In this study we discuss the use of the simplex method to solve allocation problems whose flow matrices are doubly stochastic. Although these problems can be solved via a 0 - 1 integer programming method, H. W. Kuhn [...In this study we discuss the use of the simplex method to solve allocation problems whose flow matrices are doubly stochastic. Although these problems can be solved via a 0 - 1 integer programming method, H. W. Kuhn [1] suggested the use of linear programming in addition to the Hungarian method. Specifically, we use the existence theorem of the solution along with partially total unimodularity and nonnegativeness of the incidence matrix to prove that the simplex method facilitates solving these problems. We also provide insights as to how a partition including a particular unit may be obtained.展开更多
Gambling is a useful analog to thermodynamics. When all players use the same dice, loaded or not, on the average no one wins. In thermodynamic terms, when the system is homogeneous—an assumption made by Boltzmann in ...Gambling is a useful analog to thermodynamics. When all players use the same dice, loaded or not, on the average no one wins. In thermodynamic terms, when the system is homogeneous—an assumption made by Boltzmann in his H-Theorem—entropy never decreases. To reliably win, one must cheat, for example, use a loaded dice when everyone else uses a fair dice;in thermodynamics, one must use a heterogeneous statistical strategy. This can be implemented by combining within a single system, different statistics such as Maxwell-Boltzmann’s, Fermi-Dirac’s and Bose-Einstein’s. Heterogeneous statistical systems fall outside of Boltzmann’s assumption and therefore can bypass the second law. The Maxwell-Boltzmann statistics, the equivalent of an unbiased fair dice, requires a gas column to be isothermal. The Fermi-Dirac and Bose-Einstein statistics, the equivalent of a loaded biased dice, can generate spontaneous temperature gradients when a field is present. For example, a thermoelectric junction can produce a spontaneous temperature gradient, an experimentally documented phenomenon. A magnetic field parallel to, and an electric field perpendicular to a surface produce a spontaneous current along the surface, perpendicular to both fields (Reciprocal Hall Effect). Experimental data collected by several independent researchers is cited to support the theory.展开更多
文摘In this study we discuss the use of the simplex method to solve allocation problems whose flow matrices are doubly stochastic. Although these problems can be solved via a 0 - 1 integer programming method, H. W. Kuhn [1] suggested the use of linear programming in addition to the Hungarian method. Specifically, we use the existence theorem of the solution along with partially total unimodularity and nonnegativeness of the incidence matrix to prove that the simplex method facilitates solving these problems. We also provide insights as to how a partition including a particular unit may be obtained.
文摘Gambling is a useful analog to thermodynamics. When all players use the same dice, loaded or not, on the average no one wins. In thermodynamic terms, when the system is homogeneous—an assumption made by Boltzmann in his H-Theorem—entropy never decreases. To reliably win, one must cheat, for example, use a loaded dice when everyone else uses a fair dice;in thermodynamics, one must use a heterogeneous statistical strategy. This can be implemented by combining within a single system, different statistics such as Maxwell-Boltzmann’s, Fermi-Dirac’s and Bose-Einstein’s. Heterogeneous statistical systems fall outside of Boltzmann’s assumption and therefore can bypass the second law. The Maxwell-Boltzmann statistics, the equivalent of an unbiased fair dice, requires a gas column to be isothermal. The Fermi-Dirac and Bose-Einstein statistics, the equivalent of a loaded biased dice, can generate spontaneous temperature gradients when a field is present. For example, a thermoelectric junction can produce a spontaneous temperature gradient, an experimentally documented phenomenon. A magnetic field parallel to, and an electric field perpendicular to a surface produce a spontaneous current along the surface, perpendicular to both fields (Reciprocal Hall Effect). Experimental data collected by several independent researchers is cited to support the theory.
