New Coins from Old, Smoothly

Given a (known) function f:[0,1]→(0,1), we consider the problem of simulating a coin with probability of heads f(p) by tossing a coin with unknown heads probability p, as well as a fair coin, N times each, where N may be random. The work of Keane and O’Brien (ACM Trans. Model. Comput. Simul. 4(2):213–219, 1994) implies that such a simulation scheme with the probability ℙ _p (N<∞) equal to 1 exists if and only if f is continuous. Nacu and Peres (Ann. Appl. Probab. 15(1A):93–115, 2005) proved that f is real analytic in an open set S⊂(0,1) if and only if such a simulation scheme exists with the probability ℙ _p (N>n) decaying exponentially in n for every p∈S. We prove that for α>0 noninteger, f is in the space C ^α [0,1] if and only if a simulation scheme as above exists with ℙ _p (N>n)≤C(Δ _n (p)) ^α , where \varDelta _n(x):=max{?{x(1-x)/n},1/n}\varDelta _{n}(x):=\max\{\sqrt{x(1-x)/n},1/n\}. The key to the proof is a new result in approximation theory: Let B⁺_n\mathcal{B}^{+}_{n} be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree n. We show that a function f:[0,1]→(0,1) is in C ^α [0,1] if and only if f has a series representation ?_n=1^￥F_n\sum_{n=1}^{\infty}F_{n} with F_n ? B⁺_nF_{n}\in \mathcal{B}^{+}_{n} and ∑ _k>n F _k (x)≤C(Δ _n (x)) ^α for all x∈[0,1] and n≥1. We also provide a counterexample to a theorem stated without proof by Lorentz (Math. Ann. 151:239–251, 1963), who claimed that if some j_n ? B⁺_n\varphi_{n}\in\mathcal{B}^{+}_{n} satisfy |f(x)−φ _n (x)|≤C(Δ _n (x)) ^α for all x∈[0,1] and n≥1, then f∈C ^α [0,1].     

Time-resolved emission of flavin adenine dinucleotide in water and water-methanol mixtures

Time-resolved fluorescence decay of flavin adenine dinucleotide (FAD) was studied at room temperature in water and water-methanol mixtures by a fluorescence upconversion technique. The observations were focused on the most initial decay phase (200 ps), before the residual fluorescence assumes a single exponential decay, typical for an extended conformation of the fluorophore. Within the first few picoseconds, where most of the electron transfer coupled quenching takes place, the emission decay curves could be fitted by a stretched exponent, compatible with the inhomogeneous distance dependent electron transfer model. This implies that the population of the excited FAD molecules exhibits a large number of non-identical states, each with its own separation between the donor (adenine) and acceptor (isoalloxazine) moieties, having its own rate of electron transfer. To evaluate the distribution of the separation between the donor-acceptor pair, we carried out molecular dynamics simulations of closed conformation of the FAD in water and water-methanol mixtures, sampling the structure at 10 fs intervals. The analysis of the dynamics reveals that within the 4 ps time frame, where most of the nonexponential fluorescence relaxation takes place, the relative motion of the donor-acceptor pair is consistent with a one-dimensional Brownian motion, where the diffusion coefficient and the shape of the confining potential well are solvent dependent. The presence of methanol enhances the diffusion constant and widens the width of the potential well. On the basis of these parameters, the relaxation dynamics was accurately reconstructed as an electron transfer reaction in an inhomogeneous system where the reactants are diffusing within the time frame of the observation.     

Modeling the nonradiative decay rate of electronically excited thioflavin T

A computational model of nonradiative decay is developed and applied to explain the time-dependent emission spectrum of thioflavin T (ThT). The computational model is based on a previous model developed by Glasbeek and co-workers (van der Meer, M. J.; Zhang, H.; Glasbeek, M. J. Chem. Phys. 2000, 112, 2878) for auramine O, a molecule that, like ThT, exhibits a high nonradiative rate. The nonradiative rates of both auramine O and ThT are inversely proportional to the solvent viscosity. The Glasbeek model assumes that the excited state consists of an adiabatic potential surface constructed by adiabatic coupling of emissive and dark states. For ThT, the twist angle between the benzothiazole and the aniline is responsible for the extensive mixing of the two excited states. At a twist angle of 90°, the S(1) state assumes a charge-transfer-state character with very small oscillator strength, which causes the emission intensity to be very small as well. In the ground state, the twist angle of ThT is rather small. The photoexcitation leads first to a strongly emissive state (small twist angle). As time progresses, the twist angle increases and the oscillator strength decreases. The fit of the experimental results by the model calculations is good for times longer than 3 ps. When a two-coordinate model is invoked or a solvation spectral-shift component is added, the fit to the experimental results is good at all times.     

Excited-state proton transfer in N-methyl-6-hydroxyquinolinium salts: solvent and temperature effects

The excited-state proton transfer (ESPT) reaction of the "super"photoacid N-methyl-6-hydroxyquinolinium (MHQ) was studied using both fluorescence upconversion and time-correlated single photon counting (TCSPC) techniques. The ultrafast ESPT kinetics were investigated in various alcohols and water and determined to be solvent-controlled. The ESPT temperature dependence of MHQ was also studied in various alcohols and compared to that observed for another "super"photoacid, 5,8-dicyano-2-naphthol (DCN2). A full set of kinetic and thermodynamic parameters describing the ESPT was obtained. The protolytic photodissociation rate constant for MHQ was higher than that for DCN2, while the ESPT activation energies of MHQ were smaller. These findings are attributed to the approximately 3 orders of magnitude differences in excited-state acidities of MHQ and DCN2.     

MRI rotating frame relaxation measurements for articular cartilage assessment

In the present work we introduced two MRI rotating frame relaxation methods, namely adiabatic T_1ρ and Relaxation Along a Fictitious Field (RAFF), along with an inversion-prepared Magnetization Transfer (MT) protocol for assessment of articular cartilage. Given the inherent sensitivity of rotating frame relaxation methods to slow molecular motions that are relevant in cartilage, we hypothesized that adiabatic T_1ρ and RAFF would have higher sensitivity to articular cartilage degradation as compared to laboratory frame T₂ and MT. To test this hypothesis, a proteoglycan depletion model was used. Relaxation time measurements were performed at 0 and 48 h in 10 bovine patellar specimens, 5 of which were treated with trypsin and 5 untreated controls were stored under identical conditions in isotonic saline for 48 h. Relaxation times measured at 48 h were longer than those measured at 0 h in both groups. The changes in T₂ and MT relaxation times after 48 h were approximately 3 times larger in the trypsin treated specimens as compared to the untreated group, whereas increases of adiabatic T_1ρ and RAFF were 4 to 5 fold larger. Overall, these findings demonstrate a higher sensitivity of adiabatic T_1ρ and RAFF to the trypsin-induced changes in bovine patellar cartilage as compared to the commonly used T₂ and MT. Since adiabatic T_1ρ and RAFF are advantageous for human applications as compared to standard continuous-wave T_1ρ methods, adiabatic T_1ρ and RAFF are promising tools for assessing cartilage degradation in clinical settings.     

Mixing of the upper triangular matrix walk

We study a natural random walk over the upper triangular matrices, with entries in the field ${\mathbb{Z}_2}$ , generated by steps which add row i + 1 to row i. We show that the mixing time of the lazy random walk is O(n ²) which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields ${\mathbb{Z}_q}$ for q prime.     

Markov type and threshold embeddings

For two metric spaces X and Y, say that X threshold-embeds into Y if there exist a number K > 0 and a family of Lipschitz maps ${\{\varphi_{\tau} : X \to Y : \tau > 0\}}$ such that for every ${x,y \in X}$ , $$d_X(x, y) \geq \tau \implies d_Y(\varphi_\tau (x),\varphi_\tau (y)) \geq \|{\varphi}_\tau\|_{\rm Lip}\tau/K,$$ where ${\|{\varphi}_{\tau}\|_{\rm Lip}}$ denotes the Lipschitz constant of ${\varphi_{\tau}}$ . We show that if a metric space X threshold-embeds into a Hilbert space, then X has Markov type 2. As a consequence, planar graph metrics and doubling metrics have Markov type 2, answering questions of Naor, Peres, Schramm, and Sheffield. More generally, if a metric space X threshold-embeds into a p-uniformly smooth Banach space, then X has Markov type p. Our results suggest some non-linear analogs of Kwapien’s theorem. For instance, a subset ${X \subseteq L_1}$ threshold-embeds into Hilbert space if and only if X has Markov type 2.     

Superconducting and ferromagnetic phases in SrTiO3/LaAlO3 oxide interface structures: possibility of finite momentum pairing

We introduce a model to explain the observed ferromagnetism and superconductivity in LAO/STO oxide interface structures. Because of the polar catastrophe mechanism, 1/2 charge per unit cell is transferred to the interface layer. We argue that this charge localizes and orders ferromagnetically via exchange with the conduction electrons. Ordinarily, this ferromagnetism would destroy superconductivity, but, due to strong spin-orbit coupling near the interface, the magnetism and superconductivity can coexist by forming a Fulde-Ferrell-Larkin-Ovchinikov-type condensate of Cooper pairs at finite momentum, which is surprisingly robust in the presence of strong disorder.