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17

The problem is that physicists are more influential than mathematicians. They routinely consider zero to be a nonfinite quantity, probably because they are thinking logarithmically. If you hang around physicists, you will hear expressions like “very small but finite”. But the concept of infinity is a mathematical one, not physical, and certainly ...


13

There is really no point in insisting that a definition in a dictionary has any implication on the mathematical meaning of the word. Germs have nothing to do with real world germs, and cardinals have absolutely nothing to do with the catholic church. Normal spaces are not those which are not irrational, and real numbers might not really exist (e.g. if the ...


9

It seems to be not unheard-of to speak of small but finite quantities in applied mathematical fields. At least in this context, "finite" is obviously meant to mean "nonzero", or perhaps "not infinitesimal". Note that the dictionary definition you link to doesn't claim that 0 cannot be finite, period. It lists three different mathematical usages of the ...


7

In French the meaning "bottle" is definitely never used, in particular because it is syntactically impossible! The meaning in French of flasque is exactly flabby and the terminology is very appropriate: any section on an open subset of a flabby sheaf can be extended to the whole space. A tougher sheaf would never tolerate that: just try with the sheaf ...


5

No, "isomorphy" is very uncommon in contemporary mathematical English. You should just say, e.g. "The objects A,B,C,D,E,F are all isomorphic". I don't really understand how your suggested first sentence could capture your message any better than this. If I myself understand what you intend "isomorphy" to mean, all of the suggested sentences mean exactly ...


2

Normally we only consider the continuity as given, and then the property that ($f^{-1}[U]$ open implies $U$ is open) is what makes $f$ a quotient map (by definition!). So a continuous onto $f: X \rightarrow Y$ is defined to be a quotient map, iff for all $U \subset Y$: $f^{-1}[U]$ open in $X$ implies $U$ open. Or equivalently $f: X \rightarrow Y$ (onto ...


2

Yes, chiral comes from the Greek word χέρι for hand. But in a way chirality and chiral symmetry are opposites: Both your hands together are an object with chiral symmetry because "mirror image of two hands" looks the same as "two hands" - though the reflection interchanges the hands, so to speak. On the other, erm, hand, a single hand (your right hand, say) ...


2

I believe you are mistaken. The adjoint of the matrix A is the transpose of the matrix A. One major confusion here is that there are two definitions for the word adjoint. The adjoint of a matrix is its conjugate transpose. Another definition, now often called the "classical adjoint" of a matrix is the matrix of its cofactors, which is what I think you ...


2

It depends if you are considering the quotient set, or the quotient group. Define: $$x \sim_L y \iff xy^{-1} \in H$$ We can also write this as $x \equiv_L y \mod H $. Here, $L$ is for "left". You can define it the same way for right. Then, we have the class: $$\bar{x}_L = \{y \in G \mid x \equiv_L y \mod H\}$$ Then, we define $G/H = \{ \bar{x}_L \mid x \in ...


2

$0$ is doubtless finite. I'd say that the paradox' root is the imprecise word 'lot'. I think that a 'lot' is a quantity that can't be perceived at glance. I'm sure that most people don't think in $3$ or $4$ when hear the word 'lot', because if there are $4$ stones, we don't need count them to know. But I insist. $0$ is never an infinite number.


1

Zero can be considered to be an infinitely small number, in some cases this is the natural thing to do. It is quite typical for many natural phenomena to be discontinuous when certain effects become exactly zero. E.g., if the viscosity of a fluid is exactly zero, then that's qualitatively different from being small but larger than zero, as in the latter case ...


1

Flasque, when used as an adjective, say describing sheaves, means flabby, flaccid, soft, easily deformed in an algebraic geometry context. Only when used as a noun (feminine) is flasque the French term for English flask. That is NOT the usage here though. Even the very brief definition for flasque in English language Wiktionary provides the most common ...



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