The positive semidefinite Grothendieck problem with rank constraint

Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of size m x m, the positive semidefinite Grothendieck problem with rank-n-constraint (SDP_n) is maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, ..., x_m \in S^{n-1}. In this paper we design a polynomial time approximation algorithm for SDP_n achieving an approximation ratio of \gamma(n) = \frac{2}{n}(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)})^2 = 1 - \Theta(1/n). We show that under the assumption of the unique games conjecture the achieved approximation ratio is optimal: There is no polynomial time algorithm which approximates SDP_n with a ratio greater than \gamma(n). We improve the approximation ratio of the best known polynomial time algorithm for SDP_1 from 2/\pi to 2/(\pi\gamma(m)) = 2/\pi + \Theta(1/m), and we show a tighter approximation ratio for SDP_n when A is the Laplacian matrix of a graph with nonnegative edge weights.Comment: (v3) to appear in Proceedings of the 37th International Colloquium on Automata, Languages and Programming, 12 page

Grothendieck inequalities for semidefinite programs with rank constraint

Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: a difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constrained is dropped. For instance, the integrality gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen as a Grothendieck inequality. In this paper we consider Grothendieck inequalities for ranks greater than 1 and we give two applications: approximating ground states in the n-vector model in statistical mechanics and XOR games in quantum information theory.Comment: 22 page

Fourier analysis, linear programming, and densities of distance avoiding sets in {RnR^n}

In this paper we derive new upper bounds for the densities of measurable sets in R^n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2,..., 24. This gives new lower bounds for the measurable chromatic number in dimensions 3,..., 24. We apply it to get a new, short proof of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson, Weiss and Bourgain and Falconer about sets avoiding many distances

New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry

In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l_3^p$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. These bounds give strong indications that some of the lattice packings of superballs found in 2009 by Jiao, Stillinger, and Torquato are indeed optimal among all translative packings. We improve Zong's recent upper bound for the maximal density of translative packings of regular tetrahedra from $0.3840\ldots$ to $0.3745\ldots$, getting closer to the best known lower bound of $0.3673\ldots$. We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of packings of round spheres. The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization

A recursive theta body for hypergraphs

The theta body of a graph, introduced by Grötschel, Lovász, and Schrijver (in 1986), is a tractable relaxation of the independent-set polytope derived from the Lovász theta number. In this paper, we recursively extend the theta body, and hence the theta number, to hypergraphs. We obtain fundamental properties of this extension and relate it to the high-dimensional Hoffman bound of Filmus, Golubev, and Lifshitz. We discuss two applications: triangle-free graphs and Mantel’s theorem, and bounds on the density of triangle-avoiding sets in the Hamming cube

Estudo anatomico do ligamento popliteo obliquo

OBJECTIVE:To study the anatomy of the oblique popliteal ligament, as regards its dimensions, expansion and anatomical relationships.METHODS:Eleven cadaver knees were dissected in order to study the anatomy and take mea-surements of anatomical structures and relationships of the oblique popliteal ligament. The dissection was for posterior access to the proper exposure of the oblique popliteal ligament, the semimembranosus muscle and its expansions. For measurement of dimensions, 40 × 12 needles were used for marking the specific points and a caliper. The angles were calculated using the software ImagePro Plus(r) .RESULTS:The distance from the origin of the oblique popliteal ligament to the tibial plateau was 7.4 mm, the thickness at its origin was 7.3 mm, length was 33.6 mm and the tibial plateau angle 34.8°. The length of the expansion of the proximal oblique popliteal ligament was 39.2 mm, thickness 7.8 mm and angle of the oblique popliteal ligament with its expansion 32.2°.CONCLUSION:The oblique popliteal ligament is thick, rises in the semimembranosus and protrudes proximally forming an acute angle with the joint interline, crossing the popliteal fossa. In some cases it has a proximal expansion.OBJETIVO:Estudar a anatomia do ligamento poplíteo oblíquo no que se refere às suas dimensões, expansões e relações anatômicas.MÉTODOS:Onze joelhos de cadáveres foram dissecados com o intuito de se estudar a anatomia e fazer medições das estruturas e das relações anatômicas do ligamento poplíteo oblíquo. A dissecção foi por acesso posterior até a exposição adequada do ligamento poplíteo oblíquo, do músculo semimembranoso e de suas expansões. Para aferição das medidas, foram usados agulhas 40x12 na marcação dos pontos específicos e um paquímetro. Os ângulos foram calculados com o auxílio do software ImagePro Plus(r).RESULTADOS:A distância da origem do ligamento poplíteo oblíquo ao platô tibial foi de 7,4 mm, a espessura na sua origem foi de 7,3 mm, o comprimento foi de 33,6 mm e o ângulo com o platô tibial foi de 34,8°. O comprimento da expansão proximal do ligamento poplíteo oblíquo foi de 39,2 mm, a espessura foi de 7,8 mm e o ângulo do ligamento poplíteo oblíquo com sua expansão foi de 32,2°.CONCLUSÃO:O ligamento poplíteo oblíquo é espesso, nasce no músculo semimembranoso, projeta-se proximalmente, forma um ângulo agudo com a interlinha articular e cruza a fossa poplítea. Em alguns casos apresenta uma expansão proximal.Universidade Federal do ParanaUniversidade Federal de São Paulo (UNIFESP)Universidade Federal de São Paulo (UNIFESP) Escola Paulista de MedicinaPontificia Universidade Catolica do ParanaUNIFESP, EPMSciEL

Ciprofibrate therapy in patients with hypertriglyceridemia and low high density lipoprotein (HDL)-cholesterol: greater reduction of non-HDL cholesterol in subjects with excess body weight (The CIPROAMLAT study)

BACKGROUND: Hypertriglyceridemia in combination with low HDL cholesterol levels is a risk factor for cardiovascular disease. Our objective was to evaluate the efficacy of ciprofibrate for the treatment of this form of dyslipidemia and to identify factors associated with better treatment response. METHODS: Multicenter, international, open-label study. Four hundred and thirty seven patients were included. The plasma lipid levels at inclusion were fasting triglyceride concentrations between 1.6–3.9 mM/l and HDL cholesterol ≤ 1.05 mM/l for women and ≤ 0.9 mM/l for men. The LDL cholesterol was below 4.2 mM/l. All patients received ciprofibrate 100 mg/d. Efficacy and safety parameters were assessed at baseline and at the end of the treatment. The primary efficacy parameter of the study was percentage change in triglycerides from baseline. RESULTS: After 4 months, plasma triglyceride concentrations were decreased by 44% (p < 0.001). HDL cholesterol concentrations were increased by 10% (p < 0.001). Non-HDL cholesterol was decreased by 19%. A greater HDL cholesterol response was observed in lean patients (body mass index < 25 kg/m(2)) compared to the rest of the population (8.2 vs 19.7%, p < 0.001). In contrast, cases with excess body weight had a larger decrease in non-HDL cholesterol levels (-20.8 vs -10.8%, p < 0.001). There were no significant complications resulting from treatment with ciprofibrate. CONCLUSIONS: Ciprofibrate is efficacious for the correction of hypertriglyceridemia / low HDL cholesterol. A greater decrease in non-HDL cholesterol was found among cases with excess body weight. The mechanism of action of ciprofibrate may be influenced by the pathophysiology of the disorder being treated

Pervasive gaps in Amazonian ecological research

Biodiversity loss is one of the main challenges of our time,1,2 and attempts to address it require a clear un derstanding of how ecological communities respond to environmental change across time and space.3,4 While the increasing availability of global databases on ecological communities has advanced our knowledge of biodiversity sensitivity to environmental changes,5–7 vast areas of the tropics remain understudied.8–11 In the American tropics, Amazonia stands out as the world’s most diverse rainforest and the primary source of Neotropical biodiversity,12 but it remains among the least known forests in America and is often underrepre sented in biodiversity databases.13–15 To worsen this situation, human-induced modifications16,17 may elim inate pieces of the Amazon’s biodiversity puzzle before we can use them to understand how ecological com munities are responding. To increase generalization and applicability of biodiversity knowledge,18,19 it is thus crucial to reduce biases in ecological research, particularly in regions projected to face the most pronounced environmental changes. We integrate ecological community metadata of 7,694 sampling sites for multiple or ganism groups in a machine learning model framework to map the research probability across the Brazilian Amazonia, while identifying the region’s vulnerability to environmental change. 15%–18% of the most ne glected areas in ecological research are expected to experience severe climate or land use changes by 2050. This means that unless we take immediate action, we will not be able to establish their current status, much less monitor how it is changing and what is being lostinfo:eu-repo/semantics/publishedVersio