Improved Sparse Multivariate Polynomial Interpolation Algorithms*
... This is a transposed Vandermonde system of equations and can be solved for the P̂i (uj ) in O(t2 ) arithmetic operations. From the values at v1 , . . . , vd+1 , each P̂i (xk+1 ) can now be recovered by univariate interpolation. This step needs O(d2 t) arithmetic operations as there are at most t pol ...
... This is a transposed Vandermonde system of equations and can be solved for the P̂i (uj ) in O(t2 ) arithmetic operations. From the values at v1 , . . . , vd+1 , each P̂i (xk+1 ) can now be recovered by univariate interpolation. This step needs O(d2 t) arithmetic operations as there are at most t pol ...
6 Roots, Surds and Discriminant
... There are no real roots as the function is never equal to zero. ...
... There are no real roots as the function is never equal to zero. ...
Fermat`s Last Theorem
... one of a and b is even and the other odd. This implies c2 is odd and, hence, c is odd. The above result rules out all triples (a, b, c), with c even, as a possible PPT. It is enough to consider c odd. But that still does not say all odd c are allowed. For instance, there is no choice of (a, b) ∈ N s ...
... one of a and b is even and the other odd. This implies c2 is odd and, hence, c is odd. The above result rules out all triples (a, b, c), with c even, as a possible PPT. It is enough to consider c odd. But that still does not say all odd c are allowed. For instance, there is no choice of (a, b) ∈ N s ...
development of simultaneous spectrophotometric method of
... wavelength of radiation and the structure of compound. The absorption of radiation is due to subtraction of energy from the radiation beam when electrons in orbital of lower energy are excited into orbital of higher energy. Since this is a electron transition phenomen ...
... wavelength of radiation and the structure of compound. The absorption of radiation is due to subtraction of energy from the radiation beam when electrons in orbital of lower energy are excited into orbital of higher energy. Since this is a electron transition phenomen ...
Trig Identities - A Site for Mathematical Minds
... equation if f(x) = g(x) for every value of x for which both functions are defined • An equation that is not an identity is called a conditional equation ( x 1) 2 ( x 1)( x 1) x 2 2 x 1 ...
... equation if f(x) = g(x) for every value of x for which both functions are defined • An equation that is not an identity is called a conditional equation ( x 1) 2 ( x 1)( x 1) x 2 2 x 1 ...
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.