On the spectrum of a Landau Hamiltonian with a periodic electric potential $V\in L^p_{\mathrm {loc}}(\mathbb{R}^2)$, $p>1$

We consider the two-dimensional Shrödinger operator $\widehat{H}_B+V$ with a homogeneous magnetic field $B\in {\mathbb R}$ and with an electric potential $V$ which belongs to the space $L^p_{\Lambda } ({\mathbb R}^2;{\mathbb R})$ of $\Lambda $-periodic real-valued functions from the space $L^p_{\mathrm {loc}} ({\mathbb R}^2)$, $p>1$. The magnetic field $B$ is supposed to have the rational flux $\eta =(2\pi )^{-1}Bv(K) \in {\mathbb Q}$ where $v(K)$ denotes the area of the elementary cell $K$ of the period lattice $\Lambda \subset {\mathbb R}^2$. Given $p>1$ and the period lattice $\Lambda $, we prove that in the Banach space $(L^p_{\Lambda } ({\mathbb R}^2;\mathbb R),\| \cdot \| _{L^p(K)})$ there exists a typical set $\mathcal O$ in the sense of Baire (which contains a dense $G_{\delta}$-set) such that the spectrum of the operator $\widehat H_B+V$ is absolutely continuous for any electric potential $V\in {\mathcal O}$ and for any homogeneous magnetic field $B$ with the rational flux $\eta \in {\mathbb Q}$.

Virological outcome measures during analytical treatment interruptions in chronic HIV-1 infected patients

Background Analytical treatment interruptions (ATI) are essential in research on HIV cure. However, the heterogeneity of virological outcome measures used in different trials hinders the interpretation of the efficacy of different strategies. Methods A retrospective analysis of viral load (VL) evolution in 334 ATI episodes in chronic HIV-1 infected patients collected from 11 prospective studies. Quantitative [baseline VL, set point, delta set point, VL and delta VL at given weeks after ATI, peak VL, delta peak VL, and area under the rebound curve], and temporal parameters [time to rebound (TtR), set point, peak, and certain absolute and relative VL thresholds] were described. Pairwise correlations between parameters were analyzed, and potential confounding factors (sex, age, time of known HIV infection, time on ART, and immunological interventions) were evaluated. Results Set point was lower than baseline VL (median delta set point -0.26. p< 0.001). This difference was >1 log10 copies/mL in 13.9% of the cases. Median TtR was 2 weeks; no patients had undetectable VL at week 12. Median time to set point was 8 weeks: by week 12, 97.4% of the patients had reached the set point. TtR and baseline VL were correlated with most temporal and quantitative parameters. The variables independently associated with TtR were baseline VL and the use of immunological interventions. Conclusions TtR could be an optimal surrogate marker of response in HIV cure strategies. Our results underline the importance of taking into account baseline VL and other confounding factors in the design and interpretation of these studies.

The computation of factorization invariants for affine semigroups

We present several new algorithms for computing factorization invariant values over affine semigroups. In particular, we give (i) the first known algorithm to compute the delta set of any affine semigroup, (ii) an improved method of computing the tame degree of an affine semigroup, and (iii) a dynamic algorithm to compute catenary degrees of affine semigroup elements. Our algorithms rely on theoretical results from combinatorial commutative algebra involving Gröbner bases, Hilbert bases, and other standard techniques. Implementation in the computer algebra system GAP is discussed.

Delta sets for nonsymmetric numerical semigroups with embedding dimension three

We present a fast algorithm to compute the Delta set of a nonsymmetric numerical semigroup with embedding dimension three. We also characterize the sets of integers that are the Delta set of a numerical semigroup of this kind.

Shifts of generators and delta sets of numerical monoids

Let S be a numerical monoid with minimal generating set 〈n1, …, nt〉. For m ∈ S, if [Formula: see text], then [Formula: see text] is called a factorization length of m. We denote by ℒ(m) = {m1, …, mk} (where mi < mi+1 for each 1 ≤ i < k) the set of all possible factorization lengths of m. The Delta set of m is defined by Δ(m) = {mi+1 - mi | 1 ≤ i < k} and the Delta set of S by Δ(S) = ⋃m∈SΔ(m). In this paper, we expand on the study of Δ(S) begun in [C. Bowles, S. T. Chapman, N. Kaplan and D. Reiser, On delta sets of numerical monoids, J. Algebra Appl. 5 (2006) 1–24] in the following manner. Let r1, r2, …, rt be an increasing sequence of positive integers and Mn = 〈n, n + r1, n + r2, …, n + rt〉 a numerical monoid where n is some positive integer. We prove that there exists a positive integer N such that if n > N then |Δ(Mn)| = 1. If t = 2 and r1 and r2 are relatively prime, then we determine a value for N which is sharp.

Porosity and ε‎-Fr échet differentiability

This chapter demonstrates that the results about smallness of porous sets, and so also of sets of irregularity points of a given Lipschitz function, can be used to show existence of points of (at least) ε‎-Fréchet differentiability of vector-valued functions. The approach involves combining this new idea with the basic notion that points of ε‎-Fréchet differentiability should appear in small slices of the set of Gâteaux derivatives. The chapter obtains very precise results on existence of points of ε‎-Fréchet differentiability for Lipschitz maps with finite dimensional range. The main result applies when every porous set is contained in the unions of a σ‎-directionally porous (and hence Haar null) set and a Γ‎ₙ-null Gsubscript Small Delta set.