Singularity of orbital measures in SU(n)

We show that the minimalk such that μ^κ ∈L ¹(SU(n)) for all central, continuous measures μ on SU(n) isk=n. We do this by exhibiting an elementg∈SU(n) for which the (n−1)-fold product of its conjugacy class has zero Haar measure. This ensures that if μ _g is the corresponding orbital measure supported on the conjugacy class, then μ _g ^n-1 is singular toL ¹. This research is supported in part by NSERC. The hospitality of the University of Waterloo is gratefully acknowledged.     

Simultaneous structural and elemental nano-imaging of human brain tissue

Examining chemical and structural characteristics of micro-features in complex tissue matrices is essential for understanding biological systems. Advances in multimodal chemical and structural imaging using synchrotron radiation have overcome many issues in correlative imaging, enabling the characterization of distinct microfeatures at nanoscale resolution in ex vivo tissues. We present a nanoscale imaging method that pairs X-ray ptychography and X-ray fluorescence microscopy (XFM) to simultaneously examine structural features and quantify elemental content of microfeatures in complex ex vivo tissues. We examined the neuropathological microfeatures Lewy bodies, aggregations of superoxide dismutase 1 (SOD1) and neuromelanin in human post-mortem Parkinson''s disease tissue. Although biometals play essential roles in normal neuronal biochemistry, their dyshomeostasis is implicated in Parkinson''s disease aetiology. Here we show that Lewy bodies and SOD1 aggregates have distinct elemental fingerprints yet are similar in structure, whilst neuromelanin exhibits different elemental composition and a distinct, disordered structure. The unique approach we describe is applicable to the structural and chemical characterization of a wide range of complex biological tissues at previously unprecedented levels of detail.

Structural and chemical characterisation of microfeatures in unadulterated Parkinson''s disease brain tissue using synchrotron nanoscale XFM and ptychography.     

Time-resolved infrared spectroscopy of the lowest triplet state of thymine and thymidine

Vibrational spectra of the lowest energy triplet states of thymine and its 2′-deoxyribonucleoside, thymidine, are reported for the first time. Time-resolved infrared (TRIR) difference spectra were recorded over seven decades of time from 300 fs to 3 μs using femtosecond and nanosecond pump-probe techniques. The carbonyl stretch bands in the triplet state are seen at 1603 and 1700 cm⁻¹ in room-temperature acetonitrile-d₃ solution. These bands and additional ones observed between 1300 and 1450 cm⁻¹ are quenched by dissolved oxygen on a nanosecond time scale. Density-functional calculations accurately predict the difference spectrum between triplet and singlet IR absorption cross sections, confirming the peak assignments and elucidating the nature of the vibrational modes. In the triplet state, the C4O carbonyl exhibits substantial single-bond character, explaining the large (70 cm⁻¹) red shift in this vibration, relative to the singlet ground state. Femtosecond TRIR measurements unambiguously demonstrate that the triplet state is fully formed within the first 10 ps after excitation, ruling out a relaxed ¹nπ^* state as the triplet precursor.     

A new type of Littlewood-Paley partition

We define a partition of Z into intervals {I _j} and prove the Littlewood-Paley inequality ‖f‖ _p ≦C _p‖Sf‖ _p , 2≦p<∞. Heref is a function on [o, 2π) and . This is a new example of a partition having the Littlewood-Paley property since the {I _j} are not of the type obtained by iterating lacunary partitions finitely many times.     

A new type of Littlewood-Paley partition

We define a partition of Z into intervals {I _j} and prove the Littlewood-Paley inequality ‖f‖ _p ≦C _p‖Sf‖ _p , 2≦p<∞. Heref is a function on [o, 2π) and \(Sf = (\sum |\Delta _j |^2 )^{1/2} , \hat \Delta j = \hat f\chi _{Ij} \) . This is a new example of a partition having the Littlewood-Paley property since the {I _j} are not of the type obtained by iterating lacunary partitions finitely many times.