Active set algorithms for isotonic regression; A unifying framework

In this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods. The active set approach provides a unifying framework for studying algorithms for isotonic regression, simplifies the exposition of existing algorithms and leads to several new efficient algorithms. We also investigate the computational complexity of several algorithms.In this paper we consider the isotonic regression problem with respect to a complete order where eachw _i is strictly positive and eachy _i is an arbitrary real number. We show that the Pool Adjacent Violators algorithm (due to Ayer et al., 1955; Miles, 1959; Kruskal, 1964), is a dual feasible active set method and that the Minimum Lower Set algorithm (due to Brunk et al., 1957) is a primal feasible active set method of computational complexity O(n ²). We present a new O(n) primal feasible active set algorithm. Finally we discuss Van Eeden's method and show that it is of worst-case exponential time complexity.This work was supported by the National Science and Engineering Research Council of Canada under Research Grant A8189 and an Ontario Graduate Scholarship.     

Studies on the exchange characteristics of [Co(NH3)6]3+ in amberlite IRC-50

Summary The exchange of Co(NH₃)₆]³⁺-ions on amberlite IRC-50 resin has been studied at room temperature. For this exchange process the cations are effective in the order: Cs⁺<Rb⁺<K⁺<Na⁺<Li⁺<NH₄ ⁺<Mg²⁺ <Ca²⁺<H⁺ and (C₂H₅)₄N<(CH₃)₄N⁺ ≪Cetyltrimethylammonium-ion <Cetylpyridinium-ion. The logarithm of the selectivity coefficient gives linear graphs when plotted against the radius of the hydrated ions or the reciprocals of theDebye-Hückel parameter?.     

On the relation between the fermi energies in the presence and absence of magnetic quantization in degenerate semiconductors having non-parabolic energy bands

An investigation is made of the influence of band non-parabolicity on the relation between the Fermi energies in the presence and absence of magnetic quantization in degenerate semi-conductors through derivation of a simplified form of the relation.On leave of absence from theDept. of Physics, B. N. College, Patna University, Patna 800004.     

Influence of alloy composition on the diffusivity-mobility ratio in n-channel inversion layers on ternary semiconductors

An expression is derived for the diffusivity-mobility ratio of the carriers in n-channel inversion layers on semiconductors like the ternary compounds which have strongly non-parabolic energy bands. The dependence of the ratio on alloy composition is also studied under the weak-field limit taking n-channel Hg_1–xCd_x.Te as an example.On leave of absence fromthe Department of Physics, Patna University, Patna, India.     

Evaluation of thermodynamic quantities etc. in the cation exchange of Mg2+ from magnesium saturated bentonite and vermiculite

Summary Cation exchange studies have been carried out between Mg-saturated bentonite and vermiculite and a number of monovalent electrolytes. According to the distribution and selectivity coefficients the cations are preferred in the order: Li Na43)₄N (C₂H₅)₄N for betonite and Li < Na < NH₄ < K < Rb < Cs < H and (C₂H₅)₄N < (CH₃)₄N for vermiculite. The selectivity coefficient of H⁺ is found to be much greater than other inorganic ions.TheG-values of the cation exchange reaction have been calculated by an equation ofBarrer. The experimental data were analyzed by the theoretical approach ofPauley too.     

Messung der elektrostatischen Aufladung von Teppichen

Ohne Zusammenfassung     

Effect of temperature on the validity of the einstein relation in heavily doped semiconductors

An analytical expression of the modified form of the Einstein relation in heavily doped semiconductors in which Gaussian band tails are formed near the lower limit of heavy doping is derived for studying the temperature dependence of the diffusivity-mobility ratio of the carriers in such semiconductors. It is found that, with increasing temperature from relatively low values, the ratio first increases in a nonlinear manner and then decreases till, at high temperatures, it approaches its value corresponding to the non-degenerate condition resulting in a peak over a narrow range of temperatures.     

Influence of magnetic quantization on the debye screening length in degenerate semiconductors having parabolic energy bands

The Debye screening length in degenerate semiconductors having parabolic energy bands is shown for the first time to have an oscillatory dependence on magnetic field under the influence of Landau quantization when the higher lying sub-bands are occupied.The authors are indebted to Professor J. N.Bhar for his keen interest in the work.     

On the theory of electron diffusion under a temperature gradient in degenerate semiconductors having non-parabolic energy bands

The theory of electron diffusion under a temperature gradient (Soret effect) in degenerate semiconductors having non-parabolic energy bands and spherical constant-energy surfaces is worked out for the first time. An expression is given for the Soret coefficient which may be determined from the knowledge of the energy dependence of the relaxation time and can be related to the thermoelectric power.