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You're treating the word "axiom" as you were probably taught in high school. That an axiom is something which is "simply true as an assumption". Modern mathematics has changed that definition to "an assumption made in a certain context". Not every axiom is called an axiom, some axioms are proved as theorems, and sometimes lemmas are used for axioms. But in ...


3

The existence of a model of a statement does not mean that statement is "true" (whatever that means; see below). For example, the Poincare disk is a model of Euclid's first four postulates plus the negation of the parallel postulate; this does not mean that the parallel postulate is "false." What having a model of a set of statements does mean, is: that set ...


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$\renewcommand{ket}[1]{|#1\rangle}$ $\renewcommand{bra}[1]{\langle#1|}$ There are several common terminologies: As you might guess based on the fact that $V^*$ is called the "dual" of $V$, in mathematics and mathematical physics the covector of a vector $v$ is called the "dual" of $v$. In physics, vectors are often denoted e.g. $\ket{v}$. The $v$ is a ...


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Echoing @Lærne, $v^c$ is the dual of $v$. If you want a reference, how about Halmos's "Finite-Dimensional Vector Spaces"? In the second edition, sections 13, 14, and 15 along with sections 67, 68, and 69 may be what you're looking for.


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"There exists" does not mean there is only one, i.e. "There exists" is different from "There uniquely exists." So since there is no specification that there is exactly one such object (until proven otherwise), you can read the "there exists" as "for some" if you like.


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The symbols $\gg$ and $\ll$ don't have a formal definition. Usually they are used to compare two extremely big numbers, for example $\mathrm{Graham's \; number} \ll \mathrm{TREE}(3)$ or something like that. They are only used because someone wants to make clear that one of them is so much greater. The numbers are indeed just small compared to everyday ...


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It is typical, at least in my experience, that "random walks" unless otherwise stated are sums of iid random variables. You can form two random walks with iid copies of normal distributions with means $\mu, \tilde{\mu}$ and variances $\sigma^2$ respectively. Then by the law of large numbers, the averages of their partial sums will converge to the their ...


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It means the same thing. It's just for esthetic reasons that authors try to vary their vocabulary.


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It is the same whether you say there exists or for some.


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The term anti-invariant seems to be in use for what I believe is this general idea. The Wikipedia glossary of invariant theory defines an anti-invariant as A relative invariant transforming according to a character of order 2 of a group such as the symmetric group. which, if my understanding of the relevant terms is correct, is a special case of what I ...



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