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Near-infrared (NIR) emission is less affected by dust than UV and optical emission and is therefore useful for studying the properties of dust-obscured galaxies. Although rest-frame NIR observations of high-redshift galaxies have long been made using space telescopes, their structures were unresolved due to the lack of angular resolution. This letter reports the early results from the analysis of high-resolution Pa$\beta$ imaging of the Spiderweb protocluster at $z=2.16$ with the JWST Near Infrared Camera. We investigate radial profiles of Pa$\beta$ lines and rest-frame NIR continua from luminous H$\alpha$-emitting galaxies (HAEs) in the protocluster. Particularly, we compare those of 11 HAEs (N-HAEs) on the star-forming main sequence with those of 8 HAEs (X-HAEs) with X-ray AGNs. Resultant composite Pa$\beta$ line images of N-HAEs indicate significant star formation in galactic disks. In contrast, X-HAEs are dominated by point source components rather than outer star formation, as inferred from our earlier work based on multi-wavelength SED fitting. Given their higher stellar potentials suggested from their rest-frame NIR images, the different characteristics may be driven by the impact of AGN feedback., Comment: 6 pages, 6 figures; Accepted for publication in MNRAS Letters

We report the results of the deep and wide Atacama Large Millimeter/submillimeter Array (ALMA) 1.2 mm mapping of the Spiderweb protocluster at $z=2.16$. The observations were divided into six contiguous fields covering a survey area of 19.3\,arcmin$^2$. With $\sim$13h on-source time, the final maps in the six fields reach the 1$\sigma$ rms noise in a range of $40.3-57.1 \mu$Jy at a spatial resolution of $0.5-0.9$ arcsec. By using different source extraction codes and careful visual inspection, we detect 47 ALMA sources at a significance higher than 4$\sigma$. We construct the differential and cumulative number counts down to $\sim0.2$ mJy after the correction for purity and completeness obtained from Monte Carlo simulations. The ALMA 1.2 mm number counts of dusty star-forming galaxies (DSFGs) in the Spiderweb protocluster are overall two times that of general fields, some fields/regions showing even higher overdensities (more than a factor of 3). This is consistent with the results from previous studies over a larger scale using single-dish instruments. Comparison of the spatial distributions between different populations indicates that our ALMA sources are likely drawn from the same distribution as CO(1-0) emitters from the COALAS large program, but distinct from that of H$\alpha$ emitters. The cosmic SFR density of the ALMA sources is consistent with previous results (e.g. LABOCA 870 $\mu$m observations) after accounting for the difference in volume. We show that molecular gas masses estimates from dust measurements are not consistent with the ones derived from CO(1-0) and thus have to be taken with caution. The multiplicity fraction of single-dish DSFGs is higher than that of the field. Moreover, two extreme concentrations of ALMA sources are found on the outskirts of the Spiderweb protocluster, with an excess of more than 12 times that of general fields., Comment: 22 pages, 13 figures, 6 tables. Accepted for publication in A&A

We report the initial result of our Pa$\beta$ narrowband imaging on a protocluster with the JWST Near Infrared Camera (NIRCam). As NIRCam enables deep narrowband imaging of rest-frame NIR lines at $z>1$, we target one of the most studied protoclusters, the Spiderweb protocluster at $z=2.16$, in which previous studies have confirmed more than a hundred member galaxies. The NIRCam F405N narrowband filter covers in Pa$\beta$ line the protocluster redshift given by known protocluster members, allowing the search for new member candidates. The weight-corrected color-magnitude diagram obtained 57 sources showing narrowband excesses, 41 of which satisfy further color selection criteria for limiting the sample to Pa$\beta$ emitter candidates at $z\sim2.16$, and 24 of them do not have H$\alpha$ emitter counterparts. The Pa$\beta$ emitter candidates appear to follow the spatial distribution of known protocluster members; however, follow-up spectroscopic confirmation is required. Only 17 out of 58 known H$\alpha$-emitting cluster members are selected as Pa$\beta$ emitters in the current data, albeit the rest fall out of the narrowband selection owing to their small Pa$\beta$ equivalent widths. We derive Pa$\beta$ luminosity function in the Spiderweb protocluster, showing a normalization density of $\log{\phi^\ast}=-2.53\pm0.15$ at a characteristic Pa$\beta$ luminosity of $\log{L^\ast}=42.33\pm0.17$. Furthermore, we examine the possibility of detecting faint line emitters visible only in the narrow-band image, but find no promising candidates., Comment: 19 pages, 12 figures, 2 tables; Accepted for publication in ApJ

We study the new $Z'$ boson as a portal for the production of Higgs bosons $h_0$ and $H_0$ predicted by the Bestest Little Higgs Model through the Higgs-strahlung processes $\mu^+\mu^- \to (Z, Z') \to Zh_0, ZH_0$. We focus on the resonance and non-resonance effects of the $Zh_0, ZH_0$ signals. In our analysis, we consider the center-of-mass energies of $\sqrt{s}=3, 4, 5, 6, 7, 10, 30$ TeV and integrated luminosities of ${\cal L}=2, 4, 6, 10, 30$ ${\rm ab^{-1}}$ projected for a future muon collider. The possibility of performing precision measurements for the Higgs bosons $h_0$ and $H_0$ is very promising at the future muon collider. Furthermore, our results may be helpful to the High Energy Physics community. Complementarily, we generate and provide the Feynman rules necessary for studying the processes $\mu^+\mu^- \to (Z, Z') \to Zh_0, ZH_0$., Comment: 35 pages, 11 figures

An overwhelming number of theoretical predictions for hadron colliders require parton distribution functions (PDFs), which are an important ingredient of theory infrastructure for the next generation of high-energy experiments. This whitepaper summarizes the status and future prospects for determination of high-precision PDFs applicable in a wide range of energies and experiments, in particular in precision tests of the Standard Model and in new physics searches at the high-luminosity Large Hadron Collider and Electron-Ion Collider. We discuss the envisioned advancements in experimental measurements, QCD theory, global analysis methodology, and computing that are necessary to bring unpolarized PDFs in the nucleon to the N2LO and N3LO accuracy in the QCD coupling strength. Special attention is given to the new tasks that emerge in the era of the precision PDF analysis, such as those focusing on the robust control of systematic factors both in experimental measurements and theoretical computations. Various synergies between experimental and theoretical studies of the hadron structure are explored, including opportunities for studying PDFs for nuclear and meson targets, PDFs with electroweak contributions or dependence on the transverse momentum, for incisive comparisons between phenomenological models for the PDFs and computations on discrete lattice, and for cross-fertilization with machine learning/AI approaches. [Submitted to the US Community Study on the Future of Particle Physics (Snowmass 2021).], Comment: 83 pages, 27 figures, contribution to Snowmass 2021; v.3: journal version

In a recent paper an Inexact Restoration method for solving continuous constrained optimization problems was analyzed from the point of view of worst-case functional complexity and convergence. On the other hand, the Inexact Restoration methodology was employed, in a different research,to handle minimization problems with inexact evaluation and simple constraints. These two methodologies are combined in the present report, for constrained minimization problems in which both the objective function and the constraints, as well as their derivatives, are subject to evaluation errors. Together with a complete description of the method, complexity and convergence results will be proved., Comment: 38 pages, Math. Comp., 2023

At each iteration of a Block Coordinate Descent method one minimizes an approximation of the objective function with respect to a generally small set of variables subject to constraints in which these variables are involved. The unconstrained case and the case in which the constraints are simple were analyzed in the recent literature. In this paper we address the problem in which block constraints are not simple and, moreover, the case in which they are not defined by global sets of equations and inequations. A general algorithm that minimizes quadratic models with quadratric regularization over blocks of variables is defined and convergence and complexity are proved. In particular, given tolerances $\delta>0$ and $\varepsilon>0$ for feasibility/complementarity and optimality, respectively, it is shown that a measure of $(\delta,0)$-criticality tends to zero; and the the number of iterations and functional evaluations required to achieve $(\delta,\varepsilon)$-criticality is $O(\varepsilon^2)$. Numerical experiments in which the proposed method is used to solve a continuous version of the traveling salesman problem are presented.

A general framework for solving nonlinear least squares problems without the employment of derivatives is proposed in the present paper together with a new general global convergence theory. With the aim to cope with the case in which the number of variables is big (for the standards of derivative-free optimization), two dimension-reduction procedures are introduced. One of them is based on iterative subspace minimization and the other one is based on spline interpolation with variable nodes. Each iteration based on those procedures is followed by an acceleration step inspired in the Sequential Secant Method. The practical motivation for this work is the estimation of parameters in Hydraulic models applied to dam breaking problems. Numerical examples of the application of the new method to those problems are given.

We report a detailed CO(1-0) survey of a galaxy protocluster field at $z=2.16$, based on 475 hours of observations with the Australia Telescope Compact Array. We constructed a large mosaic of 13 individual pointings, covering an area of 21 arcmin$^2$ and $\pm6500$ km/s range in velocity. We obtain a robust sample of 46 CO(1-0) detections spanning $z=2.09-2.22$, constituting the largest sample of molecular gas measurements in protoclusters to date. The CO emitters show an overdensity at $z=2.12-2.21$, suggesting a galaxy super-protocluster or a protocluster connected to large-scale filaments with ~120 cMpc size. We find that 90% CO emitters have distances $>0'.5-4'$ to the center galaxy, indicating that small area surveys would miss the majority of gas reservoirs in similar structures. Half of the CO emitters have velocities larger than escape velocities, which appears gravitationally unbound to the cluster core. These unbound sources are barely found within the $R_{200}$ radius around the center, which is consistent with a picture in which the cluster core is collapsed while outer regions are still in formation. Compared to other protoclusters, this structure contains relatively more CO emitters with relatively narrow line width and high luminosity, indicating galaxy mergers. We use these CO emitters to place the first constraint on the CO luminosity function and molecular gas density in an overdense environment. The amplitude of the CO luminosity function is 1.6$\pm$0.5 orders of magnitudes higher than observed for field galaxy samples at $z\sim2$, and one order of magnitude higher than predictions for galaxy protoclusters from semi-analytical SHARK models. We derive a high molecular gas density of $0.6-1.3\times10^{9}$ $M_\odot$ cMpc$^{-3}$ for this structure, consistent with predictions for cold gas density of massive structures from hydro-dynamical DIANOGA simulations., Comment: 21 pages + Appendices, 9 figures, 6 tables. Accepted for publication in A&A

Sequential Residual Methods try to solve nonlinear systems of equations $F(x)=0$ by iteratively updating the current approximate solution along a residual-related direction. Therefore, memory requirements are minimal and, consequently, these methods are attractive for solving large-scale nonlinear systems. However, the convergence of these algorithms may be slow in critical cases; therefore, acceleration procedures are welcome. In this paper, we suggest to employ a variation of the Sequential Secant Method in order to accelerate Sequential Residual Methods. The performance of the resulting algorithm is illustrated by applying it to the solution of very large problems coming from the discretization of partial differential equations.

Coordinate descent methods have considerable impact in global optimization because global (or, at least, almost global) minimization is affordable for low-dimensional problems. Coordinate descent methods with high-order regularized models for smooth nonconvex box-constrained minimization are introduced in this work. High-order stationarity asymptotic convergence and first-order stationarity worst-case evaluation complexity bounds are established. The computer work that is necessary for obtaining first-order $\varepsilon$-stationarity with respect to the variables of each coordinate-descent block is $O(\varepsilon^{-(p+1)/p})$ whereas the computer work for getting first-order $\varepsilon$-stationarity with respect to all the variables simultaneously is $O(\varepsilon^{-(p+1)})$. Numerical examples involving multidimensional scaling problems are presented. The numerical performance of the methods is enhanced by means of coordinate-descent strategies for choosing initial points.

We present new gas kinematic observations with the OSIRIS instrument at the GTC for galaxies in the Cl1604 cluster system at z=0.9. These observations together with a collection of other cluster samples at different epochs analyzed by our group are used to study the evolution of the Tully-Fisher, velocity-size and stellar mass-angular momentum relations in dense environments over cosmic time. We use 2D and 3D spectroscopy to analyze the kinematics of our cluster galaxies and extract their maximum rotation velocities (Vmax). Our methods are consistently applied to all our cluster samples which make them ideal for an evolutionary comparison. Up to redshift one, our cluster samples show evolutionary trends compatible with previous observational results in the field and in accordance with semianalytical models and hydrodynamical simulations concerning the Tully-Fisher and velocity-size relations. However, we find a factor 3 drop in disk sizes and an average B-band luminosity enhancement of 2 mag by z=1.5. We discuss the role that different cluster-specific interactions may play in producing this observational result. In addition, we find that our intermediate-to-high redshift cluster galaxies follow parallel sequences with respect to the local specific angular momentum-stellar mass relation, although displaying lower angular momentum values in comparison with field samples at similar redshifts. This can be understood by the stronger interacting nature of dense environments with respect to the field., Comment: 26 pages, 13 figures. Accepted for publication in A&A

In order to understand the role of the different processes that drive galaxy evolution in clusters, we need comprehensive studies that simultaneously examine several of the most important characteristics of galaxies. In this work we examine the physical properties of cluster galaxies at z=0.45 attending to the degree of distortion of their rotation curves. Regular rotating galaxies were included in our Tully-Fisher analysis while the distorted ones were used to study the role of cluster-specific interactions with respect to star formation and AGN activity. We report a higher fraction of galaxies with irregular gas kinematics in the cluster environment than in the field. Cluster galaxies with regular rotation display a moderate brightening in the B-band Tully-Fisher relation compatible with the gradual evolution of the stellar populations with lookback time, and no significant evolution in the stellar-mass Tully-Fisher relation, in line with previous studies at similar redshift. Average specific star formation rate (sSFR) values are slightly lower in our cluster sample (-0.15 dex) with respect to the main sequence of star-forming galaxies, confirming the role of the environment in the early quenching of star formation in clusters. Finally, we carried out an exploratory observational study on the stellar-to-halo mass relation finding that cluster galaxies tend to have slightly lower stellar mass values for a fixed halo mass compared to their field counterparts., Comment: 25 pages, 11 figures. Accepted for publication in A&A (06.02.2020)

Algencan is a well established safeguarded Augmented Lagrangian algorithm introduced in [R. Andreani, E. G. Birgin, J. M. Mart\'{\i}nez and M. L. Schuverdt, On Augmented Lagrangian methods with general lower-level constraints, SIAM Journal on Optimization 18, pp. 1286-1309, 2008]. Complexity results that report its worst-case behavior in terms of iterations and evaluations of functions and derivatives that are necessary to obtain suitable stopping criteria are presented in this work. In addition, the computational performance of a new version of the method is presented, which shows that the updated software is a useful tool for solving large-scale constrained optimization problems.

We observed star-forming galaxies at z~1.5 selected from the HyperSuprimeCam Subaru Strategic Program. The galaxies are part of two significant overdensities of [OII] emitters identified via narrow-band imaging and photometric redshifts from grizy photometry. We used VLT/KMOS to carry out Halpha integral field spectroscopy of 46 galaxies in total. Ionized gas maps, star formation rates and velocity fields were derived from the Halpha emission line. We quantified morphological and kinematical asymmetries to test for potential gravitational (e.g. galaxy-galaxy) or hydrodynamical (e.g. ram-pressure) interactions. Halpha emission was detected in 36 targets. 34 of the galaxies are members of two (proto-)clusters at z=1.47, confirming our selection strategy to be highly efficient. By fitting model velocity fields to the observed ones, we determined the intrinsic maximum rotation velocity Vmax of 14 galaxies. Utilizing the luminosity-velocity (Tully-Fisher) relation, we find that these galaxies are more luminous than their local counterparts of similar mass by up to ~4 mag in the rest-frame B-band. In contrast to field galaxies at z<1, the offsets of the z~1.5 (proto-)cluster galaxies from the local Tully-Fisher relation are not correlated with their star formation rates but with the ratio between Vmax and gas velocity dispersion sigma_g. This probably reflects that, as is observed in the field at similar redshifts, fewer disks have settled to purely rotational kinematics and high Vmax/sigma_g ratios. Due to relatively low galaxy velocity dispersions (sigma_v < 400 km/s) of the (proto-)clusters, gravitational interactions likely are more efficient, resulting in higher kinematical asymmetries, than in present-day clusters. (abbr.), Comment: Accepted for publication in A&A. 11 pages, 8 figures, 1 table

The successful operation of the Large Hadron Collider (LHC) and the excellent performance of the ATLAS, CMS, LHCb and ALICE detectors in Run-1 and Run-2 with $pp$ collisions at center-of-mass energies of 7, 8 and 13 TeV as well as the giant leap in precision calculations and modeling of fundamental interactions at hadron colliders have allowed an extraordinary breadth of physics studies including precision measurements of a variety physics processes. The LHC results have so far confirmed the validity of the Standard Model of particle physics up to unprecedented energy scales and with great precision in the sectors of strong and electroweak interactions as well as flavour physics, for instance in top quark physics. The upgrade of the LHC to a High Luminosity phase (HL-LHC) at 14 TeV center-of-mass energy with 3 ab$^{-1}$ of integrated luminosity will probe the Standard Model with even greater precision and will extend the sensitivity to possible anomalies in the Standard Model, thanks to a ten-fold larger data set, upgraded detectors and expected improvements in the theoretical understanding. This document summarises the physics reach of the HL-LHC in the realm of strong and electroweak interactions and top quark physics, and provides a glimpse of the potential of a possible further upgrade of the LHC to a 27 TeV $pp$ collider, the High-Energy LHC (HE-LHC), assumed to accumulate an integrated luminosity of 15 ab$^{-1}$., Comment: Report from Working Group 1 on the Physics of the HL-LHC, and Perspectives at the HE-LHC

The high-performance computing resources and the constant improvement of both numerical simulation accuracy and the experimental measurements with which they are confronted, bring a new compulsory step to strengthen the credence given to the simulation results: uncertainty quantification. This can have different meanings, according to the requested goals (rank uncertainty sources, reduce them, estimate precisely a critical threshold or an optimal working point) and it could request mathematical methods with greater or lesser complexity. This paper introduces the Uranie platform, an Open-source framework which is currently developed at the Alternative Energies and Atomic Energy Commission (CEA), in the nuclear energy division, in order to deal with uncertainty propagation, surrogate models, optimisation issues, code calibration... This platform benefits from both its dependencies, but also from personal developments, to offer an efficient data handling model, a C++ and Python interpreter, advanced graphical tools, several parallelisation solutions... These methods are very generic and can then be applied to many kinds of code (as Uranie considers them as black boxes) so to many fields of physics as well. In this paper, the example of thermal exchange between a plate-sheet and a fluid is introduced to show how Uranie can be used to perform a large range of analysis. The code used to produce the figures of this paper can be found in https://sourceforge.net/projects/uranie/ along with the sources of the platform., Comment: 35 pages, submitted to CPC (elsevier)

This Report summarizes some of the activities of the HiggsTools Initial Training Network working group in the period 2015-2017. The main goal of this working group was to produce a document discussing various aspects of state-of-the-art Higgs physics at the Large Hadron Collider (LHC) in a pedagogic manner. The first part of the Report is devoted to a description of phenomenological searches for New Physics at the LHC. As the experimental measurements become more and more precise, there is a pressing need for a consistent framework in which deviations from the SM predictions can be computed precisely. We critically review the use of the \k{appa}-framework, fiducial and simplified template cross sections, effective field theories, pseudo-observables and phenomenological Lagrangians. In the second part of the Report, we propose $\varphi_{\eta}^*$ as a new and complementary observable for studying Higgs boson production at large transverse momentum in the case where the Higgs boson decays to two photons. We make a detailed study of the phenomenology of the $\varphi_{\eta}^*$ variable, contrasting the behaviour with the Higgs transverse momentum distribution using a variety of theoretical tools including event generators and fixed order perturbative computations., Comment: 142 pages + references

We took spatially resolved slit FORS2 spectra of 19 cluster galaxies at z=1.4, and 8 additional field galaxies at 1

We present ultraviolet, optical and infrared photometry and optical spectroscopy of the type Ic superluminous supernova (SLSN) Gaia16apd (= SN 2016eay), covering its evolution from 26 d before the $g$-band peak to 234.1 d after the peak. Gaia16apd was followed as a part of the NOT Unbiased Transient Survey (NUTS). It is one of the closest SLSNe known ($z = 0.102\pm0.001$), with detailed optical and ultraviolet (UV) observations covering the peak. Gaia16apd is a spectroscopically typical type Ic SLSN, exhibiting the characteristic blue early spectra with O II absorption, and reaches a peak $M_{g} = -21.8 \pm 0.1$ mag. However, photometrically it exhibits an evolution intermediate between the fast- and slowly-declining type Ic SLSNe, with an early evolution closer to the fast-declining events. Together with LSQ12dlf, another SLSN with similar properties, it demonstrates a possible continuum between fast- and slowly-declining events. It is unusually UV-bright even for a SLSN, reaching a non-$K$-corrected $M_{uvm2} \simeq -23.3$ mag, the only other type Ic SLSN with similar UV brightness being SN 2010gx. Assuming that Gaia16apd was powered by magnetar spin-down, we derive a period of $P = 1.9\pm0.2$ ms and a magnetic field of $B = 1.9\pm0.2 \times 10^{14}$ G for the magnetar. The estimated ejecta mass is between 8 and 16 $\mathrm{M}_{\odot}$ and the kinetic energy between 1.3 and $2.5 \times 10^{52}$ erg, depending on opacity and assuming that the entire ejecta is swept up into a thin shell. Despite the early photometric differences, the spectra at late times are similar to slowly-declining type Ic SLSNe, implying that the two subclasses originate from similar progenitors., Comment: 14 pages; 8 figures. Updated on June 5 2017; now published in MNRAS after revision

We have studied a sample of 11 blue, luminous, metal-poor galaxies at redshift 0.744 < z < 0.835 from the DEEP2 redshift survey. They were selected by the presence of the [OIII]4363 auroral line and the [OII]3726,3729 doublet together with the strong emission nebular [OIII] lines in their spectra from a sample of around 6000 galaxies within a narrow redshift range. All the spectra have been taken with DEIMOS, which is a multi-slit, double-beam spectrograph which uses slitmasks to allow the spectra from many objects to be imaged at the same time. The selected objects present high luminosities (20.3 < MB < 18.5), remarkable blue color index, and total oxygen abundances between 7.69 and 8.15 which represent 1/3 to 1/10 of the solar value. The wide spectral coverage (from 6500 to 9100 angstroms) of the DEIMOS spectrograph and its high spectral resolution, R around 5000, bring us an opportunity to study the behaviour of these star-forming galaxies at intermediate redshift with high quality spectra. We put in context our results together with others presented in the literature up to date to try to understand the luminosity-metallicity relation this kind of objects define. The star-forming metal-poor galaxies would be of special relevance in showing the diversity among galaxies of similar luminosities and could serve to understand the processes of galaxy evolution., Comment: 22 pages, 7 figures, Master Thesis, Universidad Autonoma de Madrid

We give the normal and anti-normal order expressions of the number operator to the power $k$ by using the commutation relation between the annihilation and creation operators. We use those expressions to give general formulae for functions of the number operator in normal and anti-normal order.

We comment on some misleading and biased statements appearing in the manuscript arXiv:1209.0298 ("Thermal fluctuations of magnetic nanoparticles") about the use of the damped Landau-Lifshitz equation and the kinetic Langer theory for the calculation of the relaxation rate of magnetic nanoclusters. We reiterate simple scientific arguments, part of which is well known to the whole community, demonstrating that the authors' criticisms are unfounded and that they overstate the issue of damping in the Landau-Lifshitz equation with no unanimous experimental evidence., Comment: 3 pages

We investigate the dynamics of a magnetic system consisting of two magnetic moments coupled by either exchange, dipole-dipole, or Dzyalozhinski-Moriya interaction. We compare the switching mechanisms and switching rates as induced by the three couplings. For each coupling and each configuration of the two anisotropy axes, we describe the switching modes and, using the kinetic theory of Langer, we provide (semi-)analytical expressions for the switching rate. We then compare the three interactions with regard to their efficiency in the reversal of the net magnetic moment of the dimer. We also investigate how the energy barriers vary with the coupling. For the dipole-dipole interaction we find that the energy barrier may either increase or decrease with the coupling depending on whether the latter is weak or strong. Finally, upon comparing the various switching rates, we find that the dipole-dipole coupling leads to the slowest magnetic dimer, as far as the switching of its net magnetic moment is concerned., Comment: 20 pages, 18 Figures, 2 tables

We show that in the trapped ion-laser interaction all the regimes may be considered analytically. We may solve not only for different laser intensities, but also away from resonance and from the Lamb-Dicke regime. It is found a dispersive Hamiltonian for the high intensity regime, that, being diagonal, its evolution operator may be easily calculated.

We show set of transformations which allow to obtain analytic solutions in several quantum-optical problems. We start with the ion-laser (time dependent) interaction, continue with the problem of a slow atom interacting with a quantized field to end with a master equation that describes losses. In all cases it is shown that one may find useful transformations that simplify the problems., Comment: 6 pages, no figures

We solve analytically master equations that describe a cavity filled with a kerr medium, taking into account the dissipation induced by the environment, and parametric down conversion processes. We use superoperator techniques., Comment: 5 pages, 1 table, no figures

In meteorological and oceanological studies the classical approach for finding the numerical solution of the regional model consists in formulating and solving the Cauchy-Dirichlet problem. The related boundary conditions are obtained by linear interpolation of data available on a coarse grid (global data), to the boundary of regional model. Errors, in boundary conditions, appearing owing to linear interpolation may lead to increasing errors in numerical solution during integration. The methods developed to reduce these errors deal with continuous dynamic assimilation of known global data available inside the regional domain. Essentially, this assimilation procedure performs a nudging of large-scale component of regional model solution to large-scale global data component by introducing the relaxation forcing terms into the regional model equations. As a result, the obtained solution is not a valid numerical solution of the original regional model. In this work we propose the optimization approach which is free from the above-mentioned shortcoming. The formulation of the joint problem of finding the regional model solution and data assimilation, as a PDE-constrained optimization problem, gives the possibility to obtain the exact numerical solution of the regional model. Three simple model examples (ODE Burgers equation, Rossby-Oboukhov equation, Korteweg-de Vries equation) were considered in this paper. The result of performed numerical experiments indicates that the optimization approach can significantly improve the precision of the sought numerical solution, even in the cases in which the solution of Cauchy-Dirichlet problem is very sensitive to the errors in the boundary condition., Comment: Paper contents 14 pages and 12 figures. Some results of this work was presented on the sixth international conference on optimization, OPTIMIZATION2007, Porto, Portugal, July 22-25,2007

We present a phase formalism that passes the Barnett-Pegg acid test, i.e. phase fluctuations for a number state are the expected value $\pi^2/3$ which are the fluctuations for a classical random phase distribution. The formalism is shown to have consistency subjected to different approaches., Comment: 5 pages, no figures