^n$$ for $|x| < R$. Let $c \in \mathbb{R}$ and let $f_c: ...
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85 views

Can we say that each monotonic sequence is a bitonic?

Is it mathematically correct to state that every monotonic is actually a bitonic with 0 number of elements in other half? I.e., if it is increasing then we can say it has 0 elements in decreasing ...
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58 views

A single word to represent a sequence of repeating numbers

I asked this question on english.stackexchange too. What is a noun to represent a sequence of repeating numbers? For example: 777777777
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137 views

Definition of a $n$ - form on a manifold

I am confused about the definition of a differential form on a manifold. The definition I have comes from Bott and Tu and is as follows: A differential form, $\omega$, on a manifold $M$ is a ...
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2answers
112 views

Meaning of $\{ a,b \}$, and comparison with $(a,b)$

What does $\{a,b\}$ mean in real analysis? I'm also little bit confused about set definition Can you tell me the main difference between $(a,b)$ and $\{a,b\}$? Thank you.
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43 views

What is the lower bound of the subset $2^n,\; n\in\mathbb{N}$

Let: $$ A = \{2^n,\; n\in\mathbb{N}\},\quad A\subset \mathbb{R} $$ Is the lower bound: $(-\infty,0]$ $(-\infty,1]$ $(-\infty,1)$ ? I think it can be the first because ...
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1answer
123 views

Function being continuous at a point

I've been looking at the $\epsilon-\delta$ arguments for determining whether a function is continuous at a point. I'm really stuck on how to choose your $\epsilon$. Specifically lets look at the ...
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1answer
80 views

Generator of all congruence classes

Is it possible for $\langle a \rangle =\mathbb Z/n\mathbb Z^*$ for $a\in \mathbb Z/n\mathbb Z^*$? I think I recall hearing it is, what is the name this element takes? I remember hearing generator ...
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1answer
42 views

Terminology for a generalization of multilinearity to any algebraic structure.

Let $S$ and $T$ be sets with the same algebraic structure. Let $\Phi:S^n\to T$ such that for any $i\leq n$, and $s_1,\ldots,s_n\in S$, the aplication $s\in S\mapsto ...
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3answers
117 views

Why is $\infty-\infty$ undefined in measure theory?

Some additions to the title: I stumbled over this problem going through my measure theory lecture notes; the author explicitly mentions that he leaves $\infty-\infty$ undefined. I would like to know ...
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58 views

Definition of a function's domain and co-domain with subscript in name

I want to define a function that takes a parameter (lets say a real number) and returns a number (lets say a natural number). However, the function makes use of a 'global environment constant ...
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48 views

Median based on number of entries instead of values

I’m writing a computer program that provides some useful statistical information about files. Calculating the mean is trivial, and the mode at least has a simple definition, but the median is proving ...
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71 views

Analogous to convolution

A convolution of two functions $f$ and $g$ is defined as $$[f*g] = \int_{-\infty}^{\infty} f(\tau) g(t-\tau) d\tau.$$ I am interested on an analogous transformation of the form $$[f\star g] = ...
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78 views

Is it true that the definition of an open subset in a metric space is different from the combination of the definitions of subsets and opens sets?

Dear reader of this post, I have a question concerning the equivalence of two definitions of open subsets (in metric spaces). To avoid confusion, I will state the two definitions and then ask my ...
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2answers
130 views

A definition of algebraic expression

Definition of algebraic expression An algebraic expression is a collection of symbols; it may consist of one or more than one terms separated by either a $+$ or $-$ sign. If by symbol we only mean ...
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1answer
310 views

Perpendicular Symbol as Matrix Superscript

If $A$ is a matrix that is not (necessarily) square, then what is $A^\perp$? What I do know is: $A^\perp$ is a matrix, not the orthogonal complement It is related to the QR Decomposition. And ...
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74 views

What is the difference between a reflexive relation and an identitive relation

Given a set $X$ and a relation $R$ over $X$, we say that $R$ is reflexive if \begin{equation} xRx\ \forall\ x\in X. \end{equation} What does 'identitive' mean? Is it the same as antisymmetry? Seen ...
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182 views

Anonymous graphs and graph embeddedness

What are anonymous graphs, what is graph embeddedness, and how do they relate to each other? Very confused - I could not find short answer. Thanks.
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57 views

question about Morse theory in Hilbert space

This is sade to be the Morse theory in Hilbert space ,and i want to know the definition (or where i can find it ) of : The qth singular relative homology groupe The qth critical group Please; ...
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83 views

Ordinary differential equations with double resonance

I want to know what is the definition of "resonance, double resonance" in ordinary differential equations with double resonance for exemple this : what it means the probleme is resonant in infity ? ...
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502 views

Why is a CDF right-continuous at “a” in [a,b), when property Pr(a<X≤b) doesn't even require point “a” to exist, and “b” could carry baggage?

c.f. wikipedia:Cumulative distribution function properties "Every cumulative distribution function F is (not necessarily strictly) monotone non-decreasing (see monotone increasing) and ...
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97 views

Good source for Triebel-Lizorkin spaces?

I'm trying to look into different types of function spaces. At the moment, at least for function spaces involving integration, I only have $L^p$ and $W^{k,p}$. The next function spaces I thought I'd ...
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43 views

Are these two statements(theorems) equivalent?

I am given this theorem: Let $H$ be a check matrix for a linear code $C$. Then $C$ has minimum distance $d$ iff. there exists a set of $d$, but no set of $d-1$, linearly dependent columns in ...
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36 views

A follow up question on completeness of filter (generalised to p.o.s)

Now I'm looking at the following generalisation of $\kappa$-closedness of a filter on a set to a filter in a partial order: Let $\kappa$ be a regular uncountable cardinal. A partial order $P$ is ...
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35 views

$\tau_1$ coarser topology than $\tau_2$

I know the following definition of coarser topologies: If $\tau_1$ and $\tau_2$ are two topologies on $X$, we say $\tau_1$ is coarser than $\tau_2$ if $\tau_1\subseteq\tau_2$. In my book about ...
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123 views

Prüfer sequence for an order-2 tree?

All the algorithms for constructing a Prüfer sequence state that the input is a tree, but none give any output corresponding to an order-2 tree. And Wikipedia gives this definition:" A Prüfer ...
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84 views

Elementary definition: what's a parallel volume-form?

This is a very elementary question, What is the definition for a volume form (or $n$-form) to be parallel with respect to the metric? To find out more about the concept, what kind of topic do I need ...
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1answer
137 views

Differentiable manifold in dimension 1 and its critical point

Please, I want to know how to define a differentiable manifold in dimension 1, and if the circle is a differentiable manifold in dimension $1$, and what is its critical point. Thank you.
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2answers
1k views

What is a direct correlation?

I have two contrary definitions of for the direct correlation between two variables $X$ and $Y$ Their correlation coefficient is close to $1$. There is a direct causal relationship between the ...
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43 views

What is an evaluation operator and what is its use?

In a lecture about numerical mathematics, mostly about ODEs, we were given the following definition: The two-parametric family $\mathbf \Phi^{s,t}$ of maps $\mathbf \Phi^{s,t}: D \mapsto D$ is ...
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121 views

Precise differences in meaning of Power Series, Taylor Series

Being an physicist/artist, not a real mathematician, I often toss around the terms "Taylor Series" and "Power Series" without any concern. Are these terms be considered interchangeable by ...