Nonconvex Optimal Control Problems with Nonsmooth Mixed State
... and (λ0 , . . . , λn ), (u0 , . . . , un ) are regarded as control variables. This is a problem with a scalar mixed constraint to which the proposition, as proved in the last step, applies. We can now write the optimality conditions for this problem work on them to get the result for the original pr ...
... and (λ0 , . . . , λn ), (u0 , . . . , un ) are regarded as control variables. This is a problem with a scalar mixed constraint to which the proposition, as proved in the last step, applies. We can now write the optimality conditions for this problem work on them to get the result for the original pr ...
The Factor Theorem and a corollary of the
... When you have access to Mathematica itself, you should carry out the evaluations one-by-one. As you work through this notebook in Mathematica, if you come across a Mathematica function you don't understand, try using the ? information command to find out something about it. (And follow the hyperlink ...
... When you have access to Mathematica itself, you should carry out the evaluations one-by-one. As you work through this notebook in Mathematica, if you come across a Mathematica function you don't understand, try using the ? information command to find out something about it. (And follow the hyperlink ...
2.1: The Derivative and Tangent Line Problem
... provided the limit exists. For all x for which the limit exists, f is a function of x. The process of finding a derivative is called differentiation. Written as an imperative verb, it would be differentiate. A Side Note on “Notation” In addition to f ( x) , which is read “f prime of x,” other not ...
... provided the limit exists. For all x for which the limit exists, f is a function of x. The process of finding a derivative is called differentiation. Written as an imperative verb, it would be differentiate. A Side Note on “Notation” In addition to f ( x) , which is read “f prime of x,” other not ...
Lecture 5: Universal One-Way Function and Computational Number
... It will be assumed that students have sufficient background knowledge in number theory. Students who do not have this background knowledge may want to consult the lecture notes on the course website. The lecture notes cover just enough number theory for a student to understand the material in this c ...
... It will be assumed that students have sufficient background knowledge in number theory. Students who do not have this background knowledge may want to consult the lecture notes on the course website. The lecture notes cover just enough number theory for a student to understand the material in this c ...
Totient Theorem
... favourite is his proof of the two square theorem by infinite descent) or at the time seemed like his most important piece of work at the time but this, in number theory at least, is probably his most useful piece of mathematics to the world today. This proof has given me a chance to link up some of ...
... favourite is his proof of the two square theorem by infinite descent) or at the time seemed like his most important piece of work at the time but this, in number theory at least, is probably his most useful piece of mathematics to the world today. This proof has given me a chance to link up some of ...
Quick Review of the Definition of the Definite Integral
... function is the function itself. If we combine this with part 1 of the Fundamental Theorem of Calculus which says that the derivative of the integral of a function is the function itself, we see that integration and di¤erentiation are inverse processes. One undoes what the other one does. If we comb ...
... function is the function itself. If we combine this with part 1 of the Fundamental Theorem of Calculus which says that the derivative of the integral of a function is the function itself, we see that integration and di¤erentiation are inverse processes. One undoes what the other one does. If we comb ...
Recently Littlewood and Offord1 proved the following lemma Let x1
... Let now A,(1), A2(1), • • - , A Z ') be the A's containing n-j-y elements . By our lemma there exist sets Ai ( I ) , i=1, 2, . . • , x ; 1=1, 2, . - • , 2j+2y+i, such that A i ~2 i+2 9+11 has n+j+y elements and A i cal C A i t a+11 and all the A's are different . Clearly not all the sets Ai ( t) , 1 ...
... Let now A,(1), A2(1), • • - , A Z ') be the A's containing n-j-y elements . By our lemma there exist sets Ai ( I ) , i=1, 2, . . • , x ; 1=1, 2, . - • , 2j+2y+i, such that A i ~2 i+2 9+11 has n+j+y elements and A i cal C A i t a+11 and all the A's are different . Clearly not all the sets Ai ( t) , 1 ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.