Sub differential Monotonicity as Characterization of Convex Functions

We prove that a lower semicontinuous function defined on a Banach space is convex if and only if its subdifferential ismonotone.     

The Flip-Graph of the 4-Dimensional Cube is Connected

Flip-graph connectedness is established here for the vertex set of the 4-dimensional cube. It is found as a consequence that this vertex set has 92,487,256 triangulations, partitioned into 247,451 symmetry classes.     

Gold(I) complexes bearing mixed-donor ligands derived from N-heterocyclic carbenes

The new 2-phenylthiocarbamoyl-1,3-dimesitylimidazolium inner salt (IMes·CSNPh) reacts with [AuCl(L)] in the presence of NH(4)PF(6) to yield [(L)Au(SCNPh·IMes)](+) (L = PMe(3), PPh(3), PCy(3), CNBu(t)). The carbene-containing precursor [(IDip)AuCl] reacts with IMes·CSNPh under the same conditions to afford the complex [(IDip)Au(SCNPh·IMes)](+) (IDip = 1,3-bis(2,6-diisopropylphenyl)imidazol-2-ylidene). Treatment of the diphosphine complex [(dppm)(AuCl)(2)] with one equivalent of IMes·CSNPh yields the digold metallacycle, [(dppm)Au(2)(SCNPh·IMes)](2+), while reaction of [L(2)(AuCl)(2)] with two equivalents of IMes·CSNPh results in [(L(2)){Au(SCNPh·IMes)}(2)](2+) (L(2) = dppb, dppf, or dppa; dppb = 1,4-bis(diphenylphosphino)butane, dppf = 1,1'-bis(diphenylphosphino)ferrocene, dppa = 1,4-bis(diphenylphosphino)acetylene). The homoleptic complex [Au(SCNPh·IMes)(2)](+) is formed on reaction of [AuCl(tht)] (tht = tetrahydrothiophene) with two equivalents of the imidazolium-2-phenylthiocarbamoyl ligand. This product reacts with AgOTf to yield the mixed metal compound [AuAg(SCNPh·IMes)(2)](2+). Over time, the unusual trimetallic complex [Au(AgOTf)(2)(SCNPh·IMes)(2)](+) is formed. The sulfur-oxygen mixed-donor ligands IMes·COS and SIMes·COS (SIMes = 1,3-bis(2,4,6-trimethylphenyl)imidazolin-2-ylidene) were used to prepare [(L)Au(SOC·IMes)](+) and [(L)Au(SOC·SIMes)](+) from [(L)AuCl] (L = PPh(3), CN(t)Bu). The bimetallic examples [(dppf){Au(SOC·IMes)}(2)](2+) and [(dppf){Au(SOC·SIMes)}(2)](2+) were synthesized from the reaction of [(dppf)(AuCl)(2)] with the appropriate ligand. Reaction of [(tht)AuCl] with one equivalent of IMes·COS or SIMes·COS yields [Au(SOC·IMes)(2)](+) and [Au(SOC·SIMes)(2)](+), respectively. The compounds [(Ph(3)P)Au(SCNPh·IMes)]PF(6), [(Cy(3)P)Au(SCNPh·IMes)]PF(6) and [Au(AgOTf)(2)(SCNPh·IMes)(2)]OTf were characterized crystallographically.     

Synthesis of Spin‐Crossover Nano‐ and Micro‐objects in Homogeneous Media

New methods are proposed for the synthesis of spin‐crossover nano‐ and micro‐objects. Several nano‐objects that are based upon the spin‐crossover complex [Fe(hptrz)₃](OTs)₂ (hptrz=4‐heptyl‐1,2,4‐triazole, Ts=para‐toluenesulfonyl) were prepared in homogeneous media. The use of various reagents (Triton X‐100, PVP, TOPO, and PEGs of different molecular weights) as stabilizing agents yielded materials of different size (6 nm–2 μm) and morphology (nanorods, nanoplates, small spherical particles, and nano‐ and micro‐crystals). In particular, when Triton X‐100 was used, a variation in the morphology from nanorods to nanoplates was observed by changing the nature of the solvent. Interestingly, the preparation of the nanorods and nanoplates was always accompanied by the formation of small spherical particles. Alternatively, when PEG was used, 200–400 nm crystals of the complex were obtained. In addition, a very promising polymer‐free synthetic method is discussed that was based on the preparation of relatively stable Fe^II–triazole oligomers in CHCl₃. Their specific treatment led to micro‐crystals, small nanoparticles, or gels. The size and morphology of all of these objects were characterized by TEM and by dynamic light scattering (DLS) where possible. Their spin‐crossover behavior was studied by optical and magnetic measurements. The spin‐transition features for large particles (>100 nm) were very similar to that of the bulk material, that is, close to room temperature with a hysteresis width of up to 8 K. The effects of the matrix and/or size‐reduction led to modification of the transition temperature and an abruptness of the spin transition for oligomeric solutions and small nanoparticles of 6 nm in size.     

Spin state dependence of electrical conductivity of spin crossover materials

We studied the spin state dependence of the electrical conductivity of the spin crossover compound [Fe(Htrz)(2)(trz)](BF(4)) (Htrz = 1H-1,2,4-triazole) by means of dc electrical measurements. The low spin state is characterized by higher conductance and lower thermal activation energy of the conductivity, when compared to the high spin state.     

The thermodynamics of solid solutions of argon and methane

The primary concern of this paper is with the estimation of the excess Gibbs energy G^E,S for solid solutions of two molecularly simple components which are completely miscible in the solid state. The method depends on combining information on the excess thermodynamic functions of liquid mixtures of the two components with a knowledge of the liquidus and solidus lines on the temperature-composition phase diagram. It is applied to the particular case of argon-methane. For this system, unit cell sizes and some heat of fusion measurements are also available, from which V^E,S and H^E,S have been calculated.A solid solution of argon and methane departs much more from ideality than does a liquid mixture of the same composition at the same temperature, the ratio r, = G^E,S/G^E,L, being about 3. Moreover, the concentration dependence of G^E,S is less symmetrical than that of G^E,L, and the ratio r increases markedly with increasing argon mole fraction. A dilute solution of methane (which has the larger molecules) in argon has a larger G^E,S than the corresponding dilute solution of argon in methane.For a solid solution at 71 K with an argon mole fraction of 0.60, H^E,S is ≈4801 Jmol^?1. This gives TS^E,S ≈ 220 J mol^?1, which is about the same as G^E,S. The solid solutions cannot therefore be regarded as even approximating to regular solutions.From the calculated G^E,S results, it is predicted that the face-centred cubic solid solutions of argon and methane should separate into two phases on cooling. The calculated coordinates of the upper critical solution point are T = 67 K and an argon mole fraction of 0.63, in reasonable agreement with the experimental values of 63 K and 0.65 respectively.     

Primal-Lower-Nice Property of Value Functions in Optimization and Control Problems

The paper studies value functions associated with optimization problems and with Mayer-type control problems. Using methods belonging to proximal analysis and control theory, we establish new results for the primal-lower-nice (pln) property of the value functions for these problems.     

Pyrromethene–BF2 complexes as laser dyes:1.

Condensations between 3-X-2,4-dimethylpyrroles (X = H, CH₃, C₂H₅, and CO₂C₂H₅) and acyl chlorides gave derivatives of 3,5,3′,5′-tetramethylpyrromethene (isolated as their hydrochloride salts): 6-methyl, 6-ethyl, 4,4′,6-trimethyl, 4,4′-diethyl-6-methyl, and 4,4′-dicarboethoxy-6-ethyl derivatives for conversion on treatment with boron trifluoride to 1,3,5,7-tetramethylpyrromethene–BF₂ complex (TMP–BF₂) and its 8-methyl (PMP–BF₂), 8-ethyl, 2,6,8-trimethyl (HMP–BF₂),2,6,-diethyl-8-methyl (PMDEP–BF₂), and 2,6-dicarboethoxy-8-ethyl derivatives. Chlorosulfonation converted, 1,3,5,7,8-pentamethylpyrromethene–BF₂ complex to its 2,6-disulfonic acid isolated as the lithium, sodium (PMPDS–BF₂), potassium, rubidium, cesium, ammonium, and tetramethylammonium disulfonate salts and the methyl disulfonate ester. Sodium 1,3,5,7-tetramethyl-8-ethylpyrromethene-2,6-disulfonate–BF₂ complex was obtained from the 8-ethyl derivative of TMP–BF₂. Nitration and bromination converted PMP–BF₂ to its 2,6-dinitro-(PMDNP–BF₂) and 2,6-dibromo- derivatives. The time required for loss of fluorescence by irradiation from a sunlamp showed the following order for P–BF₂ compounds (10⁻³ to 10⁻⁴ M) in ethanol: PMPDS–BF₂, 7 weeks; PMP–BF₂, 5 days; PMDNP–BF₂, 72 h; HMP–BF₂, 70 h; and PMDEP–BF₂, 65 h. Under similar irradiation PMPDS–BF₂ in water lost fluorescence after 55 h. The dibromo derivative was inactive, but each of the other pyrromethene–BF₂ complexes under flashlamp excitation showed broadband laser activity in the region λ 530–580 nm. In methanol PMPDS–BF₂ was six times more resistant to degradation by flashlamp pulses than was observed for Rhodamine-6G (R-6G). An improvement (up to 66%) in the laser power efficiency of PMPDS–BF₂ (10⁻⁴ M in methanol) in the presence of caffeine (a filter for light <300 nm) was dependent on flashlamp pulse width (2.0 to 7.0 μsec).