On the peripheral spectrum of positive elements

Let A be an ordered Banach algebra with a unit \(\mathbf{e}\) and a cone \(A^+\). An element p of A is said to be an order idempotent if \(p^2 = p\) and \(0 \le p\le \mathbf{e}\). An element \(a\in A^+\) is said to be irreducible if the relation \((\mathbf{e}-p)ap = 0\), where p is an order idempotent, implies \(p = 0\) or \(p = \mathbf{e}\). For an arbitrary element a of A the peripheral spectrum \(\sigma _\mathrm{per}(a)\) of a is the set \(\sigma _\mathrm{per}(a) = \{\lambda \in \sigma (a):|\lambda | = r(a)\}\), where \(\sigma (a)\) is the spectrum of a and r(a) is the spectral radius of a. We investigate properties of the peripheral spectrum of an irreducible element a. Conditions under which \(\sigma _\mathrm{per}(a)\) contains or coincides with \(r(a)H_m\), where \(H_m\) is the group of all \(m^\mathrm{th}\) roots of unity, and the spectrum \(\sigma (a)\) is invariant under rotation by the angle \(\frac{2\pi }{m}\) for some \(m\in {\mathbb N}\), are given. The correlation between these results and the existence of a cyclic form of a is considered. The conditions under which a is primitive, i.e., \(\sigma _\mathrm{per}(a) = \{r(a)\}\), are studied. The necessary assumptions on the algebra A which imply the validity of these results, are discussed. In particular, the Lotz–Schaefer axiom is introduced and finite-rank elements of A are defined. Other approaches to the notions of irreducibility and primitivity are discussed. Conditions under which the inequalities \(0 \le b < a\) imply \(r(b) < r(a)\) are studied. The closedness of the center \(A_\mathbf{e}\), i.e., of the order ideal generated by \(\mathbf{e}\) in A, is proved.     

Self-Assembly of Low-Molecular-Weight Asymmetric Linear Triblock Terpolymers: How Low Can We Go?

The synthesis of two (2) novel triblock terpolymers of the ABC type and one (1) of the BAC type, where A, B and C are chemically different segments, such as polystyrene (PS), poly(butadiene) (PB_1,4) and poly(dimethylsiloxane) (PDMS), is reported; moreover, their corresponding molecular and bulk characterizations were performed. Very low dimensions are evident from the characterization in bulk from transmission electron microscopy studies, verified by small-angle X-ray data, since sub-16 nm domains are evident in all three cases. The self-assembly results justify the assumptions that the high Flory–Huggins parameter, χ, even in low molecular weights, leads to significantly well-ordered structures, despite the complexity of the systems studied. Furthermore, it is the first time that a structure/properties relationship was studied for such systems in bulk, potentially leading to prominent applications in nanotechnology and nanopatterning, for as low as sub-10 nm thin-film manipulations.     

Photochemical Strain‐Release‐Driven Cyclobutylation of C(sp3)‐Centered Radicals

A new photoredox‐catalyzed decarboxylative radical addition approach to functionalized cyclobutanes is described. The reaction involves an unprecedented formal Giese‐type addition of C(sp³)‐centered radicals to highly strained bicyclo[1.1.0]butanes. The mild photoredox conditions, which make use of a readily available and bench stable phenyl sulfonyl bicyclo[1.1.0]butane, proved to be amenable to a diverse range of α‐amino and α‐oxy carboxylic acids, providing a concise route to 1,3‐disubstituted cyclobutanes. Furthermore, kinetic studies and DFT calculations unveiled mechanistic details on bicyclo[1.1.0]butane reactivity relative to the corresponding olefin system.     

Theory of the Overhauser effect in the pulsed mode of EPR pumping: exploiting the advantages of coherent electron spin motion

A theoretical approach is proposed to describe Overhauser-type Dynamic Nuclear Polarization (DNP) for pulsed EPR pumping by application of a train of short pulses with a duration on the nanosecond time scale. We obtained an elegant general expression for the NMR enhancement provided by the DNP effect. The expression for the enhancement is similar to that known for cw-pumping except for the saturation factor, which is re-defined as the deviation of the electron spin magnetization from its equilibrium value averaged over the cycle of the pulse sequence. It is shown that one can achieve the maximal theoretically allowed NMR enhancement for pulsed pumping even when the duty cycle of pumping is low. This becomes possible because coherent motion of the electron spins in the B(1)-field is exploited, a key feature of the pulsed DNP experiment also enabling optimization of the achievable NMR enhancement. The dependence of the effect on the duty cycle, pulse duration and electron spin relaxation times has been studied in detail. Once the lines in the EPR spectrum are inhomogeneously broadened, higher DNP effects are expected in the pulsed pumping mode than in the cw-mode for the same total power of microwave irradiation. The theoretical results are in good agreement with experimental data obtained for the pumping frequencies of 300 MHz and 1.4 GHz.     

New approach to unsymmetrical 1,3-diazatriphenylenes through intramolecular oxidative cyclodehydrogenation

An efficient synthetic route towards previously inaccessible dibenzo[f,h]quinazolines and [1]benzothieno[3,2-f]benzo[h]quinazolines through FeCl₃-mediated intramolecular oxidative cyclodehydrogenation of readily available 5-([1,1′-biphenyl]-2-yl)pyrimidines and 5-(2-phenylbenzo[b]thiophen-3-yl)pyrimidines is described. Molecular orbital calculations (DFT), as well as redox and photophysical measurements for all new compounds have been performed. The data show that the reported polycyclic systems have a potential to use in organic electronic applications.     

Undetectable quantum transfer through a continuum

We demonstrate that a quantum particle, initially prepared in a quantum well, can propagate through a reservoir with a continuous spectrum and reappear in a distant well without being registered in the reservoir. It is shown that such a passage through the reservoir takes place even if the latter is continuously monitored. We discuss a possible experimental realization of such a teleportation phenomenon in mesoscopic systems.