# Tag Info

0

An interesting challenge! A couple of additional ones I didn't see in previous answers: strong > weak > light (adverb 'lightly') strong > weak > delicate (adverb 'delicately') strong > weak > wimpy (adverb 'wimpily') - sounds less formal but has precedents in technical areas. strong > weak > tentative (adverb 'tentatively') - likely less useful, has ...

1

In PDE theory, the term "very weak solution" is in wide use. Google results.

1

The usual definitions are: $\alpha$ is an acute angle if $0°<\alpha<90°$ $\alpha$ is an obtuse angle if $90°<\alpha<180°$ $\alpha$ is a reflex angle if $180°<\alpha<360°$ So your angle $\theta=(3+\frac{1}{6})\times 360°$ does not fint any (usual) definition. Obviously the riesidue angle $\theta-3\times 360°$ is acute.

3

I would say "pathetic" or "puny".

0

I'm not sure what you mean by a reflex angle, but it is certainly not acute. Acute angles are only between $0^\circ$ and $90^\circ$. The measure of the angle in question is $3\cdot 360^\circ +60^\circ=1140^\circ$.

4

I can't resist adding this one to the list: what do you call a principle weaker than Weak Konig's Lemma? Funny you should ask . . . Ayup, we're a creative bunch. :P Oh my goodness: page 18, after proposition 9.1. And heaven forbid we be at a loss to describe something not as weak as weak! (Page 8, definition 4.5.)

8

I don't think there is a standard adjective to describe this. If there is, we would need to know the context of the terms stronger and weaker to answer. It sounds like you are defining this weaker-er notion in your paper (since you have to introduce a new term), so it is really on you to give it a name. Now to compile a list of suggestions: subweak ...

0

A 2-place real function is a function whose domain is a subset of $(\overline {\mathbb{R}})^2$ and whose range is a subset of $\mathbb{R}$. ... says the book you are reading.

0

In one of the comments to the original question, Will says: "@GiuseppeNegro In Terence Tao's preamble [reference to this blog entry] what does he mean by "A vector space V over a field F can be described either by the set of vectors inside V, or dually by the set of linear functionals λ:V→F from V to the field F (or equivalently, the set of vectors inside ...

2

I would call them hourglass hexagons.                     (Image source: link.)

3

This would be the corona of $K_n$ and $K_1$, usually denoted $K_n \circ K_1$. The original definition was by Harary and Frucht in 1970 in their paper "On the Corona of Two Graphs" See this question for an application of it to more general graphs: Eccentricity in corona product

1

You have asked about the distinction between $a$ and $x$, but its possible that secretly, your real question is: "how can I write in such a way as to avoid these kinds of nonsense distinctions?" This is the question I will attempt to answer. Firstly, a few conventions that I find useful: Background. Given a set $X$, write $X_\bot$ for the result ...

1

Note that, perhaps somewhat counterintuitively, there is no actual difference between "a fixed number $z$" and "a variable $z$". Both mean exactly the same thing: for an arbitrary element $z$ in the domain of $f$, $f'(z)$ is defined to be such-and-such real number. Logically "fixed numbers" and "variables" have the same semantics and correspond to universal ...

1

If you think back to the time when you were in 9th grade learning to solve quadratic equations, what you saw was that $$\text{if } ax^2+bx+c=0\text{ then }x=\frac{-b\pm\sqrt{b^2-4ac\,{}}}{2a}$$ (not to be confused with $\dfrac{-b\pm\sqrt{b^2-4a} c}{2a}$ or $\dfrac{-b\pm\sqrt{b^2-4} ac}{2a}$ or $\dfrac{-b\pm\sqrt{b^2-{}} 4ac}{2a}$, etc., all of which I've ...

1

It looks like the quantile function. It's domain is usually $(0, 1)$ and here are some simple properties: $F^{- 1} (x) \leqslant t$ iff $x \leqslant F (t)$. $F^{- 1}$ is increasing and left-continuous. If $F$ is continuous, then $F \circ F^{- 1} = \text{id}$.

-2

Say you have a program p that returns the input plus one. it could be defined like this: p(int i){ i=i+1; return i; } p(int i) is like f'(x): it's a function that takes one input. Now, say you have another program that calls p at some point. for instance, q(int i, int j){ s=i+j; return p(s); } p(s) is like f'(a). It's a ...

2

Usually the first letters $a,b$ are used to indicate constant values, that are not specified but are intended to be fixed. The last letters $x,y,z$ are used to indicate variables, that is a symbol for a number that can have any value. In an equation we usually want to find the value of the variables, considering the other terms as fixed. But this rule is ...

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There's no real difference between $x$ and $a$ used as variable names. We distinguish between variables, parameters, constants - and yet this distinction does not really exist. This hierachy is just a customary order among variables, as it is customary to denote "more variable" variables with $x$ and "more constant" variables with $a$, say. Or to use $n,m,k$ ...

12

In mathematics, $x$ usually denotes a variable and $a$ denotes a (fixed) constant (however, any constant). The idea that the author wanted to give is that if you can calculate the derivative at any point, then you can consider the function that sends each point $x$ to the derivative of $f$ at $x$ (function known precisely as the derivative of $f$).

2

$r$ is the base of a Cunningham chain of the second kind. A Sophie-Germain prime is the base of a Cunningham chain of the first kind.

1

Notations include $q^{-1}\mathbb{Z}$,$\frac{1}{q}$,$\mathbb{Z}/p$. I would probably call it "Integers over p" or the likes.

2

I don't think there is a generally-accepted name for such sets. Sometimes they are written as dilations of $\mathbb{Z}$: $\dfrac12\mathbb{Z}$ or the like. Do not confuse the set $\dfrac12\mathbb{Z}=\{\dfrac n2:\ n\in\mathbb{Z}\}$ with the half-integers $\{n+\dfrac12:\ n\in\mathbb{Z}\}$.

1

The decomposition with the stronger condition is called effective. It appears in connection with pseudo-Riemannian symmetric spaces, where one deals with a special case of the symmetric decomposition $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{m}$, namely that the decomposition is effective, i.e., that it is minimal, which precisely means that ...

3

Similarly to Micah's answer, but showing an ismorphism to $\Bbb{R}\times \Bbb{R}$, note that \begin{align} (a(1-x)+bx)(c(1-x)+dx) &= ac(1^2-2x+x^2)+(ad+bc)(x-x^2)+bdx^2 \\ &= ac(1-2x+x)+(ad+bc)(x-x)+bdx \\ &=ac(1-x)+bdx. \end{align} Therefore the mapping to $\Bbb{R}\times \Bbb{R}$: $\varphi(a(1-x)+bx)=(a,b)$ preserves multiplication. Showing ...

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If $x^2=x$, then $$(2x-1)^2=4x^2-4x+1=4x-4x+1=1$$ So you actually have the split-complex numbers in disguise.

3

Let $\Bbb{F}$ denote the base field of $V$. If we assume that $\star$ is a binary operation on $V$ that turns $V$ into an $\Bbb{F}$-algebra, i.e., Left distributivity: $x \star (y + z) = x \star y + x \star z$ for every $x,y,z \in V$, Right distributivity: $(x + y) \star z = x \star z + y \star z$ for every $x,y,z \in V$, and ...

1

I think this paper gives the definition you want, if I understand you correctly: Definition 2.9. A poset $P$ will be called locally ranked if all its principal lower ideals are ranked.

1

In principle, "$F$ is a left adjoint" is fine because the adjunction is determined uniquely up to unique isomorphism by $F$. This seems to contradict your intuition that "$F$ is a left adjoint" sounds wrong, but it's perfectly consistent with standard usage such as talking about "the" limit of $G$: this sort of usage works fine except in contexts where it's ...

0

They are all the same sort of thing on different levels of abstraction/generalization. Setting a value creates a more specialized (less general) version of the mathematical object (function, optimization problem, etc.), and replacing a formerly exactly defined value by a symbol creates a generalized problem (covering a whole family of the specific problems). ...

0

A variable is, of course, a quantity that is allowed to vary over its range of definition. For example, $f(x) = 3x + 5$ is a function, where x ranges over the real numbers. Now, I think the difference between constants and parameters is a bit more subtle. First, constants: A constant is just something that doesn't vary. 3 is a constant value, $\pi$ is a ...

2

A constant is something like a "number". It doesn't change as variables change. For example $3$ is a constant as is $\pi$. A parameter is a constant that defines a class of equations. $$\left(\frac xa\right)^2 + \left(\frac yb\right)^2 = 1$$ is the general equation for an ellipse. $a$ and $b$ are constants in this equation, but if we want to talk about ...

1

I think the second definition is better. $\frac{dy}{dx}=0$ would be considered a differential equation. A differential equation doesn't necessarily need to involve the function itself.

1

I think it must mean that although the assertion might not be true of the sequence in question, it is true of some subsequence of it. For an example where the phrase is clearly used this way, see this answer. This usage is not quite the same as the one described in the Wikipedia article: any equivalence relation that considered all sequences equivalent to ...

-2

A mathematical object is an object which is completely determined by its mathematical properties, that is, when you know everything you can say about it in the language of mathematics, you know the object. For example, the natural number $0$ is a mathematical object, because you can say it is the unique element in the set of natural numbers which is not the ...

0

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $X:\Omega\to\mathbb R$ a random variable. The distribution function of X is $$F(x) = \mathbb P\circ X^{-1}(-\infty,x] = \mathbb P(X \leqslant x).$$ If $F$ is a continuous function, in which case $F$ is actually absolutely continuous (due to monotonicity and boundedness), so F is ...

2

I haven't heard "cumulative density" used before. People are saying that it is a contradiction. To see why, it helps to look at the definitions of cumulative distribution function (CDF) and probability density function (PDF). Assume $X$ is a continuous random variable. The CDF is $$F(x) = P(X\leq x) = \int_{-\infty}^x f(x) \, dx.$$ It is the integral of ...

2

The terms diagonal and anti-diagonal are descriptive and "culturally apt". The first is standard in geometry and topology (the diagonal embedding of a topological space $X$ is the inclusion $X \hookrightarrow X \times X$ defined by $x \mapsto (x, x)$), and I'm almost positive I've seen the second in connection with the normal bundle of a diagonal embedding ...

6

Aha, I found it! The $f$ function I'm using is Cantor's pairing function.

2

There are two things to point out here. $2^{\aleph_0}=\aleph_1$ is an assertion called "The Continuum Hypothesis". It cannot be proved, nor disproved, from the standard axioms of set theory. Note that in $\Bbb N$ not every number has a logarithm. For this you need to extend to the real numbers. For example $\log_25$ is not a natural number at all. So the ...

1

The logarithm of an infinite cardinal number $\alpha$ is defined as follows: $$\log\alpha=\min\{\beta\ |\ 2^\beta\ge\alpha\}.$$ This definition is given on p. 74 of Cardinal Functions in Topology by István Juhász, and I am not aware of any competing definitions. According to this definition, $$\log\aleph_0=\log\aleph_1=\aleph_0.$$ It should be noted that ...

2

I am german, and I would call a function, regardless of its domain of definition, which takes real values 'eine reellwertige Funktion' (which is pretty much the literal translation from the english, taking into account the fact that the german language often concatenates two or more words, hence the transition from 'real valued' to 'reellwertig'). Similarly, ...

0

Doing a quick search with google reveals, that most people use "reelle Funktion" for a function $f : \mathbb{R} \rightarrow \mathbb{R}$, others use it for a function $f : D \rightarrow C$ with $D,C \subseteq \mathbb{R}$, where $D$ or $C$ may or may not be open or closed intervals and some use it for $f : D \rightarrow \mathbb{R}$, where $D$ can be any set. ...

0

Bingo! I still don't have an formal name per se, but I do know how to solve the function. For anyone who might run across this: The answer is far easier than it seems. Like all good math, the solution is completely obvious when you see it. You need to solve the phasor as a sphere, not a circle. If you wrap the clothoid function (Euler spiral) onto a ...

2

The thing is, that the plane itself doesn't have a name at all, but the way to define the plane has. The Hessian Normal Form has the form $x\cdot n+d=0$, where $d$ is the distance and $n$ is a normalized normal vector. However, the same plane can also be defined using the Normal Form I gave you in my first comment. So to answer your question, if the normal ...

1

There is a distance between random variables. Taking a separating class $\mathcal H$ you can define the distance between $F$ and $G$ via $$d_\mathcal H(F,G) = \sup\{|E[h(F)]-E[h(G)]|: h \in \mathcal H\}$$ You can use for example $\mathcal H = \{ \mathbb 1_B: B \in \mathcal B(\mathbb R^d) \}$ $\mathcal H = \{ h: h\text{ is Borel measurable and bounded}\}$ ...

2

What is triangle $\triangle ABC$ called in relation to triangle $\triangle DEF$? According to Wikipedia, the inner triangle is called the Gergonne triangle, contact triangle or intouch triangle of the outer. What is triangle $\triangle DEF$ called in relation to triangle $\triangle ABC$? I don't know an answer to this yet, but searching the web ...

0

By definition: $$X^{-1}(A)=\{\omega\in\Omega_1\mid X(\omega)\in A\}$$ In words the preimage of set $A$ under function $X$. So $X^{-1}(A)=\varnothing$ if and only if no elements of $\Omega_1$ are sent to an element in set $A$. Then consequently $X^{-1}(A^c)=\Omega_1$. If that is the case then here $X$ is a measurable function because ...

1

This is very easy; it's apparent that if $\phi_k(n)=0$ if $k$ does not divide $n$, and if it does divide $n$, then $$\phi_k(n) = \varphi\left(\frac nk\right)$$ because $\gcd(ka, kb) = k\cdot\gcd(a, b)$. This gives $\phi_1 = \varphi$ as desired. So it's probably too simple to have a separate name.

2

Candidate quantities can be found in inequalities for triangles in which equality occurs just for equilateral triangles. For example, the isoperimetric inequality for triangles asserts that the ratio $p^2/A$ (where $p$ is the perimeter and $A$ the area) is minimal for the equilateral triangle (among triangles). This quantity, then, measures deviation from ...

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