Silica-supported copper-iron catalysts for hydrogenolysis reactions characterized by Mössbauer spectroscopy

A special preparation procedure has been developed to provide a very uniform distribution of iron throughout the copper precursor supported on silica. The best performance for hydrogenolysis of methyl acetate has been obtained for a catalyst prepared in a one-step procedure, in which iron(III) and copper(II) were injected together into a suspension of silica at a constant pH level. It is found that the main part of the iron has been incorporated into the formed copper hydrosilicate. After reduction and under reaction conditions, only about 6% of the iron atoms are in the Fe⁰ state. This Fe⁰ is expected to be responsible for the enhancement of the catalyst activity in hydrogenation and hydrogenolysis reactions.     

Fast and exact flow measurements with the fast Fourier flow technique

The fast Fourier flow technique offers a convenient way for extremely exact flow measurements. In combination with ECG gating, arterial flow can be studied. The total acquisition time for a single measurement is about 10 s, an ECG-gated cine sequence can be acquired in about 4 min. Fast Fourier flow can, therefore, be easily combined with a conventional imaging examination.     

Error function inequalities

We present various inequalities for the error function. One of our theorems states: Let α?≥?1. For all x,y?>?0 we have $$ \delta_{\alpha} < \frac{ \mbox{erf} \left( x+ \mbox{erf}(y)^{\alpha}\right) +\mbox{erf}\left( y+ \mbox{erf}(x)^{\alpha}\right) } {\mbox{erf}\left( \mbox{erf}(x)+\mbox{erf}(y)\right) } < \Delta_{\alpha} $$ with the best possible bounds $$ \delta_{\alpha}= \left\{ \begin{array}{ll} 1+\sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha=1$,}\\ \sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha>1$,}\\ \end{array}\right. \quad{\mbox{and} \,\,\,\,\, \Delta_{\alpha}=1+\frac{1}{\mbox{erf}(1)}.} $$      

Descrizioni geometriche sintetiche di geometrie proiettive con caratteristicap >0

Sunto è ben noto che, per gli spazi proiettivi basati sopra un corpo commutativo, vale il teorema di Pappo e che, viceversa, per mezzo di questo teorema tali spazi possono essere coordinatizzati con un corpo commutativo K. Il teorema di Fano ci dice quando K non può essere di caratteristica p=2. Un altro teorema è: In piani proiettivi sopra corpi K con p=2, e soltanto in questi, si ha che: Le tangenti d'una conica, costruita nel modo sintetico consueto, passano per un punto fisso. Teoremi del genere per p ≠2 sembrano mancare finora, ma sono necessari per una costruzione pure sintetica della geometria algebrica. Il lavoro seguente conterrà siffatti teoremi. Entrata in Redazione il 14 giugno 1976.     

Über die Stabilität der stationären Punkte eines Verbandes mit unstetigen Ableitungen der potentiellen Energie

Zusammenfassung Ein autonomer und konservativer Verband mit einem Freiheitsgrad wird betrachtet. Die potentielle Energie des Verbandes wird mit Hilfe der Heaviside Funktion ausgedrückt und ihre Ableitungen besitzen Unstetigkeitsstellen. Mit der potentiellen EnergieU wird eine DistributionU gebildet und die ersten zwei Ableitungen dieser Distribution bestimmt. In den DistributionenU undU erscheinen Funktionen und Werte welche mitV, K undR bezeichnet werden und welche es erlauben die Natur der Unstetigkeitsstellen zu bestimmen. In einer Ebene mit den Achsen OV und OR werden die Bereiche der stabilen und instabilen Gleichgewichtslagen dargestellt. Dir kritischen Fälle an den Grenzen der Bereiche werden mit Hilfe vonK untersucht. Abschließend wird ein Beispiel gebracht.

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Summary An undamped autonomous system with one degree of freedom is considered. The potential energy of the system is expressed with the help of the Heaviside function and its derivatives possess points of discontinuity. With the potential energyU a distributionU is determined and the first and second differentiation of the distributionU are calculated. In the distributionsU andU appear functions and values denoted byV, K, R which permit to establish the nature of the points of discontinuity. In a plane with the axes OV and OR the regions of stable and instable equilibrium positions are represented. Critical cases on the boundaries between the different regions are analysed with the help ofK. An example is also considered.

     

Global optimization of concave functions subject to quadratic constraints: An application in nonlinear bilevel programming

When the follower's optimality conditions are both necessary and sufficient, the nonlinear bilevel program can be solved as a global optimization problem. The complementary slackness condition is usually the complicating constraint in such problems. We show how this constraint can be replaced by an equivalent system of convex and separable quadratic constraints. In this paper, we propose different methods for finding the global minimum of a concave function subject to quadratic separable constraints. The first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex, the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints, we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.Parts of the present paper were prepared while the second author was visiting Georgia Tech and the University of Florida.     

Phase transition to hyperon matter in neutron stars

Recent progress in the understanding of the high density phase of neutron stars advances the view that a substantial fraction of the matter consists of hyperons. The possible impacts of a highly attractive interaction between hyperons on the properties of compact stars are investigated. We find that a hadronic equation of state with hyperons allows for a first order phase transition to hyperonic matter. The corresponding hyperon stars can have rather small radii of R approximately equal 8 km.     

On the Continuity of Pathwise Solutions to Langevin Equations in Infinite Dimensions

We fix a rich probability space (,F,P). Let (H,) be a separable Hilbert space and let be the canonical cylindrical Gaussian measure on H. Given any abstract Wiener space (H,B,) over H, and for every Hilbert–Schmidt operator T: HBH which is (|{}|,)-continuous, where |{}| stands for the (Gross-measurable) norm on B, we construct an Ornstein–Uhlenbeck process : (,F,P)×[0,1](B,|{}|) as a pathwise solution of the following infinite-dimensional Langevin equation d _t =db _t +T( _t )dt with the initial data ₀=0, where b is a B-valued Brownian motion based on the abstract Wiener space (H,B,). The richness of the probability space (,F,P) then implies the following consequences: the probability space is independent of the abstract Wiener space (H,B,) (in the sense that (,F,P) does not depend on the choice of the Gross-measurable norm |{}|) and the space C _B consisting of all continuous B-valued functions on [0,1] is identical with the set of all paths of . Finally, we present a way to obtain pathwise continuous solutions :d _t =

Inequalities for the harmonic numbers

We present various inequalities for the harmonic numbers defined by ${H_n=1+1/2 +\ldots +1/n\,(n\in{\bf N})}$ . One of our results states that we have for all integers n ???2: $$\alpha \, \frac{\log(\log{n}+\gamma)}{n^2} \leq H_n^{1/n} -H_{n+1}^{1/(n+1)} < \beta \, \frac{\log(\log{n}+\gamma)}{n^2}$$ with the best possible constant factors $$\alpha= \frac{6 \sqrt{6}-2 \sqrt[3]{396}}{3 \log(\log{2}+\gamma)}=0.0140\ldots \quad\mbox{and} \quad\beta=1.$$ Here, ?? denotes Euler??s constant.     

Some open categorical problems inTop

In this paper, a pendant to a recent survey paper, the authors discuss several open problems in categorical topology. The emphasis is on topology-oriented problems rather than on more general category-oriented ones. In fact, most problems deal with full subconstructs or superconstructs of the constructTop of topological spaces and continuous maps.