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5

The definition of antisymmetry does cover relations like $<$. If you examine that definition carefully, you’ll see that in order for a relation $R$ to violate it, there must be elements $a,b\in S$ such that $R(a,b)$, $R(b,a)$, and $a\ne b$. If you can’t even find elements $a,b\in S$ such that $R(a,b)$ and $R(b,a)$, then you certainly can’t find elements ...


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Theorems are things proved from a theory. Meta-theorems are things proved about the theory. The statement: If $\sf ZF$ is consistent, then $\sf ZF$ does not prove $\sf AC$. Is a meta-theorem about the theory $\sf ZF$. It quantifies over all proofs that we can write from the axioms of $\sf ZF$. If you like to think about it semantically, proofs are not ...


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Usually it's referred to as the triangle inequality.


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It means that if you consider $f(t,y_1,\dots,y_n)$, the partial derivatives $\partial f/\partial y_i$ exist and are continuous for all $i=1,\dots,n$. The first line of your question is really wrong. You are considering a composite function $F(t) = f(t,y(t))$; this is a function of $t$ only.


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According to my dictionary Peturb To disturb greatly in mind; disquiet To throw in confusion; disorder To cause (a moving body, celestial object, etc.) to deviate from a theoretically (orbital) motion. I think we can ignore the first definition here but the other two are relevant . If you were to nudge a system slightly you are perturbing it and we may ...


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I would call it a cylinder — a right cylinder if the movement is normal to the plane of the original two-dimensional region. However, cylinder is used in enough different senses that I’d probably explain my usage first.


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In varying contexts, I've seen this type of object called a (generalized) cylinder; a tube; or a prism. If you're looking for a term to use in some piece of writing, the main things are to use something simple and familiar, and to be explicit about your meaning if the term is non-standard (or there's other danger of ambiguity).


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A hypothesis is an educated guess, often a statement you want to show or prove (or disprove). A good example is the Riemann Hypothesis, which Riemann hypothesized in the mid 1800s. It says that the real parts of the nontrivial zeroes of the function $$\zeta(s) = \displaystyle \sum_{n \geq 1} \frac{1}{n^s}$$ all have real part $1/2$. We do not currently know ...


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Googling for equalizer family morphisms (also in Google Books and Google Scholar) returned some hits. It seems that some people use the name multiple equalizer. Adámek, Herrlich, Strecker: Abstract and Concrete Categories. The Joy of Cats mention this notion in Example 11.4(2), p. 194. Another book using this notion is Castellini: Categorical Closure ...


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Discrete just have to be separated from one another, finite means the total number must be $<\infty$. So $f(x)=x$ has no discontinuities at all, and $f(x)={1\over x}$ has one discontinuity at $0$ and again, this number is finite. On the other hand the floor function $f(x)=[x]$ (definition below if you haven't seen it) has infinitely many of them--one at ...


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They're called abstract sets. Personally, I would simply call this a "set," or perhaps a "mere set" or "unstructured set" to emphasize that there's no further structure around.



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