Differentiable circle maps with a flat interval

We study weakly order preserving circle maps with a flat interval, which are differentiable even on the boundary of the flat interval. We obtain estimates on the Lebesgue measure and the Hausdorff dimension of the non-wandering set. Also, a sharp transition is found from degenerate geometry to bounded geometry, depending on the degree of the singularities at the boundary of the flat interval.Partially supported by KBN grant Iteracje i Fraktale #210909101.     

Cramér theorem on symmetric spaces of noncompact type

We prove the Cramér theorem forK-invariant Gaussian measures on irreducible symmetric spacesX=G/K withG semisimple noncompact. To do this we use a kind of Abel transform ofK-invariant measures onX.This research is supported by KBN Grant.     

Critical circle maps near bifurcation

We estimate harmonic scalings in the parameter space of a one-parameter family of critical circle maps. These estimates lead to the conclusion that the Hausdorff dimension of the complement of the frequency-locking set is less than 1 but not less than 1/3. Moreover, the rotation number is a Hölder continuous function of the parameter.Partially supported by KBN grant Iteracje i Fraktale #210909101.Partially supported by NSF Grant #DMS-9206793 and the Sloan Research Fellowship.     

Inapplicability of the Antiperiplanar Lone Pair Hypothesis to C-P Bond Breaking and Formation in Some S-C-P(+) Systems

Both C(2)-P bond breaking and formation in the S-C-P(+) system do not occur according to the antiperiplanar lone pair hypothesis. Experiments using 2-phosphonio derivatives of 5-tert-butyl-1,3-dithiane and cis-4,6-dimethyl-1,3-dithiane are against the participation of higher-energy boat conformers as reactive intermediates. The results obtained support a possibility of conformational adjustment in the course of the reaction. Stereoelectronic control of the C(2)-P bond breaking and formation results from interplay of several factors. The role of the n(S)-sigma(C(2))(-)(P) and sigma(C(4,6))(-)(S)-sigma(C(2))(-)(P) hyperconjugation, as well as of the repulsive interactions between lone electron pairs pi(S) of endocyclic sulfur atoms and pi-electrons of the phenyl ring(s) connected with phosphorus, is discussed.     

Evolution of synthetic routes towards homochiral Tapentadol

Analysis of the literature shows that the development of synthetic approaches to Tapentadol, an agonist of the μ-opioid receptor and norepinephrine reuptake inhibitor, has followed four, partially overlapping, phases. The most advanced approaches, based on stereoselective syntheses, appeared only after 2010. The majority of the chemistry examples presented in this review come from patent applications and as such have not been subjected to rigorous peer review, but may serve well as an inspiration to organic chemists for solving analogous synthetic problems.     

Reactions of solute species at an electrode modified with titanocene functionalized polypyrrole film: ferrocene and titanocene dichloride

We have studied for the first time the ability of a conducting polymer film, p(Tc3Py), representing a polypyrrole matrix with covalently attached titanocene dichloride (TcCl₂) centers, to serve as an intermediator for the electron charge transport between the electrode and the reaction sites of solute reactants. The standard potential of the first of these electroactive species, ferrocene (Fc), is in the range where the polymer matrix is in its slightly oxidized state so that solute Fc species give a reversible response at the surface of this modified electrode. Another solute reactant, TcCl₂, was studied in solutions in which it demonstrates a (quasi)reversible behavior at bare electrode surfaces, THF+TBAPF₆ and AN+TEACl. The standard redox potential of this species belongs to the range of the electroactivity of immobilized TcCl₂ centers (where the matrix is in its non-conducting state) so that the electron charge has to be transported via stepwise redox reactions between neighboring centers inside the film. The combination, solute reactant+film, results in a greater CV current compared to the response of the film in background solution or of the solute species at the bare electrode surface. This current for THF solution even exceeds the sum of separate currents for the film and the reactant. This finding is attributed to a catalytic effect of solute species as redox intermediators for the transformation of immobilized electroactive centers leading to a greater degree of the film reduction. The presence of solute TcCl₂ species results in a much greater stability of immobilized centers (compared to the corresponding reactant-free solution), both in the course of CV with the passage of the range of their response and in experiments with the film holding at the potential within this range. This holding leads to an almost constant current related to the reaction of solute species at the film/solution interface. Our estimate shows that immobilized centers undergo above 10,000 reversible transformations (without an observed tendency to the degradation) to ensure the passage of this current. The conclusion has been drawn that immobilized TcCl₂ centers are able to serve as sufficiently stable redox intermediators for the electron charge transport across the film, a prerequisite for the catalytic applications of such films.Abbreviations AN acetonitrile - THF tetrahydrofuran - Cp cyclopentadienyl, C₅H₅ - Cp cyclopentadienyl radical, C₅H₄ - Fc ferrocene, Cp₂Fe - TcCl₂ titanocene=bis(cyclopentadienyl)titanium dichloride, Cp₂TiCl₂ or its radical CpCpTiCl₂ - PPy polypyrrole - Tc3Py titanocene-propyl-pyrrole, Cl₂TiCpCp(CH₂)₃NC₄H₄ - p(Tc3Py) polymer obtained from Tc3Py - TBAPF₆ tetrabuthylammonium hexafluorophosphate - TEACl tetraethylammonium chlorideDedicated to Zbigniew Galus on the occasion of his 70th birthday.     

Iron‐Catalyzed C(sp2)H and C(sp3)H Methylations of Amides and Anilides

The so‐called magic methyl effect significantly boosts the bioactivities and physical properties of pharmacologically active drugs. Direct introduction of the methyl group by C?H activation was accomplished with a versatile iron catalyst, which enabled the C?H methylation of (hetero)benzamides, anilides, alkenes, and even alkanes by triazole assistance in a chemo‐, site‐ and diastereo‐selective fashion.     

Wishart exponential families on cones related to tridiagonal matrices

Annals of the Institute of Statistical Mathematics - Let G be the graph corresponding to the graphical model of nearest neighbor interaction in a Gaussian character. We study Natural Exponential...     

Space and time inversions of stochastic processes and Kelvin transform

Let X be a standard Markov process. We prove that a space inversion property of X implies the existence of a Kelvin transform of X‐harmonic, excessive and operator‐harmonic functions and that the inversion property is inherited by Doob h‐transforms. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. We treat in details the examples of free scaled power Bessel processes, non‐colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.     

On Hausdorff Dimension of Unimodal Attractors

There exists a universal constant σ<1 such that every attractor of every C⁴ unimodal map with a non-degenerate critical point is an analytic manifold or its Hausdorff dimension is equal to or less than σ.