18. Cyclotomic polynomials II
... Thus, 2ab − b2 = 0. This requires either b = 0 or 2a − b = 0. Certainly b cannot be 0, or 5 would be the square of a rational number (which we have long ago seen impossible). Try 2a = b. Then, supposedly, 5 = a2 − 2(2a)2 = −3a2 ...
... Thus, 2ab − b2 = 0. This requires either b = 0 or 2a − b = 0. Certainly b cannot be 0, or 5 would be the square of a rational number (which we have long ago seen impossible). Try 2a = b. Then, supposedly, 5 = a2 − 2(2a)2 = −3a2 ...
Classical Yang-Baxter Equation and Its Extensions
... equivalent to a linear map r : g∗ → g satisfying [r(x), r(y)] = r(ad∗ r(x)(y) − ad∗ r(y)(x)), ∀x, y ∈ g∗ , ...
... equivalent to a linear map r : g∗ → g satisfying [r(x), r(y)] = r(ad∗ r(x)(y) − ad∗ r(y)(x)), ∀x, y ∈ g∗ , ...
Perturbation theory for infinite-dimensional
... vanishes sufficiently fast as s$0. (For further discussion, see w and Remark 3.29 below. See also [DZW, w As is well known, the NLS equation is completely integrable, and we view the problem at hand as an example of the perturbation theory of infinite-dimensional integrable systems on the line. For ...
... vanishes sufficiently fast as s$0. (For further discussion, see w and Remark 3.29 below. See also [DZW, w As is well known, the NLS equation is completely integrable, and we view the problem at hand as an example of the perturbation theory of infinite-dimensional integrable systems on the line. For ...
THE ISOMORPHISM PROBLEM FOR CYCLIC ALGEBRAS AND
... reduces to the isomorphism problem. A detailed example is given, which leads to the construction of an explicit noncrossed product division algebra. This paper solves the isomorphism problem for cyclic algebras. A cyclic algebra is a central-simple algebra that contains a maximal subfield which is a ...
... reduces to the isomorphism problem. A detailed example is given, which leads to the construction of an explicit noncrossed product division algebra. This paper solves the isomorphism problem for cyclic algebras. A cyclic algebra is a central-simple algebra that contains a maximal subfield which is a ...
04-Powers_Roots_and_Logarithms-
... EXAMPLES: After using the circular method, you see are you back to solving exponential equations. log4 3=x 3=4x log 3=log 4x log 3=x log 4 log 3=x log 4 ...
... EXAMPLES: After using the circular method, you see are you back to solving exponential equations. log4 3=x 3=4x log 3=log 4x log 3=x log 4 log 3=x log 4 ...
A NOTE ON THE METHOD OF MULTIPLE SCALES*
... applicable to more complicated problems for which the exact solution cannot be obtained, and it yields results that are at least an improvement over the usual methods of multiple scales. In each case, the procedure is equivalent: expand the available parameters in the equation(s) and use the unknown ...
... applicable to more complicated problems for which the exact solution cannot be obtained, and it yields results that are at least an improvement over the usual methods of multiple scales. In each case, the procedure is equivalent: expand the available parameters in the equation(s) and use the unknown ...
An Introduction to Linear Algebra
... Note that it is impossible to have more than n orthonormal vectors in an n-dimensional space. For instance, in three dimensions, one can only have 3 vectors which are orthogonal, and thus, only three vectors can be orthonormal (although there are an infinite number of sets of 3 orthogonal vectors). ...
... Note that it is impossible to have more than n orthonormal vectors in an n-dimensional space. For instance, in three dimensions, one can only have 3 vectors which are orthogonal, and thus, only three vectors can be orthonormal (although there are an infinite number of sets of 3 orthogonal vectors). ...
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.