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I'm not quite sure what you mean with the properties remarks at the end, but spaces $Y$ in which $X$ embeds as a dense subspace are sometimes called "extensions" of $X$. Special cases are compactifications and connectifications (where $Y$ is also demanded to be compact or connected resp.) A book that treats extensions in some generality (mostly in ...

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Apparently they are called (left) distributive magmas http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.lnl/1235423706&page=record

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I'm going to hazard an answer that is pure motivation and contains no historical work--but maybe it's not too hair-brained. A physicist probably was implicitly thinking of the quadratic form as the second fundamental form associated to a parametrized surface $z=f(x,y)$ in $\mathbb{R}^3$. Roughly speaking, this quadratic form defines the curvature at a ...

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In terms of graph products the $n \times m$ grid will be the strong product of the path graphs $P_n$ and $P_m$, usually written as $P_n \boxtimes P_m$.

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This is the King's graph. For all your Kingly needs.

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At least for finitely generated modules these are exactly the modules which have a unique maximal submodule. As such I believe they are called local modules, although I have also heard the term quasi-local modules and am not sure what the distinction is between those two terms.

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Well, if you don't mind a mild loss of grammatical correctness, the way I learnt to say this was, "Limit n (tends to)/(approaches) infinity n-1 whole divided by n-2 equals 1" Sometimes I've noticed people (including myself) even omitting the word "divided"

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There is no such relation between these symbols, but if you enjoy messing with such things, here is a challenge: Find the smallest set of symbols that can generate the entire English alphabet (where you are allowed to rotate and flip). Here is a somewhat reasonable answer with $13$ symbols: b, c, e, f, h, i, k, L, m, o, r, s, x. For completeness, the ...

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I would say "The limit as $n$ approaches infinity of $n$ minus one all over $n$ minus two equals one".

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I tend to read $\lim\limits_{x\to a}f(x)$ as 'The limit as $x$ approaches $a$ of the function $f(x)$ is . . .' As for dealing with the ambiguity when verbalising a quotient like this, I find that the use of the word 'all' is helpful to distinguish between the possible numerators: $n - 1$ and $1$. I would say '$n$ minus one all divided by $n$ minus ...

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I'd always use the full word "limit," but I'd shorten some other terms when I'm writing it on a board. I'd perhaps read it as The limit of the quotient as $n$ tends to infinity or to be slightly more wordy, The limit of $n - 1$ (brief pause) over $n - 2$ as $n$ tends to infinity But especially if I'm writing it on a board, I'd rather just point ...

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Yes and no. There is a notation alrady. As you already write, it is: $$((n!)!)!$$ (The question is, why it matters? It won't harm if you invent one notation.)

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If $\gamma(n)=n!$, then $\gamma^k(n)=(((\ldots n!)!)\ldots )$. So we could use the Gamma function shifted by $1$.

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I don't think there's any well-known notation for iterated factorials. They don't seem to be a very common concept -- googling "iterated factorial" doesn't seem to find anything except warning that this is not what the double factorial means, and programming talk about computing factorials using iteration. Of course, if you're writing something where you ...

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The curved symbol (in my experience) means that the difference is positive definite (so, with zero it means that the matrix is positive definite). The $>$ depends on the context.

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Not only does adding 1 infinitely times in a field not make sense, but also if adding 1 a positive number of times never yields zero, then certainly adding it zero times yields zero, thus it only seems natural to refer to such a field as "characteristic zero".

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Much like programming, you have functions with parameters. You can call these functions and expect an output. Lets assume we define a function $f$: $f: \mathbb Z \to \mathbb Z$ meaning we expect this function to spit out integers when it is fed integers. Lets furtherly narrow it down, and define the function $g$ as: $g(x) = x-1$ Now we write x-1, using ...

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A variable is something that can vary, in the context you're interested in. For $y=ax^2+bx+c$, you could have $x,y$ be variables, with $a,b,c$ constants. But in another problem you might want to hold $x,y$ constant, and vary $a,b,c$ -- then $a,b,c$ would be the variables. The term "parameter" has several meanings. Sometimes we call constants parameters; ...

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Say nothing over the whole domain. Think of a function of one variable that have negative second derivative at all but finite points and at these points the function is not differentiable (it has kinks), the first derivative changes sign. This function is not quasi-concave nor quasi-convex but you can divide its domain in regions where it is concave. But ...

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Let $\mathbf K$ be the class of all lattices not containing $M_3$ (the $5$-element nondistributive modular lattice) as a sublattice; in other words, lattices in which every modular sublattice is distributive. It is easy to see that $\mathbf K$ can be characterized as the class of all lattices satisfying the following sentence $\varphi$:$$\forall u\forall ... 2 Rings of the form {\mathbf Z}[\sqrt{d}] are not the most general type of order in a quadratic field. To distinguish them, a term that I like to use is pure quadratic ring, where "pure" refers to having a ring generator that is a pure square root \sqrt{d} (as opposed to something like generator (1+\sqrt{5})/2). I call cubic orders of the form {\mathbf ... 3 Non-maximal subrings of rings of algebraic integers have been called "orders" (from German, I think) for a long time, although obviously this is not a wonderfully distinguishing choice, the word being used (in English) for many other purposes. Still, it is standard, so you can say "the order \mathbb Z[\sqrt{5}] in the ring of algebraic integers \mathbb ... 1 The description you give is that you start at some digit between 1 and 4 and go from that digit to another one, and so on, not repeating a digit until the last step at which you return to the starting digit. As a function f from the set S=\{1,2,3,4\} to itself, this means that if your starting number is a, then the sequence ... 1 When we write (for example) "we say that an integer is odd if it is not divisible by 2", the statement does not explicitly deny that we might describe as odd a number that is divisible by 2. The reason for ruling out the latter circumstance is that it would make the definition pointless. Thus, in definitions, there is an implicit message "either this ... 0 To solve an equation on an interval means to find all solutions to the equation that lie in that interval; this is quite consistent across exams and textbooks. As Gerry Myerson points out in a comment, your concept of "solve" is solving a related, but different, problem. 1 What you have written is a "summation identity" (or, less precisely, a "summation formula"). You can find many lists of summation identities with a bit of internet searching. 1 "Window length" is used without comment on the Wikipedia page for moving averages and it's been there for quite a while without being edited out (despite the page getting other edits), so I think that suggests that "window length" is a fine way to refer to this idea. 2 Since you said you wanted to generalize the unit interval, you should probably refer to the "unit n-cube". You can see that this terminology is used with a google search, which pulls up papers like this one. 1 As mentioned (and linked) in the wikipedia page for Bipyramid and as mentioned in the properties box on the wikipedia for Trapezohedron, one term for the property is simply "face-transitive". When you click the link on the page for the Bipyramid, you learn another term for the property: "isohedral", and a term for the polyhedra with this property: ... 2 Equation means equality. They are both related to the word equal. If such an equality is true for all values of the variable, it is called an identity, e.g., \sin^2x+\cos^2x=1 is true for all x. If however the equation in question only holds for some values, which one is supposed to determine, then it's called conditional, and its variable is termed an ... 3 The rect- in rectum is related with the English right or (st)raight, as well as the neologism correct. And latus means side, but also wide, or width. The expression simply means straight side. Which is also why it's probably left untranslated, since (at least word-wise) it's synonymous with the notion of straight line, which however bears different ... 3 Latus is Latin for "side". In anatomy it is the flank of the body. I'm not aware of any use of "latus" in mathematics other than "latus rectum", but "lateral" is derived from it, and thus "quadrilateral" etc. 1 In an algebra, if A,B \in \mathcal A, then A \setminus B \in \mathcal A since A \setminus B = A \cap B^c. So we may take n=1 and C_1 = A \setminus B to get the semialgebra property. 2 You seem to be looking for an adjective meaning the opposite of disjoint. Given two sets that aren't disjoint, I would just call them intersecting sets, where intersecting is an adjectival participle derived from the verb to intersect. 0 Axiom scheme is similar concept but I have never seen it introduced as a function. 0 The 'shape' of the graph is not so important - where the lines lie in two-space is merely an abstraction for visual convenience. It is, in a sense, arbitrary. However, it is possible that you are referring to a graph composed of several cycles. A cyclic graph is a graph that forms a 'closed shape' because it forms a closed walk, and it seems possible to me ... 0 Arguendo is close, and might better fit proofs by contradiction, or derivations from a conjecture. 0 I would call your "continuous polygons" 1-dimensional manifolds. http://en.wikipedia.org/wiki/Manifold 2 suppose you slightly modify your definition to:$$ g(r)=\frac1{2\pi}\int_{0}^{2\pi}f(r,\theta)d\theta $$does that offer any suggestion? 0 A set of n independent solutions is traditionally called a "fundamental set of solutions" (link 1 link 2 link 3) to the differential equation. However, there is not really a standard way to refer to the rest. You could say "n-1 other solutions to form a fundamental set", or use terms of Linear Algebra and say something like "extend this solution to a ... 0 I'll restrict this answer to the case where f is a real function defined on the whole real line, (f\colon \mathbb R\to \mathbb R). Since the domain of f is \Bbb R, its input are real numbers. Therefore, the symbol f(y) is meaningless if y is not a real number. In your example you set f=\text{id}_\mathbb R. Let y be a real number. If it is ... 0 I would say that these things are all "curves". Some are closed (like rectangles and circles), and some are not. The property that interests you, I think, is "smoothness". More precisely, you are interested in the presence or absence of "corners" where the (unit) tangent vector is discontinuous. So, circles and lines and Bezier curves are "smooth" (they ... 2 You have written the definition of the derivative of a function f(x) in that formula. A differential is an infinitesimal interval; dx for example. We use differentials in expressions like \dfrac{dy}{dx}. As a new calculus student, you want to focus on finding derivatives of functions. A derivative is a rate of change, just like the slope of a line, ... 2 I assume that you know what is the meaning of being differentiable at a point x. Let the function f is differentiable at x (f'(x)<\infty). Then we can have-as that lmit tells- the following identity:$$\Delta f=f(x+\Delta x)-f(x)=f'(x)\Delta x+\epsilon\Delta such that this $\epsilon$ is a function with respect to $\Delta x$ and of course ...

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Sounds like lattice theory. They have similar notions of separation that does not involve topology. Check out this paper, bottom of the second page: http://www.emis.de/journals/HOA/IJMMS/Volume14_2/395496.pdf. If I find out more I'll edit this answer.

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I think the common term is curvilinear polygon. In this context, an ordinary polygon could be called a linear polygon or a straight polygon.

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You can call your 'continuous polygons' topological polygons. Edit: As for your 'concrete polygons', I would call them simply 'polygons'.

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Let $\mathcal{U}$ be the category with a single object $U$ and having as morphisms all functions $U\to U$, and consider the monoid $(\mathbb{N},+)$ as a category in the standard way. Then the category you described can be viewed as the functor category $Funct(\mathbb{N},\mathcal{U})$. Any functor $\mathbb{N}\to\mathcal{U}$ is arises from and is determined by ...

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Perhaps you mean the inscribed angle theorem? In French, it's called arc capable, which is similar to the Spanish and Portuguese arco capaz.

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"Elementary proof" can mean several things: 1. A proof that doesn't use big machinery; 2. A proof that is short and sweet; 3. A proof by a crackpot claiming to have solved some famous conjecture; 4. A proof that is too fiddly to write down, so the author says it is "elementary" so that he/she doesn't need to spend time figuring out a good way to write it ...

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