10 Rings
... notation for√them.) For instance, how does 2√factor into the ring of all algebraic integers? It clearly factors as ( 2)2 , which further factors as ( 4 2)4 . We can repeat this process indefinitely, so there is no minimal factorization. (Also none of these factors are units, because their reciprocal ...
... notation for√them.) For instance, how does 2√factor into the ring of all algebraic integers? It clearly factors as ( 2)2 , which further factors as ( 4 2)4 . We can repeat this process indefinitely, so there is no minimal factorization. (Also none of these factors are units, because their reciprocal ...
The Multivariate Resultant is NP-hard in any Characteristic
... use a sparse encoding for our two polynomials (their degree can therefore be exponential in the input size). The first proof does not carry over to positive characteristic since the NP-hardness of Partition relies in an essential way on the fact that the data are integers (in fact, in any finite fie ...
... use a sparse encoding for our two polynomials (their degree can therefore be exponential in the input size). The first proof does not carry over to positive characteristic since the NP-hardness of Partition relies in an essential way on the fact that the data are integers (in fact, in any finite fie ...
Instability of standing waves for non-linear Schrödinger
... A case of space-dependent/arises in optical waveguides, (see [1], [15] and [11]). In the last section of this paper we shall apply theorem 1 to conclude a new result on the instability of certain waves, as announced in [11]. The basic idea in the proof is to apply a dynamical systems point of view t ...
... A case of space-dependent/arises in optical waveguides, (see [1], [15] and [11]). In the last section of this paper we shall apply theorem 1 to conclude a new result on the instability of certain waves, as announced in [11]. The basic idea in the proof is to apply a dynamical systems point of view t ...
Nilpotent Jacobians in Dimension Three
... H is homogeneous of degree 3 the case n = 3 was solved affirmatively by Wright in [11] and the case n = 4 by Hubbers in [7]. Then in [5] (see also [6], Theorem 7.1.7) the second author found the first counterexample in dimension three (see below). On the other hand, recently de Bondt and van den Ess ...
... H is homogeneous of degree 3 the case n = 3 was solved affirmatively by Wright in [11] and the case n = 4 by Hubbers in [7]. Then in [5] (see also [6], Theorem 7.1.7) the second author found the first counterexample in dimension three (see below). On the other hand, recently de Bondt and van den Ess ...
0. Matlab instructions
... those cases, you can try to find solutions using fzero (short for "find zero") command. In order to use the command, first you need to write equation in the form f(x)=0. Thus, put all the terms of th equations on one side leaving just zero on the other. To find a solution near the x-value x=a, you ...
... those cases, you can try to find solutions using fzero (short for "find zero") command. In order to use the command, first you need to write equation in the form f(x)=0. Thus, put all the terms of th equations on one side leaving just zero on the other. To find a solution near the x-value x=a, you ...
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.