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Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science.Informally, a quantified statement "such that …" can be viewed as a question "When is there an such that …?", and the statement without quantifiers can be viewed as the answer to that question.
Next, we insert another source of electromotive force, E 1, in series with Z e, where E 1 has the same magnitude as E but is opposed in direction (see Figure 2c). The current, I 1, can be determined as follows: it is the current that would result from E 1 acting alone, with all other sources (within the active network and the external network ...
Two rings R and S (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over R, R-Mod, and the category of (left) modules over S, S-Mod. It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are
An elementary embedding of a structure N into a structure M of the same signature σ is a map h: N → M such that for every first-order σ-formula φ(x 1, …, x n) and all elements a 1, …, a n of N, N φ(a 1, …, a n) if and only if M φ(h(a 1), …, h(a n)).
A relation R is both left and right Euclidean, if, and only if, the domain and the range set of R agree, and R is an equivalence relation on that set. [8] A right Euclidean relation is always quasitransitive, [9] as is a left Euclidean relation. [10] A connected right Euclidean relation is always transitive; [11] and so is a connected left ...
An invertible measure-preserving transformation on a standard probability space that obeys the 0-1 law is called a Kolmogorov automorphism. [clarification needed] All Bernoulli automorphisms are Kolmogorov automorphisms but not vice versa. The presence of an infinite cluster in the context of percolation theory also obeys the 0-1 law.
The standard Gaussian measure on . is a Borel measure (in fact, as remarked above, it is defined on the completion of the Borel sigma algebra, which is a finer structure);; is equivalent to Lebesgue measure: , where stands for absolute continuity of measures;
This is equivalent to calculating the Thevenin resistance. When there are dependent sources, the more general method must be used. The voltage at the terminals is calculated for an injection of a 1 ampere test current at the terminals. This voltage divided by the 1 A current is the Norton impedance R no (in ohms). This method must be used if ...