On the near periodicity of eigenvalues of Toeplitz matrices

Let $A$ be an infinite Toeplitz matrix with a real symbol $f$ defined on $[-\pi, \pi]$. It is well known that the sequence of spectra of finite truncations $A_N$ of $A$ converges to the convex hull of the range of $f$. Recently, Levitin and Shargorodsky, on the basis of some numerical experiments, conjectured, for symbols $f$ with two discontinuities located at rational multiples of $\pi$, that the eigenvalues of $A_N$ located in the gap of $f$ asymptotically exhibit periodicity in $N$, and suggested a formula for the period as a function of the position of discontinuities. In this paper, we quantify and prove the analog of this conjecture for the matrix $A^2$ in a particular case when $f$ is a piecewise constant function taking values $-1$ and $1$.

Dynamics of shear-induced orientation transitions in block copolymers

This review discusses liquid crystal phase formation by biopolymers in solution. Lyotropic mesophases have been observed for several classes of biopolymer including DNA, peptides, polymer/peptide conjugates, glycopolymers and proteoglycans. Nematic or chiral nematic (cholesteric) phases are the most commonly observed mesophases, in which the rod-like fibrils have only orientational order. Hexagonal columnar phases are observed for several systems (DNA, PBLG, polymer/peptide hybrids) at higher concentration. Lamellar (smectic) phases are reported less often, although there are examples such as the layer arrangement of amylopectin side chains in starch. Possible explanations for the observed structures are discussed. The biological role of liquid crystal phases for several of these systems is outlined. Commonly, they may serve as a template to align fibrils for defined structural roles when the biopolymer is extruded and dried, for instance in the production of silk by spiders or silkworms, or of chitin in arthropod shells. In other cases, liquid crystal phase formation may occur in vivo simply as a consequence of high concentration, for instance the high packing density of DNA within cell nuclei.

Developments in block copolymer science and technology

Focuses on recent advances in research on block copolymers, covering chemistry (synthesis), physics (phase behaviors, rheology, modeling), and applications (melts and solutions). Written by a team of internationally respected scientists from industry and academia, this text compiles and reviews the expanse of research that has taken place over the last five years into one accessible resource. Ian Hamley is the world-leading scientist in the field of block copolymer research Presents the recent advances in the area, covering chemistry, physics and applications. Provides a broad coverage from synthesis to fundamental physics through to applications Examines the potential of block copolymers in nanotechnology as self-assembling soft materials

Nonspherical assemblies generated from polystyrene-b-poly(L-lysine) polyelectrolyte block copolymers

This report describes the aqueous solution self-assembly of a series of polystyrene(m)-b-poly(L-lysine)n block copolymers (m = 8-10; n = 10-70). The polymers are prepared by ring-opening polymerization of epsilon-benzyloxycarbonyl-L-lysine N-carboxyanhydride using amine terminated polystyrene macroinitiators, followed by removal of the benzyloxycarbonyl side chain protecting groups. The critical micelle concentration of the block copolymers determined using the pyrene probe technique shows a parabolic dependence on peptide block length exhibiting a maximum at n = approximately 20 (m = 8) or n = approximately 60 (m = 10). The shape and size of the aggregates has been studied by dynamic and static light scattering, small-angle neutron scattering (SANS), and analytical ultracentrifugation (AUC). Surprisingly, Holtzer and Kratky analysis of the static light scattering results indicates the presence of nonspherical, presumably cylindrical objects independent of the poly(L-lysine)n block length. This is supported by SANS data, which can be fitted well by assuming cylindrical scattering objects. AUC analysis allows the molecular weight of the aggregates to be estimated as several million g/mol, corresponding to aggregation numbers of several 10s to 100s. These aggregation numbers agree with those that can be estimated from the length and diameter of the cylinders obtained from the scattering results.

Morphologies of block copolymer melts

Morphology formation by block copolymers in the melt is reviewed, considering both theoretical and experimental aspects. Comprehensive tables provide information on morphology identification for many block copolymer systems. A particular focus is on recent structural studies on ABC triblocks and rod–coil copolymers.

Highly asymmetric phase diagram of a poly(1,2-octylene oxide)-poly(ethylene oxide) diblock copolymer system comprising a brush-like poly(1,2-octylene oxide) block

The phase diagram of a series of poly(1,2-octylene oxide)-poly(ethylene oxide) (POO-PEO) diblock copolymers is determined by small-angle X-ray scattering. The Flory-Huggins interaction parameter was measured by small-angle neutron scattering. The phase diagram is highly asymmetric due to large conformational asymmetry that results from the hexyl side chains in the POO block. Non-lamellar phases (hexagonal and gyroid) are observed near f(PEO) = 0.5, and the lamellar phase is observed for f(PEO) >= 0.5.

Molecular t-matrices for low-energy electron diffraction (TMOL v1.1)*1

We describe a FORTRAN-90 program that computes scattering t-matrices for a molecule. These can be used in a Low-Energy Electron Diffraction program to solve the molecular structural problem very efficiently. The intramolecular multiple scattering is computed within a Dyson-like approach, using free space Green propagators in a basis of spherical waves. The advantage of this approach is related to exploiting the chemical identity of the molecule, and to the simplicity to translate and rotate these t-matrices without performing a new multiple-scattering calculation for each configuration. FORTRAN-90 routines for rotating the resulting t-matrices using Wigner matrices are also provided.

A novel heteroditopic terpyridine-pincer ligand as building block for mono- and heterometallic Pd(II) and Ru(II) complexes

A palladium-catalyzed Stille coupling reaction was employed as a versatile method for the synthesis of a novel terpyridine-pincer (3, TPBr) bridging ligand, 4'-{4-BrC6H2(CH2NMe2)(2)-3,5}-2,2':6',2 ''-terpyridine. Mononuclear species [PdX(TP)] (X = Br, Cl), [Ru(TPBr)(tpy)](PF6)(2), and [Ru(TPBr)(2)](PF6)(2), synthesized by selective metalation of the NCNBr-pincer moiety or complexation of the terpyridine of the bifunctional ligand TPBr, were used as building blocks for the preparation of heterodi- and trimetallic complexes [Ru(TPPdCl)(tpy)](PF6)(2) (7) and [Ru(TPPdCl)(2)]-(PF6)(2) (8). The molecular structures in the solid state of [PdBr(TP)] (4a) and [Ru(TPBr)(2)](PF6)(2) (6) have been determined by single-crystal X-ray analysis. Electrochemical behavior and photophysical properties of the mono-and heterometallic complexes are described. All the above di- and trimetallic Ru complexes exhibit absorption bands attributable to (MLCT)-M-1 (Ru -> tpy) transitions. For the heteroleptic complexes, the transitions involving the unsubstituted tpy ligand are at a lower energy than the tpy moiety of the TPBr ligand. The absorption bands observed in the electronic spectra for TPBr and [PdCl(TP)] have been assigned with the aid of TD-DFT calculations. All complexes display weak emission both at room temperature and in a butyronitrile glass at 77 K. The considerable red shift of the emission maxima relative to the signal of the reference compound [Ru(tpy)(2)](2+) indicates stabilization of the luminescent (MLCT)-M-3 state. For the mono- and heterometallic complexes, electrochemical and spectroscopic studies (electronic absorption and emission spectra and luminescence lifetimes recorded at room temperature and 77 K in nitrile solvents), together with the information gained from IR spectroelectrochemical studies of the dimetallic complex [Ru(TPPdSCN)(tpy)](PF6)(2), are indicative of charge redistribution through the bridging ligand TPBr. The results are in line with a weak coupling between the {Ru(tpy)(2)} chromophoric unit and the (non)metalated NCN-pincer moiety.

Using the Cayley-Hamilton theorem with N-partitioned matrices

A definition is given for the characteristic equation of anN-partitioned matrix. It is then proved that this matrix satisfies its own characteristic equation. This can then be regarded as a version of the Cayley-Hamilton theorem, of use withN-dimensional systems.

Double-gyroid morphology of a polystyrene-block-poly(ferrocenylethylmethylsilane) diblock copolymer: a route to ordered bicontinuous nanoscale architectures

A polystyrene-block-poly(ferrocenylethylmethylsilane) diblock copolymer, displaying a double-gyroid morphology when self-assembled in the solid state, has been prepared with a PFEMS volume fraction phi(PFMS)=0.39 and a total molecular weight of 64 000 Da by sequential living anionic polymerisation. A block copolymer with a metal-containing block with iron and silicon in the main chain was selected due to its plasma etch resistance compared to the organic block. Self-assembly of the diblock copolymer in the bulk showed a stable, double-gyroid morphology as characterised by TEM. SAXS confirmed that the structure belonged to the Ia3d space group.

Signal detection for orthogonal space-time block coding over time-selective fading channels: the H(i) systems

One major assumption in all orthogonal space-time block coding (O-STBC) schemes is that the channel remains static over the length of the code word. However, time-selective fading channels do exist, and in such case conventional O-STBC detectors can suffer from a large error floor in the high signal-to-noise ratio (SNR) cases. As a sequel to the authors' previous papers on this subject, this paper aims to eliminate the error floor of the H(i)-coded O-STBC system (i = 3 and 4) by employing the techniques of: 1) zero forcing (ZF) and 2) parallel interference cancellation (PIC). It is. shown that for an H(i)-coded system the PIC is a much better choice than the ZF in terms of both performance and computational complexity. Compared with the, conventional H(i) detector, the PIC detector incurs a moderately higher computational complexity, but this can well be justified by the enormous improvement.

Orthogonal space–time block codes for free-space IM/DD optical links

Use of orthogonal space-time block codes (STBCs) with multiple transmitters and receivers can improve signal quality. However, in optical intensity modulated signals, output of the transmitter is non-negative and hence standard orthogonal STBC schemes need to be modified. A generalised framework for applying orthogonal STBCs for free-space IM/DD optical links is presented.

Robust non-orthogonal space-time block codes over highly correlated channels: a generalisation

Little has so far been reported on the robustness of non-orthogonal space-time block codes (NO-STBCs) over highly correlated channels (HCC). Some of the existing NO-STBCs are indeed weak in robustness against HCC. With a view to overcoming such a limitation, a generalisation of the existing robust NO-STBCs based on a 'matrix Alamouti (MA)' structure is presented.

Signal detection for orthogonal space-time block coding over time-selective fading channels: a PIC approach for the Gi Systems

One major assumption in all orthogonal space-time block coding (O-STBC) schemes is that the channel remains static over the entire length of the codeword. However, time selective fading channels do exist, and in such case the conventional O-STBC detectors can suffer from a large error floor in the high signal-to-noise ratio (SNR) cases. This paper addresses such an issue by introducing a parallel interference cancellation (PIC) based detector for the Gi coded systems (i=3 and 4).