Multiplicative stochastic heat equations on the whole space

We carry out the construction of some ill-posed multiplicative stochastic heat equations on unbounded domains. The two main equations our result covers are, on the one hand the parabolic Anderson model on R³, and on the other hand the KPZ equation on R via the Cole-Hopf transform. To perform these constructions, we adapt the theory of regularity structures to the setting of weighted Besov spaces. One particular feature of our construction is that it allows one to start both equations from a Dirac mass at the initial time

Alignment of cylindrical colloids near chemically patterned substrates induced by critical Casimir torques

Recent experiments have demonstrated a fluctuation-induced lateral trapping of spherical colloidal particles immersed in a binary liquid mixture near its critical demixing point and exposed to chemically patterned substrates. Inspired by these experiments, we study this kind of effective interaction, known as the critical Casimir effect, for elongated colloids of cylindrical shape. This adds orientational degrees of freedom. When the colloidal particles are close to a chemically structured substrate, a critical Casimir torque acting on the colloids emerges. We calculate this torque on the basis of the Derjaguin approximation. The range of validity of the latter is assessed via mean-field theory. This assessment shows that the Derjaguin approximation is reliable in experimentally relevant regimes, so that we extend it to Janus particles endowed with opposing adsorption preferences. Our analysis indicates that critical Casimir interactions are capable of achieving well-defined, reversible alignments both of chemically homogeneous and of Janus cylinders.Comment: 24 pages, 12 figures; v2: 22 pages, 12 figure

Liquid bridging of cylindrical colloids in near-critical solvents

Within mean field theory, we investigate the bridging transition between a pair of parallel cylindrical colloids immersed in a binary liquid mixture as a solvent which is close to its critical consolute point $T_c$. We determine the universal scaling functions of the effective potential and of the force between the colloids. For a solvent which is at the critical concentration and close to $T_c$, we find that the critical Casimir force is the dominant interaction at close separations. This agrees very well with the corresponding Derjaguin approximation for the effective interaction between the two cylinders, while capillary forces originating from the extension of the liquid bridge turn out to be more important at large separations. In addition, we are able to infer from the wetting characteristics of the individual colloids the first-order transition of the liquid bridge connecting two colloidal particles to the ruptured state. While specific to cylindrical colloids, the results presented here provide also an outline for identifying critical Casimir forces acting on bridged colloidal particles as such, and for analyzing the bridging transition between them.Comment: 23 pages, 12 figure

Mixed Integer Linear Programming for Feature Selection in Support Vector Machine

This work focuses on support vector machine (SVM) with feature selection. A MILP formulation is proposed for the problem. The choice of suitable features to construct the separating hyperplanes has been modelled in this formulation by including a budget constraint that sets in advance a limit on the number of features to be used in the classification process. We propose both an exact and a heuristic procedure to solve this formulation in an efficient way. Finally, the validation of the model is done by checking it with some well-known data sets and comparing it with classical classification methods.Comment: 37 pages, 20 figure

Ultradeep Infrared Array Camera Observations of sub-L* z~7 and z~8 Galaxies in the Hubble Ultra Deep Field: the Contribution of Low-Luminosity Galaxies to the Stellar Mass Density and Reionization

We study the Spitzer Infrared Array Camera (IRAC) mid-infrared (rest-frame optical) fluxes of 14 newly WFC3/IR-detected z=7 z_{850}-dropout galaxies and 5 z=8 Y_{105}-dropout galaxies. The WFC3/IR depth and spatial resolution allow accurate removal of contaminating foreground light, enabling reliable flux measurements at 3.6 micron and 4.5 micron. None of the galaxies are detected to [3.6]=26.9 (AB, 2 sigma), but a stacking analysis reveals a robust detection for the z_{850}-dropouts and an upper limit for the Y_{105}-dropouts. We construct average broadband SEDs using the stacked ACS, WFC3, and IRAC fluxes and fit stellar population synthesis models to derive mean redshifts, stellar masses, and ages. For the z_{850}-dropouts, we find z=6.9^{+0.1}_{-0.1}, (U-V)_{rest}=0.4, reddening A_V=0, stellar mass M*=1.2^{+0.3}_{-0.6} x 10^9 M_sun (Salpeter IMF). The best-fit ages ~300Myr, M/L_V=0.2, and SSFR=1.7Gyr^{-1} are similar to values reported for luminous z=7 galaxies, indicating the galaxies are smaller but not younger. The sub-L* galaxies observed here contribute significantly to the stellar mass density and under favorable conditions may have provided enough photons for sustained reionization at 7<z<11. In contrast, the z=8.3^{+0.1}_{-0.2} Y_{105}-dropouts have stellar masses that are uncertain by 1.5 dex due to the near-complete reliance on far-UV data. Adopting the 2 sigma upper limit on the M/L(z=8), the stellar mass density to M_{UV,AB} < -18 declines from rho*(z=7)=3.7^{+1.0}_{-1.8} x 10^6 M_sun Mpc^{-3} to rho*(z=8) < 8 x 10^5 M_sun Mpc^{-3}, following (1+z)^{-6} over 3<z<8. Lower masses at z=8 would signify more dramatic evolution, which can be established with deeper IRAC observations, long before the arrival of the James Webb Space Telescope.Comment: 6 pages, 3 figures, 2 tables, emulateapj, accepted for publication in ApJ

Spitzer IRAC confirmation of z_850-dropout galaxies in the Hubble Ultra Deep Field: stellar masses and ages at z~7

Using Spitzer IRAC mid-infrared imaging from the Great Observatories Origins Deep Survey, we study z_850-dropout sources in the Hubble Ultra Deep Field. After carefully removing contaminating flux from foreground sources, we clearly detect two z_850-dropouts at 3.6 micron and 4.5 micron, while two others are marginally detected. The mid-infrared fluxes strongly support their interpretation as galaxies at z~7, seen when the Universe was only 750 Myr old. The IRAC observations allow us for the first time to constrain the rest-frame optical colors, stellar masses, and ages of the highest redshift galaxies. Fitting stellar population models to the spectral energy distributions, we find photometric redshifts in the range 6.7-7.4, rest-frame colors U-V=0.2-0.4, V-band luminosities L_V=0.6-3 x 10^10 L_sun, stellar masses 1-10 x 10^9 M_sun, stellar ages 50-200 Myr, star formation rates up to ~25 M_sun/yr, and low reddening A_V<0.4. Overall, the z=7 galaxies appear substantially less massive and evolved than Lyman break galaxies or Distant Red Galaxies at z=2-3, but fairly similar to recently identified systems at z=5-6. The stellar mass density inferred from our z=7 sample is rho* = 1.6^{+1.6}_{-0.8} x 10^6 M_sun Mpc^-3 (to 0.3 L*(z=3)), in apparent agreement with recent cosmological hydrodynamic simulations, but we note that incompleteness and sample variance may introduce larger uncertainties. The ages of the two most massive galaxies suggest they formed at z>8, during the era of cosmic reionization, but the star formation rate density derived from their stellar masses and ages is not nearly sufficient to reionize the universe. The simplest explanation for this deficiency is that lower-mass galaxies beyond our detection limit reionized the universe.Comment: 4 pages, 3 figures, emulateapj, Accepted for publication in ApJ Letter

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