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Questions regarding the well-definedness of a function often arise when the defining equation of a function refers not only to the arguments themselves, but also to elements of the arguments, serving as representatives. This is sometimes unavoidable when the arguments are cosets and when the equation refers to coset representatives. The result ...
Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules: . For a singularity at a finite number b + [() + + ()] with < < and where b is the difficult point, at which the behavior of the function f is such that = for any < and = for any >.
The proposition in probability theory known as the law of total expectation, [1] the law of iterated expectations [2] (LIE), Adam's law, [3] the tower rule, [4] and the smoothing theorem, [5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then
On the other hand, if neither such a symmetry nor another form of equivalence can be established, then the use of "without loss of generality" is incorrect and can amount to an instance of proof by example – a logical fallacy of proving a claim by proving a non-representative example. [3]
For example, we can prove by induction that all positive integers of the form 2n − 1 are odd. Let P(n) represent "2n − 1 is odd": (i) For n = 1, 2n − 1 = 2(1) − 1 = 1, and 1 is odd, since it leaves a remainder of 1 when divided by 2. Thus P(1) is true.
In law, knowledge is one of the degrees of mens rea that constitute part of a crime.For example, in English law, the offence of knowingly being a passenger in a vehicle taken without consent requires that the prosecution prove not only that the defendant was a passenger in a vehicle and that it was taken by the driver without consent, but also that the defendant knew that it was taken without ...
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A form book is a tool used by attorneys in the United States to aid in the filing of pleadings, motions and other legal documents with a court or similar decision-making body. [1] A form book may be a bound volume or binder containing loose-leaf pages, containing forms, clauses and model documents that the attorney might use when preparing a ...