This paper discusses how the infinite set of real numbers between 0 and 1 could be represented by a countably infinite tree structure which would avoid Cantor’s diagonalization argument that the set of real numbers i...This paper discusses how the infinite set of real numbers between 0 and 1 could be represented by a countably infinite tree structure which would avoid Cantor’s diagonalization argument that the set of real numbers is not countably infinite. Likewise, countably infinite tree structures could represent all real numbers, and all points in any number of dimensions in multi-dimensional spaces. The objective of this paper is not to overturn previous research based on Cantor’s argument, but to suggest that this situation may be treated as a definitional or axiomatic choice. This paper proposes a “non-Cantorian” branch of cardinality theory, representing all these infinities with countably infinite tree structures. This approach would be consistent with the Continuum Hypothesis.展开更多
For a class of control systems with disc uncertainties, robust performance anaysis is developed. On the strictly positive realness and H-infinity-norm of uncertain systems, from the geometric point of view, two new su...For a class of control systems with disc uncertainties, robust performance anaysis is developed. On the strictly positive realness and H-infinity-norm of uncertain systems, from the geometric point of view, two new sufficient and necessary conditions are given. The largest H-infinity-norm bound, containing the coefficients of only stable polynomials and centered at a nominal stable point in the coecient space is found. The results obtained in the paper are tractable and concise, which is illustrated by some numerical examples.展开更多
For a conservation law with convex condition, consider its Caucby problemut+f（u）x=0, （1）u|t=0=u0. （2）Suppose that u （x, t） is the solution of （1）, （2）. We proved in [1] that the number of discontinuity lin...For a conservation law with convex condition, consider its Caucby problemut+f（u）x=0, （1）u|t=0=u0. （2）Suppose that u （x, t） is the solution of （1）, （2）. We proved in [1] that the number of discontinuity lines might be undenumerable, thus negating Oleinik’s assertion of discontinuity lines being at most denumerable.展开更多
文摘This paper discusses how the infinite set of real numbers between 0 and 1 could be represented by a countably infinite tree structure which would avoid Cantor’s diagonalization argument that the set of real numbers is not countably infinite. Likewise, countably infinite tree structures could represent all real numbers, and all points in any number of dimensions in multi-dimensional spaces. The objective of this paper is not to overturn previous research based on Cantor’s argument, but to suggest that this situation may be treated as a definitional or axiomatic choice. This paper proposes a “non-Cantorian” branch of cardinality theory, representing all these infinities with countably infinite tree structures. This approach would be consistent with the Continuum Hypothesis.
文摘For a class of control systems with disc uncertainties, robust performance anaysis is developed. On the strictly positive realness and H-infinity-norm of uncertain systems, from the geometric point of view, two new sufficient and necessary conditions are given. The largest H-infinity-norm bound, containing the coefficients of only stable polynomials and centered at a nominal stable point in the coecient space is found. The results obtained in the paper are tractable and concise, which is illustrated by some numerical examples.
文摘For a conservation law with convex condition, consider its Caucby problemut+f（u）x=0, （1）u|t=0=u0. （2）Suppose that u （x, t） is the solution of （1）, （2）. We proved in [1] that the number of discontinuity lines might be undenumerable, thus negating Oleinik’s assertion of discontinuity lines being at most denumerable.
