A Lighting-Invariant Point Processor for Shading

2020 ◽  
Author(s):  
Kathryn Heal ◽  
Jialiang Wang ◽  
Steven J. Gortler ◽  
Todd Zickler
Keyword(s):  
Invariant Point

RECIPROCAL SALT PAIRS, INVOLVING THE CATIONS Li2, Na2, AND K2, THE ANIONS SO4, AND Cl2, AND WATER, AT 25 °C

1958 ◽  
Vol 36 (11) ◽  
pp. 1511-1517 ◽  
Author(s):  
A. N. Campbell ◽  
E. M. Kartzmark ◽  
E. G. Lovering
Keyword(s):  
Sodium Chloride ◽  
Potassium Chloride ◽  
Mole Fraction ◽  
Lithium Chloride ◽  
Invariant Point ◽  
Double Salt ◽  
Potassium Sulphate ◽  
Lithium Sulphate ◽  
Solid Phases ◽  
Chloride Monohydrate

In the reciprocal salt pair Li2, K2, Cl2, SO4, and water, at 25 °C there are large areas in which potassium sulphate and potassium lithium sulphate (KLiSO4) are separately in equilibrium with solution. Two incongruent invariant points exist. At one of these the composition of the solution is 0.917 mole fraction chloride, 0.437 mole fraction lithium, and 19.4 moles of water per total mole of salt, the equilibrium solid phases being potassium chloride, potassium sulphate, and the double salt. At the second, the composition of the solution is 0.967 mole fraction chloride, 0.870 mole fraction lithium, and 13.8 moles of water per mole of salt, the solid phases being potassium chloride, double salt, and lithium sulphate monohydrate. One congruent invariant point exists, at which the composition of the solution is 1.00 mole fraction chloride, 0.960 mole fraction lithium, and 9.6 moles of water per mole of salt, the solid phases being lithium sulphate monohydrate, lithium chloride monohydrate, and potassium chloride.In the reciprocal salt pair Li2, Na2, Cl2, SO4, and water, at 25 °C there is an incongruent invariant point at which the composition of the solution is 0.873 mole fraction chloride, 0.668 mole fraction lithium, and 15.1 moles water per total mole of salt, the solid phases being sodium chloride, solid solution of sodium and lithium sulphates, and lithium sulphate monohydrate. A congruent invariant point exists, at which the composition of the solution is practically entirely lithium chloride, the solid phases present being lithium chloride monohydrate, lithium sulphate monohydrate, and sodium chloride.

Solubility in the potassium stannate – potassium hydroxide – water system at 0 to 95.0 °C

1968 ◽  
Vol 46 (16) ◽  
pp. 2715-2719 ◽  
Author(s):  
P. Rosenblum
Keyword(s):  
Potassium Hydroxide ◽  
Invariant Point ◽  
Water System ◽  
Phase Relations ◽  
Polarizing Microscope ◽  
Solid Phases ◽  
Hydroxide Water

Solubilities and phase relations have been determined in the K2Sn(OH)6–KOH–H2O system between 0 and 95.0 °C. The isotherms consist of only one major solubility branch, that of K2Sn(OH)6, while the solubility branch of KOH is indistinguishable, since the isothermal invariant point is extremely close to the solubility of potassium hydroxide. Solid phases have been determined by Schreinemakers' method and checked with the polarizing microscope.

scholarly journals Similarity Solutions for the Complex Burgers’ Hierarchy

2019 ◽  
Vol 74 (7) ◽  
pp. 597-604 ◽  
Author(s):  
Amlan K. Halder ◽  
A. Paliathanasis ◽  
S. Rangasamy ◽  
P.G.L. Leach
Keyword(s):  
Differential Equations ◽  
Invariant Point ◽  
Order Differential Equation ◽  
Reduction Process ◽  
Similarity Solutions ◽  
First Order ◽  
Similarity Transformations ◽  
Point Transformations ◽  
Burgers Hierarchy ◽  
Detailed Application

AbstractA detailed analysis of the invariant point transformations for the first four partial differential equations which belong to the complex Burgers’ hierarchy is performed. Moreover, a detailed application of the reduction process through the Lie-point symmetries is presented while we construct the similarity solutions. We conclude that the differential equations of our consideration are reduced to first-order equations such as the Abel, Riccati, and to a linearisable second-order differential equation by using similarity transformations.

scholarly journals Phase Equilibria and Phase Diagrams for the Ternary Aqueous System Containing Lithium, Sodium, and Pentaborate Ions at 298.15 and 323.15 K and 101.325 kPa

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Panpan Li ◽  
Kaiyu Zhao ◽  
Shangqing Chen ◽  
Jiayin Hu ◽  
Yafei Guo ◽  
...  
Keyword(s):  
Refractive Index ◽  
Ternary System ◽  
Phase Equilibria ◽  
Phase Diagrams ◽  
Invariant Point ◽  
Aqueous System ◽  
Eutectic Type ◽  
Two Temperatures ◽  
Sodium Pentaborate Pentahydrate ◽  
Experimental Values

Phase equilibria and phase diagrams for the ternary aqueous system containing lithium, sodium, and pentaborate ions at 298.15 and 323.15 K and 101.325 kPa were investigated by the methods of isothermal dissolution equilibrium. From the experimental data, the phase diagrams and the diagrams of physicochemical properties versus composition of lithium pentaborate in the equilibrium systems were plotted, respectively. The phase diagrams of the ternary system LiB5O8 + NaB5O8 + H2O at two temperatures contain one invariant point, two univariant curves, and two crystallization regions corresponding to sodium pentaborate pentahydrate (NaB5O8·5H2O) and lithium pentaborate pentahydrate (LiB5O8·5H2O). Due to the different dissolution behaviors of pentaborate salts in the aqueous systems, the component of LiB5O8 has a relatively strong effect on the solubility of NaB5O8. It was found that this system belongs to a simple eutectic type at two temperatures, and neither double salts nor solid solutions were formed. The densities and refractive indices in the ternary system at 298.15 and 323.15 K are as similar as changing regularly with the increase of LiB5O8 concentration. On the basis of empirical equations of the density and refractive index in electrolytes, the calculated values of density and refractive index agreed well with the experimental values at two temperatures.

Phase Diagram and Research for the Ternary System of NaCl-CsCl-H2O at 50°C

2013 ◽  
Vol 706-708 ◽  
pp. 193-196 ◽  
Author(s):  
Yong Ming Zhang ◽  
Li Chun Ma ◽  
Cheng Lin Liu ◽  
Li Fang Chen ◽  
Dong Liang Shen ◽  
...  
Keyword(s):  
Experimental Data ◽  
Phase Diagram ◽  
Ternary System ◽  
Invariant Point ◽  
Equilibrium Phase ◽  
Ternary Systems ◽  
Practical Significance ◽  
Eutectic Type ◽  
Comprehensive Utilization ◽  
Isothermal Evaporation

The equilibria phase of the ternary systems NaCl-CsCl-H2O was studied at 50°C by the isothermal evaporation method. The equilibrium phase diagrams were plotted in line with the experimental data. It suggest that the phase diagram of NaCl-CsCl-H2O is the simple eutectic type where one invariant point is found to have the compositions of 12.3% NaCl, 54.1% CsCl and 33.6% H2O by mass, respectively. The study provide fundamental thermodynamic data of brine system that contains sodium and cesium, and has practical significance for the brine’s comprehensive utilization.

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