Peirce gradings and Peirce inner ideals in JBW*-triple factors

A Peirce inner ideal J in an anisotropic Jordan*-triple A gives rise to a Peirce grading (J ₀, J ₁, J ₂) of A by defining

Unifying quantum state transfer and state amplification

We present a Hamiltonian that can be used for amplifying the signal from a quantum state, enabling the measurement of a macroscopic observable to determine the state of a single spin. We prove a general mapping between this Hamiltonian and an exchange Hamiltonian for arbitrary coupling strengths and local magnetic fields. This facilitates the use of existing schemes for perfect state transfer to give perfect amplification. We further prove a link between the evolution of this fixed Hamiltonian and classical cellular automata, thereby unifying previous approaches to this amplification task. Finally, we show how to use the new Hamiltonian for perfect state transfer in the scenario where total spin is not conserved during the evolution, and demonstrate that this yields a significantly different response in the presence of decoherence.     

Specific detection of DNA through coupling of a TaqMan assay with surface enhanced Raman scattering (SERS)

We have combined the benefits of a TaqMan assay with surface enhanced Raman scattering (SERS), to generate a novel DNA detection method which provides increased sensitivity, with clear applications for disease identification through clinical testing. Target DNA detection limits by SERS were shown to be lower than conventional fluorescence detection and clinically relevant samples of methicillin-resistant Staphylococcus aureus were detected with high specificity.     

Sub-micro-second time-resolved absorption spectroscopy of a polar carotenoid analogue, 2-(all-trans-retinylidene)indan-1,3-dione; formation of the dication by direct triplet-excited sensitization

Sub-micro-second time-resolved difference absorption spectra of a polar carotenoid analogue, 2-(all-trans-retinylidene)indan-1,3-dione (hereafter, we will call RetInd), were recorded in tetrahydrofuran at room temperature upon anthracene-sensitized triplet excitation. In addition to the typical Tn <-- T1 absorption spectrum of anthracene followed by that of RetInd, a novel transient species, which peaked at 670 nm, was detected. The lifetime and the population of the 670 nm species was not affected by the presence of oxygen but was quenched by the cation scavenger, triethylamine. Therefore, we have identified this species as a "cation". The transient 670 nm species was not generated by direct photoexcitation of RetInd in the absence of a triplet sensitizer. Therefore, this species was not generated via the T1 species of RetInd but rather via an "invisible state" of RetInd, which is generated by direct energy or electron transfer from T1 anthracene. This proposed pathway was confirmed by a singular-value decomposition followed by a global fitting analysis. The "cation" of RetInd shows vibrational structure in its absorption spectrum, and its lifetime was determined to be 15 micros. Chemical oxidation of RetInd in 2,2,2-trifluoroethanol (dichloromethane) produced a broad absorption band around 880 (1013) nm, which could be transformed into a shoulder around 640 (675) nm upon addition of increasing amounts of the oxidant, FeCl3. The former absorption band can be assigned to a radical cation, while the latter to a dication. Because of the spectral similarity, the 670 nm species can be assigned to the dication, and the "invisible state" is ascribed to the radical cation of RetInd. This is the first direct evidence for the production of a dication of a biological polyene moiety generated in non-halogenated solution following anthracene-sensitized excitation.     

On finite simple images of triangle groups

For a simple algebraic group G in characteristic p, a triple (a, b, c) of positive integers is said to be rigid for G if the dimensions of the subvarieties of G of elements of order dividing a, b, c sum to 2 dim G. In this paper we complete the proof of a conjecture of the third author, that for a rigid triple (a, b, c) for G with p > 0, the triangle group T_a,b,c has only finitely many simple images of the form G(p^r). We also obtain further results on the more general form of the conjecture, where the images G(p^r) can be arbitrary quasisimple groups of type G.     

Hopf bifurcation near a double singular point

The nonlinear equation f(x,λ,) = 0, f:X × R²→X, where X is a Banach space and f satisfies a Z₂-symmetry relation is considered. Interest centres on a certain type of double singular point, where the solution x is symmetric and f_x has a double zero eigenvalue, with one eigenvector symmetric and one antisymmetric.

We show that under certain nondegeneracy conditions, which are stated both algebraically and geometrically, there exists a path of Hopf bifurcations or imaginary Hopf bifurcations passing through the double singular point, and for which x is not symmetric except at the double singular point. An easy geometrical test is found to decide which type of phenomenon occurs. A biproduct of the analysis is that explicit expressions are obtained for quantities which help to provide a reliable numerical method to compute these paths. A pseudo-spectral method was used to obtain numerical results for the Brusselator equations to illustrate the theory.