# Tag Info

0

(J. Stewart. Calculus pp 391) I believe Stewart defines an antiderivative as an indefinite integral.

2

A theorem is an important statement in mathematics on its own, while a lemma's main/sole purpose is proving a theorem. The Pythagorean Theorem is an important result in mathematics, and we should not call it a lemma just because it has many profound consequences. Quite the contrary, in fact, that is a reason we should call it a theorem.

2

I think it is called the component group of $G$, as the group $\pi_0(G)$ of connected components of $G$ is naturally isomorphic to $G/H$.

1

It's called a Poisson binomial distribution. You can find useful information about it on Wikipedia.

0

"Idempotent" means that repeating the operation doesn't change the outcome. That doesn't mean that it doesn't do changes. I.e., asignment is idempotent (asign 10 five times in a row, the result will always be 10); increment isn't (doing i++ once or twice in C gives different results). Perhaps you mean it has no side effects, or is a "pure" function (each ...

2

There is a construction of a "universal morphism" in Brown's Topology and Groupoids, chapter 8.1. We assume that $G$ is a groupoid, $\sigma:Ob(G)\to X$ is a set map. Then we can construct a groupoid $U$ whose object set is exactly $X$, and a morphism $\barσ:G\to U$ whose object function is $σ$. The idea is similar to the construction of the free product of ...

1

Edit Based on OP Comments Actually, I realised that the sum of two bernoulli rvs with different ps will result in *under*dispersion, so perhaps underdispersed? If you don't want to be associated with dispersion models, then why not "Heterogeneous Binomial Sum", its clearer than generalized binomial, as there are several ways you could generalize it.

3

Apparently it's called a (commutative) inverse monoid. For further details, see Wikipedia1, 2, 3 or Lawson's Inverse Semigroups4. (I haven't proven that the sets of axioms are equivalent. You may want to reserve the bounty for someone who does so.)

3

I think that everything you've said is fine. It is trivially true that for any operation, $x$ and $y$ commute whenever $x=y$, so we wouldn't normally bother to say that. But the property of commuting is a local property. For example, in a group, the center of the group is the set of elements that commute with every element of the group. Since we always ...

6

It is not incorrect to refer to a non-commutative operation as being commutative when operating on certain elements. In fact, this notion is very important! For instance, the center of a group is the set of elements that commute with every element of the group. So the answer is a resounding "no". The algebraic structure need not be generally commutative to ...

7

All your examples are perfectly fine. Another example is the set of $(n\times n)$-matrices equipped with multiplication. This multiplication is not commutative, but we say two matrices commute iff $AB=BA$. Or we could say "Matrix multiplication is commutative on diagonal matrices".

0

My answer to your question, in short, is: Strictly speaking, no, ‘if’ and ‘iff’ are not interchangeable, but nevertheless there is a very general tacit convention for using ‘if’ in definitions where ‘iff’ is meant. The Tarski quote referred to by Tony Piccolo establishes (by authority) that using ‘if’ where ‘if, and only if’ is (or should be) meant, is a ...

1

Short answer: Because it mainly deals with function spaces. As opposed to calculus, where all the norms of Euclidean spaces are equivalent (and all Hausdorff topologies which respect their linear nature), in the case of infinite dimensional linear spaces norms are not equivalent. And Functional analysis is the "analysis" study of infinite (in general) ...

1

"rationalizing the denominator" still applies. To make explicit that the process is using $k$ as its base field, "$k$-rationalizing the denominator" (or rationalizing "over $k$" / "relative to $k$"). The suggestion is therefore $\mathbb{R}$-rationalizing the denominator "Realize" and "decomplexify" have accepted meanings very different from this. ...

1

As you can see in this link (page 5, col.2), Hoerl (presumably the inventor of ridge regression) "gave the name "ridge regression" to his procedure because of the similarity of its mathematics to methods he used earlier i.e., "ridge analysis," for graphically depicting the characteristics of second order response surface equations in many predictor ...

0

Sensitivity, or sensitive dependence on the input. Moreover, if your "system" consists of a repeated operation, e.g. $$x_n\to x_{n+1}=f(x_n)$$ the sensitive dependence of $x_n$ on the initial condition $x_0$ is denoted Chaos.

0

It's easier to work out if you have a specific example: Let A:I am a parent B:I have a child I am a parent if and only if I have a child has two parts: I am a parent if I have a child can be rephrased: If I have a child, then I am a parent. B => A I am a parent only if I have a child can be understood to mean: if I do not have a child, then ...

1

I don't know precisely what these are called, but you can always use the OEIS (Online Encyclopedia of Integer Sequences) to lookup a given sequence to see its name (if it exists) and some cool properties. If you're working base $n$, I think you should include 0 on your circle. While it's true that loops beginning with 0 are never interesting ...

1

"Decomplexizing the denominator" would not be as correct as "realizing the denominator", but it is less ambiguous and it sounds better for me.

0

Today I also faced the same problem for reading these very first concepts. My understanding is as follows. Path: A path is a simple graph whose vertices can be ordered so that two vertices are adjoint iff they are constitutive in the list. Walk: it is a list of vertices and edges $v_0, e_1, v_1, \dots, e_k, v_k$ for $1\le i \le k$, $e_i$ has an endpoints ...

1

First of all, there are two similar symbols: U+00D8 Ø latin capital letter o with stroke U+2205 ∅ empty set The first is a letter used in Danish, Norwegian, and Faroese languages. The second is the empty set symbol. The rendering of "\emptyset" on this site looks like the first, and ought to look like the second, in my opinion. Now, after that diversion, ...

1

You can see Ronald Lewis Graham & Donald E.Knuth, Concrete Mathematics (2nd ed, 1994), Ch.2 : Sums [page 21] : The symbol $$\sum_{i=0}^n a_i$$ was introduced by Fourier in 1820. The sigma-notation is called "summation". The quantity $a_i$ after the "big" $\Sigma$ is called the summand. The index variable $i$ is bound to the ...

6

In "A Logical Approach to Discrete Math" by David Gries & Fred Schneider authors call them all quantifiers (actual quantifiers, summation, multiplication and others). There is a whole section in which they develop general theory for those structures.

0

The intrnal part is written before the decimal and it is called the characteristic while the fractional part written after the decimal is called the mantissa..

2

Personally, I have never heard it called that (of course, it is always possible that some in subfield of mathematics I'm not familiar with, it is an accepted name). However, non-Americans will usually call $\mathbb{Z}$ (the integers) "zed", perhaps you misunderstood?

0

$\left(\begin{array}{c}1 \\ 3\\\end{array}\right)$ is the basis vector, the whole vector space (line) is the span of it.

2

I believe you are talking about a hereditary property. From Wikipedia: “In mathematics, a hereditary property is a property of an object, that inherits to all its subobjects, where the term subobject depends on the context.”

1

If $x=1$, then $y=3\cdot 1=3$.

1

A triangle $\{e_1,e_2,e_3\}⊆V(LG)$ is said to be an odd triangle if there exists a vertex $e∈V(G)$ incident to exactly one or all of $\{e_1,e_2,e_3\}$, and it is said to be even otherwise. [source]

0

As has been mentioned in the comments, the "graph paper" case (either finite or infinite) is a (square) grid/lattice/mesh graph. The generalization you mention at the end can be referred to with the same terminology: triangular/hexagonal grid/lattice/mesh graph. Wolfram has chosen "grid graph" to mean "square grid graph", but wikipedia mentioned all three ...

2

You can call it "normalization". The idea is that you are taking a range of values and scaling and translating them to lie in a standard ("normal") range of $0$ to $1$. Though I am not very familiar with statistics, I believe this is the standard terminology. For example, see this Wikipedia page, or just google normalizing data 0 to 1.

2

Reference: Jamie Mulholland http://www.sfu.ca/~jtmulhol/math302/notes/302notes.pdf p. 25: Definition of n-cycle = cycle of order n. p. 253-254: Definition of orbit.

1

I wondered if the integral of $f^2$ itself has any significance. As you say, this is the $L^2$ norm of $f$ in the space $L^2(λ)$, where $λ$ is the Lebesgue measure. This is also $E(f(X))$, where the random variable $X$ has density $f$. As such, this is the $L^1$ norm of $f$ in the space $L^1(μ)$ where $μ$ is the measure with density $f$. Is it used ...

1

A lemma are those minor results which are used into proving a definite results of a theorem.

2

Probably asking for $$f_1(x,y)=x+2y\\ f_2(x,y)= 3x+4y$$

0

I think if there were a single expression it must just simply be "test set." I came to this conclusion because I see it used both in Noncommutative generalizations of theorems of Cohen and Kaplansky by Reyes, and also in A note on prime ideal which test injectivity by Beachy and Weakley. If you only want to restrict your attention to the commutative case, ...

0

I am also very new to this concept, but this is how I think about filters. It is really an informal idea, so you don't have to agree with it. Let $X$ be a topological space, and consider the collection of all the neighborhoods of $p \in X$, say $O(p)$. This $O(p)$ is clearly a filter, but it is intuitive to think that the point $p$ "filters" open sets. ...

1

Let's call $a=2m+3$, $b=4m-2$ and $c=-m-1$. Now the equation becomes: $$ax^2+bx+c=0,$$ and the roots are: $$x_1=\frac{-b+\sqrt{b^2-4ac}}{2a},$$ $$x_2=\frac{-b-\sqrt{b^2-4ac}}{2a}.$$ Now you have to impose $x_1x_2=1,$ or, rather: $$\frac{-b+\sqrt{b^2-4ac}}{-b-\sqrt{b^2-4ac}}=1,$$ Therefore: $$-b+\sqrt{b^2-4ac}=-b-\sqrt{b^2-4ac},$$ So: $$... 5 By going to the see-also section of the wikipedia page for linear independence, I was reminded of the term "matroid", which is almost certainly what you're looking for. 2 Based on my previous answer, what about partial isometries? 2 Such Riemannian manifold (M,g) is called "locally flat" or "locally Euclidean". Its metric is locally isometric to the standard metric on R^n where n is the dimension of the manifold. Without further hypothesis, this does not tell you much. Suppose however, that the metric is complete (e.g. the manifold is compact). Then (M,g) is the quotient of the ... 0 Even though quite a bit has been said already, i wanted to add something. The numbers which you normally use in school (-1, \frac{2}{3},  \pi, etcetera) are called the real numbers. The set of real numbers is denoted by \mathbb{R}. Now the square root of any number b is normally considered to be any number x that satisfies x^2 = b, or ... 0 The usual convention is that the square root sign gives the positive square root. When the negative/both is/are required, use the minus/plusminus sign. 0 \sqrt{\mathrm{a}^{2}}=|a|= +a OR -a.Remember here OR is used.AND is not used as a function cannot have two values at same time. 2 But the truth being that the squarte root function is always associated with absolute value function. That is \sqrt {a^2} = \vert a \vert  This is very very important to remember and be careful while using the  \sqrt{}  . Note that if it is  {(a^{2}})^{\frac{1}{2}}  then we get answer as \pm a  but note that \sqrt {a^2} = \vert a \vert  ... 0 I am not sure if there is a generic name. However, certain forms of radicals had specific names in Euclid's time. Generally, depending on the use of the radicals, they were called, first binomial, second binomial, etc up to fifth binomial and similarly first through fifth apotome. Sixth was used for those beyond fifth. See ... 0 Well, I'd call that a function, say f, with domain [a,b],$$f(x)=\frac{x-a}{b-a} It is exactly the same thing if you place $x_{min}$ for $a$ and $x_{max}$ for $b$. And yes, it describes the ratio of the distances you're saying.

0

There's no "canonical" name for such functions but E.M.Stein calls them Riesz systems in all of his books. V.P.Havin had a nice name for them (which I used too): "harmonic vector field". The reason is that for any vector field with zero curl and divergence (in any connected domain) the component functions turn out to be harmonic. This is true for any ...

3

Read it as "$\kappa$ arrows $\alpha$-$m$-$\lambda$", or some small variant of this, such as "$\kappa$ arrows $\alpha$ super $m$ sub $\lambda$". Nothing too enlightening. That this is indeed the suggested reading is indicated in section 8.2 (pg. 53) of the Erdős-Hajnal-Máté-Rado book, Combinatorial set theory: Partition relations for cardinals. (The ...

2

An element $a$ of $C^*$-algebra is called an isometry if $a^* a = 1$. This coincides with the usual notion when applied to $B(H)$. Maybe we can just generalize this terminology to the setting of dagger categories. So call a morphism $f$ an isometry if $f^{\dagger} f = 1$. I don't know if this is standard.

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