Separating Solution of a Quadratic Recurrent Equation

In this paper we consider the recurrent equation $$\Lambda_{p+1}=\frac1p\sum_{q=1}^pf\bigg(\frac{q}{p+1}\bigg)\Lambda_{q}\Lambda_{p+1-q}$$ for $p\ge 1$ with $f\in C[0,1]$ and $\Lambda_1=y>0$ given. We give conditions on $f$ that guarantee the existence of $y^{(0)}$ such that the sequence $\Lambda_p$ with $\Lambda_1=y^{(0)}$ tends to a finite positive limit as $p\to \infty$.Comment: 13 pages, 6 figures, submitted to J. Stat. Phy

Entropy and the variational principle for actions of sofic groups

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group C*-algebra.Comment: 44 pages; minor changes; to appear in Invent. Mat

Effect of Tetramethylpyrazine on Atherosclerosis and SCAP/SREBP-1c Signaling Pathway in ApoE −/−

Lipid metabolism dysregulation plays a crucial role in the occurrence of atherosclerosis (As). SCAP/SREBP signaling is the main pathway for regulating lipid metabolism. Tetramethylpyrazine (TMP), a Traditional Chinese Medicine (TCM) for treating angina pectoris, has antiatherosclerotic effects and ameliorates blood lipids disturbance. However, its precise mechanism remains unclear. This study investigated the mechanism of TMP in ameliorating As in mice model. After six weeks of high-fat diet, 30 ApoE−/− mice were randomized (n=10) and treated with Lipitor, TMP, or distilled water for six weeks. The serum blood lipids and insulin levels were measured. The expressions of PAQR3, Insig-1, SCAP, SREBP-1c, IRS-1, PI3K, Akt, and mTORC-1 in the adipose tissues were determined. The results showed that TMP could significantly decrease blood lipids levels, insulin, and corrected plaque area of the ApoE−/− mice as compared to the untreated mice (P<0.05, P<0.01). Moreover, TMP could significantly downregulate the expressions of SCAP, SREBP-1c, PAQR3, IRS-1, PI3K, Akt, and mTORC1 (P<0.01). Thus, TMP may ameliorate lipid metabolism disorder and As by downregulating PAQR3 and inhibiting SCAP/SREBP-1c signaling pathway. In addition, PI3K/Akt/mTORC1 signaling pathway may be involved in this process

Shen-Yuan-Dan Capsule Attenuates Atherosclerosis and Foam Cell Formation by Enhancing Autophagy and Inhibiting the PI3K/Akt/mTORC1 Signaling Pathway

Background and Aim: The phosphoinositide 3-kinase (PI3K)/Akt/mammalian target of rapamycin complex 1 (mTORC1) signaling pathway plays a crucial role in autophagy and inflammation. Our previous studies demonstrated that Shen-Yuan-Dan Capsule (SYDC), a Chinese medicine used for treating angina pectoris, has anti-atherosclerotic and anti-inflammatory effects in mice. However, its effects on autophagy and the PI3K/Akt/mTORC1 signaling pathway remain unclear. This study aimed to explore the effects of SYDC on autophagy and PI3K/Akt/mTORC1 signaling in the apolipoprotein E knockout (ApoE−/−) mouse model and in macrophage-derived foam cells to delineate the underlying mechanism.Methods: After 6 weeks of high-fat diet, ApoE–/– mice were randomly grouped into control, Lipitor, low-SYDC (SYDC-L), middle-SYDC (SYDC-M), and high-SYDC (SYDC-H) groups (n = 10). The mice were intragastrically administered the respective treatment for 6 weeks. Murine RAW264.7 cells were stimulated with oxidized low-density lipoprotein (ox-LDL) (80 µg/ml) for 24 h and then pretreated with SYDC freeze-dried powder for another 24 h. Cells treated with SYDC were co-cultured for 24 h with LY294002, tricirbine, and rapamycin to investigate the effects on the PI3K/Akt/mTORC1 signaling pathway.Results: SYDC ameliorated blood lipid levels, reduced the atherosclerotic index and plaque areas in the aortic root in mice, and inhibited total cholesterol (TC) levels and cholinesterase (ChE)/TC ratios in ox-LDL stimulated macrophages. Moreover, SYDC up-regulated Beclin1 and LC3II/I proteins in mice and in the ox-LDL–stimulated macrophages. Moreover, SYDC inhibited AKT phosphorylation at Ser473 and mTOR phosphorylation at Ser2448 in mice and in ox-LDL–stimulated macrophages. Furthermore, SYDC’s inhibitory of ChE/TC ratios in ox-LDL–stimulated macrophages was not changed by selective inhibition of the PI3K/Akt/mTORC1 pathway.Conclusions: Our results highlight that SYDC treatment attenuates foam cell formation by promoting autophagy via inhibiting activation of the PI3K/Akt/mTORC1 signaling pathway. This study provides new insights into the molecular mechanism underlying SYDC’s therapeutic potential for treating atherosclerosis

Entire solutions of hydrodynamical equations with exponential dissipation

We consider a modification of the three-dimensional Navier--Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as \ue ^{|k|/\kd} at high wavenumbers $|k|$. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than \ue ^{-C(k/\kd) \ln (|k|/\kd)} for any $C<1/(2\ln 2)$. The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with $C= C_\star =1/\ln2$. The same behavior with a universal constant $C_\star$ is conjectured for the Navier--Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier--Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.Comment: 29 pages, 3 figures, Comm. Math. Phys., in pres

Localized behavior in the Lyapunov vectors for quasi-one-dimensional many-hard-disk systems

We introduce a definition of a "localization width" whose logarithm is given by the entropy of the distribution of particle component amplitudes in the Lyapunov vector. Different types of localization widths are observed, for example, a minimum localization width where the components of only two particles are dominant. We can distinguish a delocalization associated with a random distribution of particle contributions, a delocalization associated with a uniform distribution and a delocalization associated with a wave-like structure in the Lyapunov vector. Using the localization width we show that in quasi-one-dimensional systems of many hard disks there are two kinds of dependence of the localization width on the Lyapunov exponent index for the larger exponents: one is exponential, and the other is linear. Differences, due to these kinds of localizations also appear in the shapes of the localized peaks of the Lyapunov vectors, the Lyapunov spectra and the angle between the spatial and momentum parts of the Lyapunov vectors. We show that the Krylov relation for the largest Lyapunov exponent $\lambda\sim-\rho\ln\rho$ as a function of the density $\rho$ is satisfied (apart from a factor) in the same density region as the linear dependence of the localization widths is observed. It is also shown that there are asymmetries in the spatial and momentum parts of the Lyapunov vectors, as well as in their $x$ and $y$-components.Comment: 41 pages, 21 figures, Manuscript including the figures of better quality is available from http://www.phys.unsw.edu.au/~gary/Research.htm