Matrix Lax Polynomials, Geometry of Prym Varieties and Systems of Hess–Appel’rot Type

We consider a four-dimensional generalization of Hess–Appel’rot system and costruct its Lax pair. Both classical and algebro-geometric integration procedure are proceeded. The algebro-geometric integration is based on deep facts from geometry of Prym varieties such as the Mumford relation and Mumford-Dalalyan theory. The integration is similar to the integration of Lagrange bitop which has recetly been performed by the authors.     

Gas-phase structure of protonated histidine and histidine methyl ester: combined experimental mass spectrometry and theoretical ab initio study

Gas-phase H/D exchange experiments with CD3OD and D2O and quantum chemical ab initio G3(MP2) calculations were carried out on protonated histidine and protonated histidine methyl ester in order to elucidate their bonding and structure. The H/D exchange experiments show that both ions have three equivalent fast hydrogens and one appreciably slower exchangeable hydrogen assigned to the protonated amino group participating in a strong intramolecular hydrogen bond (IHB) with the nearest N(sp2) nitrogen of the imidazole fragment and to the distal ring NH-group, respectively. It is taken for granted that the proton exchange in the IHB is much faster than the H/D exchange. Unlike in other protonated amino acids (glycine, proline, phenylalanine, tyrosine, and tryptophan) studied earlier, the exchange rate of the carboxyl group in protonated histidine is slower than that of the amino group. The most stable conformers and the enthalpies of neutral and protonated histidine and its methyl ester are calculated at the G3(MP2) level of theory. It is shown that strong intramolecular hydrogen bonding between the amino group and the imidazole ring nitrogen sites is responsible for the stability and specific properties of the protonated histidine. It is found that the proton fluctuates between the amino and imidazole groups in the protonated form across an almost vanishing barrier. Proton affinity (PA) of histidine calculated by the G3(MP2) method is 233.2 and 232.4 kcal mol(-1) for protonation at the imidazole ring and at the amino group nitrogens, respectively, which is about 3-5 kcal mol(-1) lower than the reported experimental value.     

1,8-bis(dimethylethyleneguanidino)naphthalene: tailoring the basicity of bisguanidine "proton sponges" by experiment and theory

1,8-bis(dimethylethyleneguanidino)naphthalene (DMEGN), the second example of a peralkyl guanidine "proton sponge" based on the 1,8-naphthalene backbone, was prepared and fully characterized. The crystal structure analysis of monoprotonated DMEGN reveals an unsymmetrical intramolecular hydrogen bridge. A decrease in the basicity with respect to the noncyclic parent 1,8-bis(tetramethylguanidino)naphthalene was found. Nevertheless, a new proton sponge provides a new crossbar in the ladder of highly basic neutral organic compounds. A detailed theoretical study of DMEGN and related cyclic guanidines explains this surprising experimental result. Homodesmotic reactions reveal that the intramolecular hydrogen bond contributes effectively 10 kcal/mol to proton affinity of DMEGN.     

Approximation of Functions by the Szász-Mirakjan-Kantorovich Operator

We characterize the approximation of functions in the L_p-norm by the Szász-Mirakjan-Kantorovich operator. We prove a direct and a strong converse inequality of type B in terms of an appropriate K-functional.     

Constrained‐Geometry Bisphosphazides Derived from 1,8‐Diazidonaphthalene: Synthesis,Spectroscopic Characteristics,Structural Features,and Theoretical Investigations

Investigations on the Staudinger reaction between 1,8‐diazidonaphthalene and phosphorous(III) building blocks, a key step in the synthesis of superbasic bisphosphazene proton sponges, yielded a set of bisphosphazides with a constrained geometry 1,8‐disubstituted naphthalene backbone. This compound class has attracted our interest not only due to their surprisingly high stability, but in particular because of their theoretically predicted basicity in the range of their bisphosphazene analogues that can be referred to the constrained geometry interaction of two highly basic nitrogen atoms. Eleven new bisphosphazides bearing simple P‐amino groups as well as P‐guanidino substituents, azaphosphatrane moieties, P₂ building blocks, or chiral P‐amino substituents derived from L ‐proline are presented. They were studied concerning their spectroscopic properties and partly also their chromophoric and structural features. In the case of the pyrrolidino‐substituted TPPN(2N₂) (TPPN=1,8‐bis(trispyrrolidinophosphazenyl)naphthalene), the stepwise nitrogen elimination is investigated theoretically and experimentally, which led to the isolation and structural characterization of TPPN(1N₂) bearing a phosphazide and a phosphazene functionality in one molecule. Attempts to protonate the obtained bisphosphazides and to prove the computationally predicted pK_BH⁺ values through NMR titration reactions resulted in their decay, which again was rationalized by theoretical calculations. Altogether we present the so far most extensive spectroscopic, structural and theoretical investigation of constrained geometry bisphosphazides and their Brønsted and Lewis basic properties.     

Mono-Phosphazenyl Phosphines (R2N)3P=N–P(NR2)2 – Strong P-Bases,P-Donors,and P-Nucleophiles for the Construction of Chelates

We present a convenient three-step synthesis of amino substituted phosphazenyl phosphines of the general formula (R₂N)₃P=N–P(NR₂)₂ [NR₂ = N(CH₂)₄, N(CH₂)₅, N(CH₂)₆]. These easily accessible mixed valent compounds display a surprisingly high proton affinity and basicity in the same range as the corresponding Schwesinger diphosphazene (Me₂N)₃P=N–P=NEt(NMe₂)₂ (Et-P₂) and Verkade's proazaphosphatrane superbases. Within the central [P^III–N=P^V] scaffold, the phosphine P^III and not the phosphazene N^III atom is the center of highest proton affinity, basicity and donor strength. As P-bases, the title compounds display calculated proton affinities between 265.8 (NR₂ = NMe₂) and 274.7 kcal · mol^–1 [NR₂ = N(CH₂)₄] and pK_BH⁺ values between 26.4 (NR₂ = NMe₂) and 31.5 [NR₂ = N(CH₂)₄] on the acetonitrile scale. As P-nucleophiles, they are key intermediates in the synthesis of hyperbasic bis(diphosphazene) proton sponges, chiral bis(diphosphazene) proton pincers, bisphosphazides, and superbasic P₂-bisylides. Their Staudinger reactions as nucleophile towards 1,8-diazidonaphthalene leading to 1,8-naphthalene-bisphosphazides is described in detail. The donor strength of the title compounds towards fragments [Se] and [Ni(CO)₃] is in the same range as that of N-heterocyclic carbenes.     

An extension of the Landau-Kolmogorov inequality. Solution of a problem of Erdös

For any fixed finite interval [a, b] on the real line, an arbitrary natural numberr and σ>0, we describe the extremal function to the problem $\left\| {f^{(k)} } \right\|L_p \left[ {a,b} \right]^{ \to \sup } \left( {1 \leqslant k \leqslant r - 1, 1 \leqslant p< \infty } \right)$ over all functionsf ∈W _∞ ^r such that |f ^(r)(x)| ≤σ, |f(x)|≤1 on (?∞, ∞). Similarly, we solve the problem, raised by Paul Erdös, of characterizing the trigonometric polynomial of fixed uniform norm whose graph has maximal arc length over [a, b].