New lower bound for the Hilbert number in low degree Kolmogorov systems

Our main goal in this paper is to study the number of small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point a class of polynomial Kolmogorov systems. We denote by $\mathcal M_{K}(n)$ the maximum number of limit cycles bifurcating from the equilibrium point via a degenerate Hopf bifurcation for a polynomial Kolmogorov vector field of degree $n$. In this work, we obtain another example such that $\mathcal M_{K}(3)\geq 6$. In addition, we obtain new lower bounds for $\mathcal M_{K}(n)$ proving that $\mathcal M_{K}(4)\geq 13$ and $\mathcal M_{K}(5)\geq 22$

Currents and Moduli in the (4,0) theory

We consider black strings in five dimensions and their description as a (4,0) CFT. The CFT moduli space is described explicitly, including its subtle global structure. BPS conditions and global symmetries determine the spectrum of charged excitations, leading to an entropy formula for near-extreme black holes in four dimensions with arbitrary charge vector. In the BPS limit, this formula reduces to the quartic E(7,7) invariant. The prospects for a description of the (4,0) theory as a solvable CFT are explored.Comment: 40 pages; v2: refs adde

The stability of the O(N) invariant fixed point in three dimensions

We study the stability of the O(N) fixed point in three dimensions under perturbations of the cubic type. We address this problem in the three cases $N=2,3,4$ by using finite size scaling techniques and high precision Monte Carlo simulations. It is well know that there is a critical value $2<N_c<4$ below which the O(N) fixed point is stable and above which the cubic fixed point becomes the stable one. While we cannot exclude that $N_c<3$, as recently claimed by Kleinert and collaborators, our analysis strongly suggests that $N_c$ coincides with 3.Comment: latex file of 18 pages plus three ps figure

Weak-foci of high order and cyclicity

Agraïments: This work was done when H. Liang was visiting the Department of Mathematics of Universitat Autònoma de Barcelona. He is very grateful for the support and hospitality. The first author is supported by the NSF of China (No. 11201086 and No. 11401255) and the Excellent Young Teachers Training Program for colleges and universities of Guangdong Province, China (No. Yq2013107).Agraïments: The second author is partially supported by UNAB13-4E-1604.A particular version of the 16th Hilbert's problem is to estimate the number, M(n), of limit cycles bifurcating from a singularity of center-focus type. This paper is devoted to finding lower bounds for M(n) for some concrete n by studying the cyclicity of different weak-foci. Since a weak-focus with high order is the most current way to produce high cyclicity, we search for systems with the highest possible weak-focus order. For even n, the studied polynomial system of degree n was the one obtained by QiuYan2009 where the highest weak-focus order is n^2 n-2 for n=4,6, 18. Moreover, we provide a system which has a weak-focus with order (n-1)^2 for n 12. We show that Christopher's approach Chr2006, aiming to study the cyclicity of centers, can be applied also to the weak-focus case. We also show by concrete examples that, in some families, this approach is so powerful and the cyclicity can be obtained in a simple computational way. Finally, using this approach, we obtain that M(6) 39, M(7) 34 and M(8) 63

Modular Invariants for Lattice Polarized K3 Surfaces

We study the class of complex algebraic K3 surfaces admitting an embedding of H+E8+E8 inside the Neron-Severi lattice. These special K3 surfaces are classified by a pair of modular invariants, in the same manner that elliptic curves over the field of complex numbers are classified by the J-invariant. Via the canonical Shioda-Inose structure we construct a geometric correspondence relating K3 surfaces of the above type with abelian surfaces realized as cartesian products of two elliptic curves. We then use this correspondence to determine explicit formulas for the modular invariants.Comment: 29 pages, LaTe

Supersymmetry Breaking from a Calabi-Yau Singularity

We conjecture a geometric criterion for determining whether supersymmetry is spontaneously broken in certain string backgrounds. These backgrounds contain wrapped branes at Calabi-Yau singularites with obstructions to deformation of the complex structure. We motivate our conjecture with a particular example: the $Y^{2,1}$ quiver gauge theory corresponding to a cone over the first del Pezzo surface, $dP_1$. This setup can be analyzed using ordinary supersymmetric field theory methods, where we find that gaugino condensation drives a deformation of the chiral ring which has no solutions. We expect this breaking to be a general feature of any theory of branes at a singularity with a smaller number of possible deformations than independent anomaly-free fractional branes.Comment: 32 pages, 6 figures, latex, v2: minor changes, refs adde

A Direct Multigrid Poisson Solver for Oct-Tree Adaptive Meshes

We describe a finite-volume method for solving the Poisson equation on oct-tree adaptive meshes using direct solvers for individual mesh blocks. The method is a modified version of the method presented by Huang and Greengard (2000), which works with finite-difference meshes and does not allow for shared boundaries between refined patches. Our algorithm is implemented within the FLASH code framework and makes use of the PARAMESH library, permitting efficient use of parallel computers. We describe the algorithm and present test results that demonstrate its accuracy.Comment: 10 pages, 6 figures, accepted by the Astrophysical Journal; minor revisions in response to referee's comments; added char

Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice

We present exact calculations of the partition function of the q-state Potts model for general q and temperature on strips of the square lattice of width L_y=3 vertices and arbitrary length L_x with periodic longitudinal boundary conditions, of the following types: (i) (FBC_y,PBC_x)= cyclic, (ii) (FBC_y,TPBC_x)= M\"obius, (iii) (PBC_y,PBC_x)= toroidal, and (iv) (PBC_y,TPBC_x)= Klein bottle, where FBC and (T)PBC refer to free and (twisted) periodic boundary conditions. Results for the L_y=2 torus and Klein bottle strips are also included. In the infinite-length limit the thermodynamic properties are discussed and some general results are given for low-temperature behavior on strips of arbitrarily great width. We determine the submanifold in the {\mathbb C}^2 space of q and temperature where the free energy is singular for these strips. Our calculations are also used to compute certain quantities of graph-theoretic interest.Comment: latex, with encapsulated postscript figure