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  • In 1975 Pippenger and Golumbic proved that any graph on $n$ vertices admits at most $2e(n/k)^k$ induced $k$-cycles. This bound is larger by a multiplicative factor of $2e$ than the simple lower bound obtained by a blow-up construction. Pippenger and Golumbic conjectured that the latter lower bound is essentially tight. In the present paper we establish a better upper bound of $(128e/81) \cdot (n/k)^k$. This constitutes the first progress towards proving the aforementioned conjecture since it was posed. Read More
  • We have conducted a spectroscopic analysis of the far ultraviolet archival spectra of four symbiotic variables, EG And, AE Ara, CQ Dra and RW Hya. RW Hya and EG And have never had a recorded outburst while CQ Dra and AE Ara have outburst histories. We analyze these systems while they are in quiescence in order to help reveal the physical properties of their hot components via comparisons of the observations with optically thick accretion disk models and NLTE model white dwarf photospheres. We have extended the wavelength coverage down to the Lyman Limit with FUSE spectra. We find that the hot component in RW Hya is a low mass white dwarf with a surface temperature of 160,000K. We re-examine whether or not the symbiotic system CQ Dra is a triple system with a red giant transferring matter to a hot component made up of a cataclysmic variable in which the white dwarf has a surface temperature as low as $\sim$20,000K. The very small size of the hot component contributing to the shortest wavelengths of the FUSE spectrum of CQ Dra agrees with an optically thick and geometrically thin ($\sim$4\% of the WD surface) hot ($\sim 120,000$K) boundary layer. Our analysis of EG And reveals that its hot component is a hot, bare, low mass white dwarf with a surface temperature of 80-95,000K, with a surface gravity $\log(g)= 7.5$. For AE Ara, we also find that a low gravity ($\log(g) \sim 6$) hot ($T \sim 130,000$K) WD accounts for the hot component. Read More
  • The thermal contribution to the chiral vortical effect is believed to be tied to the axial anomaly in external gravitational fields. We use the universality of the spin-gravity interaction to extend this idea to a wider set of phenomena. We consider the Kubo formula at weak coupling for the spin current of a vector field and derive a novel anomalous effect caused by the medium rotation: chiral vortical effect for bosons. The effect consists in a spin current of vector bosons along the angular velocity of the medium. We argue that it has the same anomalous nature as in the fermionic case and show that this effect provides a mechanism for helicity transfer, from flow helicity to magnetic helicity. Read More
  • Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem limited. We explore how generally we can apply Banach's fixed point theorem to establish the convergence of iterative methods when pairing it with carefully designed metrics. Our first result is a strong converse of Banach's theorem, showing that it is a universal analysis tool for establishing uniqueness of fixed points and for bounding the convergence rate of iterative maps to a unique fixed point. In other words, we show that, whenever an iterative map globally converges to a unique fixed point, there exists a metric under which the iterative map is contracting and which can be used to bound the number of iterations until convergence. We illustrate our approach in the widely used power method, providing a new way of bounding its convergence rate through contraction arguments. We next consider the computational complexity of Banach's fixed point theorem. Making the proof of our converse theorem constructive, we show that computing a fixed point whose existence is guaranteed by Banach's fixed point theorem is CLS-complete. We thus provide the first natural complete problem for the class CLS, which was defined in [Daskalakis-Papadimitriou 2011] to capture the complexity of problems such as P-matrix LCP, computing KKT-points, and finding mixed Nash equilibria in congestion and network coordination games. Read More
  • The solar active region photospheric magnetic field evolves rapidly during major eruptive events, suggesting appreciable feedback from the corona. The new high-cadence (90 s or 135 s) vector magnetogram dataset from the Helioseismic and Magnetic Imager (HMI) is suited for investigating these "magnetic imprints". Observations of an archetypical event, SOL2011-02-15T01:56, show the following trends. Firstly, the horizontal magnetic field component ($B_h$) exhibits permanent, step-like changes with a time scale of several minutes, whereas the radial component ($B_r$) varies less. Secondly, $B_h$ near the main polarity inversion line increases significantly during the earlier phase of the associated flare, whereas $B_h$ in the periphery decreases at later times with smaller magnitudes. Thirdly, transient artifacts coincide with enhanced flare emission, where the Stokes profiles are no longer adequately modeled under standard settings, and the inferred magnetic field becomes unreliable. Our results corroborate previous findings, remove certain ambiguities that arise from line-of-sight only or lower-cadence vector observations, and provide insights on the momentum processes during solar eruption. The dataset may also be useful to the study of sunquakes and data-driven modeling of the solar corona. Read More
  • We consider the electromagnetic radiation from newborn binary black holes (BBHs) formed by the evolution of isolated massive stellar binaries. Before the formation of a BBH, the binary consists of a primary black hole (BH) and a secondary Wolf-Rayet star. We investigate two types of transients from the birth of a secondary BH: one powered by the Bondi-Hoyle-Lyttleton accretion onto the primary BH, and the other induced by accretion onto the secondary BH. In the former scenario, when the secondary collapses to a BH, it may eject a fraction of its outer material, which forms a disk around the primary BH and induces an ultrafast outflow. This companion-induced outflow can lead to week-scale optical transients with a kinetic energy of $\sim10^{47}$ -- $3\times10^{48}$~erg, ejecta velocity of $10^8$ -- $10^9\rm~cm~s^{-1}$, and absolute magnitude ranging from about $-10$ to $-12$. In the latter scenario, assuming that the tidal torque synchronizes the spin period of the secondary to the orbital period of the primary, the accretion of the stellar material is expected to form a disk around a newborn BH, following its core-collapse. This disk may produce an energetic outflow with a kinetic energy of $\sim10^{52}$~erg and the outflow velocity of $\sim10^{10}\rm~cm~s^{-1}$, resulting in an optical transient of absolute magnitude from $\sim -13$ to $\sim-14$ with a duration of a few days. While dimmer than ordinary supernovae, their light curves and late-time spectra are distinctive, and dedicated optical transient surveys could detect these two types of transients, the second type also leading to detectable radio signals. Read More