Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then

1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B _n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B _n . Among others, we obtain
2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.

     

Conditions under which the order of two non-negative Hermitian matrices is preserved when taking integral powers

Let A and B be two n×n non-negative matrices. We write A ? B iff

$\text{u}{}^1\text{(A ? B)u ? 0}$

for all column vectors u in Cⁿ. Here u¹ is the conjugate transpose of u. In this paper are stated equivalent conditions under which A^k ? B^k for all natural numbers k. The result is then generalized to Hermitian operators in a Hilbert space.     

Cerium(IV)‐Driven Water Oxidation Catalyzed by a Manganese(V)–Nitrido Complex

The study of manganese complexes as water‐oxidation catalysts (WOCs) is of great interest because they can serve as models for the oxygen‐evolving complex of photosystem II. In most of the reported Mn‐based WOCs, manganese exists in the oxidation states III or IV, and the catalysts generally give low turnovers, especially with one‐electron oxidants such as Ce^IV. Now, a different class of Mn‐based catalysts, namely manganese(V)–nitrido complexes, were explored. The complex [Mn^V(N)(CN)₄]²⁻ turned out to be an active homogeneous WOC using (NH₄)₂[Ce(NO₃)₆] as the terminal oxidant, with a turnover number of higher than 180 and a maximum turnover frequency of 6 min⁻¹. The study suggests that active WOCs may be constructed based on the Mn^V(N) platform.     

Rheological parameters analysis: Different technical approach

Conclusion In clinical states, we did not found any correlation between the techniques using these two viscometers which appear complementary; probably in erythrocyte dysfunction, they allow a better approach of the environment and the influence, in some cases, of plasmatic factors upon erythrocyte deformability. From this point of view, we actually consider that different techniques are necessary to establish in vitro a schedule of rheological behavior.

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Analyse von rheologischen Parametern: verschiedene Techniken

     

Interpretations of the differential cross-sections for pp → π−π+

The differential cross-sections for $\text{p}$p → π^?π⁺ in the momentum range 0.8 to 2.4 GeV/c (centre-of-mass energy 2.02 to 2.57 GeV) are analysed in terms of the possible existence of s-channel resonances. The systematics of the dip structures and the s-dependence of dσ/dt are presented.     

Organocatalytic asymmetric Michael-type reaction between β,γ-unsaturated α-keto ester and α-nitro ketone

A Michael-type reaction of β,γ-unsaturated α-keto ester and α-nitro ketone was established. With a thiourea catalyst derived from cinchona alkaloid, the reactions afford products in 47-94% yields with 68-96% ee.     

Metallo-macrocycles with labile coordination sites through self assembly: binuclear phenyl bridged bisbipyridine manganese complexes

Self assembly of phenyl bridged bisbipyridines with manganese perchlorate gave structurally different metallo-macrocycles having cis-labile coordination sites which can catalyse olefin epoxidation with peracetic acid in good yield.     

Ultracompact and fabrication-tolerant integrated polarization splitter

Design and fabrication of a 2×2 two-mode interference (TMI) coupler based on-chip polarization splitter is presented. By changing the angle between the access waveguides, one can tune the effective TMI length for the mode with less optical confinement (transverse magnetic, TM) to coincide with the target TMI length for a desired transmission of the mode with higher optical confinement (transverse electric, TE). The fabricated 0.94?μm long 2×2 TMI splits the input power into TM (bar) and TE (cross) outputs with splitting ratio over 15?dB over 50?nm bandwidth. Fabrication tolerance analysis shows that the device is tolerant to fabrication errors as large as 60?nm.