Lingering random walks in random environment on a strip

We consider a recurrent random walk (RW) in random environment (RE) on a strip. We prove that if the RE is i. i. d. and its distribution is not supported by an algebraic subsurface in the space of parameters defining the RE then the RW exhibits the "(log t)-squared" asymptotic behaviour. The exceptional algebraic subsurface is described by an explicit system of algebraic equations. One-dimensional walks with bounded jumps in a RE are treated as a particular case of the strip model. If the one dimensional RE is i. i. d., then our approach leads to a complete and constructive classification of possible types of asymptotic behaviour of recurrent random walks. Namely, the RW exhibits the $(\log t)^{2}$ asymptotic behaviour if the distribution of the RE is not supported by a hyperplane in the space of parameters which shall be explicitly described. And if the support of the RE belongs to this hyperplane then the corresponding RW is a martingale and its asymptotic behaviour is governed by the Central Limit Theorem

Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation $T$ of a probability space $(X,\mathcal F,\mu)$ we investigate almost sure and distributional convergence of random variables of the form $$x \to \frac{1}{C_n} \sum_{i_1<n,...,i_d<n} f(T^{i_1}x,...,T^{i_d}x),\, n=1,2,...,$$ where $f$ (called the \emph{kernel}) is a function from $X^d$ to $\R$ and $C_1, C_2,...$ are appropriate normalizing constants. We observe that the above random variables are well defined and belong to $L_r(\mu)$ provided that the kernel is chosen from the projective tensor product $$L_p(X_1,\mathcal F_1, \mu_1) \otimes_{\pi}...\otimes_{\pi} L_p(X_d,\mathcal F_d, \mu_d)\subset L_p(\mu^d)$$ with $p=d\,r,\, r\ \in [1, \infty).$ We establish a form of the individual ergodic theorem for such sequences. Next, we give a martingale approximation argument to derive a central limit theorem in the non-degenerate case (in the sense of the classical Hoeffding's decomposition). Furthermore, for $d=2$ and a wide class of canonical kernels $f$ we also show that the convergence holds in distribution towards a quadratic form $\sum_{m=1}^{\infty} \lambda_m\eta^2_m$ in independent standard Gaussian variables $\eta_1, \eta_2,...$. Our results on the distributional convergence use a $T$--\,invariant filtration as a prerequisite and are derived from uni- and multivariate martingale approximations

Interleukin-2 therapy in patients with HIV infection

BACKGROUND Used in combination with antiretroviral therapy, subcutaneous recombinant interleukin-2 raises CD4+ cell counts more than does antiretroviral therapy alone. The clinical implication of these increases is not known. METHODS We conducted two trials: the Subcutaneous Recombinant, Human Interleukin-2 in HIV-Infected Patients with Low CD4+ Counts under Active Antiretroviral Therapy (SILCAAT) study and the Evaluation of Subcutaneous Proleukin in a Randomized International Trial (ESPRIT). In each, patients infected with the human immunodeficiency virus (HIV) who had CD4+ cell counts of either 50 to 299 per cubic millimeter (SILCAAT) or 300 or more per cubic millimeter (ESPRIT) were randomly assigned to receive interleukin-2 plus antiretroviral therapy or antiretroviral therapy alone. The interleukin-2 regimen consisted of cycles of 5 consecutive days each, administered at 8-week intervals. The SILCAAT study involved six cycles and a dose of 4.5 million IU of interleukin-2 twice daily; ESPRIT involved three cycles and a dose of 7.5 million IU twice daily. Additional cycles were recommended to maintain the CD4+ cell count above predefined target levels. The primary end point of both studies was opportunistic disease or death from any cause. RESULTS In the SILCAAT study, 1695 patients (849 receiving interleukin-2 plus antiretroviral therapy and 846 receiving antiretroviral therapy alone) who had a median CD4+ cell count of 202 cells per cubic millimeter were enrolled; in ESPRIT, 4111 patients (2071 receiving interleukin-2 plus antiretroviral therapy and 2040 receiving antiretroviral therapy alone) who had a median CD4+ cell count of 457 cells per cubic millimeter were enrolled. Over a median follow-up period of 7 to 8 years, the CD4+ cell count was higher in the interleukin-2 group than in the group receiving antiretroviral therapy alone--by 53 and 159 cells per cubic millimeter, on average, in the SILCAAT study and ESPRIT, respectively. Hazard ratios for opportunistic disease or death from any cause with interleukin-2 plus antiretroviral therapy (vs. antiretroviral therapy alone) were 0.91 (95% confidence interval [CI], 0.70 to 1.18; P=0.47) in the SILCAAT study and 0.94 (95% CI, 0.75 to 1.16; P=0.55) in ESPRIT. The hazard ratios for death from any cause and for grade 4 clinical events were 1.06 (P=0.73) and 1.10 (P=0.35), respectively, in the SILCAAT study and 0.90 (P=0.42) and 1.23 (P=0.003), respectively, in ESPRIT. CONCLUSIONS Despite a substantial and sustained increase in the CD4+ cell count, as compared with antiretroviral therapy alone, interleukin-2 plus antiretroviral therapy yielded no clinical benefit in either study. (ClinicalTrials.gov numbers, NCT00004978 [ESPRIT] and NCT00013611 [SILCAAT study].

Quasi-Two-Dimensional Dynamics of Plasmas and Fluids

In the lowest order of approximation quasi-twa-dimensional dynamics of planetary atmospheres and of plasmas in a magnetic field can be described by a common convective vortex equation, the Charney and Hasegawa-Mirna (CHM) equation. In contrast to the two-dimensional Navier-Stokes equation, the CHM equation admits "shielded vortex solutions" in a homogeneous limit and linear waves ("Rossby waves" in the planetary atmosphere and "drift waves" in plasmas) in the presence of inhomogeneity. Because of these properties, the nonlinear dynamics described by the CHM equation provide rich solutions which involve turbulent, coherent and wave behaviors. Bringing in non ideal effects such as resistivity makes the plasma equation significantly different from the atmospheric equation with such new effects as instability of the drift wave driven by the resistivity and density gradient. The model equation deviates from the CHM equation and becomes coupled with Maxwell equations. This article reviews the linear and nonlinear dynamics of the quasi-two-dimensional aspect of plasmas and planetary atmosphere starting from the introduction of the ideal model equation (CHM equation) and extending into the most recent progress in plasma turbulence.U. S. Department of Energy DE-FG05-80ET-53088Ministry of Education, Science and Culture of JapanFusion Research Cente

Canonical description of ideal magnetohydrodynamic flows and integrals of motion

In the framework of the variational principle the canonical variables describing ideal magnetohydrodynamic (MHD) flows of general type (i.e., with spatially varying entropy and nonzero values of all topological invariants) are introduced. The corresponding complete velocity representation enables us not only to describe the general type flows in terms of single-valued functions, but also to solve the intriguing problem of the ``missing'' MHD integrals of motion. The set of hitherto known MHD local invariants and integrals of motion appears to be incomplete: for the vanishing magnetic field it does not reduce to the set of the conventional hydrodynamic invariants. And if the MHD analogs of the vorticity and helicity were discussed earlier for the particular cases, the analog of Ertel invariant has been so far unknown. It is found that on the basis of the new invariants introduced a wide set of high-order invariants can be constructed. The new invariants are relevant both for the deeper insight into the problem of the topological structure of the MHD flows as a whole and for the examination of the stability problems. The additional advantage of the proposed approach is that it enables one to deal with discontinuous flows, including all types of possible breaks.Comment: 16 page

Multidimensional Gaussian sums arising from distribution of Birkhoff sums in zero entropy dynamical systems

A duality formula, of the Hardy and Littlewood type for multidimensional Gaussian sums, is proved in order to estimate the asymptotic long time behavior of distribution of Birkhoff sums $S_n$ of a sequence generated by a skew product dynamical system on the $\mathbb{T}^2$ torus, with zero Lyapounov exponents. The sequence, taking the values $\pm 1$, is pairwise independent (but not independent) ergodic sequence with infinite range dependence. The model corresponds to the motion of a particle on an infinite cylinder, hopping backward and forward along its axis, with a transversal acceleration parameter $\alpha$. We show that when the parameter $\alpha /\pi$ is rational then all the moments of the normalized sums $E((S_n/\sqrt{n})^k)$, but the second, are unbounded with respect to n, while for irrational $\alpha /\pi$, with bounded continuous fraction representation, all these moments are finite and bounded with respect to n.Comment: To be published in J. Phys.

Development of singularities for the compressible Euler equations with external force in several dimensions

We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singularity formation, "the best sufficient condition", in the sense that one can explicitly construct a global in time smooth solution for which this condition is not satisfied "arbitrary little". Also compactly supported perturbation of nontrivial constant state is considered. We generalize the known theorem by Sideris on initial data resulting in singularities. Finally, we investigate the influence of frictional damping and rotation on the singularity formation.Comment: 23 page

Discontinuation of Prophylaxis against Mycobacterium avium Complex Disease in HIV-Infected Patients Who Have a Response to Antiretroviral Therapy

BACKGROUND Several agents are effective in preventing Mycobacterium avium complex disease in patients with advanced human immunodeficiency virus (HIV) infection. However, there is uncertainty about whether prophylaxis should be continued in patients whose CD4+ cell counts have increased substantially with antiviral therapy. METHODS We conducted a multicenter, double-blind, randomized trial of treatment with azithromycin (1200 mg weekly) as compared with placebo in HIV-infected patients whose CD4+ cell counts had increased from less than 50 to more than 100 per cubic millimeter in response to antiretroviral therapy. The primary end point was M. avium complex disease or bacterial pneumonia. RESULTS A total of 520 patients entered the study; the median CD4+ cell count at entry was 230 per cubic millimeter. In 48 percent of the patients, the HIV RNA value was below the level of quantification. The median prior nadir CD4+ cell count was 23 per cubic millimeter, and 65 percent of the patients had had an acquired immunodeficiency syndrome-defining illness. During follow-up over a median period of 12 months, there were no episodes of confirmed M. avium complex disease in either group (95 percent confidence interval for the rate of disease in each group, 0 to 1.5 episodes per 100 person-years). Three patients in the azithromycin group (1.2 percent) and five in the placebo group (1.9 percent) had bacterial pneumonia (relative risk in the azithromycin group, 0.60; 95 percent confidence interval, 0.14 to 2.50; P=0.48). Neither the rate of progression of HIV disease nor the mortality rate differed significantly between the two groups. Adverse effects led to discontinuation of the study drug in 19 patients assigned to receive azithromycin (7.4 percent) and in 3 assigned to receive placebo (1.1 percent; relative risk, 6.6; P=0.002). CONCLUSIONS Azithromycin prophylaxis can safely be withheld in HIV-infected patients whose CD4+ cell counts have increased to more than 100 cells per cubic millimeter in response to antiretroviral therapy