the power of the continuum - Biblical Christian World View
... With the properties of closure and equality under our belt, let us now consider irrational numbers as a whole. How many are there? Can we count them like we can count rational numbers? That is, are they denumerable like the set of rational numbers? So far, we have come across irrational numbers haph ...
... With the properties of closure and equality under our belt, let us now consider irrational numbers as a whole. How many are there? Can we count them like we can count rational numbers? That is, are they denumerable like the set of rational numbers? So far, we have come across irrational numbers haph ...
Global exact controllability in infinite time of Schrödinger equation
... infinite time. Application of this result gives global exact controllability in infinite time in H 3 for d = 1. We show also that the system is not exactly controllable in finite time in lower Sobolev spaces. Let us recall some previous results on the controllability problem of Schrödinger equation. ...
... infinite time. Application of this result gives global exact controllability in infinite time in H 3 for d = 1. We show also that the system is not exactly controllable in finite time in lower Sobolev spaces. Let us recall some previous results on the controllability problem of Schrödinger equation. ...
EFFECTIVE RESULTS FOR DISCRIMINANT EQUATIONS OVER
... an ideal representation for A. For A to be an integral domain of characteristic 0 it is necessary and sufficient that I be a prime ideal of Z[X1 , . . . , Xr ] with 1 6∈ QI. This can be checked for instance by means of [1, Proposition 4.10] and [20, §4]. By a representation for an element y of A we ...
... an ideal representation for A. For A to be an integral domain of characteristic 0 it is necessary and sufficient that I be a prime ideal of Z[X1 , . . . , Xr ] with 1 6∈ QI. This can be checked for instance by means of [1, Proposition 4.10] and [20, §4]. By a representation for an element y of A we ...
Square Roots and Cube Roots
... Write an expression to describe the total number of photos on the pages. ...
... Write an expression to describe the total number of photos on the pages. ...
A Conjecture on Integer Arithmetic which Implies that there is an
... where a(i1 , . . . , ik ) denote integers. Theorem 2. ([3]) The Conjecture implies that if a Diophantine equation (1) has only finitely many solutions in integers (non-negative integers, rationals), then their heights are bounded from above by a computable function of ...
... where a(i1 , . . . , ik ) denote integers. Theorem 2. ([3]) The Conjecture implies that if a Diophantine equation (1) has only finitely many solutions in integers (non-negative integers, rationals), then their heights are bounded from above by a computable function of ...
Full text
... [7] M. Bicknell-Johnson and D. C. Fielder, The Number of Representations of N Using Distinct Fibonacci Numbers, Counted by Recursive Formulas, The Fibonacci Quarterly, 37.1 (1999), 47–60. [8] M. Bicknell-Johnson and D. C. Fielder, The Least Number Having 331 Representations as a Sum of Distinct Fibo ...
... [7] M. Bicknell-Johnson and D. C. Fielder, The Number of Representations of N Using Distinct Fibonacci Numbers, Counted by Recursive Formulas, The Fibonacci Quarterly, 37.1 (1999), 47–60. [8] M. Bicknell-Johnson and D. C. Fielder, The Least Number Having 331 Representations as a Sum of Distinct Fibo ...
Vector fields and infinitesimal transformations on
... use v1 and Ui to denote the contravariant and covariant components of a vector field v respectively. Moreover, if we multiply, for example the components di/ of a covariant tensor by the components b^ of a contravariant tensor, it will always be understood thatj is to be summed. Let 91 be the set { ...
... use v1 and Ui to denote the contravariant and covariant components of a vector field v respectively. Moreover, if we multiply, for example the components di/ of a covariant tensor by the components b^ of a contravariant tensor, it will always be understood thatj is to be summed. Let 91 be the set { ...
Solving linear inequalities
... Like an equation, we can solve an inequality by adding or subtracting the same value to both sides of the inequality sign. We can also multiply or divide both sides of the inequality by a positive value. For example, solve 4x – 7 > 11 – 2x add 7 to both sides: ...
... Like an equation, we can solve an inequality by adding or subtracting the same value to both sides of the inequality sign. We can also multiply or divide both sides of the inequality by a positive value. For example, solve 4x – 7 > 11 – 2x add 7 to both sides: ...
Equation
In mathematics, an equation is an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. In this situation, variables are also known as unknowns and the values which satisfy the equality are known as solutions. An equation differs from an identity in that an equation is not necessarily true for all possible values of the variable.There are many types of equations, and they are found in all areas of mathematics; the techniques used to examine them differ according to their type.Algebra studies two main families of equations: polynomial equations and, among them, linear equations. Polynomial equations have the form P(X) = 0, where P is a polynomial. Linear equations have the form a(x) + b = 0, where a is a linear function and b is a vector. To solve them, one uses algorithmic or geometric techniques, coming from linear algebra or mathematical analysis. Changing the domain of a function can change the problem considerably. Algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.Geometry uses equations to describe geometric figures. The objective is now different, as equations are used to describe geometric properties. In this context, there are two large families of equations, Cartesian equations and parametric equations.Differential equations are equations involving one or more functions and their derivatives. They are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model real-life processes in areas such as physics, chemistry, biology, and economics.The ""="" symbol was invented by Robert Recorde (1510–1558), who considered that nothing could be more equal than parallel straight lines with the same length.