Diassociative algebras and Milnor's invariants for tangles

We extend Milnor's mu-invariants of link homotopy to ordered (classical or virtual) tangles. Simple combinatorial formulas for mu-invariants are given in terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves corresponds to axioms of Loday's diassociative algebra. The relation of tangles to diassociative algebras is formulated in terms of a morphism of corresponding operads.Comment: 17 pages, many figures; v2: several typos correcte

On homotopies with triple points of classical knots

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point $p$ of the cylinder is called {\em coherent} if all three branches intersect at $p$ pairwise with the same index. A {\em triple unknotting} of a classical knot $K$ is a homotopy which connects $K$ with the trivial knot and which has as singularities only coherent triple points. We give a new formula for the first Vassiliev invariant $v_2(K)$ by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that there are triple unknottings which are not homotopic as triple unknottings even if we allow more complicated singularities to appear in the homotopy of the homotopy.Comment: 10 pages, 13 figures, bugs in figures correcte

Dynamic sampling schemes for optimal noise learning under multiple nonsmooth constraints

We consider the bilevel optimisation approach proposed by De Los Reyes, Sch\"onlieb (2013) for learning the optimal parameters in a Total Variation (TV) denoising model featuring for multiple noise distributions. In applications, the use of databases (dictionaries) allows an accurate estimation of the parameters, but reflects in high computational costs due to the size of the databases and to the nonsmooth nature of the PDE constraints. To overcome this computational barrier we propose an optimisation algorithm that by sampling dynamically from the set of constraints and using a quasi-Newton method, solves the problem accurately and in an efficient way

Mirror Descent and Convex Optimization Problems With Non-Smooth Inequality Constraints

We consider the problem of minimization of a convex function on a simple set with convex non-smooth inequality constraint and describe first-order methods to solve such problems in different situations: smooth or non-smooth objective function; convex or strongly convex objective and constraint; deterministic or randomized information about the objective and constraint. We hope that it is convenient for a reader to have all the methods for different settings in one place. Described methods are based on Mirror Descent algorithm and switching subgradient scheme. One of our focus is to propose, for the listed different settings, a Mirror Descent with adaptive stepsizes and adaptive stopping rule. This means that neither stepsize nor stopping rule require to know the Lipschitz constant of the objective or constraint. We also construct Mirror Descent for problems with objective function, which is not Lipschitz continuous, e.g. is a quadratic function. Besides that, we address the problem of recovering the solution of the dual problem

Fast Primal-Dual Gradient Method for Strongly Convex Minimization Problems with Linear Constraints

In this paper we consider a class of optimization problems with a strongly convex objective function and the feasible set given by an intersection of a simple convex set with a set given by a number of linear equality and inequality constraints. A number of optimization problems in applications can be stated in this form, examples being the entropy-linear programming, the ridge regression, the elastic net, the regularized optimal transport, etc. We extend the Fast Gradient Method applied to the dual problem in order to make it primal-dual so that it allows not only to solve the dual problem, but also to construct nearly optimal and nearly feasible solution of the primal problem. We also prove a theorem about the convergence rate for the proposed algorithm in terms of the objective function and the linear constraints infeasibility.Comment: Submitted for DOOR 201

East African HIV care: depression and HIV outcomes.

Importance:Depression is a common co-morbidity for people living with HIV (PLWH) and is associated with elevated plasma HIV RNA levels. While depression correlates with deficits in antiretroviral (ARV) adherence, little data exist to inform the relationship between depression and HIV vial load more broadly. Objective:To examine the relationship between depression and viral load in the African Cohort Study (AFRICOS) independently of ARV adherence. Design:PLWH in Kenya, Uganda and Tanzania underwent screening for depression using the Center for Epidemiologic Studies Depression Scale (CESD) upon enrollment at AFRICOS HIV care sites. Setting:AFRICOS is an ongoing prospective longitudinal cohort study enrolling HIV-infected adults at HIV care centers including sites in Kenya, Tanzania and Uganda. These sites are administered by President's Emergency Plan For AIDS Relief programs. Participants:HIV+ individuals were eligible if they were at least 18 years old, receiving HIV care at the enrolling clinic and consented to data and specimen collection. Main outcome measure:CESD. Results:Among 2307 participants, 18-25% met the CESD threshold for depression. Depression was associated with decreased ARV adherence (OR 0.59, p =  0.01). Higher scores on three CESD items were significantly associated with 209-282% higher viral load, independently of ARV adherence among participants on ARVs ⩾6 months. Conclusions:PLWH had high prevalence of depression on the CESD. Diverse depression symptoms were independently associated with increases in viral load, underscoring the need for comprehensive treatment of depression

Counting and computing regions of DD-decomposition: algebro-geometric approach

New methods for $D$-decomposition analysis are presented. They are based on topology of real algebraic varieties and computational real algebraic geometry. The estimate of number of root invariant regions for polynomial parametric families of polynomial and matrices is given. For the case of two parametric family more sharp estimate is proven. Theoretic results are supported by various numerical simulations that show higher precision of presented methods with respect to traditional ones. The presented methods are inherently global and could be applied for studying $D$-decomposition for the space of parameters as a whole instead of some prescribed regions. For symbolic computations the Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure

Multiple effects of silymarin on the hepatitis C virus lifecycle

Silymarin, an extract from milk thistle (Silybum marianum), and its purified flavonolignans have been recently shown to inhibit hepatitis C virus (HCV) infection, both in vitro and in vivo. In the current study, we further characterized silymarin's antiviral actions. Silymarin had antiviral effects against hepatitis C virus cell culture (HCVcc) infection that included inhibition of virus entry, RNA and protein expression, and infectious virus production. Silymarin did not block HCVcc binding to cells but inhibited the entry of several viral pseudoparticles (pp), and fusion of HCVpp with liposomes. Silymarin but not silibinin inhibited genotype 2a NS5B RNA-dependent RNA polymerase (RdRp) activity at concentrations 5 to 10 times higher than required for anti-HCVcc effects. Furthermore, silymarin had inefficient activity on the genotype 1b BK and four 1b RDRPs derived from HCV-infected patients. Moreover, silymarin did not inhibit HCV replication in five independent genotype 1a, 1b, and 2a replicon cell lines that did not produce infectious virus. Silymarin inhibited microsomal triglyceride transfer protein activity, apolipoprotein B secretion, and infectious virion production into culture supernatants. Silymarin also blocked cell-to-cell spread of virus. CONCLUSION: Although inhibition of in vitro NS5B polymerase activity is demonstrable, the mechanisms of silymarin's antiviral action appear to include blocking of virus entry and transmission, possibly by targeting the host cell