# Tag Info

4

Based on a very cursory examination of a paper that uses this term, here's my guess. Let $V$ be a vector space. The symmetric algebra $S(V)$ of $V$ is a simple version of what in physics is called bosonic Fock space, and the exterior algebra $\Lambda(V)$ of $V$ is a simple version of what in physics is called fermionic Fock space. The names come from the ...

0

I include this example merely to illustrate. The usage of the terms bosonic and fermionic are far more broad. In the study of $N=1$ supersymmetry one finds that the super Poincare group has a natural action on what is known as $\mathbb{R}^{4|4}$ superspace. This space supports 4 commuting coordinates and 4 anticommuting coordinates. To keep manifest the ...

2

In particle physics, bosons and fermions are classes of elementary particles. Every elementary particle is either a boson or a fermion - for example, electrons are fermions and photons are bosons. Fermions are characterized by the property that only one fermion may occupy a given quantum state, whereas many bosons may occupy the same quantum state. ...

3

The action of a group $G$ on $X$ (in particular a set, but it could be an object in any category, for instance a topological space) can be encoded as a group homomorphism $G\to{\rm Aut}(X)$. In group theory we say a homomorphism $G\xrightarrow{\phi} A$ "factors through $H$" if there exists more group homomorphisms $G\xrightarrow{\psi}H$ and ...

4

If the other group is a quotient group of the first $G$, it means that the action can be written as composition of the canonical projection (group morphism) from $G$ to the quotient, followed by the action of the quotient (group morphism to a general linear group). It implies that the kernel of the projection acts trivially.

0

Yes, singular $n$-simplices are maps. We are effectively trying to probe our spaces with maps from the $n$-simplex. Note that these are topologically the same as solid spheres, and their boundary is a sphere. Homology is trying to capture the idea of holes in the space, and so this is a natural thing to consider. We want to see if we can throw a sphere into ...

0

make the change of variable $x=\sin(t)$ then your integral becomes $$\int_{0}^{k}dt(1-t^{2})^{1/2}t$$ here $k$ is the upper limit corresponding to $\infty =sin(t)$ which can not be consistently defined however if we impose that the integral MUST be real then we can only integrate from $0$ to $1$ so $$\int_{0}^{1}dt(1-t^{2})^{1/2}t$$ otherwise the ...

7

There is no difference between "to" and "into" here. In Gelbaum and Olmstead's Counterexamples in Analysis, they use this terminology in a way that makes it seem slightly more reasonable (though I personally still wouldn't ever use "into").

14

Both expressions say the same thing. But note that saying "$f: A \to B$ is a function from $A$ into $B$ does not imply that $f$ is into but not onto (i.e., it does not rule out that $f$ might be onto, so it is not the "converse" of, or the negation of, the descriptor "onto" or "surjective"). Indeed, it says nothing more, and nothing less, than the ...

7

Any function $f : A \to B$ is said to map $A$ into $B$ or to be a mapping from $A$ into $B$. The term into is used in general for any function, it doesn't relate to any specific kind of functions.

11

Both mean the same thing. There is no such thing as an "into" function.

0

Let $R$ be a ring. A chain complex is a sequence (of left $R$-modules) $A = \cdots\to A_{n}\xrightarrow{d} A_{n-1}\xrightarrow{d}\cdots$ Such that $d^2 = 0$. This allows you to define $H_n(A) = \ker{d_n}/\mathrm{im}~d_{n+1}$. An exact complex is a chain complex such that $H_n(A) = 0$ for all $n$. Exact complexes are usually called acyclic complexes. A ...

2

A small remark: the sequence $...\rightarrow A_n\rightarrow A_{n+1}\rightarrow A_{n+2}\rightarrow A_{n+3}\rightarrow ...$ can be "exact" or a "complex" (it depends on the nature of the maps, as confirmed in the answer by @exitingcorpse), but not a "short exact sequence". Short exact sequences (of $R$-modules, for example, denoting by $R$ a given ring) are ...

0

To say that the sequence is a chain complex is a less imposing condition: it simply says that if you compose any two of the maps in the sequence, you get $0$. But to say the sequence is exact says more: it says this is precisely (or, if you rather, exactly) the only way you get something mapping to zero. The first statement says that the image of the arrow ...

2

When talking about functions between sets, there is no such thing as an "inverse" in the first place if the map is not bijective. Take a look at the relevant Wikipedia page. Moreover, when you refer to isomorphism is bijective homeomorphism I think you're thinking of isomorphism is bijective homomorphism which, by the way, happens to be true in ...

0

Sounds like a poor translation from another language... your best bet might be to just google raster scan...

0

Following up on the comment by Arturo Magidin G and S are the first and last entries in the main diagonal of the submatrix determined by I and Q I'll point out that (i) two selected entries should not be in the same row or same column; (ii) if the entries K and R are picked instead, then the corresponding pair N, P is on the secondary diagonal of the ...

1

I’d advise against using it in this context. It’s etymologically correct, true; but it’s neither common nor totally transparent, so non-native speakers of English may be confused by it, and even native speakers may do a double-take. Since there’s a clearer and more usual alternative, “taking adjoints”, you should avoid tripping the reader up with language ...

1

Based on Adeel's comment above, I've been calling $B$ the "factor domain."

5

Would $$t\mapsto(a\cos t, a\sin t, bt)$$ fit the bill? Its image is a helix. The parameters $a$ and $b$ can be any non-zero constants. They determine the rise/rotation and handedness of the helix.

5

There is: $$\begin{array}{rcl}\mathbb{R}&\longrightarrow&\mathbb{R}^3\\t&\longmapsto&(\sin t,\cos t,t)\end{array}$$

5

This is called computing the Smith normal form, and the application you describe features in this question: Determining the Smith Normal Form

1

I think this is an Application of the main structure theorem of finitely generated modules over a P.I.D applied to the case of abelian groups. If you have Jacobson algebra book go to page 188 (Ch. 3.10 "application to abelian groups and to linear transformations") I don't see a specific name for this method here, but it is well explained.

1

By applying RK4 to an ODE, one solves the equation numerically: that is, a numeric solution is obtained. It is implicitly understood that a numeric solution is a finite vector of floating-point reals, rather than a map from an interval $(a,b)$ to $\mathbb R$; it's a rather different kind of object. Both are called solutions, even though they are not the ...

4

Let $X$ be an integral scheme. Then, for any open subset $U$, the ring $\mathscr{O}_X (U)$ embeds as a subring of the function field $K (X)$ in a canonical way, and so for every point $x$ of $X$, the local ring $\mathscr{O}_{X, x}$ is canonically a subring of $K (X)$. Definition. An element $f$ of $K (X)$ is regular at $x$ if it is in $\mathscr{O}_{X, x}$, ...

3

"$f$ is regular on $U$" just means $f\in\mathcal{O}(U)$.

2

Alexander and Serkan point out that the individual elements are just special types of cosets: $xHy = (xy) H^y = H^{x^{-1}} (xy)$. This is a special subset of $G$ that is both a left and right coset (of possibly different but conjugate subgroups). I don't believe it has a special name. A different idea that is better suited to the entire collection $\{ xHy ... 2 As has been pointed out in the comments, this is perfectly fine punctuation. 0 the number before the decimal point is called the whole number while the number after the decimal point is called the part of a whole ... 0 Regressive function appears to be a common term in set theory: e.g. Regressive function on an ordinal and set of infinite cardinals admits an injective regressive function. It requires strict inequality, but if you use it outside of set theory, you are going to explain the term anyway, which will give you an opportunity to say that only$\leq$is required. ... 1 This is a CW write-up of the comments of Arturo Magidin in order to remove this question from the unanswered tab. An "isomorphism of sets" (that is, an isomorphism in the category of Sets) is just a bijection. So what they are saying is that$F$can be bijected with$\mathbb{F}^r_p$(since they are both vector spaces over$\mathbb{F}_p$of the same ... 0 Sure why not, right? Who's it going to hurt? 4 Converting comment to answer to get this off the Unanswered list: I’d describe$C$simply as$\{a\cup b:a\in A\land b\in B\}$, which is of course equivalent to your$\{a\cup b:\langle a,b\rangle \in A\times B\}$. I see nothing simpler. 5 An excerpt from Gregory H. Moore's The evolution of the concept of homeomorphism: The evolution of the concept of “homeomorphism” was essentially complete by$1935$when Pavel Aleksandrov (Paul Alexandroff) at the University of Moscow and Heinz Hopf at the Eidgenossische Technische Hochschule in Zurich published their justly famous book Topologie, aiming to ... 2$Y$is the direct product of$X$with its opposite, the semigroup denoted by$X^{\text{op}}$: $$Y = X \times X^{\text{op}}$$ If$X$is a semigroup, we can define a semigroup$X^{\text{op}}$such that$X, X^{\text{op}}$are opposite semigroups: As Jack notes in the comments, a left action of$X^{\text{op}}$is the same as a right action of$X$. For some ... 2 Here is something which you may be interested in: open (closed) cover: A cover$\mathcal U$of$X$is called an open cover (or a closed cover) if each member of$\mathcal U$is open (closed) in$X$. Note that closed cover is not often appeared in the general topology. We always consider open covers of$X$. Lindelof: A regular space$X$is a Linfdelof ... 3 You might be looking for Lindelöf spaces. In those, every open cover contains a countable subcover. In$T_1$spaces, if we try to apply the definition for "compact set" and replace "open" with "closed," we run into a problem, since singleton's are closed. For any infinite set$X$, $$\bigcup_{x \in X} \{ x \}$$ is a "closed cover" of$X$with no finite ... 3 First consider unweighted directed graphs. I'd say that your$\mathcal{N}_v^+$is the set of out-neighbors of$v$and that your$\mathcal{N}_v^-$is the set of in-neighbors. Their cardinalities are the in-degree and out-degree respectively. (Or in- and out-valency.) But if in a paper I came across$\mathcal{N}_v^+$, I would not know if it referred to ... 2 Note that the equiconsistency of a theory$T + \phi$relative to$T$says, for theories that include or interpret arithmetic, that if$T + \phi$proves$0=1$then$T$proves$0=1$. We can broaden this notation in a certain way. Let$C$be a set of formulas. We say that a theory$T'$is conservative over a theory$T$for formulas in$C$if, for all$\phi ...

0

There is no reason for introducing two different terms. Apparently, somebody introduced one term, and somebody else introduced a different term, either because he didn't like the first guy's term, or he hadn't heard of it. Or maybe it was the same guy and he changed his mind, or forgot what he called it before. How would I know, I'm not a historian (nor a ...

5

Models are structures, and structures are models. But when we say "model" we mean that there is a particular theory which holds in the structure, and when we say "structure" we are mainly interested in an arbitrary interpretation of the language.

6

A structure is a set with some interpretable symbols(constants, relations and functions) within a fixed language. You do not ask for more from a structure. However... A model (of a theory) is a structure which satisfies the axioms of the theory. It makes more "structural sense"... Maybe an example brings more clarification: Consider the theory of groups. ...

3

Let $P$ be a set of (different) presents, and let $K$ be a set of kids. You can think of a function $f:P \to K$ as a way of distributing all the presents among the kids. (We say that $f(x)=y$ if present $x$ was given to kid $y$.) The function $f$ is onto if every kid gets at least one present. The function $f$ is one to one if no kid gets $2$ or more ...

0

A function has a domain, the set of possible inputs, and a codomain, the set of possible outputs. "Onto" means that every possible output is actually produced by some input. For example, let us consider the function $M$ that takes each person $p$ to their mother, and let's say that the codomain is the set of all women. Then this is not an onto function, ...

0

A function $f$ going from a set $X$ to a set $Y$ is onto if and only if for every member $y$ of the set $Y$ there is some $x$ in the other set $X$ for which $f(x) = y$. That is, $f$ doesn't miss out on any of the possible things in $Y$ that it could hit; it hits them all. For example, suppose $f$ maps a list of your best friends $X=\{\text{John}, ... 0 A function relates members of one set X to members of a second set Y by assigning each member of set X to a single member of Y. However, this does not require that every member of Y has received an assignment. A function which does have every member of Y receiving such an assignment is said to be "onto". (I read from your comment that you already ... 1 A function$f:A\to B$is onto if its image is equal to$B$, that is,$\operatorname{Im}(f)=B$. For example,$f:\{0,1\}\to \{3,4\}$given by$f(x)=x+3$is onto. Remember that the image of a function$f:A\to B$is $$\operatorname{Im}(f)=\{b\in B\mid \exists a\in A; f(a)=b\}.$$ 2 Exponential growth and exponential decay when$x\to+\infty$are often defined by the fact that $$\lim_{x\to+\infty}\frac{\log f(x)}x$$ exists and is not zero. In this context, every function$f:x\mapsto cx^a(\log x)^br^x$with$c\gt0$and$r\gt0$,$r\ne1$, has exponential growth or exponential decay when$x\to+\infty\$.

1

A function that is continuous and open is an embedding of a quotient of the original space. This is a very interesting notion, just like subquotients of groups. For instance, if you restrict a covering map to a subset of your domain, you (usually) get a continuous open map that is not one-to-one or surjective. This comes up a lot in geometry, for instance ...

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