A generalization of carathéodory''s theorem

The following theorem is proved. If the sets and a εⁿ⁺¹_i=1 conv V_i, then there exist elements v_i ε V_i (i=1…,n+1) such that a ε conv{v₁,…,v_n+1}. This is generalization of Carathéodory's theorem. By applying this and similar results some open questions are answered.     

Identification of novel chiral aluminium containing oxonium cations in the enantioselective hydrogenation of ethyl pyruvate catalyzed using cinchonidine modified Pt-alumina in acetic acid

Summary The first identification of aluminium containing oxonium cations having chiral atomic groups is described which may play a role in enantioselective hydrogenation of ethyl pyruvate using Pt-alumina catalyst in AcOH. The structure these cations at m/z = 509 and m/z = 567 is: [O{Al(OAc)₂}₂Al(OPy.OLt)]⁺ and [O(AlOAc.OLt)₂Al(OPy)₂]⁺     

Modeling rate-controlling solvent effects. The pericyclic meisenheimer rearrangement of N-propargylmorpholine N-oxide

The activation parameters of the pericyclic Meisenheimer rearrangement and a competitive rearrangement of N-propargylmorpholine N-oxide were determined by experimental and computational methods. A number of aprotic and protic solvents of different polarities and hydrogen bond-forming abilities and the roles of electron-pair acceptor additives were investigated. The reaction kinetics were followed by means of NMR. In protic solvents, isotope-labeling experiments revealed a novel inverse secondary kinetic isotope effect (k(H)/k(D) about 0.8) for the rate-determining cyclization step, probably occurring because of a C(sp) --> C(sp(2)) change in hybridization at the reaction center. In molecular computations at the B3LYP/6-31++G(d,p) level of theory, implicit, explicit, and joint explicit-implicit solvent models were used. The explicit-implicit model and molecular dynamic simulations gave the most accurate results. The components of the rate-controlling solvent effect are discussed, and general equations are proposed for accurate prediction of the solvent-dependent activation parameters.     

Sum-free sets in abelian groups

An IP system is a functionn taking finite subsets ofN to a commutative, additive group Ω satisfyingn(α∪β)=n(α)+n(β) whenever α∩β=ø. In an extension of their Szemerédi theorem for finitely many commuting measure preserving transformations, Furstenberg and Katznelson showed that ifS _i ,1≤i≤k, are IP systems into a commutative (possibly infinitely generated) group Ω of measure preserving transformations of a probability space (X, B, μ, andA∈B with μ(A)>0, then for some ø≠α one has μ(? _i=1 ^k S _i({α})A>0). We extend this to so-called FVIP systems, which are polynomial analogs of IP systems, thereby generalizing as well joint work by the author and V. Bergelson concerning special FVIP systems of the formS(α)=T(p(n(α))), wherep:Z ^t →Z ^d is a polynomial vanishing at zero,T is a measure preservingZ ^d action andn is an IP system intoZ ^t . The primary novelty here is potential infinite generation of the underlying group action, however there are new applications inZ ^d as well, for example multiple recurrence along a wide class ofgeneralized polynomials (very roughly, functions built out of regular polynomials by iterated use of the greatest integer function).     

A note on the size of the largest ball inside a convex polytope

Summary Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>m>1$ be an integer, $B_m$ the set of all unit vectors of $\Bbb R^m$ pointing in the direction of a nonzero integer vector of the cube $[-1,\,1]^m$. Denote by $s_m$ the radius of the largest ball contained in the convex hull of $B_m$. We determine the exact value of $s_m$ and obtain the asymptotic equality $s_m\sim\frac{2}{\sqrt{\log m}}$.     

The Randomized Integer Convex Hull

Let K R^d be a sufficiently round convex body (the ratio of the circumscribed ball to the inscribed ball is bounded by a constant) of a sufficiently large volume. We investigate the randomized integer convex hull I_L(K) = conv (K L), where L is a randomly translated and rotated copy of the integer lattice Z^d. We estimate the expected number of vertices of I_L(K), whose behaviour is similar to the expected number of vertices of the convex hull of Vol K random points in K. In the planar case we also describe the expectation of the missed area Vol (K \ I_L(K)). Surprisingly, for K a polygon, the behaviour in this case is different from the convex hull of random points.     

Kreislagerungen auf Flächen konstanter Krümmung

Ohne Zusammenfassung Vorgelegt von L. Fejes Tóth     

Characterizations of locally testable events

Let Σ be a finite alphabet, Σ^* the free monoid generated by Σ and χ the length of χ ∈ Σ^*. For any integer k0, f_k(χ) (t_k(χ)) is χ if χ < k + 1, and it is the prefix (suffix) of χ of length k, othewise. Also let m_k+1(χ) = {νχ = uνw and ν = k+1}. For χ, y ε Σ^* define χ ～ _k+1y iff f_k(χ) = f_k(y), t_k(χ) = t_k(y) and m_k+1(χ) = m_k+1(y). The relation ～_k+1 is a congruence of finite index over Σ^*. An event E ? Σ^* is (k+1)-testable iff it is a union of congruence classes of ～_k+1. E is locally testable (LT) if it is k+1-testable for some k. (This definition differs from that of [6] but is equivalent.)We show that the family of LT events is a proper sub-family of star-free events of dot-depth 1. LT events and k-testable events are characterized in terms of (a) restricted star-free expressions based on finite and cofinite events; (b) finite automata accepting these events; (c) semigroups; and (d) structural decomposition of such automata. Algorithms are given for deciding whether a regular event is (a) LT and (b) k+1-testable. Generalized definite events are also characterized.     

Nachweis und volumetrische Bestimmung von Nitrosoderivaten im Mikromaßstab

Zusammenfassung Nitrosoverbindungen lassen sich in Gegenwart von Rhodanidionen durch Eisen(II)-sulfat quantitativ reduzieren. Auf dieser Reaktion beruht ein neuer Nachweis der Nitrosogruppe auf Grund der Farbe des Eisen(III)-rhodanidkomplexes, sowie ihre Bestimmung durch Titration des Eisen(III) mit Quecksilber(I)-nitrat-Maßlösung. Diese Methode ermöglicht die Analyse auch in Gegenwart von Nitroverbindungen.

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Detection and volumetric microdetermination of nitroso compounds

Summary Nitroso compounds can be quantitatively reduced by iron(II) sulphate in the presence of thiocyanate ions. The reaction provides a new method for the detection of the nitroso group, based on the colour developed by the iron(III) thiocyanate complex. The determination of nitroso group can also be carried out by titrating the iron(III) with mercury(I) nitrate, even if nitro compounds are present.