Affine Schemes
... Thus far, we have described classical (affine) algebraic geometry over an algebraically closed field. For lots of reasons, coming from number theory over even classical geometry it is useful to work over arbitrary fields or even more general rings. In the last chapter, we showed that reduced affine ...
... Thus far, we have described classical (affine) algebraic geometry over an algebraically closed field. For lots of reasons, coming from number theory over even classical geometry it is useful to work over arbitrary fields or even more general rings. In the last chapter, we showed that reduced affine ...
Sum-free sets. DEFINITION 1: A subset A of an abelian group (G,+)
... The quantity E[X k ]/k! in this expression is called the k:th moment of the random variable X. From (1.3) we see that the variance of X involves its second moment, hence the name. A rough analogy to studying the 2nd moment of a random variable is to study the second derivative of a smooth function i ...
... The quantity E[X k ]/k! in this expression is called the k:th moment of the random variable X. From (1.3) we see that the variance of X involves its second moment, hence the name. A rough analogy to studying the 2nd moment of a random variable is to study the second derivative of a smooth function i ...
Logarithmic Functions and Their Graphs
... The ear is sensitive to an extremely wide range of sound intensities. We take as a reference intensity I 0 = 10 –12 W/m 2 (watts per square meter) at a frequency of 1000 hertz, which measures a sound that is just barely audible (the threshold of hearing). The psychological sensation of loudness vari ...
... The ear is sensitive to an extremely wide range of sound intensities. We take as a reference intensity I 0 = 10 –12 W/m 2 (watts per square meter) at a frequency of 1000 hertz, which measures a sound that is just barely audible (the threshold of hearing). The psychological sensation of loudness vari ...
Mathematica
... Apply[f,expr] replaces the head of expr by f and evaluates it Map[f,list] applies the function f to each element of list Nest[f,expr,n] gives an expression with f applied n times to expr NestList[f,expr,n] does the same but returns the list of all intermediate results ...
... Apply[f,expr] replaces the head of expr by f and evaluates it Map[f,list] applies the function f to each element of list Nest[f,expr,n] gives an expression with f applied n times to expr NestList[f,expr,n] does the same but returns the list of all intermediate results ...
Function of several real variables
In mathematical analysis, and applications in geometry, applied mathematics, engineering, natural sciences, and economics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The ""input"" variables take real values, while the ""output"", also called the ""value of the function"", may be real or complex. However, the study of the complex valued functions may be easily reduced to the study of the real valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real valued functions will be considered in this article.The domain of a function of several variables is the subset of ℝn for which the function is defined. As usual, the domain of a function of several real variables is supposed to contain an open subset of ℝn.