Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles

Given a simple graph $G=(V,E)$, a subset of $E$ is called a triangle cover if it intersects each triangle of $G$. Let $\nu_t(G)$ and $\tau_t(G)$ denote the maximum number of pairwise edge-disjoint triangles in $G$ and the minimum cardinality of a triangle cover of $G$, respectively. Tuza conjectured in 1981 that $\tau_t(G)/\nu_t(G)\le2$ holds for every graph $G$. In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding small triangle covers. These algorithms imply new sufficient conditions for Tuza's conjecture on covering and packing triangles. More precisely, suppose that the set $\mathscr T_G$ of triangles covers all edges in $G$. We show that a triangle cover of $G$ with cardinality at most $2\nu_t(G)$ can be found in polynomial time if one of the following conditions is satisfied: (i) $\nu_t(G)/|\mathscr T_G|\ge\frac13$, (ii) $\nu_t(G)/|E|\ge\frac14$, (iii) $|E|/|\mathscr T_G|\ge2$. Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs, Combinatorial algorithm

COHOMOLOGY OF CONGRUENCE SUBGROUPS OF SL4(Z). III

In two previous papers we computed cohomology groups for a range of levels , where is the congruence subgroup of consisting of all matrices with bottom row congruent to mod . In this note we update this earlier work by carrying it out for prime levels up to . This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20 million nonzero entries. We also make two conjectures concerning the contributions to for prime coming from Eisenstein series and Siegel modular forms

Cohomology of congruence subgroups of SL(4, Z) II

In a previous paper [3] we computed cohomology groups H5(..0(N),C), where ..0(N) is a certain congruence subgroup of SL(4,Z), for a range of levels N. In this note we update this earlier work by extending the range of levels and describe cuspidal cohomology classes and additional boundary phenomena found since the publication of [3]. The cuspidal cohomology classes in this paper are the first cuspforms for GL(4) concretely constructed in terms of Betti cohomology

Effective Hamiltonians in quantum optics: a systematic approach

We discuss a general and systematic method for obtaining effective Hamiltonians that describe different nonlinear optical processes. The method exploits the existence of a nonlinear deformation of the usual su(2) algebra that arises as the dynamical symmetry of the original model. When some physical parameter, dictated by the process under consideration, becomes small, we immediately get a diagonal effective Hamiltonian that correctly represents the dynamics for arbitrary states and long times. We extend the technique to su(3) and su(N), finding the corresponding effective Hamiltonians when some resonance conditions are fulfilled.Comment: 13 Pages, no figures, submitted for publicatio

Quantum noise and mixedness of a pumped dissipative non-linear oscillator

Evolutions of quantum noise, characterized by quadrature squeezing parameter and Fano factor, and of mixedness, quantified by quantum von Neumann and linear entropies, of a pumped dissipative non-linear oscillator are studied. The model can describe a signal mode interacting with a thermal reservoir in a parametrically pumped cavity with a Kerr non-linearity. It is discussed that the initial pure states, including coherent states, Fock states, and finite superpositions of coherent states evolve into the same steady mixed state as verified by the quantum relative entropy and the Bures metric. It is shown analytically and verified numerically that the steady state can be well approximated by a nonclassical Gaussian state exhibiting quadrature squeezing and sub-Poissonian statistics for the cold thermal reservoir. A rapid increase is found in the mixedness, especially for the initial Fock states and superpositions of coherent states, during a very short time interval, and then for longer evolution times a decrease in the mixedness to the same, for all the initial states, and relatively low value of the nonclassical Gaussian state.Comment: 10 pages, 12 figure

Lymphatic Filariasis Control in Tanzania: Effect of Six Rounds of Mass Drug Administration with Ivermectin and Albendazole on Infection and Transmission.

Control of lymphatic filariasis (LF) in most countries of sub-Saharan Africa is based on annual mass drug administration (MDA) with a combination of ivermectin and albendazole, in order to interrupt transmission. We present findings from a detailed study on the effect of six rounds of MDA with this drug combination as implemented by the National Lymphatic Filariasis Elimination Programme (NLFEP) in a highly endemic rural area of north-eastern Tanzania.\ud The effect of treatment on transmission and human infection was monitored in a community- and a school-based study during an 8-year period (one pre-intervention and 7 post-intervention years) from 2003 to 2011. Before intervention, 24.5% of the community population had microfilariae (mf) in the blood, 53.3% had circulating filarial antigens (CFA) and 78.9% had specific antibodies to the recombinant filarial antigen Bm14. One year after the sixth MDA, these values had decreased considerably to 2.7%, 19.6% and 27.5%, respectively. During the same period, the CFA prevalence among new intakes of Standard 1 pupils in 10 primary schools decreased from 25.2% to 5.6%. In line with this, transmission by the three vectors (Anopheles gambiae, An. funestus and Culex quinquefasciatus) as determined by dissection declined sharply (overall vector infectivity rate by 99.3% and mean monthly transmission potential by 99.2% between pre-intervention and fifth post-intervention period). A major shift in vector species composition, from predominantly anopheline to almost exclusively culicine was observed over the years. This may be largely unrelated to the MDAs but may have important implications for the epidemiology of LF in the area. Six MDAs caused considerable decrease in all the measured indices for transmission and human infection. In spite of this, indices were still relatively high in the late period of the study, and it may take a long time to reach the recommended cut-off levels for interruption of transmission unless extra efforts are made. These should include increased engagement of the target population in the control activities, to ensure higher treatment coverage. It is expected that the recent initiative to distribute insecticide impregnated bed nets to every household in the area will also contribute towards reaching the goal of successful LF elimination

Actuation of Micro-Optomechanical Systems Via Cavity-Enhanced Optical Dipole Forces

We demonstrate a new type of optomechanical system employing a movable, micron-scale waveguide evanescently-coupled to a high-Q optical microresonator. Micron-scale displacements of the waveguide are observed for milliwatt(mW)-level optical input powers. Measurement of the spatial variation of the force on the waveguide indicates that it arises from a cavity-enhanced optical dipole force due to the stored optical field of the resonator. This force is used to realize an all-optical tunable filter operating with sub-mW control power. A theoretical model of the system shows the maximum achievable force to be independent of the intrinsic Q of the optical resonator and to scale inversely with the cavity mode volume, suggesting that such forces may become even more effective as devices approach the nanoscale.Comment: 4 pages, 5 figures. High resolution version available at (http://copilot.caltech.edu/publications/CEODF_hires.pdf). For associated movie, see (http://copilot.caltech.edu/research/optical_forces/index.htm

Singular projective varieties and quantization

By the quantization condition compact quantizable Kaehler manifolds can be embedded into projective space. In this way they become projective varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This allows for generalization to the case of singular varieties. The set-up is explained in the first part of the contribution. The second part of the contribution is of tutorial nature. Necessary notions, concepts, and results of algebraic geometry appearing in this approach to quantization are explained. In particular, the notions of projective varieties, embeddings, singularities, and quotients appearing in geometric invariant theory are recalled.Comment: 21 pages, 3 figure

Calorons and BPS monopoles with non-trivial holonomy in the confinement phase of SU(2) gluodynamics

With the help of the cooling method applied to SU(2) lattice gauge theory at non-zero $T \le T_c$ we present numerical evidence for the existence of superpositions of Kraan-van Baal caloron (or BPS monopole pair) solutions with non-trivial holonomy, which might constitute an important contribution to the semi-classical approximation of the partition function.Comment: 3 pages, 6 figures, contribution to Lattice2002(topology