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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) ...


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 ...


6

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 ...


0

Nobody seems to have picked up on definition 2(a) from the cited dictionary entry: “capable of being completely counted.”  Students are taught from childhood (see Wolfram MathWorld, Math/is Fun, Math Goodies, Purplemath, and For Dummies) that counting numbers are positive integers.  If you ask a person, “How many elephants do you have in your pockets?”, most ...


0

Maybe this is too applied for your question, but there is a context in which finite means "large" or possibly large. That is in elasticity theory, to distinguish the linear theory of small strain from the nonlinear theory of "finite" strain.


-3

My understanding is that natural number 0 (which is also the cardinal 0 and the ordinal 0) is finite, but the real number 0 is infinitesimal, hence not finite. This page: https://www.princeton.edu/~achaney/tmve/wiki100k/docs/Non-standard_analysis.html says "A non-zero element of an ordered field F is infinitesimal if and only if its absolute value is ...


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 ...


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 ...


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 ...


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 ...


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.


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 ...


0

A more general name for this kind of topology is "final." This involves a change of perspective: we imagine the topology on $Y$ is not already given, but that it's going to be defined by the quotient map $q$. We say that a quotient space has the final topology induced by $q$. "Final" here means that the topology is the finest possible one that makes $q$ ...


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 ...


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 ...


0

Since the universal quantifier in $\varphi$ is equivalent to a conjunction of $[\overline{a}/x]\varphi$ of all elements $a$ of the universe $U$ (and the same holds for the existential quantifier in terms of disjunctions), they are regarded to be generalizations of De Morgan's laws, as others answered already: $$\begin{align} \neg \forall x (\varphi) & ...


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 ...


3

Is there a similar word for rewriting ∀x.φ into ¬∃x.¬φ? Short answer: No. Or at least there surely isn't one in common usage. Making that kind of move in a proof in a lecture, I would probably have just said "By [the familiar] quantifier equivalence, ..." By the way, the observation that Regularity is more intuitively appealing in the version ...


1

Contraposition in classical logic can be "reduced" to the equivalence between : $p \rightarrow q$ and $\lnot p \lor q$. Thus, from : $\varphi \rightarrow \psi$ using Commutativity : $(\lnot \varphi \lor \psi) \leftrightarrow (\psi \lor \lnot \varphi)$, and Double Negation we get : $\lnot \psi \rightarrow \lnot \varphi$. For the ...


0

A mathematical structure is a set or sets associated with some mathematical object (s) like a binary operation,collection of its subsets etc which satisfy some axioms.the mathematical object (s) is called structure and the set is called ground or underling set.example topological structure (X, tau) here tau is structure ans x is underlying set... similarly ...


0

A mathematical structure is a set or sets associated with some mathematical object (s) like a binary operation,collection of its subsets etc which satisfy some axioms.the mathematical object (s) is called structure and the set is called ground or underling set.example topological structure (X, tau) here tau is structure ans x is underlying set... similarly ...


0

Commutatitive:Is a condition where by the number's and letter's are adding or multiply. Example addition; x+y=y+x or 2+3=3+2 and multiply; xy=yx or 2*3=3*2 Associative:Is a condition that a group of quantities connecte by operators gives the same result whatever their grouping. Example addition, a+(b+c)=(a+b)+c or multiplication, a*(bc)=(ab)*c. ...


1

Please note tha $\frac{d}{dx}$ is an operator and not a quantity, so it makes sense to write $$ \frac{d}{dx} \circ \frac{d}{dx} = \frac{d^2}{dx^2} $$ which means applying the operator twice.


3

Really, the standard second derivative symbol $d^2\over dx^2$ should be considered an abuse of notation in its own right. The first derivative symbol $d\over dx$ is already a "single symbol", so iterating it twice should yield $({d\over dx})^2$. Without parentheses that yields ${d\over dx}^2$, which is confusingly like $d^2\over dx$, which is very very ...


2

Note what $$ \frac{dy}{dx}=\frac{d}{dx}[y]=g(x) $$ where $y=y(x)$. So, it is convenient define $$ \frac{dg}{dx}=\frac{d}{dx}\left[\frac{d}{dx}[y]\right]=\frac{d^2}{dx^2}[y]=\frac{d^2 y}{dx^2} $$ to designate the second derivative of $y$ with respect to $x$.


7

$(dx)^2$ is not $d^2x^2$ because there is no quantity called $d$. Rather $dx$ can be thought of as an infinitely small increment of the variable $x$.


11

The mnemonic is the following. The operator "d" is applied twice to $y$, so $d(dy)=d^2y$. But to get the second derivative we have to divide by $dx$ twice namely the operator $d$ is applied once but the result is squared. So $dx\cdot dx=(dx)^2= dx^2$. This is not $d$ applied to $x^2$ , this is $(dx)^2$.


1

we have per definition $\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)=\dfrac{d^2y}{dx^2}$


0

They are called smooth or infinitely differentiable. See smoothness on Wikipedia.


1

Generally speaking, a cost function is a function that needs to be minimized. They are used in a wide array of bigger questions within finance and economics. If you can minimize your cost function (i.e., taking the first derivative to find where the absolute/local min points are) then you can maximize your profit if you can figure out the maximum revenue. ...


1

You can characterize such a matrix by writing it as $A=\mathbf{x}\mathbf{y}^T$ where $\mathbf{x}$ and $\mathbf{y}$ are vectors. See if you can use this to show that $\mathbf{x}$ is an eigenvector of $A$. $$A\mathbf{x}=\left(\mathbf{x}\mathbf{y}^T\right)\mathbf{x}=\ldots$$


1

This is more an English language question. Personally I do not think you can say Lens $1$ has "$2$ times bigger magnification" without ambiguity. If you said it had $50\%$ more magnification, then this would be $1.5$ times Lens $2$'s magnification, so $100\%$ more magnification would be twice and logically $200\%$ more magnification would be three times. ...


1

Without denying the truthfulness of Michael Hardy's answer, I was always under the impression that the reason for connecting products to geometry was due to the fact that the height of a straight edge triangle is the geometric mean of the projections of its other two sides onto the hypotenuse. This and the fact that geometric shapes such as rectangles ...


0

Imagine a line of length $1$, and the next of length $2$, and the next of length $4$, and then $8$, etc. The pair of lines of lengths $1$ and $2$ has the same geometric shape as the pair of lengths $2$ and $4$, and similarly $4$ and $8$, etc.


0

In robotics or computer vision the term pose is generally used to describe the position and orientation of a rigid object. Wikipedia says: The combination of position and orientation is referred to as the pose of an object. [...] The pose can be described by means of a rotation and translation transformation which brings the object from a ...


0

This is a community wiki answer intended to remove this question from the unanswered queue. As Daniel Fischer pointed out in the comments, the OP's reasoning is correct, and this was probably just a momentary lapse in consistency with definitions.


0

"Canonical" can mean simple in appearance or utility. For example, The Jordan Canonical Form is a transformation of a matrix so that it's block diagonal and all the blocks are upper triangular. It's simple in appearance (most of the elements are 0, that's good). And it's a nice form to be able to use. For example, it's simple to find a solution to $J x = ...


2

How should a half primorial be notated? $\dfrac{p\#}2$ Is there a name for a half primorial? No, there is no established term for this notion.


1

He means the minimum of the two numbers. If $\frac{\epsilon}{2|a| + 1}$ is less than $1$, then it will be that number. Essentially, the reason for the $1$ is that he's saying the number is no greater than $1$ (if $\frac{\epsilon}{2|a| + 1}$ happens to be greater than $1$, then $1$ will be the value). Note that the definition of how to compute the minimum ...


2

Spivak already introduced the $\min$ and $\max$ functions in exercise 13 in the very first chapter, where he asked the reader to prove that: $$\min(a,b)=\dfrac{a+b-|a-b|}2\quad\color{grey}{\text{and}}\quad\max(a,b)=\dfrac{a+b+|a-b|}2.\tag1$$ What the $\min$ function basically does, is it outputs the smallest number from the input. So $\min(a,b)=a$ if ...


1

Alterando is "Alternendo". And, componendo et dividendo is "componendo and dividendo". Rest are all the same in English as in Latin.


1

In my experience, the definition of a binary operation as $f\colon S\times S\to S$ is standard. Certainly you would want $f\colon S\times S\to T$. Mapping back to $S$ though is very useful because you want to be able to repeatedly apply the map (say, to form a group). I guess the thing is that we just aren't usually interested in giving a name to $f\colon ...


1

The particular relation you're thinking of is probably functional composition; that is, if the value of $\pi$ determines the value of $m$, there the should be some function $f$ such that $$f(\pi(n))=m(n)$$ which just takes the given value for $\pi$ and uses it to find the appropriate value for $m$. We would write this relation as $$f\circ \pi = m.$$


2

The Sierpinski space is a dualizing object which mediates the Stone duality between the category of topological spaces and the category of frames. http://ncatlab.org/nlab/show/dualizing+object


2

It will still be called Legendre's conjecture. What will be appended is "proved in .... by ...." and the prover will be justly celebrated. What will really matter is how many other results follow from the proof technique.


1

The wikipedia page on graph minors states "$H$ is called a minor of the graph $G$ if $H$ can be formed from $G$ by deleting edges and vertices and by contracting edges." I'd say the deleting vertex part is mostly redundant, since you can effectively delete through edge deletions and contractions: delete all of its edges except one, then contract along ...


1

A sequence of real numbers is a function from $\Bbb N$ to $\Bbb R$. That is a rule which assigns to each natural number $n$ a real number $a_n$. Given a sequence of real numbers $(a_n)$ a series is defined by the sequence of its partial sums. The $n$th partial sum of the sequence $(a_n)$ is $(S_n)$ where $$S_n = \sum_{k = 1}^n a_k$$ Now, $ \sum_{n= ...


2

A sequence, by strict definition, is a mapping from $\mathbb N$ to $\mathbb R$. Less strictly, it's just a listing of real numbers, $a_1,a_2,a_3,\dots$ For example, $$1,\frac12,\frac13,\frac14,\cdots$$ is a sequence. A series is an infinite sum of real numbers, so for example, $$1+\frac12+\frac14+\frac18+\cdots$$ is a series.


1

Yes! Its right! If your square is $a$ then the number that originate it is $\sqrt{a}$ thus the next near consecutive perfect square is $(\sqrt{a}+2)^2$, but $(\sqrt{a}+2)^2 = a + 4(\sqrt{a}+1)$


1

It is called completing the square.



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