A Hamiltonian formulation of the Einstein-Cartan-Sciama-Kibble theory of gravity

We postulate the energy-momentum functionE for the ECSK theory of gravity and formulate the functional Hamiltonian equation in terms of the energy-momentum functionE and the symplectic 2-form . The system of partial differential equations which follows from the functional Hamilton equation is equivalent to the system of variational equations of the ECSK theory. The Hamiltonian method gives rise to a natural division of these equations into 10 constraint equations and the set of dynamical equations. We discuss the geometric sense of the constraint equations and their relations to the initial value problem.     

Gauge group,degeneration, and degrees of freedom in the ECSK theory of gravity. I. The symplectic structure of the ECSK theory

This paper is the first of a two-article series in which the connections between the gauge group, degeneration, and degrees of freedom in the ECSK theory coupled to an arbitrary tensor field are discussed. In this paper a multisymplectic formulation of the ECSK theory is presented and the symplectic 2-form, which plays a leading role in our considerations, is found.     

Witt equivalence of global fields

A Simplicial derivation procedure for crossed module valued functors is established. As a consequence, a theory of higher Baer invariants and relative homology is settle on crossed modules. Then we illustrative how this method is efficient, as it extends classical formulations and results, like various known five term sequences on crossed modules by Conduché and Ellis (1989), the author (1993) and Ladra and R-Grandjeán (1994).     

An Improved Synthesis of Per(6-Deoxyhalo) Cyclodextrins Using N-Halosuccinimides —Triphenylphosphine in Dimethylformamide

Abstract

Per(6-deoxy-6-halo) cyclodextrins (bromo, chloro or iodo) have been prepared in excellent yield and high selectivity by treatment of the native cyclodextrins with the corresponding N-halosuccinimide and triphenylphosphine in N,N-dimethylformamide.     

Asymmetric Catalytic Assembly of Triple-Columned and Multilayered Chiral Folding Polymers Showing Aggregation-Induced Emission (AIE)

Unprecedented chiral multilayer folding 3D polymers have been assembled and regulated by uniform and differentiated aromatic chromophore units between naphthyl piers. Screening catalysts, catalytic systems and monomers were proven to be crucial for asymmetric catalytic Suzuki–Miyaura polycouplings for this assembly. X-ray crystallography of the corresponding dimers and trimers revealed the absolute configuration and the intermolecular packing pattern. Up to 61 960 M_w/41 900 M_n and m/z 4317 for polymers and oligomers, as confirmed by gel permeation chromatography (GPC) and MALDI-TOF MS, indicated that these frameworks were composed of multiple stacked layers. The resulting multiple π-assemblies exhibited remarkable optical properties in aggregated states (photoluminescence in solids and aggregation-induced emission in solutions), as well as reversible redox properties in electrochemical performance.     

Functions generating (m,M, $$\varvec{\Psi }$$ )-Schur-convex sums

Aequationes mathematicae - The notion of $$(m,M,\Psi )$$ -Schur-convexity is introduced and functions generating $$(m,M,\Psi )$$ -Schur-convex sums are investigated. An extension of the...     

Smoothing arithmetic error terms: the case of the Euler φ function

In many cases known methods of detecting oscillations of arithmetic error terms involve certain smoothing pro‐cedures. Usually an application of the smoothing operator does not change significantly the order of magnitude of the error under consideration. This is so for instance in the case of the classical error terms known in the prime number theory. The main purpose of this paper is to show that the situation for primes is not general. Considering the error term in the asymptotic formula for the Euler totient function we show that just one application of an integral smoothing operator changes situation dramatically: the order of magnitude of drops from x to √x (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Around Jensen’s inequality for strongly convex functions

In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen–Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen’s operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that if \(Sp\left( A \right) \subset \left( 1,\infty \right) \), then

$$\begin{aligned} {{\left\langle Ax,x \right\rangle }^{r}}\le \left\langle {{A}^{r}}x,x \right\rangle -\frac{{{r}^{2}}-r}{2}\left( \left\langle {{A}^{2}}x,x \right\rangle -{{\left\langle Ax,x \right\rangle }^{2}} \right) ,\quad r\ge 2 \end{aligned}$$

and if \(Sp\left( A \right) \subset \left( 0,1 \right) \), then

$$\begin{aligned} \left\langle {{A}^{r}}x,x \right\rangle \le {{\left\langle Ax,x \right\rangle }^{r}}+\frac{r-{{r}^{2}}}{2}\left( {{\left\langle Ax,x \right\rangle }^{2}}-\left\langle {{A}^{2}}x,x \right\rangle \right) ,\quad 0<r<1 \end{aligned}$$

for each positive operator A and \(x\in \mathcal {H}\) with \(\left\| x \right\| =1\).

     

Why and How Much Brouwer’s Fixed Point Theorem Fails in Noncompact Setting?

The celebrated Brouwer’s Fixed Point Theorem is dated in 1912. Its extension to compact set setting in Banach spaces due to Schauder appeared in 1930. Immediately it raised the question whether the Theorem can be extended to noncompact setting. The works of Kakutani, Klee, Benyamini and Sterfeld, Sternfeld and Lim solved the qualitative part of the problem. Lack of compactness makes the statement of the theorem false. However, there are some quantitative aspects of the question. The two basic are called minimal displacement problem, and optimal retraction problem. The aim of this article is to present the historical back ground and possibly, up to date state of investigations in this field. A list of open problems with comments will be discussed.     

Calix[4]arene-thiacrown-5 di(carboxylic acid) regioisomers as metal ion extractants

A regioisomer of a previously reported p-tert-butylcalix[4]arene-1,3-thiacrown-5 di(carboxylic acid) is prepared in which the thiacrown ring attachment sites are moved from distal phenolic oxygens of a cone p-tert-butylcalix[4]arene scaffold to proximal phenolic oxygens. The influence of this structural variation in the di-ionizable calixcrown ligand on competitive solvent extractions of alkali metal cations and of alkaline earth metal cations and single species solvent extractions of Hg²⁺ and Pb²⁺ from aqueous solutions into chloroform is evaluated.