Enantioselective catalytic lithiation using a chiral binaphthyl derivative as electron carrier

The lithiation, of the secondary chloride 2, catalyzed by binaphthyl derivatives, i.e. BINAM 4, BINOL 5, BINAP 6, H8-BINAP 7, Tol-BINAP 8, 2,2’-bis(pyrrolidin-1-yl)-1,1’-binaphthalene 9, and 2,2’-dimethyl-1,1’-binaphthalene 11, in the presence of different ketones has been studied, yielding the corresponding alcohol derivatives 3 and 12-16 in moderate to good yields. Binaphthyl derivative 11 has revealed to be very active as catalyst in the lithiation process at room temperature, and has allowed the preparation of the alcohol derivatives with enantioselectivities up to 50%.

Local probe of fractional edge states of S=1 Heisenberg spin chains

Spin chains are among the simplest physical systems in which electron-electron interactions induce novel states of matter. Here we propose to combine atomic scale engineering and spectroscopic capabilities of state of the art scanning tunnel microscopy to probe the fractionalized edge states of individual atomic scale S=1 spin chains. These edge states arise from the topological order of the ground state in the Haldane phase. We also show that the Haldane gap and the spin-spin correlation length can be measured with the same technique.

On some solutions of a functional equation related to the partial sums of the Riemann zeta function

In this paper, we prove that infinite-dimensional vector spaces of α-dense curves are generated by means of the functional equations f(x)+f(2x)+⋯+f(nx)=0, with n≥2, which are related to the partial sums of the Riemann zeta function. These curves α-densify a large class of compact sets of the plane for arbitrary small α, extending the known result that this holds for the cases n=2,3. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the nth power of the density approaches the Jordan content of the compact set which the curve densifies.

Developing pre-service teachers’ noticing of students’ understanding of the derivative concept

This research study examines the development of the ability of pre-service teachers to notice signs of students’ understanding of the derivative concept. It analyses preservice teachers’ interpretations of written solutions to problems involving the derivative concept before and after participating in a teacher training module. The results indicate that the development of this skill is linked to pre-service teachers’ progressive understanding of the mathematical elements that students use to solve problems. We have used these results to make some suggestions for teacher training programmes.

The Zeros of Riemann Zeta Partial Sums Yield Solutions to f(x) + f(2x) + · · · + f(nx) = 0

This paper proves that every zero of any n th , n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation f(x)+f(2x)+⋯+f(nx)=0,x∈R . The continuity of the solutions depends on the sign of the real part of each zero.

A note on the real projection of the zeros of partial sums of Riemann zeta function

This paper proves that the real projection of each simple zero of any partial sum of the Riemann zeta function ζn(s):=∑nk=11ks,n>2 , is an accumulation point of the set {Res : ζ n (s) = 0}.

Computing the zeros of the partial sums of the Riemann zeta function

In this paper, we introduce a formula for the exact number of zeros of every partial sum of the Riemann zeta function inside infinitely many rectangles of the critical strips where they are situated.

Students’ understanding of the function-derivative relationship when learning economic concepts

The aim of this study is to characterise students’ understanding of the function-derivative relationship when learning economic concepts. To this end, we use a fuzzy metric (Chang 1968) to identify the development of economic concept understanding that is defined by the function-derivative relationship. The results indicate that the understanding of these economic concepts is linked to students’ capacity to perform conversions and treatments between the algebraic and graphic registers of the function-derivative relationship when extracting the economic meaning of concavity/convexity in graphs of functions using the second derivative.

Lezioni di Geometria Algebrica : Geometria sopra una curva; superficie di Riemann, Integrali abeliani /

Errata slip inserted.

Riemann's method of integration: its extensions with an application.

Mode of access: Internet.

Multi-derivative numerical methods for the solution of stiff ordinary differential equations /

"Under contracts US AEC AT(11-1)2383 and US AEC AT(11-1)1469."

Development of double isotope derivative dilutions analyses for selected food addititives /

"Contract No. AT(30-1)-2715."

A glossary of the Cleveland dialect: explanatory, derivative, and critical.

Mode of access: Internet.