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The arithmetic and geometric adjectives come from the Pythagoreans before the Christian Era. Apparently, the expression “geometric progression” comes from the “geometric mean” (Euclidean notion) of segments of length $a$ and $b$: it is the length of the side $c$ of a square whose area is equal to the area of the rectangle of sides $a$ and $b$. The ...


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Because geometric progressions are based on multiplication, and the most important geometric notion, namely, volume, arises from multiplication (length times width times height). The term “multiplicative” is not used because it already has a special meaning in Number Theory.


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We should state the question more general: Given a commutative ring $R$ with identity and an ideal $I \subseteq R$, we can always build the quotient $R/I$. Now we can ask: Is $R/\{0\}$ equal to $R$ or are they just isomorphic? Point of view 1: Let us define $R/I = \{a + I \mid a \in R\}$ together with the usual addition and multiplication of cosets. Then ...


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Why don't you just define $\overline{e} = \frac{1}{n} \sum_{k=1}^n e_k$?


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There's no such thing as "assignment" in mathematics (it's a bit different from equivalence, e.g. $y=y-1$ is a statement, rather than an operation), unless you're speaking of a function. In such a case it would look something like (rewriting what you've said explicitly): $$ f(y) = \begin{cases} y-1, &\text{if } y>0\\ 0, & \text{otherwise}. ...


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I found this image which probably sums up the problems deaf mathematicians face: A simple answer to your questions is 'I don't know' - I would love to see a signed version of the proof of Poincaré's conjecture!


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Brian M. Scott wrote: In the topological case I'd simply call it a (pairwise) disjoint family whose union is dense in $X$; I've not seen any special term for it. In fact, I can remember seeing it only once: such a family figures in the proof that almost countable paracompactness, a property once studied at some length by M.K. Singal, is a property ...


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A "only if B" is the same as saying "B is necessary" for A which is the same as saying A could not have happened without B, but that does mean that other things do not also need to happen for A to be true. Therefore, $A \to B$ but it is not true that $B \to A$ because B being true does not guarantee A happened. There could also be other requirements for ...


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Let $A$ and $B$ be ant two sets. Then $f:A\rightarrow B$ is said to be a $\textbf{function}$, if every element of $A$ is mapped to a unique element of $B$. Here requirement for $A$ and $B$ is only arbitrary sets. Let $V_1,V_2$ be any two vector spaces. A map or a function $T:V_1 \rightarrow V_2$ is an $\textbf{operator}$. Here minimum requirement of ...


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It seems like you're really starting with an ordered set $(S,<)$. That is, if you just started with an arbitrary set with three elements $\{x,5,*\}$, there would be nothing to distinguish $\{x,*\}$ from $\{x,5\}$. It's important that you know $i_1$ comes first, followed by $i_3$, then $i_5$, right? Then what you've written down is the set of all ...


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I think expected value and expectation are the most common in use in the US, I haven't really seen expectation value that much.


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While your reasoning is correct that "every function is an operation" under that extremely general definition of "operation", I would say that a more common definition of an "operation" on a set $S$ would be a function $$\alpha: S^n\to S\quad\text{ for some }n\geq 0$$ or, to allow "partial" operations, $$\alpha: X\to S\quad\text{ where }X\subset S^n\text{ ...


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If $\alpha $ is injection then $\alpha $ must be equal to identity function.


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The given circle is the unit circle, therefore $O$ can also be called the origin. I don't think any of $A,B,C,E$ have a name, since they are depedent on $\theta$, so it can be any point on the circle (in the case of A and B) or on the x-axis (in case of C and E). I suspect that $D$ actually has a name, because it is an important point on the unit circle, but ...


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I remember using it in measure theory to show that a function $f : [a,b] \to \mathbb R$ was almost everywhere differentiable. The idea was that the clamp function was used to extend the identity $[a,b] \to [a,b]$ to the whole of $\mathbb R$, and then we could compose the clamp with $f$ to extend $f$ to the reals. This was just the first step of the proof, ...


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There's no such thing as a "formally recognized" mathematical function. Some are more well known and widely used, but there is no standard defining what is a recognized function. Yes, $\mathop{\rm clamp}(x)$ is a mathematical function.


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There is already an answer of @Chappers, but I wanted to remark on the word "compact" and why it can be considered suitable for its purpose. Perhaps you will find it relevant. In my opinion, compact is a very good term – compact spaces really are the spaces that are closely and neatly packed together, however, not in the common literal meaning of the ...


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Fréchet was the one to coin the term, but according to this page from Earliest Uses of Some Words in Mathematics, even he didn't remember why he used it (and also according to that entry, some mathematicians didn't like the term at the time). But we're stuck with it now! (I'd like to find another source on this, but I doubt there's much about: even History ...


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An algorithm is a systematic method to achieve a variable Goal with different Quantitative or Qualitative Components; A Formula is a systematic method to achieve a constant or Fixed Goal With Similar Arithmetical or Algebraic components Example: try above same examples


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It depends on the context. $c_i$ can be the unit cost (price) of input factor i . And $a_{ij}$ is a coefficent. The meaning would be: It is the amount of input factor i, which is required to produce one unit of product j$. $b_j$ is the required amount of product j. $x_i$ is the amount of input factor i$ Example: You want to produce at least 10 units ...


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It is special in the sense that it is a representation of a generating element for $C_2$, the cyclic 2 group. Which is one of the most basic groups. This means you can find a block diagonalization such that the square of the blocks are the identity matrix. For instance this could be -1 or $\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$, since ...


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There is no mathematical reason to prefer right or left modules. The symmetry is broken by the conventions of written English (or more generally, written European language) and mathematical exposition, in which a function applied to an element of a set is typically written $f(x)$ and function composition is $f(g(x))$; most of us would find the notation ...


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$(e^{2\pi i})=(\cos(2\pi)+i\sin(2\pi))=1+0i=1$ $\sqrt{1}=1$


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The first one is correct. In the second one, $\sqrt{e^{2\pi i}}=e^{\pi i}$ is wrong. Should be $\sqrt{e^{2\pi i}}=|e^{\pi i}|=1$ (This stands of course under that assumption of knowing $e^{\pi i}$ is real).


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$\sqrt{(-1)^2} = \sqrt{1} = 1$


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Typically in probability and statistics, capital letters such as $X_i$ are used to denote a random variable, while lower-case letters such as $x_i$ denote realizations of the random variable. Rigorously speaking, you are correct in pointing out that $P(x_i = c|\mathcal{N}_i)$ should have been written as $P(X_i = c|\mathcal{N}_i)$, although this distinction ...


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The paper is concerned with a graph with attributes. The attributes are $X_1,\ldots,X_n$, and are collectively known as $\mathbf{X}$, which is either the vector $X_1,\ldots,X_n$ or the set $\{X_1,\ldots,X_n\}$. Each attribute $X_i$ can take a value in the set $\mathcal{X} = \{c_1,\ldots,c_m\}$. The value of the attribute $X_i$ is denoted $x_i$. There are ...


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It seems to me that you need to mention which surface the two objects lie on, because simply saying that they both lie on some surface doesn't say very much. Saying it another way ... given any two objects, it is very often (maybe always) possible to find some surface that contains them both, so "co-surfacity" doesn't mean very much without some mention of ...


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I'm not sure I completely understand the class of objects you are trying to describe, but they sound like (connected) $2$-manifolds. One related term to "collinear" which applies to circles is concyclic. My gut tells me that if you will find any such term, you will find it in algebraic geometry - the theory of elliptic curves and other algebraic varieties ...


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In the terminology for partitions of a set, the nonempty subsets that belong to the partition (a set of sets) are called blocks, parts, or cells. So one might refer to the constructed partitions here as partitions of a set with two parts. However I call your attention to the list of 14 "splittings" given for the set $V=\{1,2,3,4\}$. Because a partition is ...


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Let your wanted function be $r_n(a)$. If $a\ge 0$, then (in your case $n=3,a\in\{29,63\}$): $$r_n(a)=a-\lfloor \sqrt[n]{a}\rfloor^n$$ Here I've used the floor function: $\lfloor x\rfloor$ is defined as the largest integer less than or equal to $x$.


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As already has been pointed out in the comments above, there will be a slight difference between American English and British English. Also, people have different ways to say the same thing. For the two above, I would personally say The quotient R mod Z (ar mod zee) Here I would say one of the following: the closed interval from 0 to 1 to the n. the ...


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Many periodic functions have this property. In measure theory a related idea involves a broader definition of negligible sets: "sets of measure zero". Functions of the kind you are asking about would be a subset of the set of functions which are non-zero "almost everywhere" (used to mean everywhere except a set of measure zero). However, there are ...


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I don't know about the name, but this class of functions all have the form $$g(x) \chi_{\mathbb{R} \setminus E}(x) $$ where $g:\mathbb{R} \to \mathbb{R} \setminus \{0\}$ is an arbitrary nonzero function, $E$ is any countable set, and $\chi_F$ is the characteristic function of the set $F$. In words, you can call these functions "non-vanishing multiples of ...


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Think about those two series (we'll call them S1 and S2) you mentioned that converge to the same number. Let say that both series get to that number after at least 50 terms in each series. So, at 50 terms, the sum you have at that point in S1 will be different than the sum of S2 at 50 terms. The series with the sum at 50 terms that is closer to the ...


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$M_\mathfrak{p}$ denotes the localization of the $S$-module $M$ at the prime ideal $\mathfrak{p}\subset S$. It's defined exactly how you'd expect: $$M_\mathfrak{p}=\{\tfrac{m}{s}:m\in M, s\notin \mathfrak{p}\}$$ For any $\mathfrak{p}$, we have that $M_\mathfrak{p}$ is a module over $S_\mathfrak{p}$. Here is the relevant Wikipedia link. Then there is a ...


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In an algebraic extension $F:\mathbb Q$, the ideal $p\mathfrak o_F$ can be written as a product of prime ideals. If the product has $[F:\mathbb Q]$ factors, $p$ splits completely; if the product has only one factor, $p$ is inert. Anything inbetween may happen (not to mention ramification, i.e., the occurance of repeated factors). Maybe you have so far ...


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They don't mean the same thing, but it is a short jump to show they are equivalent. A set of events, $A_1,...,A_n$ are mutually exclusive if the occurrence of one of them implies that the other $n-1$ events can't happen. It's immediate that the events are pairwise mutually exclusive. The other direction is immediate as well. They aren't a priori the same, ...


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A diagram in category $A$ is most commonly defined as a functor from a small category $D$ to $A$. In this view, we can say, the diagrams are automatically labeled by objects and morphisms of $D$. For your specific example, take the free category $D$ on the graph with two (different) objects and two arrows as in your picture ...


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I think the standard terminology is the (generalized) Dirichlet convolution $$ (\alpha\circ F)(x)=\sum_{n\le x} \alpha(n)F\left(\frac{x}{n}\right), $$ where $\alpha$ is an arithmetic function, and $F$ is a real or complex valued function such that $F(x)=0$ on $0<x<1$. If $F(x)=0$ for all non-integral $x$, then this product is just the Dirichlet ...


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As I understand it, the word "jet" is meant to evoke the idea of a "spray" of curves through a point, or more accurately, their equivalence classes up to $k$th order contact. Consider this section of the Wikipedia entry on jets: (The jet of a map between manifolds is then defined in terms of the jets of curves.)


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Sequences' elements are indexed with natural numbers N. Tuples' elements are indexed with positive integers N*. Sequence is use to define relations in general (i.e. where the cardinality of arbitrary sets is not clearly identified). Tuple is use to define finitary relations in particular (i.e. where the cardinality of finite sets can be directly ...


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"Dropped" just means deleted. If the numerator is $A+B+C$ and one replaces it with $A+B$, one has dropped the third term from the numerator.


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the followings archives corresponds to the doctoral thesis of Henri Lebesgue: You can search for Analysis situs, old name for the actual topology.Lebesgue and Rham (french mathematicians) wrote about that subject. sincerely Hugo Mancera Colombia


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You know, this is something that really annoys me about applied or non-mathematical subjects that invent superfluous terminology in order to clarify something to an audience whose background in careful mathematics is at best weak. Computer science texts are notorious for this, especially with the explosive growth of computational geometry the last decade or ...


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Regarding $P(V)$ All 1-dim subspaces of $\mathbb{R}^3$ are lines through the origin because zero has to belong to the subspace. The collection of lines (objects) are a vector space for appropriate definition of vector. Associate with each line its point of intersection with some plane, say $x=1$. Then these points constitute a $2$-dim vector space. The ...


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$\newcommand\Z{\mathbf{Z}}$ Call the ring $R$. You will notice that the image of $\Z$ in $R$ under the map $\phi:\Z \rightarrow R$ you constructed has the property that, if you write is as: $$\phi(a) = (a_1,a_2,a_3, \ldots )$$ then: $a_2 \in \Z/p^2$ has remainder $a_1$ when reduced modulo $p$, $a_3 \in \Z/p^3$ has remainder $a_2$ when reduced modulo ...


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Either the professor is making a joke about mathematicians (the same way that we mathematicians often joke about engineers), or that he is mis-remembering/misinterpreting. First some evidence: A quick survey of the available textbooks on my shelf behind me indicates that among mathematicians such as ... Richard Courant and David Hilbert (Methods of ...


1

First, the broad concept of numerical analysis is that we are studying algorithms to obtain some kind of approximation (numerical in nature) to problems in pure maths (particularly analysis). It has applications in huge swathes of pure and applied mathematics. A couple of examples of topics in numerical analysis are: Numerical linear algebra: This is the ...



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