# Tagged Questions

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### Definitions of hemicontinuity

can anyone see the equivalence or relation between the following two definitions of hemicontinuity that I encountered: Assume that $K$ is a closed, convex subset of Banach space $X$. Let $X^{*}$ be ...
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### Localization of the Integer Ring

Let $\mathbb{Z}$ be the ring of integers and let $p$ be a prime, then the $p$-localization of $\mathbb{Z}$ is defined as $\mathbb{Z}_{(p)}=\{\displaystyle\frac{a}{b}|a,b\in\mathbb{Z},p\nmid b\}$. I ...
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### Has anyone succeeded in formalizing Leibniz notation in such a way that the chain rule and inversion rule “work”?

The notation $\frac{\partial}{\partial x}$ is ubiquitous and totally useful, but also kind of weird. It seems to be doing the following: Bind $x$ Compute the derivative Evaluate at $x$ To ...
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### When is a binary operation bipotent?

I learnt that $\max(-,-)$ is a bipotent binary operation but I'm not able to find a definition of bipotent operation. QUESTION A binary operation $*:M\times M \rightarrow M$ is bipotent if ...
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### Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary ...
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### Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
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### Nonorientable surfaces: genus or demigenus?

The genus $g$ of a closed, orientable surface is the maximum number of disjoint simple closed curves that can be drawn on the surface without disconnecting it. In terms of the Euler characteristic, ...
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### What is the “Principle of permanence”?

While reading the book "The Number-System of Algebra (2nd edition)." term "Principle of permanence" occurred to me. I remember I had read this in the book "Beginning algebra for college students.". I ...
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### Odd and even functions.

I have a book which says: If a function $f$ satisfies $f(-x)=f(x)$ for all $x$ in its domain, then $f$ is called an even function. However, if $f(-x)=-f(x)$ for every $x$ in the domain of $f$, ...
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### Defining “Penon Infinitesimals”.

In this lecture (which is accompanied by these slides), right near the end (so page 9 in the pdf of the slides; I don't think you have to watch the lecture), P. Johnstone refers to the "Penon ...
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### 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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### Equality of positive rational numbers.

I am reading the second article Rational Numbers of the book "A Treatise on Advanced Calculus" by Philip Franklin. I have mainly 3 questions regarding this article. I am writing all these $3$ ...
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### “Preimage” of a binary relation

Consider the binary relation $R \subseteq X \times Y$. Is there a standard name and notation for the set $X' = \{x\ |\ (x, y) \in R\}$? ProofWiki calls $X'$ the preimage of $R$, denoted as ...
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### Conglomerate in mathematical literature.

I rememeber I downloaded a pdf in category theory and after talking about classes it defined something called conglomerates, but I can't remember what that is and wikipedia has no article on it. ...
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### Definition of the higher dimensional mapping tori

This is proving harder to search for than I imagined. The usual definition of a mapping torus $\mathcal{M}_h$ associated to a homeomorphism $h\colon X\rightarrow X$ on a topological space $X$ is the ...
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### what are first and second order logics? [duplicate]

The only knowledge I have on logic is due to a book I read a couple of years ago called Introduction to logic: and to the methodology of deductive sciences by Alfred Tarski. And in it he talks about ...
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### What does boson-type realization mean?

I have seen several different contexts the expression "boson-type realization", for instance in the study of algebras growth and realization of affine algebras. To be or not be a boson-type ...
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### What is a “distinguished automorphism” of a field?

Math people: The title is the question. The reason I am asking is that I am trying to determine exactly what fields can be used for an inner product. I posed that question at ...
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### What does relax mean in the mathematical context

Here is a direct citation from wikipedia: The assumptions were further relaxed in the works of Terence Tao and Van H. Vu, Friedrich GĂ¶tze and Alexander Tikhomirov. Finally, in 2010 Tao and Vu ...
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### A question on linear ordered space

A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$. And we know every space contains a dense left-separated subspace. My question ...
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### The definition of “number”

Maybe my question is very trivial. I would like to have the definition of "number". Can anyone advise me some documents online? thank you very much
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### Rate of convergence of random variables for weak convergence

Suppose $X_{n}$ be a sequence of random variable that converges to $X$ in distribution. How can we define the rate of convergence? What would be the reference?
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### Rate of convergence of double sequences

Suppose $\{X_{n,m} \}$ be a double sequence of real numbers and suppose $\lim_{n}\lim_{m}X_{n,m}=X$. What is the definition and reference for the rate of convergence of double sequences?
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### Sequences convergent to 'cycles'

Consider sequences $(x_n)_{n=1}^\infty\subset\mathbb R$. Is there a name for the following property? There exists $L\in\mathbb N$ such that: ...
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### 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 ...