New PDF release: A barrier method for quasilinear ordinary differential

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Extra resources for A barrier method for quasilinear ordinary differential equations of the curvature type

Example text

Since the composition of any two bijective functions is bijective, W is stable. Furthermore, since the identity function e on the set S is bijective, W is a submonoid of the monoid X. 1. The intersection of any family of sub-semigroups of a semigroup X is a sub-semigroup of X. The intersection of any family of submonoids of a monoid X is a submonoid of X. Proof: Let us consider an arbitrarily given indexed family, of sub-semigroups of a semigroup X and let A denote their intersection; that is, A n AI" I'EM To prove that A is a sub-semigroup of X, let u and v be arbitrary elements 2.

For an arbitrarily given homomorphism h:X - t Y of a group X into a group Y, we have hex-I) = [h(X)]-1 for every element x of x. Proof: Since h is a homomorphism, we have h(x)h(x-l ) h(xx-l ) h(ex) 44 for every x E X. III. 1), h(cx) = ey. Hence we obtain [h(X)]-1 for every x E X. 3. If h:X -+ Y is a homomorphism oj a group X into a group Y, then its image 1m (h) is a subgroup of Y and its kernel Ker (h) is a subgroup of X. 6) that 1m (h) is a submonoid of Y and Ker (h) is a submonoid of X. To prove that 1m (h) is a subgroup of Y, let y denote an arbitrary element of 1m (h).

2. If a binary operation in a set X has a neutral element, zt has only one. " If Xl, X2, ••• , Xn are the elements of X, the multiplication table of a binary operation in X has the form of a square array of elements of X, consisting of n rows and n columns both labeled by Xl, X2, ••• , x n • The element at the intersection of the row labeled by x, and the column labeled by x, is the product X'XJ • For example, let us consider the usual composition in the example (3). Assume that the set S consists of two elements a and b.

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A barrier method for quasilinear ordinary differential equations of the curvature type by Kusahara T.


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