# Tag Info

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You are right, it is the coproduct (in the commutative case). It is called a tensor product because it satisfies the usual laws from the finite tensor products. By the way, the tensor product of algebras does not coincide with the tensor product of modules - rather we endow the tensor product of the underlying modules with the structure of an algebra. The ...

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Following my answer to your previous post, we can say that a formal system is made by an alphabet (the set of symbols), a gramamr (the formation rules, defining the "correct" expressions, i.e. the set of well-formed formulas) and a proof system or deductive calculus. See Enderton, page 110 : We will introduce formal proofs but we will call them ...

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This is a special case of the tensor product of vectors known as outer product. It has some interesting properties, for example that the trace of the matrix is the square of the (Euclidean) norm of the vector. And as you point out it is always symmetric.

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An operator is a mapping from one vector space or module to another. where a function is a map from any arbitrary set to another

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This is a book by kirillov ,titled what are numbers?,it could help you http://www.math.upenn.edu/~kirillov/MATH480-S08/WN1.pdf http://www.math.upenn.edu/~kirillov/MATH480-S08/WN2.pdf

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Neither term exists, so define it however you like. A google search yields zero results for implications tuple and merely three for tuple of implications. One of those results is another Math.SE question regarding a chain of implications, which I now see you yourself started.

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This holds for arbitrary (measurable) subsets $A,B \subset \Bbb{R}^2$, because $$A^\circ \cap B^\circ \subset A \cap B$$ is an open set (finite intersection of open sets) of measure zero, thus empty, where I denoted the (topological) interior relative to $\Bbb{R}^2$ of a set $M$ by $M^\circ$.

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There are some partial orderings that have more than one maximal element. As an example, consider the set $\{\emptyset, \{ab\}, \{bc\}, \{cd\}, \{abc\}, \{bcd\} \}$, and we'll say that $A < B$ if $A \subset B$. It can be verified that $\{abc\}$ is greater than every other element except $\{bcd\}$, but that $\{abc\}$ is not smaller than $\{bcd\}$. So ...

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Since we are describing $C$ as the union of two sets, why not simply refer to $A$ as the first set in this union of two sets. But, honestly, I'd simply refer to these sets by name. If you state that $C = A\cup B$, why not go on to assert: "Set $A$ ensures...., while $B$ guarantees..."

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In set theory, operator or operation is sometimes used for those operations that, unlike functions, cannot be modelled by sets of ordered pairs, but whose collection of ordered pairs instead constitute a proper class. The most common example of this is the power set operator $\mathcal{P}(X)$.

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"regular" here means holomorphic. Each connected component of the space of quasi-characters is just a copy of the complex plane; if I understand your notation correctly, then $c(\alpha)$ is some (unitary) character in this component, and then all the other quasi-chars. in the component have the form $c(\alpha)|\alpha|^s$. The $\zeta$-function can thus be ...

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There is something you are missing for this particular situation. "By inspection", as noted by others, means "by happening to think of a solution that works", which is allowed to include mental calculation. The thing is, in different situations, there should be different things you are looking for to help you think of such a solution, and to focus your ...

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A geodesical ball $B$ centered in $p\in M$ of radius $r>0$, in a connected Riemannian manifold $(M,g)$, is a set of the form $$B:= \{x\in M \:|\: d(x,p) < r\}\:,$$ where $d(x,y) = \inf\left\{ L(\gamma)\:|\: \gamma: [0,1] \to M\:, \gamma \in C^1([0,1])\:, \gamma(0)=x\:, \gamma(1)=y \right\}$ and $L(\gamma)$ is the arch length of $\gamma$, $$L(\gamma) ... 3 I would add to the previous answer that f A \subset B or B \subset A and the application h|_A=Id_X (assuming that A \subset B) then you have what is called a \textbf{Deformation retract} (http://en.wikipedia.org/wiki/Deformation_retract) and the application is called a retraction. This has some interesting properties. The most important one is ... 4 This very closely matches the notion of homotopy. However, homotopy is slightly different, as it represents the notion of a map h\colon [0,1] \times A \rightarrow X such that h(0,-) is the identity and h(1,-) = B. This is basically a deformation of maps into X, but doesn't require that all of X be mapped at each step. Technically, you wouldn't ... 1 "If it looks like a duck, and quacks like a duck, it is a duck... or something that is so close that is good enough" In Mathematics you don't really work with objects, you work with the properties. So, we define an object (for example, a vector) in a certain context (linear algebra) as something that has a series of properties (in the example, you can add ... 0 There are a lot of good answers already. This isn't much of a deep, mathematical answer. It's just my thoughts on the matter. A "soft answer" to your "soft question", if you will :) To me, a number basically is a constant that has notions equivalent to basic addition and multiplication over the natural numbers. By that definition, the above often-mentioned ... 1 Do you know about the refutation theorem-proving technique of resolution ? The resolution rule is refutation-complete, in that a set of clauses is unsatisfiable if and only if there exists a derivation of the empty clause using resolution alone. Paramodulation is the modification of resolution system to manage equality. You can see detailed tratments of ... 1 Yes, "Mean squared [X]" refers to the first one, as in mean squared error. For the second one, I would use something that clearly disambiguates from 1st, e.g., "squared mean of [X]" or "the square of the mean of [X]". 1 Actually, the "practical" examples of numbers start to break down at \mathbb{R}. Yes, the algebraic numbers have interpretations in geometry (though these are already more abstract than owing someone a dollar or splitting a pizza), but there are many numbers in \mathbb{R} that are not algebraic and in fact aren't really describable in any ordinary way. ... 0 The question is controversial in philosophy of mathematics. Some answers: (A) Numbers are sets. According to Bertrand Russell: The number 0 is the empty set. The number 1 is the set of all single-membered sets. For natural number n, the successor of n is the set of all sets that are equinumerous with {0, 1, ... n}, where two sets are equinumerous if there ... 6 In my thinking, a number is simply an object which satisfies some set of algebraic rules. In particular, these rules are typically constructed to allow an equation a solution. Most interesting, these solutions were unavailable in the less abstract version of the number, whereas, with the extended concept of number the solutions exist. This seems to be the ... 0 Mathematics is really just a system of formal logic; you define the rules of a universe then use those rules, and only those rules, to build systems. Within that rule system, you can define what a number is, some operations you can do on numbers, and start proving results about what happens with those numbers. Eventually, you build up more complex systems, ... 2 Bertrand Russell in the early 20th century defined "number" as anything which is the number of some class and "the number of a class" as the class of all those classes that are similar to it where similarity of classes is defined by the existence of a bijection between them. (Paraphrased from Introduction to Mathematical Philosophy.) 0 There are two schools of thought here: 1 - Intuitive quantity. 2 - Arithmetic quantity. For me, I go with the Intuitive concept, which means: restrict the concept of numbers - into measuring quantity, either positive or negative. (Set of Real numbers) Complex numbers are a problem because they measure more than quantity. They have some sort of ... 1 There isn't a rigorous definition of number. It doesn't make sense to come up with a rigorous definition of number until there is some specific (and hopefully useful) concept you wish to formalize: and we already have formalizations of concepts like "real number" if that's the specific concept we wish to talk about. As an aside, I have used complex numbers ... 7 There is no concrete meaning to the word number. If you don't think about it, then number has no "concrete meaning", and if you ask around people in the street what is a number, they are likely to come up with either example or unclear definitions. Number is a mathematical notion which represents quantity. And as all quantities go, numbers have some ... 4 This is a deep and philosophical question. You are right, kids start by learning to count things like 3 apples and 7 cars but quickly build up to the real numbers, which we can make sense of by money and distances. When you start adding things like the imaginary numbers, it gets less clear. My revelation came when I learned some electricity and ... 38 A basic question is, what would be the purpose of such a definition? Would it clarify anything if we came up with a definition that, say, included quaternions and not matrices or analytic functions? Most of the usages of the term "number" are due to historical choices that have lived on. I'd be interested in seeing things that were called "numbers" ... 1 Does anyone know of any definition of number that can be generalised to complex numbers as well (and even higher order number systems like the quaternions)? There is a general consensus that no such definition could possibly exist. "Number" is inherently a fluffy concept. However, a good guideline is: if the analogy with any of the usual numbers ... 4 These questions are addressed in mathematical philosophy (e.g. see Russell's Introduction to Mathematical Philosophy). The very short answer to your question is classes. (I'll have to elaborate some other time unless someone beats me to it) 1 This doesn't exactly answer your question, but here are some characteristics all numbers I know of satisfy.(I know of counting numbers, integers, rational numbers, real numbers, complex numbers, quaternions and octonions). Number 1: All the counting numbers seem to be equipped with two binary operations which receive the name of addition and multiplication. ... 2 A connected graph with two vertices of degree one and the rest of degree two is called a path. If you want to distinguish between successor and predecessor vertices, you can direct the edges all the same direction and obtain a directed path. 5 If you're in elementary probability instead of measure-theoretic probability, the following will make very little sense. My apologies if this is the case. I'd say one reason is that we really don't look at the properties of random variables as functions from the underlying set of the probability space (usually denoted \Omega). You could change \Omega ... 3 Because we think of it as a variable that take random value intuitively. Formally they are function. Just like why we call a sequence a sequence, or call an arithmetical function an arithmetical function, when they are actually the same thing formally speaking. Just to add to the issue, calling a variable also match the notation. For example, X=Y+Z is NOT ... 3 Galois theory is the theory of the duality between profinite groups associated to fields and closed subgroups which arise as dual to field extensions of the original field. It's about the algebra of polynomials over a field and how that helps to understand other fields constructed algebraically from the original field, i.e. from roots of polynomials over a ... 3 The two things described are very different; the only "relationship" is that the term null refers to something having to do with zero, and both things are some sort of vector space. Those who use Lang's definition of a null-space would strictly refer to the second idea (what Lay calls a "null space") as the "kernel" of a matrix. 1 In rings with zero-divisors, factorization theory is much more complicated than in domains, e.g. \rm\:x = (3+2x)(2-3x)\in \Bbb Z_6[x].\: Basic notions such as associate and irreducible bifurcate into at least a few inequivalent notions, e.g. see the papers below, where three different notions of associateness are compared: \ a\sim b\  are   ... 1 Courtiel and Vaughan, Gerechte designs with rectangular regions, JCD (2012) calls them a gerechte framework. A gerechte framework is a partition of an n \times n array into n regions of n cells each. This definition comes from a gerechte design, which is a Latin square together with the entries partitioned into sets of size n such that each ... 3 That is a question for a native speaker, I fear. In German both are used to differentiate = differenzieren (determing the derivative) to derive = ableiten -> Ableitung (derivative) In English literature, I think I only saw differentiate for the operation. In German you can use "Herleitung" to stress more that it is about taking conclusions. In English ... 12 In English, I've almost always heard mathematicians say "We now differentiate f to get ...". Occasionally I've heard "derive," but in English (my native language!), that's generally used to mean "work out", as in "Ralph couldn't derive a proof of the intermediate value theorem from the information he had at hand." It's also used in generating one thing ... 1 For Spanish (Castellano) ⇔ English: Bilingual mathematics dictionary by Norman Koch Diccionario de matemáticas : castellano-inglés, inglés-castellano = Dictionary of mathematics : English-Spanish, Spanish-English by Kenneth Allen Hornak 1 The following is an excerpts from here (an inaugural lecture of some sort at Heidelberg university): "Auf der zweiten Stufe der Schönheitskriterien stehen nach Hasse die Forderungen nach Eleganz und nach Zielstrebigkeit. Die letzte bedeutet, daß man an jeder Stelle eines Beweises weiß, wo man steht, und das Ziel vor Augen sieht. Das Gegenteil eines solchen ... 3 I find that notion (or rather its German translation Mausefallenbeweis) only in the context of the German philosopher Schopenhauer, who coins that in the context of Euclid's proof of the Pythagorean theorem: It logically forces the reader to believe the result without much insight (as opposed to some other proof that uses just a picture and one immediately ... 2 At least two different terms are used in the literature for a commutative monoid in which division is a partial order: holoid and naturally partially ordered. Another possibility would be \mathcal{H}-trivial since a commutative semigroup has the required property if and only if the Green's relation \mathcal{H} is the equality in this monoid. See ... 1 There is a quote on this point from page 808 in J.L. Doob's Classical Potential Theory and Its Probabilistic Counterpart. Before martingales had been formally christened, Lévy [1, 1935; 2, 1937], Bernstein [1, 1937], and other mathematicians had analyzed some of their properties in special contexts; usually the martingales in question arose as ... 1 As far as I know, this naming has its origin in the connection between Brownian motion and harmonic functions. Let B be a d-dimensional Brownian motion. Let f:\mathbb{R}^d\to\mathbb{R} be twice continuously differentiable. By Ito's formula, we have$$ f(B_t) = f(0) + \sum_{i=1}^d\int_0^t \frac{\partial f}{\partial x_i}(B_s) dB^i_s + ...

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If the group $G$ has $24$ members and a normal subgroup $H$ has $6$ members, then $G/H$ has $24/6$ members and $24/6$ is clearly a quotient. Division is an operation on numbers defined initially by thinking about finite sets: split a set of $24$ members into parts of size $6$; then how many such parts are there? Clearly that is a quotient, in the sense of ...

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Read Julia Nicholson's paper "The development and understanding of the concept of quotient group", Historia Mathematica 20 (1993), 68--88. (If you have suitable permissions, you can read it online at http://www.sciencedirect.com/science/article/pii/S0315086083710074.) I will summarize the basic story. The idea of a quotient group arose in work of Galois, ...

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According to Young, Amer. Jnl. Math, 1893, p.130 the terminology was introduced by Hölder when he proved the uniqueness of the factor groups in composition series (Jordan-Hölder theorem) in his 1889 paper in Mathematische Annalen titled Zurückführung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen. This is often cited as the first ...

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