It is well known from differential geometry that an n-dimensional Riemannian manifold can be isometrically embedded in a Euclidean space of dimension 2n+1 [Nas54]. Though the proof by Nash is intuitive, it is not clear whether such a construction is achievable by an algorithm that only has access to a finite-size sample from the manifold. In this paper, we study Nash’s construction and develop ...

using hartree–fock (hf) and ِِdensity functional theory (dft) calculations the thermodynamic properties such as thermal energy , , thermal enthalpy , , thermal entropy , , thermal gibbs free energy , , heat capacity ,cv, and molecular structures of several species involving in keto↔enol tautomerism related to acetaldehyde (a), 5,5-dimethyl-1,3-cyclohexanedione (dimedone) and  acetylacetone (aa) h...

the complexation of sodium molybdate with para-sulfonatocalix [4] arene was studied in thetemperature range from 20 to 40 °c and ph = 5.8 by spectroscopic method in water solution. thestability constants of complexes were calculated by the multiwavelength analysis of spectral datausing squad program. the equilibrium constant, spectroscopic data, molar absorptivities andconcentrations of all spe...