And\^o dilations for a pair of commuting contractions: two explicit constructions and functional models

One of the most important results in operator theory is And\^o's \cite{ando} generalization of dilation theory for a single contraction to a pair of commuting contractions acting on a Hilbert space. While there are two explicit constructions (Sch\"affer \cite{sfr} and Douglas \cite{Doug-Dilation}) of the minimal isometric dilation of a single contraction, there was no such explicit construction of an And\^o dilation for a commuting pair $(T_1,T_2)$ of contractions, except in some special cases \cite{A-M-Dist-Var, D-S, D-S-S}. In this paper, we give two new proofs of And\^o's dilation theorem by giving both Sch\"affer-type and Douglas-type explicit constructions of an And\^o dilation with function-theoretic interpretation, for the general case. The results, in particular, give a complete description of all possible factorizations of a given contraction $T$ into the product of two commuting contractions. Unlike the one-variable case, two minimal And\^o dilations need not be unitarily equivalent. However, we show that the compressions of the two And\^o dilations constructed in this paper to the minimal dilation spaces of the contraction $T_1T_2$, are unitarily equivalent. In the special case when the product $T=T_1T_2$ is pure, i.e., if $T^{* n}\to 0$ strongly, an And\^o dilation was constructed recently in \cite{D-S-S}, which, as this paper will show, is a corollary to the Douglas-type construction. We define a notion of characteristic triple for a pair of commuting contractions and a notion of coincidence for such triples. We prove that two pairs of commuting contractions with their products being pure contractions are unitarily equivalent if and only if their characteristic triples coincide. We also characterize triples which qualify as the characteristic triple for some pair $(T_1,T_2)$ of commuting contractions such that $T_1T_2$ is a pure contraction.Comment: 24 page

Long-term impact of the low-FODMAP diet on gastrointestinal symptoms, dietary intake, patient acceptability, and healthcare utilization in irritable bowel syndrome

Background: The low-FODMAP diet is a frequently used treatment for irritable bowel syndrome (IBS). Most research has focused on short-term FODMAP restriction; however, guidelines recommend that high-FODMAP foods are reintroduced to individual tolerance. This study aimed to assess the long-term effectiveness of the low-FODMAP diet following FODMAP reintroduction in IBS patients. Methods: Patients with IBS were prospectively recruited to a questionnaire study following completion of dietitian-led low-FODMAP education. At baseline and following FODMAP restriction (short term) only, gastrointestinal symptoms were measured as part of routine clinical care. Following FODMAP reintroduction, (long term), symptoms, dietary intake, acceptability, food-related quality of life (QOL), and healthcare utilization were assessed. Data were reported for patients who continued long-term FODMAP restriction (adapted FODMAP) and/or returned to a habitual diet (habitual). Key Results: Of 103 patients, satisfactory relief of symptoms was reported in 12% at baseline, 61% at short-term follow-up, and 57% at long-term follow-up. At long-term follow-up, 84 (82%) patients continued an ‘adapted FODMAP’ diet (total FODMAP intake mean 20.6, SD 14.9\ua0g/d) compared with 19 (18%) of patients following a ‘habitual’ diet (29.4, SD 22.9\ua0g/d, P=.039). Nutritional adequacy was not compromised for either group. The ‘adapted FODMAP’ group reported the diet cost significantly more than the ‘habitual’ group (

free surface simulation data wrr

Additional data for WRR paper free surface search simulation using adaptive moving-mesh algorithm by Bin Zhang et al 2018</p

Echinacoside alleviates osteoarthritis in rats by activating the Nrf2-HO-1 signaling pathway

Osteoarthritis (OA) is a progressive disease characterized by degeneration of cartilage and echinacoside (Ech) has anti-inflammatory and antioxidant effects in various human diseases. This study aimed to reveal the effect and potential mechanism of Ech on OA. The in vitro OA model was established by rat chondrocytes treated with IL-1β, and the in vivo OA model was established by anterior cruciate ligament transaction. The effect of Ech on the viability, inflammatory response, extracellular matrix (ECM) degradation, and oxidative stress of IL-1β-treated rat chondrocytes were evaluated by Cell Counting Kit-8 assay, enzyme-linked immunosorbent assay, quantitative real-time PCR, Western blot, and immunofluorescence assay. Meanwhile, the mechanism of Ech was assessed using Western blot, Cell Counting Kit-8 assay, enzyme-linked immunosorbent assay, and immunofluorescence analysis. Moreover, the function of Ech in vivo was analyzed in rat models of OA. Functionally, Ech enhanced the viability of rat chondrocytes, repressed the inflammatory response and ECM degradation of rat chondrocytes induced by IL-1β with restrained oxidative stress. Mechanically, Ech repressed IL-1β-induced chondrocyte injury by activating the Nrf2/HO-1 signaling pathway. Meanwhile, Ech alleviated the degree of articular cartilage injury in rats and exerted protective effects on the rat model of OA in vivo. Ech alleviated OA in rats by activating the Nrf2-HO-1 signaling pathway.</p

Structural characterization of chromatin loops.

(A) Probability distribution of the end-to-end distance for the chromatin loop formed between chr1:39.56 Mb and chr1:39.73 Mb from GM12878 cells (blue) and for a random genomic pair (yellow). Two example configurations that correspond to open and closed chromatin loop structures are shown in the inset. (B) End-to-end distances of chromatin loops versus their corresponding contact probabilities. The shaded areas represent the variances in distances estimated from the simulated structural ensemble.</p

Detuning Effects on the Reverse Intersystem Crossing from Triplet Exciton to Lower Polariton

The lower polariton (LP) can reduce the energy barrier of the reverse intersystem crossing (rISC) process from T1 to harvest triplet energy for fluorescence. Based on a Tavis–Cummings model including both singlet and triplet excitons, both coupled with quantized photons, we derive here a comprehensive rISC rate formalism. We found that the latter consists of three contributions: the one originated from spin–orbit coupling as first obtained by Martinez-Martinez et al. (J. Chem. Phys.2019, 151, 054106), the one from light–matter coupling of Ou et al. (J. Am. Chem. Soc.2021, 143, 17786), and the cross-term first reported here. We apply the formalism to investigate the experimentally observed barrier-free rISC (BFrISC) process in cavity devices with DABNA-2 molecular thin film. We found it can be attributed to the detuning effect. The rISC rates can be increased by orders of magnitude through changing the detuning energy to realize the BFrISC process. In addition, the BFrISC rates exhibit a maximum as a function of the incident angle and the doping concentration. The formalism provides a solid ground for molecular design toward highly efficient cavity-promoted light-emitting materials

Dependence of chromatin state interaction energies on genomic separation.

(A) Heat maps for the interaction matrices at various genomic separations, with blue and red corresponding to attractive and repulsive interactions respectively. We subtracted out the mean of the interaction energies in order to shift different plots to the same scale. (B) Dendrogram calculated using the interaction energy matrix at 1.5 Mb to highlight the hierarchical clustering of chromatin states. The coloring scheme is the same as in part (A). (C) The eigenvectors corresponding to the largest eigenvalues of the four interaction matrices, with grey and red indicating positive and negative values respectively. (D) Pearson correlation coefficients between interaction matrices at different scales. (E) The complexity measure for different interaction matrices as a function of the index for top eigenvalues. See text for the definition of the complexity measure.</p

Rotational Friction Correlated with Moiré Patterns in Strained Bilayer Graphene: Implications for Nanoscale Lubrication

Energy conservation of graphene nanodevices requires frictional dissipation management for their high surface area to volume rates, and rotational motion besides sliding scenarios appear frequently. Here, the rotational friction between graphene layers is investigated by molecular dynamics simulations with a validated graphene-spring model. It shows that the interlayer frictional torque drops an order of magnitude and approaches superlubricity with the emergence of Moiré patterns (MPs). Further analysis reveals that the incommensurate interface of MPs tunes the local energy, which cancels each other out with relative rotation, resulting in ultralow interlayer energy barriers. Accordingly, for bilayer systems with biaxially stretched substrates, a significant torque reduction can be achieved when the size of MPs matches that of the graphene flake. This study would provide new insight into the lubrication of rotatable nanomechanical systems

Quantum Dynamical Approach to Predicting the Optical Pumping Threshold for Lasing in Organic Materials

The quantum dynamic (QD) study of organic lasing (OL) is a challenging issue in organic optoelectronics. Previously, the phenomenological method has achieved success in describing experimental observation. However, it cannot directly bridge the laser threshold (LT) with microscopic parameters, which is the advantage of the QD method. In this paper, we propose a microscopic OL model and apply time-dependent wave packet diffusion to reveal the microscopic QD process of optically pumped lasing. LT is obtained from the onset of output as a function of optical input pumping. We predict that the LT has an optimal value as a function of the cavity volume and depends linearly on the intracavity photon leakage rate. The calculated structure–property relationships between molecular parameters and the LT are in qualitative agreement with the experimental results, confirming the reliability of our approach. This work is beneficial for understanding the OL mechanism and optimizing the design of organic laser materials

Overview of the key elements of the computational model.

(A) Illustration of genome organization at various length scales that includes the formation of CTCF mediated chromatin loops, TADs, and compartments. (B) A schematic representation of the computational model that highlights the assignment of chromatin states and CTCF binding sites. Chromatin states for each bead—a 5kb long genomic segment—are derived from the combinatorial patterns of histone marks. They are shown in part (C) as a heat map with darker colors indicating higher probabilities of observing various marks.</p