Lego-like spheres and tori

Given a connected surface \({\mathbb {F}}^2\) with Euler characteristic \(\chi \) and three integers \(b>a\ge 1<k\), an \((\{a,b\};k)\)-\({\mathbb {F}}^2\) is a \({\mathbb {F}}^2\)-embedded graph, having vertices of degree only k and only a- and b-gonal faces. The main case are (geometric) fullerenes (5, 6; 3)-\({\mathbb {S}}^2\). By \(p_a\), \(p_b\) we denote the number of a-gonal, b-gonal faces. Call an \((\{a,b\};k)\)-map lego-admissible if either \(\frac{p_b}{p_a}\), or \(\frac{p_a}{p_b}\) is integer. Call it lego-like if it is either \(ab^f\)-lego map, or \(a^fb\)-lego map, i.e., the face-set is partitioned into \(\min (p_a,p_b)\) isomorphic clusters, legos, consisting either one a-gon and \(f=\frac{p_b}{p_a}\,b\)-gons, or, respectively, \(f=\frac{p_a}{p_b}\,a\)-gons and one b-gon; the case \(f=1\) we denote also by ab. Call a \((\{a,b\};k)\)-map elliptic, parabolic or hyperbolic if the curvature \(\kappa _b=1+\frac{b}{k}-\frac{b}{2}\) of b-gons is positive, zero or negative, respectively. There are 14 lego-like elliptic \((\{a,b\};k)\)-\({\mathbb {S}}^2\) with \((a,b)\ne (1,2)\). No \((\{1,3\};6)\)-\({\mathbb {S}}^2\) is lego-admissible. For other 7 families of parabolic \((\{a,b\};k)\)-\({\mathbb {S}}^2\), each lego-admissible sphere with \(p_a\le p_b\) is \(a^fb\) and an infinity (by Goldberg–Coxeter operation) of \(ab^f\)-spheres exist. The number of hyperbolic \(ab^f\,(\{a,b\};k)\)-\({\mathbb {S}}^2\) with \((a,b)\ne (1,3)\) is finite. Such \(a^f b\)-spheres with \(a\ge 3\) have \((a,k)=(3,4),(3,5),(4,3),(5,3)\) or (3, 3); their number is finite for each b, but infinite for each of 5 cases (a, k). Any lego-admissible \((\{a,b\};k)\)-\({\mathbb {S}}^2\) with \(p_b=2\le a\) is \(a^f b\). We list, explicitly or by parameters, lego-admissible \((\{a,b\};k)\)-maps among: hyperbolic spheres, spheres with \(a\in \{1,2\}\), spheres with \(p_b\in \{2,\frac{p_a}{2}\}\), Goldberg–Coxeter’s spheres and \((\{a,b\};k)\)-tori. We present extensive computer search of lego-like spheres: 7 parabolic (\(p_b\)-dependent) families, basic examples of all 5 hyperbolic \(a^fb\) (b-dependent) families with \(a\ge 3\), and lego-like \((\{a,b\};3)\)-tori.     

The mass renormalization for the translational Brownian motion

The damped harmonic oscillator is modeled as a local mode X with mass m and frequency \(\omega _{0}\) immersed in a phonon bath with spectral density function \(j_{0}(\omega \)). This function behaves as \(\omega ^{s}\, (s= 1,2,3,\ldots )\) when \(\omega \rightarrow 0\). The limit \(\omega _{0} = 0\) represents translational (free) Brownian motion. The earlier work (Hakim and Ambegaokar in Phys Rev A 32:423, 1985) concluded that the so defined limit transition is prohibited for spectral densities with \(s<2\). In the present study we demonstrate that a specially constructed preliminary excitation changing the original bath spectrum as \(j_{0}(\omega ) \rightarrow j(\omega )\) allows for treating the free damped motion of X with no restriction for the initial spectrum dimensionality. This procedure validates the finite mass renormalization (i.e. \(m\rightarrow M\) when \(\omega _{0}\rightarrow 0)\) for the conventional bath spectra with \(s=1,2\). We show that the new spectral density \(j(\omega )\) represents the momentum bilinear interaction between mode X and the environmental modes, whereas the conventional function \(j_{0}(\omega )\) is inherent to the case of bilinear coordinate interaction in terms of the same variables. The translational damping kernel is derived based on the new spectral density.     

Transition-metal dichalcogenide heterostructure solar cells: a numerical study

We evaluate the tunneling short-circuit current density \(J_{TU}\) in a p–i–n solar cell in which the transition metal dichalcogenide heterostructure (\(\hbox {MoS}_2/\hbox {WS}_2\) superlattice) is embedded in the intrinsic i region. The effects of varying well and barrier widths, Fermi energy levels and number of quantum wells in the i region on \(J_{TU}\) are examined. A similar analysis is performed for the thermionic current \(J_{TH}\) that arises due to the escape and recapture of charge carriers between adjacent potential wells in the i-region. The interplay between \(J_{TU}\) and \(J_{TH}\) in the temperature range (300–330 K) is examined. The thermionic current is seen to exceed the tunneling current considerably at temperatures beyond 310 K, a desirable attribute in heterostructure solar cells. This work demonstrates the versatility of monolayer transition metal dichalcogenides when utilized as fabrication materials for van der Waals heterostructure solar cells.     

Hierarchical enumeration of Kekulé’s benzenes,Ladenburg’s benzenes,Dewar’s benzenes,and benzvalenes by using combined-permutation representations

The group hierarchy for each skeleton of ligancy 6 is formulated to be: point group (PG \({\varvec{G}}_{\sigma }\)) \(\subseteq \) RS-stereoisomeric group (RS-SIG \({\varvec{G}}_{\sigma \widetilde{\sigma }\widehat{I}}\)) \(\subseteq \) stereoisomeric group (SIG \(\widetilde{{\varvec{G}}}_{\sigma \widetilde{\sigma }\widehat{I}}\)) \(\subseteq \) isoskeletomeric group (ISG \(\widetilde{\widetilde{{\varvec{G}}}}_{\sigma \widetilde{\sigma }\widehat{I}}\) = \({\varvec{S}}^{[6]}_{\sigma \widehat{I}}\)), where we start from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{D}}_{6h}\) for the Kekulé benzene skeleton, from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{D}}_{3h}\) for the Ladenburg benzene skeleton, from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{C}}_{2v}\) for the Dewar benzene skeleton, or from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{C}}_{2v}\) for the benzvalene skeleton. After these groups are constructed as combined-permutation representations, the calculation of the respective cycle indices with chirality fittingness (CI-CFs) and the introduction of ligand-inventory functions are conducted to give generation functions for 3D-based enumerations (for PGs and RS-SIGs) and 2D-based enumerations (for SIGs and ISGs). The enumeration results are discussed by means of isomer-classification diagrams, in which equivalence classes under enantiomerism (for PGs), RS-stereoisomerism (for RS-SIGs), stereoisomerism (for SIGs), and isoskeletomerism (for ISGs) are illustrated schematically. The implicit connotations of the conventional terms “skeletal isomerism”, “positional isomerism”, and “constitutional isomerism” are discussed, where the effects of the concept of isoskeletomerism are emphasized.     

Thermo Physical and Spectroscopic Properties of Binary Liquid Systems: Ethanoic Acid/Propanoic Acid/Butanoic Acid with 2-Methylaniline

Densities (ρ), speeds of sound (u), and viscosities (η) are reported for binary mixtures of 2-methylaniline with carboxylic acids (ethanoic acid, propanoic acid and butanoic acid) over the entire composition range of mole fraction at T?=?(303.15–318.15) K and at atmospheric pressure (0.1 MPa). The excess properties such as excess molar volume (V _m ^E ), excess isentropic compressibility (κ _S ^E ) and excess Gibbs energy of activation of viscous flow (G^*E) are calculated from the experimental densities, speeds of sound and viscosities. Excess properties are correlated using the Redlich–Kister polynomial equation. The partial molar volumes, \( \bar{V}_{\text{m,1}} \) and \( \bar{V}_{\text{m,2}} \), partial molar isentropic compressibilities, \( \bar{K}_{\text{s,m,1}} \) and \( \bar{K}_{\text{s,m,2}} \), excess partial molar volumes, \( \bar{V}_{\text{m,1}}^{\text{E}} \) and \( \bar{V}_{\text{m,2}}^{\text{E}} \), and excess partial molar isentropic compressibilities, \( \bar{K}_{\text{s,m,1}}^{\text{E}} \) and \( \bar{K}_{\text{s,m,2}}^{\text{E}} \), over whole composition range, partial molar volumes, \( \bar{V}_{\text{m,1}}^{ \circ } \) and \( \bar{V}_{\text{m,2}}^{ \circ } \), partial molar isentropic compressibilities, \( \bar{K}_{\text{s,m,1}}^{ \circ } \) and \( \bar{K}_{\text{s,m,2}}^{ \circ } \), excess partial molar volumes, \( \bar{V}_{\text{m,1}}^{{ \circ {\text{E}}}} \) and \( \bar{V}_{{{\text{m}},2}}^{{ \circ {\text{E}}}} \), and excess partial molar isentropic compressibilities, \( \bar{K}_{\text{s,m,1}}^{{ \circ {\text{E}}}} \) and \( \bar{K}_{\text{s,m,2}}^{{ \circ {\text{E}}}} \), of the components at infinite dilution have also been calculated from the analytically obtained Redlich–Kister polynomials. The excess molar volume V^E results are analyzed using the Prigogine–Flory–Patterson theory. Analysis of each of the three contributions viz. interactional V^E(int.), free volume V^E(fv.) and characteristic pressure p* to V^E showed that the interactional contributions are positive for all systems while the free volume and characteristic pressure p* contributions are negative for all the binary mixtures. The results are analyzed in terms of attractive forces between 2-methylaniline and carboxylic acids molecules. Good agreement is obtained between excess quantities and spectroscopic data.     

Nested wreath groups and their applications to phylogeny in biology and Cayley trees in chemistry and physics

Nested wreath product groups arise from looped or recursive structures that contain repeated copies of the same structure one within the other. Phylogeny trees in biology, Cayley trees, Bethe lattices, NMR graphs of non-rigid molecules, ammoniated ammonium ions are all examples of structures that exhibit such nested wreath product automorphism groups. We show that the conjugacy classes, irreducible representations and character tables of these nested group structures can be generated using multinomial generating functions cast in terms of matrix types that can be simplified into generalized cycle type polynomials. The nested wreath product groups rapidly increase in orders, for example, a simple wreath product group \(\hbox {S}_{7}[\hbox {S}_{7}]\) consists of \((7!)^{8}\) or \(4.1633\times 10^{23}\) operations, 481,890 conjugacy classes, spanning a 481,891 \(\times \) 481,891 character table that would occupy 232,217,972 pages. We have obtained powerful recursive relations for the conjugacy classes, character tables and the orders of various conjugacy cases of any nested wreath product \(\{[\hbox {S}_{\mathrm{n}}]\}^{\mathrm{m}}\) or \(\hbox {S}_{\mathrm{n}}[\hbox {S}_{\mathrm{n}}[\hbox {S}_{\mathrm{n}}{\ldots }.[\hbox {S}_{\mathrm{n}}]]{\ldots }.]\) with order \(\left( {n!}\right) ^{a_m},\,\hbox {a}_{\mathrm{m}}=(\hbox {n}^{\mathrm{m}}-1)/(\hbox {n}-1)\). We have obtained the character tables of phylogenetic trees of any order, character tables of Cayley trees of degrees 3 and 4 and for Cayley trees of larger degrees, we have derived exact analytical expressions for the conjugacy classes and IRs for up to \(\{[\hbox {S}_{7}]\}^{\mathrm{m}}\) with order \((7!)^{137257}\) for \(\hbox {m}=7\). Applications to colorings of phylogenic trees in biology are considered.     

Deposition Morphology and Magnetism of Co,Pt Adatoms and Small CoPt Adclusters on Ni(100) Substrate

Theoretical calculations of Co\(_{n-x}\)Pt\(_x\) (n = 1–3; \(x \le n\)) clusters on Ni(100) surface for their spin and orbital magnetic moments, as well as the magnetic anisotropy energy (MAE), are performed by using the density-functional theory (DFT) method including a self-consistent treatment of spin–orbit coupling (SOC). The results reveal that the ferromagnetic Co atoms in intra Co\(_{n-x}\)Pt\(_x\) adclusters couple ferromagnetically to their underlayer Ni atoms. The predominant inter-interactions between Co adatoms and Ni surface with the partly filled 3d band, together with the secondary intra-interactions between Co adatoms and Pt adatoms with fully filled 5d band, lead to a strongly quenched orbital moment (\(\mu _{\mathrm{{orb}}}^{\mathrm{{Co}}}\) = 0.18–0.14 \(\mu _B\); \(\mu _{\mathrm{{orb}}}^{\mathrm{{Pt}}} \approx \) 0.24–0.19 \(\mu _B\)) but a less quenched spin moment (\(\mu _{\mathrm{{spin}}}^{\mathrm{{Co}}} \approx \) 2.0 \(\mu _B\); \(\mu _{\mathrm{{spin}}}^{\mathrm{{Pt}}} \approx \) 0.35 \( \mu _B\)). The MAEs of CoPt adclusters exhibit a strong dependence on alloying effect rather than size effect, which is direly proportional to SOC strength and orbital moment anisotropy. The oxidations of CoPt clusters always reduce orbital magnetic moments and consequently decrease the corresponding MAEs.     

Single carbon dioxide molecules on surfaces studied by low-temperature scanning tunneling microscopy

The chemical conversion of carbon dioxide (\(\hbox {CO}_2\)) has been intensively studied because the molecule is responsible for global warming. Rational design of catalysts plays an important role in converting \(\hbox {CO}_2\) into value-added compounds. Understanding the interaction between \(\hbox {CO}_2\) and surfaces of catalysts is a prerequisite to preparing high-performance catalysts. This review focuses on the investigations of \(\hbox {CO}_2\) molecules on single crystalline surfaces studied by low-temperature scanning tunneling microscopy. Molecular adsorption, diffusion, and conversion on metal surfaces, metal oxide surfaces, and surfaces decorated by metal-organic frameworks are summarized.     

Additional Aspects of Complexation of Gold(I) with Thiomalate

Some equilibria involving gold(I) thiomalate (mercaptosuccinate, TM) complexes have been studied in the aqueous solution at 25 °C and I?=?0.2 mol·L^?1 (NaCl). In the acidic region, the oxidation of TM by \( {\text{AuCl}}_{4}^{ - } \) proceeds with the formation of sulfinic acid, and gold(III) is reduced to gold(I). The interaction of gold(I) with TM at n_TM/n_Au?≤?1 leads to the formation of highly stable cyclic polymeric complexes \( {\text{Au}}_{m} \left( {\text{TM}} \right)_{m}^{*} \) with various degrees of protonation depending on pH. In general, the results agree with the tetrameric form of this complex proposed in the literature. At n_TM/n_Au?>?1, the processes of opening the cyclic structure, depolymerization and the formation of \( {\text{Au}}\left( {\text{TM}} \right)_{2}^{*} \) occur: \( {\text{Au}}_{4} ( {\text{TM)}}_{4}^{8 - } + {\text{TM}}^{3 - } \rightleftharpoons {\text{Au}}_{ 4} ( {\text{TM)}}_{5}^{11 - } \), log₁₀ K₄₅?=?10.1?±?0.5; 0.25 \( {\text{Au}}_{4} ( {\text{TM)}}_{4}^{8 - } + {\text{TM}}^{3 - } \rightleftharpoons {\text{Au(TM)}}_{2}^{5 - } \), log₁₀ K₁₂?=?4.9?±?0.2. The standard potential of \( {\text{Au(TM)}}_{2}^{5 - } \) is \( E_{1/0}^{ \circ } = -0. 2 5 5\pm 0.0 30{\text{ V}} \). The numerous protonation processes of complexes at pH?<?7 were described with the use of effective functions.     

Binary self-similar chemical structures

The recursive series \(f_{n+1} =l\cdot f_n +m\cdot f_{n-1} \) (l, m = 1,2,3...) defines the generators of a chain of components L and S, \(\left\{ {\begin{array}{l} L\rightarrow \underbrace{LL{\ldots }L}_l\underbrace{SS{\ldots }S}_m \\ S\rightarrow L \\ \end{array}} \right\} \), such that L and S have a geometric dimension equal to \(d = 0, 1, 2, 3,{\ldots }\). If L, S are the atoms of two different elements A, B (dimension \(d = 0\)) or two self-similar structures with \(d > 0\) that are composed of these elements, then such a generator forms a self-similar binary structure with the dimension \(d+1\) (a quasicrystal), composed of \(\hbox {A}_{x^{\prime }}\hbox {B}\), where \(x^{\prime }\) depends solely on the parameters l and m. In this study, the stoichiometric coefficient \(x^{\prime }\) was calculated for about 20 of such quasicrystals. The generator was found to enable, for certain values of l and m, the formation of structures with degenerate symmetry, that is, the transition from self-similar to translational symmetry. Thus, the generated structures can be divided according to l and m into three groups: structures that contain only one type of homobonds, A–A, with self-similarity as the only permissible symmetry; structures that at least sometimes contain both types of homobonds, A–A and B–B, with self-similarity as the only permissible symmetry; and structures that sometimes show translational symmetry. All of the researched structures in the first group have the same estimated values of internal energy and configuration entropy determined by the isotopic composition. All structures in the second group have the same configuration entropy but different internal energies. In the third group, the configuration entropy of structures that show translational symmetry is lower than that of the self-similar structures. On the other hand, internal energy favours (that is, is lower in) structures with translational symmetry over self-similar structures only when the energy of the A–B bonds is higher than the mean energy of A–A and B–B bonds. In other words, self-similar systems can be energetically more stable compared to crystals as long as the energy of the A–B heterobonds is low. The method for generating self-similar objects proposed in this paper seems to be the first method in which the means of generation do not change with the geometric dimension d of the generated structure.     

Blinded predictions of host-guest standard free energies of binding in the SAMPL5 challenge

In the context of the SAMPL5 blinded challenge standard free energies of binding were predicted for a dataset of 22 small guest molecules and three different host molecules octa-acids (OAH and OAMe) and a cucurbituril (CBC). Three sets of predictions were submitted, each based on different variations of classical molecular dynamics alchemical free energy calculation protocols based on the double annihilation method. The first model (model A) yields a free energy of binding based on computed free energy changes in solvated and host-guest complex phases; the second (model B) adds long range dispersion corrections to the previous result; the third (model C) uses an additional standard state correction term to account for the use of distance restraints during the molecular dynamics simulations. Model C performs the best in terms of mean unsigned error for all guests (MUE \(3.2\,<\,3.4\,<\,3.6\,\text{kcal}\,\text{mol}^{-1}\)—95 % confidence interval) for the whole data set and in particular for the octa-acid systems (MUE \(1.7\,<\,1.9\,<\,2.1\,\text{kcal}\,\text{mol}^{-1}\)). The overall correlation with experimental data for all models is encouraging (\(R^2\, 0.65\,<\,0.70<0.75\)). The correlation between experimental and computational free energy of binding ranks as one of the highest with respect to other entries in the challenge. Nonetheless the large MUE for the best performing model highlights systematic errors, and submissions from other groups fared better with respect to this metric.     

Extending the McClelland formula for total $$\pi $$-electron energy

The McClelland formula, based on the upper bound \(\sqrt{2mn}\), is capable of reproducing over 99.5% of the total \(\pi \)-electron energy (\(E_\pi \)) of a conjugated hydrocarbon, whose molecules possess n carbon atoms and m carbon–carbon bonds. Its weak point is that it predicts equal \(E_\pi \)-values for all isomers. We now show how this failure can be overcome, offering a general strategy for extending McClelland’s formula. By means of one of these extensions, \(E_\pi \) is related with the energy of the highest occupied molecular orbital, and the error of the new formula is diminished by more than \(50\%\) relative to the standard McClelland approximation.     

On the structure of discrete spectrum of a non-selfadjoint system of differential equations with integral boundary condition

In this paper, we investigated the spectrum of the operator \(L(\lambda )\) generated in Hilbert Space of vector-valued functions \(L_{2}(\mathbb {R}_{+},C_{2})\) by the system

$$\begin{aligned} iy_{1}^{'}(x,\lambda )+q_{1}(x)y_{2}(x,\lambda )&=\lambda y_{1}(x,\lambda )\\ -iy_{2}^{'}(x,\lambda )+q_{2}(x)y_{1}(x,\lambda )&=\lambda y_{2}(x,\lambda ),x\in \mathbb {R}_{+}:=(0,\infty ), \end{aligned}$$

and the integral boundary condition of the type

$$\begin{aligned} \int _{0}^{\infty }K(x,t)y(t,\lambda ){\mathrm {dt}}+\alpha y_{2}(0,\lambda )-\beta y_{1}(0,\lambda )=0 \end{aligned}$$

where \(\lambda \) is a complex parameter, \(q_{i},\,i=1,2\) are complex-valued functions and \(\alpha ,\beta \in \mathbb {C}\). K(x, t) is vector fuction such that \(K(x,t)=(K_{1}(x,t),K_{2}(x,t)),\,K_{i}(x,t)\in L_{1}(0,\infty )\cap L_{2}(0,\lambda ),\,i=1,2\). Under the condition

$$\begin{aligned} \left| q_{i}(x)\right| \le ce^{-\varepsilon \sqrt{x}},c>0,\varepsilon >0,i=1,2 \end{aligned}$$

we proved that \(L(\lambda )\) has a finite number of eigenvalues and spectral singularities with finite multiplicities.

     

On the origins of intersecting potential energy surfaces

Techniques developed for investigating nonadiabatic processes in molecular systems are adapted to study the structure and properties of holomorphic and meromorphic functions of a complex variable, \(f(z)=\mathfrak {R}(f)+i\,\mathfrak {I}(f)\). The connection is that \(\mathfrak {R}(f)\) and \(\mathfrak {I}(f)\) are correlated two-dimensional scalar functions, interrelated by the Cauchy–Riemann equations. Exploiting this fact, it is demonstrated that \(\mathfrak {R}(f)\) and \(\mathfrak {I}(f)\) of f can be envisaged in Euclidean \({\mathbb {R}}^{3}\) space as a two-state set of constrained, intersecting two-dimensional potential energy surfaces (PESs), called the graph of f. Importantly, the analytic and algebraic properties of f dictate the geometric structure evinced in the graph of f. This parallels multi-state sets of higher-dimensional, constrained, intersecting PESs linked with correlated electronic eigenstates of the parameterized molecular Hamiltonian operator. In view of this association, the language and mathematical infrastructure devised by chemists for discussing and analyzing intersections in higher-dimensional PESs are suitably modified for f. Notably, an algorithm capable of optimizing roots and poles of f through analysis of the real, two-dimensional \(\mathfrak {R}(f)\) and \(\mathfrak {I}(f)\) functions is derived, which is based on intersection-adapted coordinate and constrained Lagrangian methodologies. As constrained, intersecting PESs are indispensible for conceptualizing and characterizing the physics governing nonadiabatic phenomena, f represents a foundational bridge to these more abstract constructions.     

Thermodynamic Properties of Ternary Ionic Liquid Mixture Containing a Common Ion: Excess Molar Volumes,Excess Isentropic Compressibilities,Excess Molar Enthalpies and Excess Heat Capacities

In the present investigations, the excess molar volumes, \( V_{ijk}^{\text{E}} \), excess isentropic compressibilities, \( \left( {\kappa_{S}^{\text{E}} } \right)_{ijk} \), and excess heat capacities, \( \left( {C_{p}^{\text{E}} } \right)_{ijk} \), for ternary 1-butyl-2,3-dimethylimidazolium tetrafluoroborate (i) + 1-butyl-3-methylimidazolium tetrafluoroborate (j) + 1-ethyl-3-methylimidazolium tetrafluoroborate (k) mixture at (293.15, 298.15, 303.15 and 308.15) K and excess molar enthalpies, \( \left( {H^{\text{E}} } \right)_{ijk} \), of the same mixture at 298.15 K have been determined over entire composition range of x _i and x _j . Satisfactorily corrections for the excess properties \( V_{ijk}^{\text{E}} \), \( \left( {\kappa_{S}^{\text{E}} } \right)_{ijk} \), \( \left( {H^{\text{E}} } \right)_{ijk} \) and \( \left( {C_{p}^{\text{E}} } \right)_{ijk} \) have been obtained by fitting with the Redlich–Kister equation, and ternary adjustable parameters along with standard errors have also been estimated. The \( V_{ijk}^{\text{E}} \), \( \left( {\kappa_{S}^{\text{E}} } \right)_{ijk} \), \( \left( {H^{\text{E}} } \right)_{ijk} \) and \( \left( {C_{p}^{\text{E}} } \right)_{ijk} \) data have been further analyzed in terms of Graph Theory that deals with the topology of the molecules. It has also been observed that Graph Theory describes well \( V_{ijk}^{\text{E}} \), \( \left( {\kappa_{S}^{\text{E}} } \right)_{ijk} \), \( \left( {H^{\text{E}} } \right)_{ijk} \) and \( \left( {C_{p}^{\text{E}} } \right)_{ijk} \) values of the ternary mixture comprised of ionic liquids.     

Thermodynamic analysis of the structure of aqueous solutions of C<Subscript>12</Subscript>–C<Subscript>18</Subscript> hydrocarbons

Thermodynamic cycles including the increments \(\Delta G_{CH_2 }^0 , \Delta H_{CH_2 }^0 \), and \(T\Delta S_{CH_2 }^0 \) were constructed for dissolution, evaporation, hydrophobic hydration of C₅–C₉ hydrocarbons, and transfer from vapor (\(\Delta G_{CH_2 }^0 \) = ?0.7 kJ·mol^?1, \(\Delta H_{CH_2 }^0 \) = 2.9 kJ·mol^?1, \(T\Delta S_{CH_2 }^0 \) = 3.6 kJ·mol^?1) and water (\(\Delta G_{CH_2 }^0 \) = ?1.4 kJ·mol^?1, \(\Delta H_{CH_2 }^0 \) = 5.8 kJ·mol^?1, \(T\Delta S_{CH_2 }^0 \) = 7.2 kJ·mol^?1) to micelles of C₁₂–C₁₈ hydrocarbons. The formation of bistable hydrated micelles of C₁₂–C₁₈ is explained by a transition between the order-disorder states in an assembly of small (nano) systems of water. The extensive parameters of small systems and critical phenomena predicted by fluctuation theory are discussed.     

Volumetric,Acoustic and Transport Properties of Metformin Hydrochloride Drug in Aqueous <Emphasis Type="SmallCaps">d</Emphasis>-Glucose Solutions at <Emphasis Type="Italic">T</Emphasis> = (298.15–318.15) K

Apparent molar volumes, apparent molar adiabatic compressibilities and viscosity B-coefficients for metformin hydrochloride in aqueous d-glucose solutions were determined from solution densities, sound velocities and viscosities measured at T = (298.15–318.15) K and at pressure p = 101 kPa as a function of the metformin hydrochloride concentrations. The standard partial molar volumes (\( \phi_{V}^{0} \)) and slopes (\( S_{V}^{*} \)) obtained from the Masson equation were interpreted in terms of solute–solvent and solute–solute interactions, respectively. Solution viscosities were analyzed using the Jones–Dole equation and the viscosity A and B coefficients discussed in terms of solute–solute and solute–solvent interactions, respectively. Adiabatic compressibility (\( \beta_{s} \)) and apparent molar adiabatic compressibility (\( \phi_{\kappa }^{{}} \)), limiting apparent molar adiabatic compressibility (\( \phi_{\kappa }^{0} \)) and experimental slopes (\( S_{\kappa }^{*} \)) were determined from sound velocity data. The standard volume of transfer (\( \Delta_{t} \phi_{V}^{0} \)), viscosity B-coefficients of transfer (\( \Delta_{t} B \)) and limiting apparent molar adiabatic compressibility of transfer (\( \Delta_{t} \phi_{\kappa }^{0} \)) of metformin hydrochloride from water to aqueous glucose solutions were derived to understand various interactions in the ternary solutions. The activation parameters of viscous flow for the studied solutions were calculated using transition state theory. Hepler’s coefficient \( (d\phi /dT)_{p} \) indicated the structure making ability of metformin hydrochloride in the ternary solutions.     

Sparse and nonnegative sparse D-MORPH regression

An underdetermined linear algebraic equation system \(\mathbf{y}={\varvec{\Phi }}\mathbf{x}\), where \({\varvec{\Phi }}\) is an \(m\times n (m<n)\) rectangular constant matrix with rank \(r\le m\) and \(\mathbf{y}\in \mathrm {Ran}({\varvec{\Phi }})\) (range of \({\varvec{\Phi }})\), has an infinite number of solutions. Diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression seeks a solution satisfying the extra requirement of minimizing a chosen cost function, \({\mathcal {K}}\). A wide variety of choices of the cost function makes it possible to achieve diverse goals, and hence D-MORPH regression has been successfully applied to solve a range of problems. In this paper, D-MORPH regression is extended to determine a sparse or a nonnegative sparse solution of the vector \(\mathbf{x}\). For this purpose, recursive reweighted least-squares (RRLS) minimization is adopted and modified to construct the cost function \({\mathcal {K}}\) for D-MORPH regression. The advantage of sparse and nonnegative sparse D-MORPH regression is that the matrix \({\varvec{\Phi }}\) does not need to have row-full rank, thereby enabling flexibility to search for sparse solutions \(\mathbf{x}\) with ancillary properties in practical applications. These tools are applied to (a) simulation data for quantum-control-mechanism identification utilizing high dimensional model representation (HDMR) modeling and (b) experimental mass spectral data for determining the composition of an unknown mixture of chemical species.