## Advances in Mathematics Research, Volume 2Mathematics has been behind many of humanity's most significant advances in fields as varied as genome sequencing, medical science, space exploration, and computer technology. But those breakthroughs were yesterday. Where will mathematicians lead us tomorrow and can we help shape that destiny? This book assembles carefully selected articles highlighting and explaining cutting-edge research and scholarship in mathematics. |

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### Contents

7 | |

15 | |

Some Recent Results and Problems for SetValued Mappings | 33 |

A Coupled Legendre PetrovGalerkin and Collocation Method for the Generalized Kortewegde Vries Equations | 53 |

Hausdorff Measures of Homogeneous Cantor Sets | 75 |

Best Approximation of a Compact Convex Set by a Ball in an Arbitrary Norm | 81 |

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associated Assume Banach space best approximation ball boundary Cantor set Caristi selection collocation method compact set complete metric space completely multiplicative consider constant continuous contradicts convergence convex compact set convex function convex set Corollary decreasing defined denote dependence structures differential equations distribution elements estimate Euclidean norm exists finite fixed point theorem follows formula function f function R(x G N(m Hausdorff Hence induced inequality intD kernel Korteweg-de Vries equation Lemma Let us observe lifelength vector limsup Math multivariate shot-noise process nonempty nonexpansive mappings nonlinear nonnegative obtain ordered set orthant orthant dependent pn(x point process Polish space polynomials positive Potential Theory probability measure problem 1.1 Proposition random vectors resp satisfies sequence set-valued mapping solution strictly quasiconvex norm Suppose system operating Theorem 1.1 unique V*-invariant Xi(t